ABSTRACTTitle of dissertation: EARLY AFTERGLOW EVOLUTIONOF X-RAY FLASHES OBSERVEDBY SWIFTDerek Hullinger, Doctor of Philosophy, 2006Dissertation directed by: Dr. Ann ParsonsNASA Goddard Space Flight CenterGamma-ray bursts (GRBs) are bright flashes of gamma-ray energy that originate in distantgalaxies and last only a matter of seconds before fading away, never to appear again. They areaccompanied by longer-wavelength “afterglows” that fade away much more gradually and can bedetected for up to several days or even weeks after the gamma-ray burst has vanished.In recent years, another phenomenon has been discovered that resembles gamma-ray burstsin almost every way, except that the radiated energy comes mostly from x-rays instead of gamma-rays. This new class of bursts has been dubbed “x-ray flashes” (XRFs). There is strong evidenceto suggest that GRBs and XRFs are closely-related phenomena.The Swift mission, launched in November of 2004, is designed to answer many questionsabout GRBs and their cousins, XRFs—where they come from, what causes them, and why gamma-ray bursts and x-ray flashes differ. The key to the Swift mission is its ability to detect and determinethe location of a burst in the sky and then autonomously point x-ray and optical telescopes atthe burst position within seconds of the detection. This allows the measurement of the afterglowwithin 1 – 2 minutes after the burst, rather than several hours later, as was necessary with pastmissions. This early afterglow measurement is an important key to distinguishing between differenttheories that seek to explain the differences between XRFs and GRBs.This dissertation describes the calibration of the Burst Alert Telescope, which measures thespectral and temporal properties of GRBs and XRFs. It also presents a study of XRFs and GRBsdetected by Swift, including the first analysis and comparison of the early afterglow properties ofthese phenomena. This study reveals interesting differences between the temporal properties ofGRB and XRF afterglows and sets strong constraints on some theories that seek to explain XRForigins.EARLY AFTERGLOW EVOLUTION OF X-RAY FLASHESOBSERVED BY SWIFTbyDerek HullingerDissertation submitted to the Faculty of the Graduate School of theUniversity of Maryland, College Park in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy2006Advisory Committee:Professor Greg Sullivan, ChairDr. Ann ParsonsDr. Elihu BoldtProfessor Kara HoffmanProfessor Christopher Reynoldsc© Copyright byDerek Hullinger2006ACKNOWLEDGMENTSWhere to begin! There are so many people to whom I owe so much!First, I am very grateful to my advisor, Ann Parsons, who has always, at every stage of mywork, been anxious for me to have a good experience, and who has ever been ready and eager tohelp me. I also owe an immense debt of gratitude to Taka Sakamoto, who suggested this study andmade it possible, and who shepherded me through every stage of the analysis and interpretation. Iwant to thank Cathy Dicks, who helped me edit my dissertation and even took a couple of chaptershome with her over Thanksgiving because I was so eager for them. I’m grateful to the entire BATscience team—Scott Barthelmy, Louis Barbier, Jay Cummings, Ed Fenimore, Hans Krimm, CraigMarkwardt, David Palmer, Ann Parsons, Taka Sakamoto, and Jack Tueller—for their individualhelp and tutelage over the last six years, and for making this a memorable experience for me. Iam very grateful to Goro Sato, who worked alongside me throughout my time at Goddard. Hiswisdom and ingenuity made the calibration work possible, and he was a delight to work with.I appreciate very much my dissertation committee for their patience with me and for readingand giving insightful help on a dissertation that ended up being rather long. I want to thankValentin Pal’shin of the Konus science team, who provided me with the spectra and responsefiles for the bursts detected by Konus, and who tutored me on how to adjust for time delaydifferences between the Swift and Wind spacecrafts. I’m grateful to the HETE-2 science team forthe collaboration and assistance they gave me in doing joint studies of those bursts observed bythe FREGATE and by BAT. I appreciate the help of Kevin Hurley in providing me with timinginformation that allowed me to check and verify the results of my own time delay calculations. I’mgrateful to David Morris of the XRT science team for providing me with the plot lc source codeand for his kind and readily available support for it.To Donald Knuth, Leslie Lamport, and the other creators of LATEX—I will forever praiseiitheir names! When I think of the work this dissertation would have taken without the assistanceof LATEX, I shudder!I am especially full of gratitude to my dear wife, Christina, who has supported me and hasbeen completely understanding as I worked from dawn to dusk on my dissertation for months, andwho let me tell her all about gamma-ray bursts and even acted interested.Finally, I would be unthinking and foolish if I didn’t thank my heavenly Father, who hassupported me and sustained me throughout this endeavor.iiiTABLE OF CONTENTSList of Tables viiList of Figures ix1 Introduction 11.1 The Discovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Extra-Galactic Origin of Gamma-Ray Bursts . . . . . . . . . . . . . . . . . . . 21.3 The Discovery of Afterglows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 The Discovery and Observations of X-Ray Flashes . . . . . . . . . . . . . . . . . . 61.5 Overview of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Observational Characteristics of Gamma-Ray Bursts and X-Ray Flashes 92.1 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Prompt Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.1 Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4.2 Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.3 Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.4 Spectral Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.5 Amati Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.6 Yonetoku Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 Afterglow Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5.1 Spectral Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5.2 Temporal Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5.3 Jet Breaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.4 Ghirlanda Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5.5 Dark Bursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Previous XRF Observations 263.1 Einstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Ginga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 BeppoSAX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 HETE-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 XRF Afterglow Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5.1 XRF 011030 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5.2 XRF 020427 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.5.3 XRF 020903 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5.4 XRR 030723 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5.5 XRF 040701 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 Gamma-Ray Burst Theory 364.1 The Prompt Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.1.1 The Fireball Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1.2 Internal Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 The Afterglow Emission and External Shocks . . . . . . . . . . . . . . . . . . . . . 414.3 Models for the “Inner Engine” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3.1 Collapsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3.2 Compact Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4 X-Ray Flash Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4.1 Models Based on Geometric Effects . . . . . . . . . . . . . . . . . . . . . . . 484.4.2 Models Based on Intrinsic Properties . . . . . . . . . . . . . . . . . . . . . . 54iv5 The Swift Observatory 595.1 Description of the Swift Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 The Burst Alert Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2.2 Physical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2.3 Coded Aperture Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.2.4 Effective Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2.5 The Trigger Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2.6 GRB Data Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3 The X-Ray Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.3.1 Purposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.3.2 Physical Description and Performance . . . . . . . . . . . . . . . . . . . . . 725.3.3 GRB Data Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.4 The UltraViolet and Optical Telescope . . . . . . . . . . . . . . . . . . . . . . . . . 755.4.1 Purposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.4.2 Physical Description and Performance . . . . . . . . . . . . . . . . . . . . . 765.4.3 GRB Data Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786 Calibration of the Burst Alert Telescope 796.1 The Physics of Gamma Ray Detection . . . . . . . . . . . . . . . . . . . . . . . . . 796.1.1 Processes of Photon Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 796.1.2 What Takes Place Inside a Semiconductor Detector . . . . . . . . . . . . . 816.2 Properties of CZT Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.3 Measuring the µτ Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.3.1 Energy Deposition Distributions . . . . . . . . . . . . . . . . . . . . . . . . 886.3.2 The Single-Detector Spectral Model . . . . . . . . . . . . . . . . . . . . . . 936.3.3 Correcting for Electronics Nonlinearity . . . . . . . . . . . . . . . . . . . . . 946.4 Determining the Energy Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.5 The Angular Dependence of the Peak Shape . . . . . . . . . . . . . . . . . . . . . . 986.6 Adjusting for Distance and Angle Variations . . . . . . . . . . . . . . . . . . . . . . 1006.7 The Full-Array Spectral Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.8 On-Orbit Calibration Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.9 The Purpose of a Response Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117 Analysis Procedures 1167.1 Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167.1.1 BAT Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167.1.2 Wind Konus Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.1.3 HETE-2 FREGATE Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217.1.4 XRT Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217.2 Spectral Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.2.1 Prompt Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.2.2 Afterglow Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.2.3 Error Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.3 Estimating Redshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.4 The Kolmogorov-Smirnov Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267.5 Definition of GRBs, XRRs, and XRFs . . . . . . . . . . . . . . . . . . . . . . . . . 1277.6 Selection Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318 BAT XRFs, XRRs, and GRBs 1328.1 XRFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1328.1.1 XRF 050416A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1328.1.2 XRF 050406 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1388.1.3 XRF 050714B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141v8.1.4 XRF 050824 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488.1.5 XRF 050528 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538.1.6 XRF 050215B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1578.2 XRRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1608.2.1 XRR 050315 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1628.2.2 XRR 050318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1688.2.3 XRR 050525A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1738.2.4 XRR 050219B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1808.3 GRBs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1838.3.1 GRB 050401 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1858.3.2 GRB 050326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1898.3.3 GRB 050603 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1938.3.4 GRB 041223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2009 Global Characteristics of BAT XRFs, XRRs, and GRBs 2069.1 Prompt Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2069.1.1 Fluence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2069.1.2 Spectral Peak Energy Eobspeak . . . . . . . . . . . . . . . . . . . . . . . . . . . 2069.1.3 Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2089.1.4 Correlation of Eobspeak with Other Quantities . . . . . . . . . . . . . . . . . . 2099.2 Afterglow Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2129.2.1 Light Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2129.2.2 Photon Index and nH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2149.2.3 Photon Index and Temporal Index . . . . . . . . . . . . . . . . . . . . . . . 2189.2.4 2 – 10 keV Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2209.2.5 2 – 10 keV Isotropic Luminosity . . . . . . . . . . . . . . . . . . . . . . . . 2209.2.6 Jet Breaks and Opening Angles . . . . . . . . . . . . . . . . . . . . . . . . . 2219.3 Prompt Emission and Afterglow Emission . . . . . . . . . . . . . . . . . . . . . . . 2239.3.1 Ratio of Prompt Emission Fluence and Afterglow Emission Fluence . . . . 2239.3.2 Prompt Emission Eiso and Afterglow Emission Liso . . . . . . . . . . . . . . 22610 Discussion 23010.1 The Continuity of XRFs, XRRs, and GRBs . . . . . . . . . . . . . . . . . . . . . . 23010.2 Differences and similarities between the afterglows of XRFs and GRBs . . . . . . . 23010.3 Implications for the off-jet model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23410.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236A Mask-Weighted Count Rates 238B Glossary of Terms 243Bibliography 248viLIST OF TABLES3.1 Properties of Pre-Swift XRF Prompt Emission . . . . . . . . . . . . . . . . . . . . 353.2 Properties of Pre-Swift XRF Afterglow Emission and Redshifts . . . . . . . . . . . 355.1 Swift Mission Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2 Burst Alert Telescope Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 645.3 BAT burst-related data products and their time delays after the burst trigger . . . 725.4 X-Ray Telescope Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.5 Summary of the XRT mode characteristics . . . . . . . . . . . . . . . . . . . . . . 755.6 UltraViolet and Optical Telescope Characteristics . . . . . . . . . . . . . . . . . . . 776.1 The mean and standard deviations of the 7 full-array model fit parameters . . . . . 1077.1 The fluence ratio for all bursts detected by BAT through 31 July 2005 . . . . . . . 1308.1 The best fit spectral parameters for the XRF 050416a prompt emission . . . . . . 1358.2 The best fit spectral parameters for the XRF 050416a x-ray afterglow . . . . . . . 1368.3 The best fit spectral parameters for the XRF 050406 prompt emission . . . . . . . 1418.4 The best fit spectral parameters for the XRF 050604 x-ray afterglow . . . . . . . . 1428.5 The best fit spectral parameters for the XRF 050714B prompt emission . . . . . . 1468.6 The best fit spectral parameters for the XRF 050714B x-ray afterglow . . . . . . . 1488.7 The best fit spectral parameters for the XRF 050824 prompt emission . . . . . . . 1508.8 The best fit spectral parameters for the XRF 050824 x-ray afterglow . . . . . . . . 1548.9 The best fit spectral parameters for the XRF 050528 prompt emission . . . . . . . 1558.10 The best fit spectral parameters for the XRF 050215B prompt emission . . . . . . 1588.11 The best fit spectral parameters for the XRF 050215B x-ray afterglow . . . . . . . 1628.12 The best fit spectral parameters for the XRR 050315 prompt emission . . . . . . . 1678.13 The best fit spectral parameters for the XRR 050315 x-ray afterglow . . . . . . . . 1698.14 The best fit spectral parameters for the XRR 050318 prompt emission . . . . . . . 1728.15 The best fit spectral parameters for the XRF 050318 x-ray afterglow . . . . . . . . 1738.16 The best fit spectral parameters for the XRR 050525A prompt emission . . . . . . 175vii8.17 The best fit spectral parameters for the XRR 050525a x-ray afterglow . . . . . . . 1788.18 The best fit spectral parameters for the XRR 050219B prompt emission . . . . . . 1828.19 The best fit spectral parameters for the XRR 050219B x-ray afterglow . . . . . . . 1848.20 The best fit spectral parameters for the GRB 050401 prompt emission . . . . . . . 1878.21 The best fit spectral parameters for the XRF 050401 x-ray afterglow . . . . . . . . 1898.22 The best fit spectral parameters for the GRB 050326 prompt emission . . . . . . . 1918.23 The best fit spectral parameters for the GRB 050326 x-ray afterglow . . . . . . . . 1948.24 The best fit spectral parameters for the GRB 050603 prompt emission . . . . . . . 1988.25 The best fit spectral parameters for the GRB 050603 x-ray afterglow . . . . . . . . 2018.26 The best fit spectral parameters for the GRB 041223 prompt emission . . . . . . . 2038.27 The best fit spectral parameters for the GRB 041223 x-ray afterglow . . . . . . . . 2059.1 K-S test results for Swift GRBs, XRRs, and XRFs . . . . . . . . . . . . . . . . . . 2089.2 Redshifts, Esrcpeak values and Eiso values for Swift bursts . . . . . . . . . . . . . . . . 2139.3 Constraints on jet break times and opening angles for XRFs . . . . . . . . . . . . . 2239.4 Constraints on jet break times and opening angles for XRRs . . . . . . . . . . . . . 2249.5 Constraints on jet break times and opening angles for GRBs . . . . . . . . . . . . . 2259.6 Ratio of 0.6 – 10 keV afterglow fluence to prompt emission fluence . . . . . . . . . 22610.1 Light curve decay indices and break times . . . . . . . . . . . . . . . . . . . . . . . 231viiiLIST OF FIGURES1.1 The distribution of BATSE burst positions in galactic coordinates. . . . . . . . . . 31.2 A plot of the number of bursts N with a peak flux greater than some value P . . . 52.1 GRB Redshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Redshifts of GRBs and XRFs detected by HETE-2 and BeppoSAX . . . . . . . . . 132.3 Durations of GRBs Measured by BATSE . . . . . . . . . . . . . . . . . . . . . . . 142.4 Durations vs. fluence ratios of GRBs detected by BATSE . . . . . . . . . . . . . . 152.5 Light curves for some gamma-ray bursts detected by BATSE . . . . . . . . . . . . 162.6 A plot of the Band spectral model . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.7 Correlation between Esrcpeak and Eiso . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.8 Correlation between Esrcpeak and Liso . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.9 Eiso and Eγ for bursts studied by Frail et al. . . . . . . . . . . . . . . . . . . . . . 232.10 Correlation between Esrcpeak and Eγ . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1 Eobspeak for Ginga GRBs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Ratio of x-ray energy to gamma-ray energy for Ginga GRBs . . . . . . . . . . . . . 294.1 The fireball and the internal/external shock scenario . . . . . . . . . . . . . . . . . 394.2 Synchrotron spectrum from a relativistic shock . . . . . . . . . . . . . . . . . . . . 434.3 Light curve due to synchrotron radiation from a spherical relativistic shock . . . . 454.4 A drawing illustrating the off-jet model for x-ray flashes . . . . . . . . . . . . . . . 494.5 The peak flux ratio and fluence ratio as a function of viewing angle . . . . . . . . . 504.6 Light curves from a jet with a thin uniform ring structure . . . . . . . . . . . . . . 534.7 Light curves for a jet with an angular structure of a thin fan . . . . . . . . . . . . 544.8 A light curve calculation for typical GRB afterglows in the “dirty fireball” model . 565.1 The Swift Observatory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2 A drawing of the Burst Alert Telescope . . . . . . . . . . . . . . . . . . . . . . . . 635.3 The mask weight values assigned to various detectors . . . . . . . . . . . . . . . . . 655.4 The “mask-weighted” effective area curve for the Burst Alert Telescope . . . . . . 67ix5.5 The “traditional” effective area curve for the Burst Alert Telescope . . . . . . . . . 685.6 Block diagram of the X-Ray Telescope . . . . . . . . . . . . . . . . . . . . . . . . . 735.7 Block diagram of the UltraViolet and Optical Telescope . . . . . . . . . . . . . . . 765.8 The effective area of the UltraViolet and Optical Telescope . . . . . . . . . . . . . 776.1 Diagram illustrating photoelectric absorption . . . . . . . . . . . . . . . . . . . . . 806.2 Diagram illustrating Compton scattering . . . . . . . . . . . . . . . . . . . . . . . . 816.3 Diagram illustrating pair production . . . . . . . . . . . . . . . . . . . . . . . . . . 826.4 Diagram illustrating charge migrating through a detector . . . . . . . . . . . . . . 836.5 The charge collection efficiency (Q/Q0), as given by the Hecht relation . . . . . . . 866.6 An example of a PHA spectrum from a CZT detector . . . . . . . . . . . . . . . . 876.7 Two examples of depth distributions generated by the Monte Carlo simulations . . 906.8 Depth distributions from two different incident angles . . . . . . . . . . . . . . . . 916.9 The geometry of the simulations used to generate the depth distribution lists . . . 916.10 Diagram illustrating the way the average depth is calculated . . . . . . . . . . . . . 926.11 A full-energy depth distribution fit to a 20th-order polynomial . . . . . . . . . . . 936.12 The dependence of the model peak shape on different values of µτe and µτh . . . . 966.13 A plot of the µτe and µτh values of the BAT array . . . . . . . . . . . . . . . . . . 976.14 Simulated spectra from 122 keV photons incident on a detector from various angles 996.15 Diagram illustrating the sub-groups of 128 detectors . . . . . . . . . . . . . . . . . 1006.16 Diagram illustrating the affect of different incident angles . . . . . . . . . . . . . . 1026.17 The positions of the 133Ba calibration spectra . . . . . . . . . . . . . . . . . . . . . 1056.18 A typical composite spectrum from a 133Ba source . . . . . . . . . . . . . . . . . . 1066.19 The peak position adjustment factor, as a function of photon energy . . . . . . . . 1086.20 The energy resolution width “sigma” as a function of photon energy . . . . . . . . 1086.21 A measurement of the Crab Nebula spectrum . . . . . . . . . . . . . . . . . . . . . 1106.22 θ verses best-fit photon index for the Crab spectrum measurements . . . . . . . . . 1126.23 θ verses best-fit 15-150 keV flux for the Crab spectrum measurements . . . . . . . 1126.24 Diagram of an m × n response matrix . . . . . . . . . . . . . . . . . . . . . . . . . 1137.1 Distribution of bursts in the Eobspeak–S(2–30 keV)/S(30–400 keV) plane . . . . . . . 128x7.2 Histogram of the fluence ratios of BAT bursts . . . . . . . . . . . . . . . . . . . . . 1298.1 The light curve of the XRF 050416a prompt gamma-ray emission . . . . . . . . . . 1338.2 The spectrum of XRF 050416a prompt gamma-ray emission . . . . . . . . . . . . . 1348.3 Eobspeak probability density distribution for XRF 050416a . . . . . . . . . . . . . . . 1358.4 The light curve of the XRF 050416A x-ray afterglow . . . . . . . . . . . . . . . . . 1368.5 The spectrum of the XRF 050416a x-ray afterglow . . . . . . . . . . . . . . . . . . 1378.6 The light curve of the XRF 050406 prompt gamma-ray emission . . . . . . . . . . 1398.7 The spectrum of XRF 050406 prompt gamma-ray emission . . . . . . . . . . . . . 1408.8 Eobspeak probability density distribution for XRF 050406 . . . . . . . . . . . . . . . . 1408.9 The light curve of the XRF 050406 x-ray afterglow . . . . . . . . . . . . . . . . . . 1428.10 The spectrum of the XRF 050406 x-ray afterglow . . . . . . . . . . . . . . . . . . . 1438.11 The light curve of the XRF 050714b prompt gamma-ray emission . . . . . . . . . . 1448.12 The spectrum of XRF 050714b prompt gamma-ray emission . . . . . . . . . . . . . 1458.13 Eobspeak probability density distribution for XRF 050714b . . . . . . . . . . . . . . . 1468.14 The light curve of the XRF 050714B x-ray afterglow . . . . . . . . . . . . . . . . . 1478.15 The spectrum of the XRF 050714B x-ray afterglow . . . . . . . . . . . . . . . . . . 1498.16 The light curve of the XRF 050824 prompt gamma-ray emission . . . . . . . . . . 1518.17 The spectrum of XRF 050824 prompt gamma-ray emission . . . . . . . . . . . . . 1528.18 Eobspeak probability density distribution for XRF 050824 . . . . . . . . . . . . . . . . 1528.19 The light curve of the XRF 050824 x-ray afterglow . . . . . . . . . . . . . . . . . . 1538.20 The spectrum of the XRF 050824 x-ray afterglow . . . . . . . . . . . . . . . . . . . 1548.21 The light curve of the XRF 050528 prompt gamma-ray emission . . . . . . . . . . 1568.22 The spectrum of XRF 050528 prompt gamma-ray emission . . . . . . . . . . . . . 1578.23 The light curve of the XRF 050215B prompt gamma-ray emission . . . . . . . . . 1598.24 The spectra of XRF 050215B prompt gamma-ray emission . . . . . . . . . . . . . . 1608.25 The light curve of the XRF 050215B x-ray afterglow . . . . . . . . . . . . . . . . . 1618.26 The spectrum of the XRF 050215B x-ray afterglow . . . . . . . . . . . . . . . . . . 1618.27 The light curve of the XRR 050315 prompt gamma-ray emission . . . . . . . . . . 1648.28 The Eobspeak confidence contour for the XRF 050315 prompt gamma-ray emission . . 165xi8.29 The spectrum of XRR 050315 prompt gamma-ray emission . . . . . . . . . . . . . 1668.30 The light curve of the XRR 050315 x-ray afterglow . . . . . . . . . . . . . . . . . . 1668.31 The spectra of the XRR 050315 x-ray afterglow . . . . . . . . . . . . . . . . . . . . 1688.32 The light curve of the XRR 050318 prompt gamma-ray emission . . . . . . . . . . 1708.33 The spectrum of XRR 050318 prompt gamma-ray emission . . . . . . . . . . . . . 1718.34 The light curve of the XRR 050318 x-ray afterglow . . . . . . . . . . . . . . . . . . 1728.35 The spectrum of the XRF 050318 x-ray afterglow . . . . . . . . . . . . . . . . . . . 1748.36 The light curve of the XRR 050525A prompt gamma-ray emission . . . . . . . . . 1768.37 The spectrum of XRR 050525A prompt gamma-ray emission . . . . . . . . . . . . 1778.38 The light curve of the XRR 050525A x-ray afterglow . . . . . . . . . . . . . . . . . 1788.39 The spectrum of the XRR 050525a x-ray afterglow . . . . . . . . . . . . . . . . . . 1798.40 The light curve of the XRR 050219B prompt gamma-ray emission . . . . . . . . . 1818.41 The spectrum of XRR 050219B prompt gamma-ray emission . . . . . . . . . . . . 1828.42 The light curve of the XRR 050219B x-ray afterglow . . . . . . . . . . . . . . . . . 1838.43 The spectrum of the XRR 050219B x-ray afterglow . . . . . . . . . . . . . . . . . . 1848.44 The light curve of the GRB 050401 prompt gamma-ray emission . . . . . . . . . . 1868.45 The spectrum of GRB 050401 prompt gamma-ray emission . . . . . . . . . . . . . 1878.46 The light curve of the GRB 050401 x-ray afterglow . . . . . . . . . . . . . . . . . . 1888.47 The spectrum of the GRB 050401 x-ray afterglow . . . . . . . . . . . . . . . . . . . 1908.48 The light curve of the GRB 050326 prompt gamma-ray emission . . . . . . . . . . 1928.49 The spectrum of GRB 050326 prompt gamma-ray emission . . . . . . . . . . . . . 1938.50 The light curve of the XRR 050326 x-ray afterglow . . . . . . . . . . . . . . . . . . 1948.51 The spectrum of the GRB 050326 x-ray afterglow . . . . . . . . . . . . . . . . . . . 1958.52 The light curve of the GRB 050603 prompt gamma-ray emission . . . . . . . . . . 1978.53 The spectrum of the GRB 050603 prompt gamma-ray emission . . . . . . . . . . . 1988.54 The light curve of the GRB 050603 x-ray afterglow . . . . . . . . . . . . . . . . . . 1998.55 The spectrum of the GRB 050603 x-ray afterglow . . . . . . . . . . . . . . . . . . . 2008.56 The light curve of the GRB 041223 prompt gamma-ray emission . . . . . . . . . . 2028.57 The spectrum of the GRB 041223 prompt gamma-ray emission . . . . . . . . . . . 2038.58 The light curve of the GRB 041223 x-ray afterglow . . . . . . . . . . . . . . . . . . 204xii8.59 The spectrum of the GRB 041223 x-ray afterglow . . . . . . . . . . . . . . . . . . . 2059.1 Fluence histograms of XRFs, XRRs, and GRBs detected by BAT . . . . . . . . . . 2079.2 Eobspeak histograms of XRFs, XRRs, and GRBs detected by BAT . . . . . . . . . . . 2079.3 Duration histograms of XRFs, XRRs, and GRBs detected by BAT . . . . . . . . . 2099.4 T50 duration vs. Eobspeak for bursts detected by BAT . . . . . . . . . . . . . . . . . . 2109.5 Fluence vs. Eobspeak for bursts detected by BAT . . . . . . . . . . . . . . . . . . . . . 2119.6 The Eobspeak-Eiso correlation, with BAT data included . . . . . . . . . . . . . . . . . 2129.7 Composite plot of Swift GRB, XRR, and XRF x-ray afterglow light curves . . . . 2149.8 Composite plot of the afterglow light curves of Swift x-ray flashes . . . . . . . . . . 2159.9 Composite plot of the afterglow light curves of Swift x-ray-rich GRBs . . . . . . . 2169.10 Composite plot of the afterglow light curves of Swift GRBs . . . . . . . . . . . . . 2179.11 Hydrogen column densities nH and photon indices I of Swift afterglows . . . . . . 2189.12 Temporal decay indices β and photon indices I of Swift afterglows at 10 hours . . 2199.13 Temporal decay indices β and photon indices I of Swift afterglows at 8000 s . . . . 2209.14 2 – 10 keV flux from Swift GRB, XRR, and XRF afterglows . . . . . . . . . . . . . 2219.15 2 – 10 keV isotropic luminosity from Swift GRB, XRR, and XRF afterglows . . . . 2229.16 2 – 10 keV isotropic luminosity at 5 hours in the source frame . . . . . . . . . . . . 2229.17 Correlation between Eiso and Liso at 10 hours . . . . . . . . . . . . . . . . . . . . . 2279.18 Correlation between Eiso and Liso at 5 hours in the source frame . . . . . . . . . . 22810.1 Pattern of XRF and GRB afterglow light curves . . . . . . . . . . . . . . . . . . . 23310.2 The light curves of an XRF afterglow, as predicted by the off-jet model . . . . . . 236xiiiChapter 1IntroductionEvery day, a few brief but very bright bursts of gamma rays appear in the sky. They arebrighter than any other gamma-ray emitting object in the heavens, but they last only a shorttime—from as little as a few milliseconds to as long as a few hundred seconds. During those fewseconds, they may vary in brightness quite a bit (growing brighter, then dimmer, then brighteragain) or there may be just a single flash. The spectra (that is, the number of photons as a functionof energy) of these bursts tend to show a gradual increase up to a certain “peak” energy, followedby a gradual decrease. They originate from distant galaxies that are billions of light years away,and they never repeat in the same place twice. These “gamma-ray bursts” have intrigued andbaffled the astrophysics community ever since their discovery over 30 years ago.1.1 The DiscoveryIn 1963, the first of a group of observatories called the Vela satellites were launched intohigh Earth orbit. Equipped with instruments that could detect x-rays (photons carrying betweenabout 0.1 keV and about 10 keV of energy each, though this definition is somewhat fuzzy), gammarays (photons carrying more than about 10 keV of energy each) and neutrons (uncharged particlesoften emitted in nuclear explosions), these satellites were designed to monitor the Earth andthe nearby solar system for signs of violations of the 1963 nuclear test ban treaty between theUnited States and the Soviet Union. In 1967, these satellites detected an intense, brief emission ofgamma-rays from outside the solar system. Between 1967 and 1972, the satellites detected sixteensuch gamma-ray bursts (GRBs). The data was classified until 1973, when Ray Klebesadel, IanStrong, and Roy Olsen of Los Alamos Scientific Laboratory (now LANL) were finally permittedto publish the discovery [51]. Prior to this announcement, no one had proposed a mechanism forsuch a phenomenon, but soon thereafter, hundreds of theories were advanced seeking to explain1the origins of these bursts.1.2 The Extra-Galactic Origin of Gamma-Ray BurstsOne of the hottest debates in the early days of discovery was whether gamma-ray burstscame from within or near our own galaxy or whether they originated much farther away. If theirorigins were local, the total gamma-ray energy generated at their source would be quite a bitsmaller than if they originated from far beyond our own galaxy. If the bursts came from distantgalaxies, the amount of gamma-ray energy given off in a few seconds’ time would have to be aboutthe same as the total mass of our sun converted into energy. It was hard to imagine anything thatcould do that. So the predominant theories hypothesized that gamma-ray bursts resulted fromexplosions or interactions on the surfaces of neutron stars within our own galaxy.An interesting breakthrough came with the launch of the Compton Gamma-Ray Observatoryin 1991. Aboard this orbiting spacecraft was an instrument called the Burst and Transient SourceExperiment (BATSE). BATSE’s primary purpose was to measure the distribution of gamma-raybursts in the sky by determining the locations of a large number of bursts to within a few degrees.It also measured the number of photons coming from these bursts as a function of time (“lightcurves”) and as a function of energy (“spectra”). BATSE consisted of 8 detector modules–one oneach corner of the satellite. Each module housed detectors that could measure gamma-ray photonswith energies ranging from 15-2000 keV [24]. Since it was able to monitor the entire sky at once(except for the portion that was blocked by the earth), BATSE was able to detect thousandsof gamma-ray bursts and measure their spatial distribution. During the course of its nine-yearmission, BATSE made measurements and found positions of over 2700 gamma-ray bursts [62].One of the greatest discoveries was that, contrary to all expectations, gamma-ray bursts didnot cluster around the plane of our galaxy (which would be expected if they came from neutronstars or other objects within our galaxy). Instead, they were distributed very evenly all over thesky (that is, “isotropically”) (see figure 1.1).An isotropic distribution of bursts could mean one of three things:2+90-90-180+1802704 BATSE Gamma-Ray Bursts10-7 10-6 10-5 10-4Fluence, 50-300 keV (ergs cm-2)Figure 1.1: The distribution of BATSE burst positions in galactic coordinates. IfGRBs originated from within our own galaxy, we would expect the bursts to clustermostly along the galactic plane (the line joining the 180◦ and the −180◦ marks). (fromhttp://www.batse.msfc.nasa.gov/batse/grb/skymap/)31. Gamma-ray bursts might still originate from within our own galaxy, but only if BATSE weredetecting bursts only from our own local vicinity within the galaxy (at a distance smallerthan the distance to the edge of the galaxy in any direction).2. Gamma-ray bursts might originate from a large spherical halo surrounding our galaxy.3. Gamma-ray bursts might be “cosmological” in origin. That is, they may originate fromdistances far beyond our galaxy or even our own local cluster of galaxies. On such dis-tance scales, the distribution of matter in the universe is homogeneous, so that an isotropicdistribution would be expected.Another piece of information that can provide a clue about burst origins is a plot of the numberof bursts N with a peak “flux” (that is, a peak number of photons passing through a cm2 of areain 1 second) greater than a certain value vs. the value of that limiting flux P . If the bursts thatBATSE detected all originated fairly close to us within our own galaxy, then we would expect thenumber of bursts to be pretty evenly distributed within the volume in which BATSE was detectingthem. This would lead to a relationship between N and P ofN ∝ P− 32 . (1.1)If this equation were plotted with both axes of the plot having a logarithmic scale, it would looklike a straight line with a slope of -3/2.It turns out that the plot of N vs. P for BATSE bursts has a slope of -3/2 for brighterbursts, but then it becomes less steep for fainter bursts (see figure 1.2). This, together with theisotropic nature of the distribution of bursts in the sky, gave an indication that gamma-ray burstsweren’t a local phenomenon, but it still left open the question of whether they originated in a haloaround our galaxy or at cosmological distances.1.3 The Discovery of AfterglowsThe brief duration of gamma-ray bursts made measurements rather difficult. Gamma-ray telescopes tend to have poor imaging capabilities, and with only a few seconds of gamma-ray4Figure 1.2: A plot of the number of bursts N with a peak flux greater than some value P .If GRBs came from our own local part of the galaxy, the curve would have a slope of -3/2throughout the entire range. A straight line with a slope of -3/2 is shown for comparison.(from http://science.nasa.gov/newhome/headlines/ast06may98 1.htm)emission to work with, it wasn’t possible to determine with high precision the position of a gamma-ray burst, and only a limited amount of information could be extracted. If a counterpart could beidentified at longer wavelengths, it would provide a wealth of additional information. Some modelspredicted a longer-lived “afterglow” at longer wavelengths [56], and an intense search began whichsought to identify such a counterpart [40]. A breakthrough came with an Italian-Dutch satellitenamed BeppoSAX. BeppoSAX bore a complement of instruments that covered a wide energy range,from 0.1 to 600 keV. These included two instruments sensitive to gamma-rays and “hard” (thatis, high energy) x-rays, and that had a wide “field of view” (that is, they could monitor a largeportion of the sky at once) along with a set of telescopes that were sensitive to photons with lowerenergies and which observed a relatively narrow field of view. With this combination, BeppoSAXwas able to monitor the sky for gamma-ray bursts, and when it detected one, the narrow-fieldinstruments could be pointed at the location of the burst to search for a lower-energy counterpart.On 28 February 1997, the first measurement of such an event took place with the detection of agamma-ray burst by the wide-field instruments. Data processing on the ground revealed a pointsource with an error circle 3 arcminutes in radius. This enabled a series of follow-ups with the5narrow field instruments that resulted in the detection of x-ray emission that gradually faded overthe course of a few weeks [18].Since this discovery, afterglows have been seen from dozens of gamma-ray bursts. Theyremain visible for hours and even days, gradually growing dimmer. They’ve been observed notonly in the x-ray band, but also in optical, infrared, and radio bands. X-ray and optical telescopescan obtain a much more precise position of a source, down to the arcsecond level and smaller–precisely enough to determine what distant galaxy the burst is associated with. Furthermore,with x-ray and optical radiation, the presence of emission or absorption features from either theafterglow itself or from its associated galaxy allow observers to obtain a measure of the redshift(z) of the burst, which is a surrogate measure for its distance from us (see section 2.3). In the caseof this first afterglow discovered with BeppoSAX, the redshift was found to be 0.695 [11], placingit far beyond our own galaxy in the outer reaches of the universe. This and subsequent afterglowmeasurements have definitively established the distant (or “cosmological”) origin of at least mostgamma-ray bursts.1.4 The Discovery and Observations of X-Ray FlashesDuring the early days of discovery, x-ray emission was sometimes observed from gamma-raybursts while the bursts were taking place. Some of the bursts observed with the Vela satellites andcontemporaneous instruments saw simultaneous x-ray emission [80, 58, 78]. This emission showedtemporal structure that was similar to that of the gamma-ray emission.Gotthelf et al. searched for transient (i.e., not steady-state) x-ray sources in data collectedby the Einstein Observatory [36]. They found 18 with emission in the 0.2 - 3.5 keV band that hadtemporal and spectral properties similar to those of gamma-ray bursts. They called these sources“x-ray flashes” (XRFs). (The term “x-ray burst” was already taken, being used to describe x-rayemission from thermonuclear explosions on the surface of neutron stars within our galaxy.)Strohmayer et al. studied the gamma-ray bursts detected by the Ginga satellite, which wassensitive to 2-400 keV photons. They found that the spectra of these bursts were consistent with6BATSE GRBs, but that the peak energy distribution extended to below 10 keV (much lower thanthe peak energies of bursts detected by BATSE), hinting that XRFs were simply a continuationof GRBs down into x-ray energies [75]. Results from the BeppoSAX and HETE-2 missions, whichboth include instruments sensitive to both x-rays and gamma-rays, have also indicated that XRFsand GRBs form a single continuum and are not distinct phenomena, with XRFs tending to havelower peak energies than GRBs [50].With the evidence that GRBs and XRFs are simply two extremes of a single population,there is great potential for understanding the nature and origins of the phenomenon by studyingthe differences between the two types of events and investigating the reasons for those differences.Several models have been proposed that seek to explain why some bursts emit mostly gamma-raysand others mostly x-rays. An important key to distinguishing between these models lies in the earlyafterglow emission of the burst. Swift is the first mission capable of observing afterglows withinseconds to minutes of the prompt emission. Previous missions required response times on the orderof hours. Swift’s prompt multi-wavelength capability allows us to gain a clearer understanding ofthe origin and nature of GRBs.1.5 Overview of This ThesisIn this thesis, I explore the properties of prompt and afterglow emission of x-ray flashesdetected by Swift, comparing them with x-ray flashes observed with other missions and withgamma-ray bursts detected by Swift. Chapter 2 reviews the observational characteristics of theprompt and afterglow emission of x-ray flashes and gamma-ray bursts. Chapter 3 describes thenotable observations and contributions made by past missions. An overview of the current stan-dard theoretical models for prompt and afterglow emission is presented in chapter 4, along witha review of different x-ray flash models. In chapter 5, I describe the Swift mission and its com-ponent instruments: the BAT, the XRT, and the UVOT. The methods and procedures involvedin calibrating the Swift Burst Alert Telescope are described in chapter 6. The analysis and datareduction procedures used are discussed in chapter 7. Chapter 8 gives detailed analysis results7of each x-ray flash and gamma-ray burst included in the study. A comprehensive overview ofthe global characteristics of Swift x-ray flashes and gamma-ray bursts is presented in chapter 9.In chapter 10, I conclude by comparing these results to predictions made by various x-ray flashmodels. In the appendices, I describe the derivation of a technique known as “mask-weighting”. Ialso provide a glossary of some of the acronyms and concepts discussed in this dissertation.8Chapter 2Observational Characteristics of Gamma-Ray Bursts and X-Ray FlashesAt a conference a few years ago, I attended a series of lectures and presentations describingmany aspects of gamma-ray burst astronomy. I found that in the case of some GRB theories orideas, most people seemed to agree. With others, everyone seemed to disagree. Even when therewas a general consensus (for example, on the cosmological nature of gamma-ray bursts), therewas always someone with an opposing opinion. Sometimes I would listen to one speaker describeevidence that supported one view, and then another speaker would stand and present evidencefor an opposing view, and I would wonder to myself which of them would end up being right.I commented on this to one of the other participants at lunch one day. With a wry smile hereplied, “What makes you think anybody’s right?” That’s the nature of observational astronomy.There are many theories and many views, but over time and with more observations, a generalunderstanding eventually emerges. Here I present a brief list of some of the observations that haveled to the current understanding of gamma-ray bursts.2.1 NomenclatureBefore describing the characteristics of gamma-ray bursts, we should say something aboutthe naming convention that has been adopted. A gamma-ray burst is designated by the dateon which its emission was detected, as measured in Universal Time (UT).1 This designation iscomposed of the last two digits of the year, followed by the month, followed by the day of themonth. Hence, GRB 990510 was detected on 10 May 1999. In cases where more than one burstwas detected on a given day, a letter is added to distinguish them. GRB 050502a, for example,was the first burst detected on 2 May 2005, and GRB 050502b was the second. Sometimes burststhat have been designated as x-ray flashes are labelled with the prefix “XRF” instead of “GRB”.1Univeral Time is the time of day as measured from a longitude of zero (for example, in Grenwich, England).9So one might see the burst detected on 3 September 2002 labelled as either GRB 020903 or asXRF 020903. An intermediate class of bursts, referred to as “x-ray rich gamma-ray bursts” aresometimes designated with the prefix “XRR” or “XRGRB”.2.2 FrequencyOver the course of its lifetime, BATSE observed about 305 gamma-ray bursts each year(nearly one per day) [62]. With BATSE’s field of view, this leads to an extrapolation of aroundthree bursts per day. BATSE was able to detect bursts that had a total fluence2 above about 10−8ergs/cm2 for photons ranging from 50 and 300 keV. Near the edge of BATSE’s sensitivity limit,bursts continued to be more numerous, indicating that an instrument able to detect bursts with asmaller fluence in this energy range would probably detect many more bursts.BATSE’s energy range did not extend below 30 keV. However, the energy range of HETE-2’s Wide-field X-ray Monitor (WXM) extended down to 2 keV, making it well-suited for detectingx-ray flashes. In its first 1.1 years of “on-time” (time during which it was actually enabled andable to detect bursts), HETE-2 detected 15 XRFs, 20 XRRs and 10 GRBs [70]. Thus, about 1/3of all bursts detected by HETE-2 were XRFs. Considering the WXM’s 60 ◦×60 ◦ field of view, thismeans about 160 XRFs, 220 XRRs, and 110 GRBs take place each year that have a fluence in theWXM’s energy range large enough for the WXM to be able to detect them. Based on BeppoSAXdata, Heise et al. estimated a rate of ∼ 100 XRFs per year [43].2.3 RedshiftWhen two observers are in relative motion, light propagating between them will have adifferent wavelength when measured in each observer’s own reference frame. Light originatingfrom a star that is moving toward us has a shorter wavelength when it is measured here on earth2Fluence is the total amount of energy per cm2 that passes through the region of space where the instrument islocated in a given amount of time. For GRBs, it’s the average intensity of the burst multiplied by the duration ofthe burst. Fluence is often given for a specific range of photon energies.10than it had in the star’s own reference frame. This is referred to as a “blueshift”. On the otherhand, Light originating from a star that is moving away from us has a longer wavelength whenmeasured on Earth than it had in the star’s own reference frame. This is known as a “redshift”.The redshift (z) of an object is defined asz =∆λλ(2.1)where λ is the wavelength of light measured at the source, and ∆λ is the difference between thewavelength measured on earth and the wavelength measured at the source. By this definition,z = 0 would correspond to a source that is neither moving away from nor toward the earth.In 1929, Edwin Hubble noticed that galaxies farther from us tend to emit light that hasbeen “redshifted” more than galaxies closer to us. This redshift can be understood to mean thatgalaxies farther from us are moving away from us at a faster rate than those nearby, which ledto the modern understanding of the expansion of the universe. At the same time, “Hubble’s law”provided a way to directly measure the distance to distant objects. In modern astronomy, weusually cite the “redshift” of a distant object when we want to specify how far away it is (muchthe same way we often say that another city is “an hour away”). Redshift is a surrogate measurefor distance.When we know the distance to a gamma-ray burst, we can calculate the energy emitted bythe burst at its source. We can then test and constrain different models of energy production. Aknowledge of GRB distances also allows us to track the evolution of GRBs in cosmic time (GRBsoriginating at greater distances took place longer ago). As such, redshifts are a valuable piece ofinformation, and obtaining them has been a high priority in GRB studies.Each element radiates at characteristic wavelengths when it absorbs energy. This radiationis like a fingerprint that can be used to identify the element. Afterglow emission at the sourceof a gamma-ray burst often interacts with atoms in the vicinity, causing them to radiate. Thesecharacteristic photons travel through space along with the smoother spectrum of photons fromthe afterglow and arrive at telescopes here on earth redshifted. These characteristic photons canbe picked out of the spectrum and, by noting how many of these photons there are what their11Figure 2.1: Redshifts of bursts detected by Swift as of 31 July 2005 and of bursts detectedbefore the launch of Swiftenergies are, they can be identified and their redshift determined. On the other hand, sometimeslight from the host galaxy provides a measure of the redshift of the burst. The host galaxy canonly be determined if an afterglow is detected, so in either case, the detection of an afterglow isessential. In the 7 years between the first afterglow detection and Swift’s launch, redshifts for 45GRBs were measured or constrained. These redshifts range from 0.0085 all the way up to 4.5.Since Swift’s launch, at least one burst with an even higher redshift has been measured.One of Swift’s primary goals is to provide precise positions very early after the detection ofthe gamma-ray burst, at a time when many more afterglows will be bright enough to detect, sothat many more gamma-ray burst redshifts can be measured. Between it’s launch in November of2004 and 31 July 2005, Swift detected 11 GRBs for which redshifts were subsequently measured.During that same period, The HETE-2 and Integral missions each detected one GRB for which theredshift was successfully measured. Although this isn’t a very large data set, it appears that theredshift distribution of Swift bursts is consistent with the redshift distributions of bursts detectedin the pre-Swift era. (see Figure 2.1).An analysis of GRBs and XRFs detected by HETE-2 and by BeppoSAX show that thedistributions of redshifts for the two populations appear to overlap [70]. The sample isn’t large12Figure 2.2: Redshifts of GRBs and XRFs detected by HETE-2 and BeppoSAXenough, though, to firmly state whether XRFs tend to appear at lower or higher redshifts onaverage than GRBs (see Figure 2.2).2.4 Prompt EmissionMost GRB characteristics can be categorized as either pertaining to the prompt burst emis-sion itself or to the longer-lived afterglow. Here we examine a few of these characteristics.2.4.1 DurationEach gamma-ray burst lasts anywhere from a few milliseconds all the way up to ten minutesor more. Several authors have noted that a plot of the number of bursts vs. the duration of thosebursts reveals two distinct groups (i.e., there is a “bimodal distribution”) [55, 52], (see Figure2.3). This indicates the likelihood of at least two distinct classes of gamma-ray bursts that arisefrom different mechanisms. It has been noted that the long class of bursts tend to be spectrally“softer” than the short class of bursts (meaning that they tend to have more low-energy photonsand fewer high-energy photons). A good way to characterize burst “hardness” is with a fluenceratio, dividing the energy detected in some high-energy band with the energy detected in somelow-energy band. Figure 2.4 shows a plot of a fluence ratio vs. duration for bursts detected by13Figure 2.3: Durations of GRBs Measured by BATSE. In this case, the duration is defined asthe time in which the burst emits between 5% and 95% of its energy, often labelled “T90”.(From http://www.batse.msfc.nasa.gov/batse/grb/duration/)BATSE. Studies have been shown that these two populations differ from each other in other ways,too, lending support to the argument that they arise from distinctly different phenomena [60].XRFs discovered to date appear to be a sub-class of the long, soft GRB population and show littleresemblance to the short, hard population.2.4.2 VariabilitySome Gamma-ray bursts show a great deal of variability within their temporal profiles,with many peaks appearing within a given burst (see Figure 2.5). Other bursts have very simpleprofiles with only a single peak. When multiple peaks are present, those peaks sometimes have verycomplicated structures, varying in individual durations and intensities. Studies have shown that thevariability of bursts is correlated with intrinsic luminosity (or total energy emitted in all directionsper second by the burst): bursts with more variability tend, on average, to be brighter than burstswith less variability (see, for example, [74]). This variability also places strong constraints on thesize of the emission region, as we shall discuss in section 4.1.14Figure 2.4: Durations vs. fluence ratios of GRBs detected by BATSE. The fluence ratio, inthis case, is the ratio of total fluence from 100 – 300 keV photons to total fluence from 50 –100 keV photons.2.4.3 IntensityAt gamma-ray wavelengths, GRBs are the brightest objects in the universe, outshining allother sources. BATSE’s results indicate that their fluences in the 50 – 300 keV energy band canbe anywhere from 10−8 to 10−4 ergs/cm2. It’s likely that bursts with fluences below the detectionthreshold of current and past missions also exist. X-ray flashes tend to have lower gamma-rayfluences simply because most of their energy falls in the x-ray band.15BATSE Trigger 7343-50 0 50 100 150Seconds Since Trigger (990123 : 35216.121)02•1044•1046•1048•104Rate (counts s-1 )Ch: (1: 4)Time Res: 0.512 sBATSE Trigger 7560-50 0 50 100 150Seconds Since Trigger (990510 : 31746.297)1.0•1042.0•1043.0•1044.0•1045.0•104Rate (counts s-1 )Ch: (1: 4)Time Res: 0.512 sBATSE Trigger 7975-20 0 20 40 60Seconds Since Trigger (000131 : 53877.497)8.0•1031.0•1041.2•1041.4•1041.6•104Rate (counts s-1 )Ch: (1: 4)Time Res: 1.024 sBATSE Trigger 7906-10 0 10 20 30 40 50Seconds Since Trigger (991216 : 58021.369)01.0•1052.0•1053.0•1054.0•105Rate (counts s-1 )Ch: (1: 4)Time Res: 0.064 sFigure 2.5: Light curves for some gamma-ray bursts detected by BATSE [33]. A light curveis the number of photons detected by an instrument per second, so it measures the brightnessof the burst as a function of time. Clockwise, from top left: GRB 990123, GRB 990510, GRB991216, and GRB 000131.162.4.4 Spectral CharacteristicsIt was first noted by Band et al. [5] that GRB spectra are well fit in general by a power law(that is, a function of the form y = axk, where k is called the “power law index”) at low energiesand another power law at high energies, joined smoothly by an exponential expression:N(E) = A(E100 keV)αexp(−E(2 + α)Eobspeak)E <((α− β)Eobspeak2 + α)= A[(α− β)Eobspeak(100 keV)(2 + α)]α−βexp (β − α)(E100 keV)βE >((α− β)Eobspeak2 + α)(2.2)whereN(E) is the number of photons/cm2/s/keV,α is the spectral index of the low-photon-energy portion of the spectrum,β is the spectral index of the high-photon-energy portion of the spectrum,Eobspeak is the photon energy at which the greatest amount of energy is radiated, andA is a normalization constant with units of photons/cm2/s/keV.Figure 2.6 shows the shape of this curve for some typical parameters. The values of these param-eters vary over a wide range from one burst to the next, with α varying from -2 to +1, β from-4 to -1.5, and Eobspeak from 30 keV to 1.5 MeV in the BATSE data set, with the greatest numberof bursts having an Eobspeak value at around 250 keV. Spectra from bursts detected by BeppoSAXand HETE-2 have the same overall shape, but with different distributions of α, β, and Eobspeak. Inparticular, while the Eobspeak values for the bursts in the BATSE data set tend to cluster around 250keV, the Eobspeak values of bursts detected by BeppoSAX and HETE-2 are distributed prominently tolower energies. This also appears to be the primary spectral difference between GRBs and XRFs:x-ray flashes tend to have the same spectral characteristics as gamma-ray bursts except that Eobspeaktends to fall at lower energies.17Figure 2.6: A plot of the Band spectral model N(E), in units of photons/s/cm2/keV, withwith α = −0.5, β = −3.0, Eobspeak = 150 keV, and A = 10−3 photon/cm2/s/keV. Above the plotof N(E) is a plot of E2N(E) (in units of ergs/cm2/s), which is proportional to the radiatedenergy as a function of photon energy. Note that the peak of this spectrum is at Eobspeak,indicated by the vertical line.2.4.5 Amati RelationWe noted earlier that Eobspeak is defined as the photon energy at which a plot of radiatedpower (as measured by the observer) vs. photon energy reaches its peak. When we correct Eobspeakfor redshift, the result is the energy of peak power in the reference frame of the source. We refer tothis value as Esrcpeak. When the redshift is known, finding Esrcpeak is simply a matter of multiplyingEobspeak by (1 + z). A number of correlations between Esrcpeak and various other quantities have beendiscovered, and we shall examine several of these.Amati et al. [1] first noted a correlation between Esrcpeak and the isotropic equivalent energyEiso (the total amount of gamma-ray energy emitted by the burst if it emitted equally in alldirections). A good way to tell how well “correlated” (or related) two quantities are is to calculatetheir “correlation coefficient”. Correlation coefficients range from -1 to 1. A coefficient close to 1indicates that when one quantity is large, the other quantity will likely be large, too. A coefficientclose to -1 indicates that when one quantity is large, the other quantity is likely to be small. Acoefficient near 0 indicates that if one quantity is large, there’s no telling what the other quantitywill be. A coefficient identically equal to 1 or -1 means that the two quantities lie exactly on a18Figure 2.7: Correlation between Esrcpeak and Eiso. The correlation coefficient between the twoquantities is 0.902 ± 0.078.straight line, so that you can predict one precisely if you know the other.In this case, the correlation coefficient for log(Esrcpeak) and log(Eiso) was found to be 0.902±0.078, indicating quite a strong correlation. The authors found thatEsrcpeak ∝ Episo, (2.3)where p = 0.52± 0.06. Calculating both Esrcpeak and Eiso requires a knowledge of the redshift of thebursts, so it has only been possible to test this relation on the limited sample of bursts with knownredshifts. Recent measurements have confirmed that XRFs, when combined with GRBs, followthe same correlation, extending the relation to lower peak energies. (see Figure 2.7) [4, 53, 69].2.4.6 Yonetoku RelationThere is also a correlation between Esrcpeak and the peak isotropic equivalent luminosity Liso(the energy per second that is emitted by the source if it emits equally in all directions) (seeFigure 2.8). This correlation was discovered by Yonetoku et al. [83], who found a correlation19Figure 2.8: Correlation between Esrcpeak and Liso. The diagonal line is a plot of equation 2.4.coefficient for log(Esrcpeak) and log(Liso) of 0.957, indicating that it is a tighter correlation than theAmati relation. The authors found that Esrcpeak and peak Liso obeyLiso1052 ergs/2= (4.29 ± 0.15)× 10−5(Esrcpeak1 keV)1.94±0.19, (2.4)if the peak Liso includes photons with energies ranging from 30 keV – 10 MeV in the source frame.The uncertainties shown reflect a 68% probability.In part because this correlation is quite strong, it can be used to estimate gamma-ray burstredshifts when the redshifts cannot be measured directly. This procedure is described in section 7.3.2.5 Afterglow Emission2.5.1 Spectral CharacteristicsNearly every GRB for which x-ray observations have been promptly conducted has exhibitedan x-ray afterglow. Usually the number of photons per given photon energy follows a power lawmodulated by some absorption from intervening hydrogen atoms:dNdE∝ E−I · exp[nH · σ(E)], (2.5)20where I is called the photon index, nH is the number of hydrogen atoms per cm2 between thedetector and the source, and σ(E) is the photo-electric cross-section of hydrogen (which is just aknown function of photon energy).The measured photon indices lie in an approximately Gaussian distribution with a meanvalue of 1.95±0.03 and a standard deviation σI of ∼ 0.4. Hydrogen column densities (nH) usuallytend to be consistent with the measured values of our own galaxy in the direction of the burst butin some cases have been significantly higher [27].There are indications that the spectral properties of XRFs are similar to those of GRBs.Piro found that the fluxes of XRFs 12 hours after the bursts are consistent with those of GRBs 12hours after the bursts [64]. XRR041006 exhibited a photon index of 2.1 and XRF030723 a photonindex of 1.9+0.3−0.2, both near the center of the distribution described by Frontera [38].2.5.2 Temporal CharacteristicsPrior to the early observations made possible by Swift, Gamma-ray burst afterglow fluxeswere observed to usually follow a power law in time (Nphoton ∝ tβ). The power law indices β forthe various bursts that have been observed are distributed in a roughly Gaussian distribution witha mean value of −1.30± 0.02 and a standard deviation σβ of about 0.35 [27].Some afterglow light curves have exhibited “breaks” (or changes in shape from one powerlaw index to another) after about 8 hours or so. Examples of these include GRB 010214, whichgave signs of a power law index β of about 1 early on and of 2.1 later, and also GRB 010222 whichactually showed signs of increasing in intensity with a power law index β of -0.8 before beginningto diminish [27].XRF afterglow light curves seem to lie firmly within this distribution. Granot cites a decayindex of 1.0 ± 0.1 for XRF 030723 between 3.2 and 13.2 days after the burst [38]. Amati found adecay index of 1.33+0.02−0.03 for XRF 020427 [2].212.5.3 Jet BreaksMany gamma-ray burst afterglow light curves have been found to exhibit a “break” wherethe power law index changes abruptly. Sometimes this break is “achromatic”, occurring at allwavelengths simultaneously, as opposed to “chromatic” breaks that would result from the evolutionof the afterglow spectrum with time [71]. These breaks often appear on the order of a day followingthe burst. The afterglow is believed to originate in a relativistic outflow of material that forms anarrow “jet” emanating from the burst site. This material emits radiation in a narrow cone dueto relativistic beaming. As the material decelerates, the emission cone grows wider. When thecone width reaches an angle comparable to the width of the jet, the observer begins to observe theedge of the jet, which causes the light curve to steepen. The relationship between the time of thissteepening (the so-called “jet break”) and the jet angle was derived by Sari et al. [72, 34]:∆θ = 0.161( tjet,d1 + z)3/8( nηγEiso,52)1/8, (2.6)where tjet,d is the jet break time (measured in days), z is the burst redshift, n is the particledensity in the region surrounding the burst, ηγ is the fraction of the burst’s kinetic energy that isemitted in gamma-rays, and Eiso,52 is the isotropic-equivalent energy of the burst, in units of 1052ergs. Frail et al. first noted these jet breaks and derived jet angles for several bursts [26]. Theyfound a correlation between the isotropic-equivalent energies Eiso and the jet angles ∆θ, leadingto the relationshipEiso ∝ ∆θ2. (2.7)Since the isotropic-equivalent energy Eiso is related to the actual total energy Eγ byEγ = Eiso(1 − cos(∆θ)), (2.8)(assuming a uniform flux distribution within the jet), where ∆θ is the half opening angle of the jet,Frail et al. found that the total gamma-ray energy emitted by the bursts Eγ fit within a relativelynarrow distribution (see Figure 2.9). Bloom et al. expanded on this data set and found that thepeak of the Eγ distribution is 1.33 × 1051 ergs with a width of 0.07 × 1051 ergs [12]. Lamb et22Figure 2.9: Eiso and Eγ for bursts studied by Frail et al. They found that when Eiso wascorrected for jet angle (resulting in Eγ), they were much more tightly distributed. Boxes witharrows in them indicate that those bursts either have an upper limit (left-pointing arrow) ora lower limit (right-pointing arrow) for Eγ .al. noted that the isotropic energies of x-ray flashes are so small that even an isotropic outflowwouldn’t be sufficient to enable their values of Eγ to be (1.33 ± 0.07)× 1051 ergs [54].2.5.4 Ghirlanda RelationGhirlanda et al. [34] discovered a correlation between the total gamma-ray energy Eγ andEsrcpeak (see Figure 2.10):Esrcpeak ∝ Epγ , (2.9)where p = 0.706 ± 0.047. They found that the correlation coefficient between log(Esrcpeak) andlog(Eγ) was 0.94, so that this correlation is stronger than the correlation found by Amati betweenlog(Esrcpeak) and log(Eiso).2.5.5 Dark BurstsAs noted earlier, it appears that for nearly every GRB, there is a corresponding x-ray af-terglow. On the other hand, it appears that only a fraction of bursts have corresponding opticalafterglows (about 50% of the well-localized bursts detected by BeppoSAX). Those without de-tected optical afterglows have been dubbed “dark bursts”. A few possible explanations have been23Figure 2.10: Correlation between Esrcpeak and Eγ . The correlation coefficient for these twoquantities is 0.94, indicating a stronger correlation than that of Esrcpeak and Eiso (see Figure 2.7).proposed for them:• There may be a significant amount of dust between the source and the observer, obscuringthe optical emission but allowing the x-ray emission to pass through.• They may lie at high redshifts (z > 7), so that their optical emission is redshifted down to awavelength at which it is susceptible to absorption by intervening hydrogen atoms (so called“Lyman-α” absorption).• They may come from regions with less material surrounding the burst.It has been found that dark bursts also tend to have relatively dim x-ray afterglows (abouta factor of 5 times less flux on average than those with optical afterglows). This argues againstthe dust absorption hypothesis, since dust has little effect on x-ray photons. Furthermore, thosedark bursts for which redshifts have been measured do not lie at significantly high redshifts [20].HETE-2 was able to begin prompt optical follow-up of XRF 021211 within 90 seconds of the burst,24and it detected a faint optical afterglow that would have been below the detection threshold if ithad been observed any later. This burst would have been characterized as a dark burst if not forthe prompt follow-up [19].25Chapter 3Previous XRF ObservationsOver the past 25 years, a number of different missions have observed X-Ray Flashes. In thischapter, we review the results of these observations.3.1 EinsteinThe Einstein observatory, which was launched into orbit in November of 1978, had thedistinction of being the first imaging x-ray telescope in space. Among the instruments aboard theobservatory was a Wolter Type I grazing incidence telescope sensitive to photons with energiesranging from 0.1 – 4 keV. This instrument included four different detectors, each of which couldbe rotated into the focal plane: an Imaging Proportional Counter, a High Resolution Imager, aSolid State Spectrometer, and a Focal Plane Crystal Spectrometer. A non-imaging instrumentcalled the Monitor Proportional Counter was co-aligned with the x-ray telescope and was sensitiveto photons with energies ranging from 1.5 – 20 keV. An Object Grating Spectrometer, used inconjunction with the High Resolution Imager, rounded out the complement of instruments. Themission ended in April of 1981.In 1996, Gotthelf et al. conducted a search of the data acquired by the Imaging ProportionalCounter to identify fast x-ray transient sources [36]. They were able to detect transients that variedon time scales on the order of 10s of seconds down to a limiting sensitivity of 10−11 ergs/cm2. Theirsearch resulted in 18 candidates (which they named ”X-Ray Flashes”) that had spectra consistentwith the x-ray counterparts of GRBs previously reported. They noted that while these XRFs werenot coincident with known GRBs, the instrument was never coincidentally pointed at a knownGRB during the lifetime of the mission.The X-ray flashes identified by this search were distributed isotropically on the sky, con-sistent with BATSE’s results, and in particular, were not preferentially found in the direction of26nearby galaxies. The logarithmic plot of fluence vs. the number of bursts with a fluence greaterthan that fluence showed a differential slope of -2.5, which the authors speculated could be con-sistent with either a local or an extragalactic population. The XRFs tended to exhibit rise timesshorter than their decay times and lasted anywhere from a few seconds to a few tens of seconds.As such, they were shorter than the prompt x-ray counterparts of GRBs previously reported. Thenumber of events detected in this search implied a rate of about 2 × 106 per year.3.2 GingaGinga was a Japanese x-ray astronomy mission launched in February of 1987 and lastinguntil November of 1991. It included three instruments: the Large Area Proportional Counter(LAC), which was sensitive to photons with energies ranging from 1.5 – 37 keV, the All SkyMonitor (ASM) sensitive in the 1 – 20 keV range, and the Gamma-ray Burst Detector (GBD).The primary purpose of the GBD was to investigate the spectra of gamma-ray bursts in the x-rayregime. It consisted of two parts: a proportional counter which was sensitive to 2 – 25 keV photons,and a scintillation counter which was sensitive to 15 – 400 keV photons. The GDB had a field ofview of π steradians, or about 1/4 of the sky.Strohmayer et al. [75] identified 22 bursts that occurred between March of 1987 and Octoberof 1991 for which the instrument measured spectra that could be reliably analyzed. They notedthat these bursts resembled BATSE bursts in duration and overall spectral shape (that is, theBand function provided a good fit to the data), but the peak energies Eobspeak extended to lowervalues than those of the BATSE bursts. In the BATSE data set, the Eobspeak distribution peaksat about 300 keV and extends down to about 30 keV. Also, α > 0 for 15% of the bursts. Inthe Ginga data set, the Eobspeak distribution peaks somewhere below 20 keV and extends down to2.5 keV (see Figure 3.1), and α > 0 for 40% of the bursts. The authors pointed out that theBATSE data set and the Ginga data set both display a correlation between Eobspeak and α. Theyalso noted that the low energy emission of the Ginga bursts lasts longer than the high energyemission, suggesting overall spectral softening, as was also seen in BATSE bursts. They suggest27Figure 3.1: Eobspeak for Ginga GRBsthat perhaps the differences between the BATSE and Ginga Eobspeak and α distributions are simplya selection effect—each mission saw the portion of the burst population to which they were mostsensitive.Although they didn’t make a distinction between gamma-ray bursts and x-ray flashes,Strohmayer et al. did define a quantity that they used to characterize the hardness of each burst:the ratio of burst energy in the x-ray band (defined as 2 – 10 keV) to burst energy in BATSE’senergy range (50 – 300 keV). Most of the bursts have ratios below about 0.1 (indicating spectrallyhard GRBs like those detected by BATSE), but the distribution also extends to higher ratios, evenabove 1.0 for one burst (see Figure 3.2).3.3 BeppoSAXIn April of 1996, an Italian-Dutch x-ray observatory called BeppoSAX was launched intoorbit. Its complement of instruments covered more than three decades of energy from 0.1 – 300keV and enabled it to provide arcminute localizations of gamma-ray bursts, permitting follow-up observations at longer wavelengths. The spacecraft carried a set of narrow-field instrumentsthat included 4 x-ray telescopes, each with a Low or Medium Energy Concentrator Spectrometer28Figure 3.2: Ratio of 2 – 10 keV x-ray energy to 50 – 300 keV gamma-ray energy for Ginga GRBs(LECS or MECS) sensitive to photons up to 10 keV, a High Pressure Gas Scintillator ProportionalCounter (HPGSPC) sensitive to 4 – 120 keV photons, and a Phoswich Detection System (PDS)which covered a range of 15 – 300 keV. In addition, the observatory carried two Wide FieldCameras (WFCs) sensitive to 2 – 30 keV photons. These cameras used a coded aperture imagingsystem and were mounted perpendicular to the axis of the narrow field instruments and pointedin opposite directions to each other. Each camera had a field-of-view of 20◦ × 20◦. These camerasmade it possible for scientists on the ground to determine the position of a gamma-ray burst to2 – 3 arcmin within 4 – 5 hours. Finally, the lateral shields of the PDS served a dual role as theGamma-Ray Burst Monitor (GRBM) and could detect photons with energies ranging from 60 –600 keV. The observatory re-entered the Earth’s atmosphere in April of 2003.Heise et al. reported that among the sources imaged by the WFCs was a class of fast x-raytransients with durations less than 1000 s that were not “triggered” (that is, detected) by theGRBM [43]. This became their working definition of x-ray flashes. Seventeen such sources wereidentified by the authors, who estimated a rate of about 100 per year for such events. Their peakfluxes tended to fall in the range of 10−8 – 10−7 ergs/cm2/s and they tended to last anywherefrom about 10 seconds to about 200 seconds, similar to the durations of gamma-ray bursts. Thespectra of x-ray flashes were generally consistent with a single power law with photon indices29ranging from 3 to 1.2, attenuated by galactic hydrogen absorption. Two of the flashes observed bythe BeppoSAX WFCs showed indications of spectral breaks between 30 and 50 keV. Kippen et al.noted that the temporal structure, spectra, and spectral evolution of these flashes were similar tothe x-ray properties of gamma-ray bursts.In some instances, these XRFs were observed simultaneously with BATSE. In these cases,a comparison of the x-ray (2 – 10 keV) fluence (from the WFCs) and the gamma-ray (50 – 300keV) fluence (from BATSE) revealed that the XRFs tended to exhibit an x-ray/gamma-ray ratioof between 0.2 to 10, with an average value of about 2, whereas gamma-ray bursts had ratios ofbetween 0.004 to about 1, with an average value of about 0.1. Joint spectral analysis with BATSEand BeppoSAX data showed that while these XRFs had significantly lower Eobspeak values thanGRBs, the distribution of their low-energy photon indices (α) was similar to that of GRBs [50].3.4 HETE-2HETE-2 was launched in October 2000 and is still in operation. It has the distinction of beingthe first satellite dedicated to observing GRBs and is able to localize bursts to anywhere from 10sof arcmin to 10s of arcsec with delay times between about 10 seconds to a few hours. Instrumentsaboard the observatory give an energy coverage of 2 – 400 keV. These include the French GammaTelescope (FREGATE), which is a cleaved NaI crystal read out by a photomultiplier tube andguarded by a graded-z shield. The FREGATE has a field of view of 3 steradians and is sensitiveto photons with energies between 6 and 400 keV. It triggers on gamma-ray bursts, alerting thecommunity to their detection. HETE-2 also carries a Wide-field X-ray Monitor (WXM) thatconsists of two units, each with a 1-dimensional position sensitive proportional counter and codedaperture mask. The two units are oriented perpendicular to each other, are sensitive to 2 – 25 keVphotons, and provide burst positions to 10 arcmin. The final instrument aboard HETE-2 is theSoft X-ray Camera (SXC), which consists of X-ray CCDs and a fine coded aperture mask. SXChas an energy range of 2 – 14 keV and a field of view of 0.9 sr.Sakamoto et al. [70] studied the properties of x-ray flashes detected and measured by HETE-302. For their analysis, they distinguished between gamma-ray bursts (GRBs), x-ray rich gamma-raybursts (XRRs) and x-ray flashes (XRFs) using the fluence ratio SX(2 – 30 keV)/Sγ(30 – 400 keV):log[SX(2 – 30 keV)/Sγ(30 – 400 keV)] ≤ −0.5 GRB−0.5 < log[SX(2 – 30 keV)/Sγ(30 – 400 keV)] ≤ 0 XRR (3.1)log[SX(2 – 30 keV)/Sγ(30 – 400 keV)] > 0 XRFThey selected bursts that were recorded between January 2001 and September 2003 by both theWXM and FREGATE which were sufficiently bright to provide adequate statistics for spectralanalysis. Forty-five events met these criteria.33% of the bursts they identified qualified as XRFs. All of the events (GRBs, XRRs, andXRFs) showed a positive correlation between S(2-30 keV) and S(30-400 keV), indicating that whenx-rays were prominent, so were γ-rays. The distribution of low-energy photon indices α for XRFsdid not differ significantly from that of GRBs, but Eobspeak tended to be lower for XRFs than forGRBs (∼ 25 keV or smaller for XRFs and 150 keV on average for GRBs). No correlation betweenα and Eobspeak was seen. The overall distribution of Eobspeak values was consistent with that of BATSE,with an excess at low energies and a deficit at high energies, resulting from the different sensitivitiesof the instruments. A plot of the number of XRFs detected vs. XRF brightness was consistentwith a homogenous distribution in space, indicating that XRFs may tend to lie at lower redshifts(z ≤ 0.2) than GRBs. The number of XRFs detected was consistent with a rate of about 160 peryear.3.5 XRF Afterglow ObservationsUntil the launch of Swift, only a handful of afterglows associated with x-ray flashes werereported. Here we review those events for which relatively good measurements were made.3.5.1 XRF 011030The first x-ray flash for which an afterglow was reported was XRF 011030. It was detectedand observed by BeppoSAX [31, 44], and its afterglow was identified with the Chandra x-ray31observatory [41].The prompt emission lasted 1400 s (longer than most GRBs) and had a spectrum charac-terized by a photon index of 1.9± 0.1. Eobspeak was constrained to less than 40 keV. The measured 2– 28 keV fluence was 9×10−7 ergs/cm2, and the measured peak flux was 8.3×10−9 ergs/cm2/s inthe same energy range [44]. These spectral parameters fit nicely within the usual range for GRBs.The x-ray afterglow was first detected by Chandra 29 days after the burst. Its spectrumwas found to be consistent with a power law with a photon index of about 1.45 (which is withinthe norm for GRBs), attenuated by a hydrogen column density consistent with galactic values. Itsintensity was measured to be 2.4 × 10−13 ergs/cm2/s in the 2 – 10 keV band 29 days after theburst [41]. Its flux was found to be decreasing over time at a rate consistent with a power law decayindex of about -2.00 [28]. Extrapolating this decay back in time to the time of the burst yields anintensity consistent with that of the prompt emission. This index is a bit steeper than most GRBdecay indices. The redshift of this burst was constrained to be < 3 based on the probable hostgalaxy, which was faint and blue, similar to most GRB host galaxies [28].3.5.2 XRF 020427XRF 020427 was detected and observed by BeppoSAX, which also identified and measuredits afterglow. Chandra provided additional follow-up observations of the afterglow, making possiblethe measurement of the decay rate [2].The prompt emission consisted of two peaks with a total duration of about 60 s. Thespectrum was consistent with a power law photon index of 2.09+0.23−0.21 with an attenuating hydrogencolumn density consistent with galactic values. By fixing α at 1, Eobspeak was constrained to beless than 5.5 keV. The 2 – 28 keV fluence and peak flux were (5.8 ± 0.4) × 10−7 ergs/cm2 and(3.0±0.4)×10−8 ergs/cm2/s, respectively, both of which are similar to those of BeppoSAX GRBs.The first measurement of the afterglow from XRF 020427 was made by BeppoSAX approx-imately 8 hours after the burst. Assuming a galactic column density, the afterglow photon indexwas found to be 2.0+2.2−1.1. Chandra measured a photon index of 1.5+0.5−0.5 about 9 days later. Both32measurements are consistent with the spectrum of the second pulse of the prompt emission, whichexhibited a photon index of 2.22+0.31−0.25. The time decay index was found to be −1.30+0.09+0.10, whichextrapolates back to the second pulse of the prompt emission. The intensity, spectrum, and decayare all consistent with normal GRBs [27].Bloom et al. [13] were able to constrain the redshift of this burst to z < 2.3 based on thespectrum of the host galaxy. If the redshift is less than 0.1 or 0.2, then the Eiso and Esrcpeak of theburst satisfy the Amati relation quite nicely.3.5.3 XRF 020903XRF 020903 was detected and observed by HETE-2. It was the first x-ray flash for which adefinitive redshift was measured. The duration of the prompt emission was about 10 seconds. Itsspectrum was consistent with a photon index of −2.4+0.5−0.6 with a galactic hydrogen column density.Eobspeak was constrained to 2.4+1.2−0.7 keV. The 2 – 10 keV fluence and peak flux were (5.9±1.4)×10−8ergs/cm2 and (14.7± 5.3)× 10−9 ergs/cm2/s, respectively [69]. An x-ray afterglow was not found,but an optical afterglow was detected during measurements that began about 21 hours after theburst [73]. The afterglow was found to have a decay index of about 1. A redshift of 0.236 wasmeasured for the host galaxy. At this redshift, the calculated Esrcpeak and Eiso are consistent withthe Amati relation.3.5.4 XRR 030723XRR 030723 was detected and observed by HETE-2 on 23 July 2003, with x-ray afterglowmeasurements made two days later using Chandra observations. Although some observers classifiedit as an x-ray flash, by the working definition used by the HETE-2 team, it technically qualifiedas an x-ray rich gamma ray burst with log[SX(2 – 30 keV)/Sγ(30 – 400 keV)] = 0.9. It was thefirst XRF (or borderline XRF) for which a joint study of prompt and x-ray afterglow emission waspresented [16].The duration of the prompt emission was approximately 23 s [65]. The spectrum wasconsistent with a power law with photon index of 1.93+0.17−0.15. It was not possible to constrain the33value of Eobspeak. The total fluence of the burst (2 – 30 keV) was 2.4 × 10−7 ergs/cm2 [70], and thepeak flux (7 – 30 keV) was > 3 × 10−8 ergs/cm2/s [65].The afterglow was first detected in the optical wavelengths 24 hours after the burst, followedby an x-ray detection 51 hours after the burst. The x-ray afterglow spectrum was well fit by apower law with a photon index of 1.9 ± 0.3 and a hydrogen absorption column density consistentwith galactic values [70]. Between 51.4 and 59 hours after the burst, the average x-ray flux was(2.2± 0.3)× 10−14 ergs/cm2/s in the 0.5 – 0.8 keV band. The afterglow decayed with a power lawindex of 1.0 ± 0.1. At optical frequencies, the afterglow decayed with an index of about 0.9 untilabout 30 – 50 hours after the burst, after which time, its decay steepened to an index of about2 [16]. The optical afterglow showed signs of rebrightening at 7 days after the burst [29]. Fynboet al. were able constrain the redshift to < 2.3 [30].3.5.5 XRF 040701Another HETE-detected x-ray flash for which an afterglow was measured was XRF 040701.Follow-up observations with Chandra detected the x-ray afterglow 8 days later [25]. The promptemission lasted roughly 60 s, with a spectrum consistent with a power law of photon index 2.394±0.3. The total fluence was reported to be (4.5 ± 0.8) × 10−7 ergs/cm2 [6]. The afterglow wasfound to decay with a power law index of ∼ 1 [25]. A redshift of 0.2146 was found from the hostgalaxy [49].34Table 3.1: Prompt emission properties of XRFs detected prior to Swift’s launchPrompt EmissionDuration Photon Index Eobspeak Fluence (ergs/cm2)XRF 011030 1400 s 1.9 ± 0.1 < 40 keV 9 × 10−7 (2 – 28 keV)XRF 020427 60 s 2.09+0.23−0.21 < 5.5 keV (5.8 ± 0.4)× 10−7 (2 – 28 keV)XRF 020903 10 s 2.4+0.6−0.5 2.4+1.2−0.7 keV (5.9 ± 1.4)× 10−8 (2 – 10 keV)XRF 030723 23 s 1.93+0.17−0.15 unconstrained 2.4 × 10−7 (2 – 30 keV)XRF 040701 60 s 2.395 ± 0.3 unconstrained (4.5 ± 0.8)× 10−7 (2 – 25 keV)Mean GRB values 50+550−48 s 1.5 to 4 (β) 30 keV to 1.5 MeV 10−4 to 10−8 (50 – 300 keV)Table 3.2: Afterglow emission properties and redshifts of XRFs detected prior to Swift’s launchAfterglow Emission RedshiftFirst Observation Photon Index Decay IndexXRF 011030 T + 29 days ∼ 1.45 2 < 3XRF 020427 T + 8 hours 1.5 ± 0.5 1.30+0.10−.09 < 2.3XRF 020903 T + 21 hours unknown ∼ 1 0.236XRF 030723 T + 24 hours 1.9 ± 0.3 1.0 ± 0.1 < 2.3XRF 040701 T + 8 days unknown ∼ 1 0.2146Mean GRB values 1.95 ± 0.4 1.30 ± 0.35 0.0085 to 4.535Chapter 4Gamma-Ray Burst Theory4.1 The Prompt EmissionThose who study gamma-ray bursts often refer to the gamma-ray burst itself as the “promptemission” (as opposed to the “afterglow emission” that starts later and lasts longer). The char-acteristics of the GRB prompt emission place some stringent constraints on GRB models. Asalready noted, whatever produces these bursts must be capable of generating an enormous outputof gamma-ray energy (∼ 1051 ergs or so) in a matter of seconds. Another important condition is“compactness”. The duration of the individual peaks in the gamma-ray emission place an upperlimit on the size of the emission region. This size must be equal to or smaller than the distancethat a photon could travel in the duration of the burst peak, c · δt. For a peak with a duration of1 ms (as is sometimes observed), the emitting region must be no larger than 300 km in size.A third requirement from observation has to do with “optical thickness” (also sometimescalled “optical depth”) τ . The optical thickness of a region is a measurement of how likely it isthat a photon might be able to pass through that region without being absorbed or scattered.A region’s optical width is defined as the thickness of the region divided by the average distancea photon can travel within that region before it gets absorbed or scattered. Thus the opticalthickness of a region could be described as its thickness in units of “average photon distances”.If τ = 3 for a particular region, then the region is three times as wide as the average distance aphoton would be able to travel before being absorbed or scattered. So usually a good dividingpoint between a region that is “optically thick” and one that is “optically thin” is about τ = 1.Since photons of different energy have different probabilities of being absorbed or scattered, theoptical thickness of a region will depend on the energy of the photons.As we’ve seen, the emission region of a gamma-ray burst must be very compact. As a36result, the gamma-ray photon density must be very high (considering the number of photons thatare emitted). At such a density, we would expect many photons to annihilate each other, creatingelectron-positron pairs in their place. This would radically reduce the number of photons above 511keV (source frame energy). This reduction isn’t observed, though. The best explanation is thatthe emitting material is moving at relativistic speeds. At such speeds, “relativistic beaming” wouldtake place (that is, all of the photons would be emitted in the direction of motion within a narrowbeam), so that the angle between emitted photons (in the observer reference frame) would be muchsmaller and the probability of pair production much lower. In order for the pair production to besuppressed to the extent that measurements demand (the pair production optical depth τ± mustbe . 1 for photons of energy ∼ 100 MeV), the Lorentz factor Γ (pronounced “Gamma”) of theemitting material must be > 100. The Lorentz factor Γ is given byΓ =1√1 − v2/c2, (4.1)where v is the speed of the material and c is the speed of light. As v gets closer and closer to c, Γgets larger and larger. A Γ greater than 100 means that the material must be traveling in excessof 0.99995 times the speed of light.4.1.1 The Fireball ModelOne model that satisfies most of these requirements is the so-called “fireball” model. Thefireball model has become the “standard” model for gamma-ray burst emission. The treatmentpresented here follows the review given by Piran [63].In the fireball model, a compact, hidden “inner engine” of some kind produces a relativistic“fireball”—a plasma of electrons, photons, and a small concentration of baryons (such as protonsand neutrons)—that propagates outward. This fireball is believed to be made up of a series of“shells”, each having its own well-defined thickness. The number of shells may vary from burst toburst. In some variations of the model, a Poynting flux is also involved. These shells undergo twodistinct phases: an initial energy-dominated phase and a later matter-dominated phase. Duringthe energy-dominated phase, the photon concentration in the shells is extremely high, resulting37in a pair-production optical thickness τ± that is much greater than 1, so that radiation cannotescape. Under these conditions, the plasma can be termed a “photosphere”. Unable to releaseany of their internal energy in the form of radiation, the plasma shells cool adiabatically as theyexpand, causing them to accelerate to ultrarelativistic speeds. During this phase, the Temperature(T ) and Lorentz factor Γ of the shells change as the radius(R) increases according to the followingrelationships:T ∝ R−1, Γ ∝ R. (4.2)Throughout this phase, the thickness of the shells remains constant (in the observer frame).Throughout this expansion and acceleration, the internal energy of the plasma (mostly inthe form of photons that are continually being converted into and from electron-positron pairs)is converted into bulk kinetic energy, mostly carried by the more massive (though relatively few)baryons. At some point, the majority of the energy of the fireball is kinetic, and Γ becomesconstant with time, settling at a value of approximately E/mc2 (where E is the energy of thefireball and m is the mass of the baryons within the fireball, which make up most of its mass),so that the fireball begins to coast along at a constant speed. This marks the beginning of thematter-dominated phase. This transition typically occurs at a radius of about 109cm, or about oneand a half times the radius of the earth. As the fireball shells coast along, variations in Γ causethem to expand, and the thicknesses of the shells begin to grow linearly with their radii.At some point, the fireball cools sufficiently that the plasma becomes optically thin. Theradius at which this takes place is known as the “photospheric radius”. The presence of the smallconcentration of baryons delays this transition until after the fireball has become matter-dominated.This radiation phase begins at a radius Rrad phase of aboutRrad phase = 6 · 1013cm√E1052ergs( η100)−1, (4.3)where E is the energy of the fireball, and η is the ratio of energy to baryon mass (E/mc2). Afterthis point, any radiation that exists or is generated inside the fireball freely escapes.The presence of baryons serves a crucial role in the fireball model. Without them, the plasmawould become optically thin much earlier (at a radius of typically 1010cm instead of 1014cm). The38Figure 4.1: The fireball and the internal/external shock scenario [33]. At a time T=0, the“internal engine” emits a series of plasma shells which begin to expand outward. When theshells reach a radius of about 1014cm, they begin colliding with one another, resulting in theprompt emission of the gamma-ray burst. At a radius of about 3×1016cm, they begin to slowdown as they interact with material in space. As they slow down, they radiate at x-ray andother wavelengths, giving rise to the afterglow.plasma would not become matter-dominated, and the radiation emitted would have a “thermal”spectrum, which looks different from the actual spectrum we observe from gamma-ray bursts. If,on the other hand, the concentration of baryons were too large, the plasma would not be able toachieve the ultrarelativistic speeds that we have shown are a necessary condition for generatinggamma-ray bursts. This is because when the fireball becomes matter-dominated, with its internalenergy entirely converted into kinetic energy, this kinetic is proportional to both the baryon massm and the Lorentz factor Γ. For a fixed kinetic energy, a greater mass m results in a smallerLorentz factor Γ. So the model requires that some baryons be present, but not too many.4.1.2 Internal ShocksFor reasons that are not well understood, the internal engines of some bursts apparently emitmany shells while others emit few. If there are multiple shells, each with a slightly different Lorentzfactor Γ, faster shells will overtake slower shells and collide with them, producing “internal shocks”that propagate forward into the slower shell and backward into the faster shell. These shocks39facilitate the conversion of some of the kinetic energy into radiation. They are designated “internal”because they result from interactions between different shells within the fireball, as opposed to“external shocks” which result from interactions between these shells and the local environmentand give rise to the afterglow emission, as we will discuss in the next section. Each collision resultsin an observed peak in the gamma-ray burst light curve with a duration t determined by thethickness w of the shells involved. The duration of the entire burst is determined by the widthof the entire complex of shells, W . The variability of the “inner engine” determines the numberof shells and their widths, and hence the number and widths of the burst peaks. In every otherway, the inner engine is completely hidden from direct observation. The radius at which the GRBemission occurs isRprompt ≈ 1014cm(∆R1010cm)(Γ100), (4.4)where ∆R is the initial separation between the shells in the observer’s reference frame, and Γ isthe Lorentz factor of the inner shell (assumed to be slightly larger but on the same order as thatof the outer shell). The typical radius of 1014 cm is a littler larger than the radius of the orbit ofJupiter. In order to explain gamma-ray bursts, the Lorentz factors of these shells must be largeenough to limit the optical depth due to pair production, yet small enough to produce the shortdurations and short timescale variability that we observe:570(ζ0.01)−14(T10s)−14≤ Γ ≤ 2800(ζ0.01)−12(T10s)−38(l1018cm2)38, (4.5)where ζ = T/∆T , T is the total duration of the burst, ∆T is the typical duration of the individualpeaks in the burst, and l is a constant known as the Sedov length which depends on the energy ofthe burst and the density of the interstellar medium. These constraints on Γ in turn constrain theradius at which the GRB can be produced by an internal shock:1015cm(ζ0.01)12(T10s)12≤ Rprompt ≤ 2.5 · 1016cm(T10s)14(l1018cm2)34. (4.6)The amount of energy converted to radiation by the forward shock and the reverse shock arecomparable to each other. The most likely conversion process is synchrotron emission, whereby arelativistic electron interacts with a magnetic field and emits radiation of a characteristic frequency.40The photon energy of typical synchrotron photons resulting from an internal shock Einternal (inthe observer’s reference frame) isEinternal =~qeBmecΓ2eΓ, (4.7)where qe is the charge of an electron, B is the strength of the magnetic field, me is the mass of anelectron, Γe is the Lorentz factor of the shocked electron, Γ is the Lorentz factor of the shell throughwhich the shock is propagating, ~ is Dirac’s constant, and c is the speed of light. For typical valuesof these variables, Einternal is about 200 keV, which is consistent with the magnitude of Eobspeakvalues observed in gamma-ray burst spectra. This internal shock process is capable of convertingabout 2 − 20% of the kinetic energy of the fireball into radiation.4.2 The Afterglow Emission and External ShocksAs we mentioned in the last section, “external” shocks result when the relativistic outflowof the fireball interacts with surrounding material (either a solar wind generated by the burstprogenitor or interstellar material). These shocks form later than the internal shocks and are themost likely cause of the afterglow emission. They generate emission with very little variability onshort timescales. In this section, I follow the derivation described by Sari et al. [71].As with internal shocks, the most likely radiation mechanism is synchrotron emission. Asgiven in Equation 4.7, the frequency at which a relativistic electron interacting with a magneticfield emits energy is given byν(Γe) = ΓΓ2eqeB2πmec. (4.8)The individual Lorentz factors of the electrons in the plasma (Γe) are assumed to follow a powerlaw distribution, down to some minimum Lorentz factor Γm:n(Γe) ∝ Γ−pe , Γe ≥ Γm, (4.9)where n(Γe) is the number of electrons with a Lorentz factor of Γe and p is a constant known as the“electron power law index”. In addition to Γm, there is another value of Γe that is significant: a41critical value below which the electron begins to lose a significant fraction of its energy to radiation:Γc =6πmecσT ΓB2t, (4.10)where σT is the Thompson cross-section and t is the time in the observer’s reference frame. Thereare two possible scenarios, each with its own predicted spectrum:1. Γm > Γc. In this situation, all of the electrons cool down to Γc. This is known as the“fast-cooling” regime. The flux in this regime at different frequencies ν is given byFν =(ν/νm)1/3Fν,max, νc > ν,(ν/νm)−1/2Fν,max, νm > ν > νc,(νm/νc)−1/2(ν/νm)−p/2Fν,max, ν > νm,(4.11)where νc ≡ ν(Γc) and νm ≡ ν(Γm) (see Figure 4.2. νc and νm are spectral break frequencies,at which the power law index of the spectrum suddenly changes. This spectrum is expectedat early times, during the first few days following the burst.2. Γm < Γc. In this case, only those electrons with Γ > Γc can cool. This is known as the“slow-cooling” regime. The flux in this regime is given byFν =(ν/νm)1/3Fν,max, νm > ν,(ν/νm)−(p−1)/2Fν,max, νc > ν > νm,(νm/νc)−(p−1)/2(ν/νc)−p/2Fν,max, ν > νc.(4.12)(see Figure 4.2. This condition is expected at late times—weeks or months after the burst.The time-dependence of the flux depends on how these break frequencies (νm and νc) evolvewith time. There are two limiting cases:1. fully radiative evolution, in which all of the internal energy of the fireball is radiated away(this is the case if the fraction of internal energy going to electrons is very large, and if fastcooling is taking place),2. fully adiabatic evolution, in which the internal energy of the fireball remains constant.4210810101012101410161018100102104ν2Aν1/3Bν−1/2Cν−p/2Dfast coolingtt0bνat0νmt−3/2νct−1/2Flux (µJ)ν (Hz)Figure 4.2: Synchrotron spectrum from a relativistic shock with a power law distribution ofelectrons. (a) The case of fast cooling, which is expected at early times (t < t0) in a γ-ray burstafterglow. t0 is the transition time between fast and slow cooling and occurs when νc = νm.The spectrum consists of four segments, identified as A, B, C, D. Self-absorption is importantbelow νa. The frequencies, νm, νc, νa, decrease with time as indicated; the scalings above thearrows correspond to an adiabatic evolution, and the scalings below, in square brackets, to afully radiative evolution. (b) The case of slow cooling, which is expected at late times (t > t0).The evolution is always adiabatic. The four segments are identified as E, F, G, H. [71]43In the fully radiative case, the break frequencies and peak flux evolve according toνc = 1.3 × 1013 ε−3/2B E−4/752 Γ4/72 n−13/141 t−2/7d Hz,νm = 1.2 × 1014 ε1/2B ε2eE4/752 Γ−4/72 n−1/141 t−12/7d Hz,Fν,max = 4.5 × 103ε1/2B E8/752 Γ−8/72 n5/141 D−228 t−3/7d µJy. (4.13)where εB is the fraction of the shock energy density that is in the form of magnetic energy, εe isthe fraction of internal energy which is carried by the electrons, n1 ≡ n/(1cm−3) is the densityof electrons in the interstellar medium, td ≡ r/(1day) is the number of days since the burst, andD28 ≡ D/(1028cm) is the luminosity distance between the burst and us. In the fully adiabaticcase, the break frequencies and peak flux evolve according toνc = 2.7 × 1012ε−3/2B E−1/252 Γ4/72 n−11 t−1/2d Hz,νm = 5.7 × 1014ε1/2B ε2eE1/252 t−3/2d Hz,Fν,max = 1.1 × 105ε1/2B E52n1/21 D−228 µJy. (4.14)The flux at a given frequency decreases with time according to a simple power law, with twofrequency-dependent breaks at which the index of the power law changes (see Figure 4.3):tc =7.3 × 10−6 ε−3B E−152 n−21 ν−215 days, adiabatic,2.7 × 10−7 ε−21/4B E−252 Γ22n−13/41 ν−2/315 days, radiative,(4.15)tm =0.69 ε1/3B ε4/3e E1/352 ν−2/315 days, adiabatic,0.29 ε7/24B ε7/6e E1/352 Γ−1/32 n−1/241 ν−7/1215 days, radiative.(4.16)At sufficiently late times and at all frequencies, the light curve decreases as t(2−3p)/4, which, for atypical value of p = 2.5, becomes t−1.4.4.3 Models for the “Inner Engine”So far, we’ve described the physics and dynamics of the relativistic fireball emanating fromthe hypothetical “inner engine”. Here we address the question of what the inner engine might be.Observations give us a few clues:4410−2100102101102103104105106t1/6 [t−1/3]Bt−1/4 [t−4/7]C t(2−3p)/4D [t(2−6p)/7]t(2−3p)/4 Hhigh frequencyν>ν0a tc tm t0Flux (µ J)10−2100102101102103104105106t1/6 [t−1/3]Bt1/2 Ft3(1−p)/4 Gt(2−3p)/4 Hlow frequencyν<ν0b t0 tm tcFlux (µ J)t (days)Figure 4.3: Light curve due to synchrotron radiation from a spherical relativistic shock. (a)The high frequency case (ν > ν0). The light curve has four segments, separated by the criticaltimes, tc, tm, t0. The labels, B, C, D, H, indicate the correspondence with spectral segments inFig. 1. The observed flux varies with time as indicated; the scalings within square brackets arefor radiative evolution (which is restricted to t < t0) and the other scalings are for adiabaticevolution. (b) The low frequency case (ν < ν0). [71]45• The inner engine must be a compact source in order to produce the observed variability.• Long gamma-ray bursts (at least) typically seem to occur in regions rich in star formation.• The rate of gamma-ray bursts is consistent with about 1 burst per 106 years per galaxy, if theenergy is emitted isotropically. We know that the energy is actually collimated into narrowbeams, increasing this rate by a factor of (4π/θ2).• The amount of energy produced is around 1051 ergs.• If we assume that the fireball model is indeed correct, it requires that ≈ 10−5 solar massesof material be accelerated to relativistic energies.• The internal shock model further requires a variable energy flow from the engine.Models that satisfy these requirements all involve the formation of a compact object (like a blackhole) and the release of its binding energy. Here we discuss the two most prominent candidates.4.3.1 CollapsarsA “Collapsar” is a rotating massive star whose center collapses into a black hole surroundedby an accretion disk. In order for the collapsar to meet the compactness requirement discussed insection 4.1, it must lose its hydrogen envelope prior to its collapse. Material accreting from thedisk onto the black hole would produce the relativistic outflow along the axis of rotation. Thisoutflow would then pass through the rotating stellar mantle (which would further collimate it) andthen give rise to the GRB. The observed variability could arise from instabilities that result duringthe interaction between the outflow and the stellar envelope.This particular inner engine model has a number of advantages:1. It is expected to produce a strongly collimated beam, which we indeed observe.2. It would naturally occur in star-forming regions.3. A good mechanism exists for the high variability we see in gamma-ray bursts.464. Numerical simulations indicate that the time scale of the collapse is on the order of tens ofseconds. This would be consistent with the durations of the long class of GRBs, but it ishard to reconcile with the durations of short GRBs.4.3.2 Compact MergersAnother prominent model for the inner engine involves two compact objects (for example,two neutron stars, or a neutron star and a black hole) in a binary orbit. Over time (108 years orso) the objects gradually lose energy to gravitational radiation causing their orbits to decay untilthey eventually spiral in and merge, resulting in a rotating black hole. Models predict that thismerging event would release about 5 × 1053 ergs of energy, mostly in the form of gravitationalenergy and neutrinos. Enough energy would be left over, though, to power a gamma-ray burst.A neutron star binary system would have a long enough lifetime to allow it to wander farfrom the environment of the supernova that created its constituents, even far enough that it couldend up outside of its host galaxy. A burst created by a merger would likely not be well-collimated.Simulations indicate that the time scales of the event would be smaller than those of long-durationgamma-ray bursts, but it may be a viable model for short GRBs [48].4.4 X-Ray Flash ModelsSeveral models have been proposed to explain what circumstances give rise to x-ray flashes asopposed to gamma-ray bursts. We may classify these models broadly into two categories: geometricand intrinsic. Those that suggest that GRBs and XRFs arise from different physical processes weshall call “intrinsic” models. Those that propose that GRBs and XRFs appear different to usonly because of our particular location or orientation in relation to them we shall call “geometric”models. We’ll begin by describing the most prominent geometric models.474.4.1 Models Based on Geometric EffectsOff-Jet ModelYamazaki et al. have proposed that x-ray flashes are gamma-ray bursts viewed from anangle outside the emission jet [81] (see Figure 4.4). Thus, an x-ray flash is an “off-jet” gamma-ray burst. According to their predictions, the peak energy in the co-moving frame of the ejectedmaterial E0 is reduced by a factor of δ when viewed from an angle θv with respect to the jet axisby an observer in the frame of the burst progenitor:Esrcpeak =E0Γ(1 − β cos(θv − ∆θ)), (4.17)where Γ is the Lorentz factor of the ultrarelativistic outflow, β is the speed of the outflow dividedby the speed of light (v/c), and ∆θ is the jet opening half-angle. The denominator in Equation 4.17is a quantity known as the “relativistic Doppler factor” δ. In this model, an observer within thejet would see a peak energy Eobspeak of:Esrcpeak =E0Γ(1 − β) , (4.18)so that the relationship between Esrcpeak as seen from off the jet and Esrcpeak,in jet as seen from withinthe jet isEsrcpeak = Esrcpeak,in jet1 − β1 − β cos(θv − ∆θ)≈Esrcpeak,in jet1 + Γ2(θv − ∆θ)2. (4.19)Thus, a gamma-ray burst with an Esrcpeak of 300 keV, a typical Γ of ∼ 300 would appear as an x-rayflash with an Eobspeak of 20 keV if viewed from just slightly outside the jet at θv − ∆θ = 0.7◦.Also according to this model, the peak fluence S is reduced by a factor of δ3 (see Figure 4.5):S =S0δ3. (4.20)With the off-jet model, it is possible to calculate a number of different measurable quantitiesand compare them to observation. The model predicts peak XRF fluxes that range from about48Figure 4.4: A drawing illustrating the off-jet model for x-ray flashes. In this model, it isproposed that x-ray flashes are typical GRBs observed from outside the jet [81].10−8 to 10−7 ergs/s/cm2. The x-ray to gamma ray fluence ratio S(2 – 10 keV)/S(50 – 300 keV)predicted by the model extends up to 20 and the peak flux ratio corresponding to the same energybands extends up to 100. The 2 – 25 keV spectrum is consistent with a power law with a photonindex ranging from 1.2 to 3, with a mean of ≈ 2. The duration of XRFs is predicted to be roughlythe same as for GRBs: between about 10 s and about 200 s. The predicted event rate is ∼ 100/yr,and the sky distribution is expected to be isotropic. All of these predictions agree with observation.Because the fluence of the emission from outside the jet is many orders of magnitude lowerthan the emission from inside the jet, XRFs as a population would be dimmer than GRBs andtherefore would not be visible at such high redshifts. We would therefore expect those XRFs wedetect to lie at smaller redshifts than GRBs. The sample of XRF redshifts measured to date isnot large enough to be able to determine if this is the case.A crucial prediction made by this model which we may test directly, thanks to Swift’srapid response capabilities, pertains to the characteristics of the early afterglow. If a gamma-rayburst is observed from anywhere within the jet opening angle, the afterglow flux would decrease49Figure 4.5: The peak flux ratio (upper panel) and fluence ratio (lower panel) of 2 – 10 keVto 50 – 300 keV photons as a function of viewing angle [81]50monotonically with time. But if the afterglow is viewed from outside the jet, the flux rises until itreaches a peak, and then it begins to diminish again at a rate consistent with that of an afterglowobserved on axis [37, 77]. If θv & ∆θ, the peak tp of the light curve would be observed at a time:tp =(5 + 2 ln( θv∆θ− 1))( θv∆θ− 1)2tj,0 [37], (4.21)where tj,0 is the time at which the jet break would be observed if the afterglow were viewed fromwithin the jet. Equation 4.21 is valid when the Lorentz factor Γ is relatively large (Γ & 300) orwhen θv & ∆θ is relatively large (θv & ∆θ & 5◦). Under these conditions, the peak would comesome time after the jet break, which has typically been on the order of a day after the burst.Another model with similar predictions was proposed by Toma et al. [76] in which multiplesub-jets explain the differences between XRFs, XRRs, and GRBs. The multiple sub-jet modelexplains the correlation between Eiso and Eobspeak discovered by Amati et al. [1] and, like the off-jetmodel, it predicts that XRFs arise when the viewing angle is outside all of the sub-jets.Structured Jet ModelAnother model that seeks to explain x-ray flashes on the basis of viewing geometry is theso-called “structured jet model”. Instead of assuming that the collimated relativistic outflow isuniform within the jet, this model assumes that the Lorentz factor Γ and the energy per unit solidangle ε are functions of angle. Rossi et al. [68] consider a case in which these parameters decreasewith angle according to a power law:Γ =Γc 0 ≤ θ ≤ θcΓc(θθc)−αΓ, αΓ > 0 θc ≤ θ ≤ θj ,(4.22)ε =εc 0 ≤ θ ≤ θcεc(θθc)−αε, αε > 0 θc ≤ θ ≤ θj ,(4.23)where θc is introduced simply to avoid Γ or ε becoming infinite at θ = 0. Like the off-jet model,this model postulates that x-ray flashes are the result of a larger viewing angle θ than gamma-raybursts.51This model is able to explain the correlation between Eiso and jet break time tb that isalluded to in section 2.5.3. Furthermore, it predicts a rising afterglow light curve with the samegeneral shape as that predicted by the off-jet model. In this model, the time of the peak of thelight curve depends only on the observing angle tp ∝ θ2.A further prediction of this model is that since the average viewing angle would be verylarge (because as θ increases, the solid angle encompassed by a given increment ∆θ also increases),the number of x-ray flashes we observe should exceed the number of gamma-ray bursts by severalorders of magnitude [54]. Observations thus far do not support this prediction: XRFs and GRBsare observed by HETE-2 in roughly the same numbers [70].Another type of structured jet—a two-component jet—has been suggested to account forsharp rebrightening features in the afterglows of XRF 030723 [46] and GRB 030329 [10]. It wassuggested that if the inner jet had a much larger Lorentz factor than the outer jet, and if our lineof sight were outside the jet structure, the beaming angle from the inner jet would reach our lineof sight much later than the outer jet, causing a rebrightening. Granot investigated the light curvearising from such a jet and concluded that the rebrightening bump it would produce is much toosmooth to explain the features seen in those two bursts [39].Other types of structured jets that have been proposed include ring-shaped and fan-shapedjets. A ring-shaped jet can be described by a jet profile ofε =dEdΩ=ε0 θc < θ < θc + ∆θ0 otherwise, (4.24)where θ is the angle from the symmetry axis of the jet, θc is the inner half-opening angle, and ∆θ isthe angular width. Granot calculated the light curve arising from this model (see Figure 4.6) [39].In the case of a thin jet (∆θ θc), observing angles θobs within the jet itself (that is, θc < θobs <θc + ∆θ) result in a light curve that peaks very early (on the order of 1 s) and then decreasesmonotonically. For angles outside the jet (either θobs < θc or θobs > θc +∆θ), the light curve peaksrelatively late, on the order of a day after the burst (see Figure 4.6). It was noted that this modelpredicts two shallow but distinct steepening epochs and cannot produce the large single-epoch5210−610−510−410−310−210−110010110210310−1610−1510−1410−1310−1210−1110−1010−910−810−710−610−510−410−310−210−1100θc = 0.1, ∆θ = 0.01, η = 103, b = −p/2E=1051 erg, next = 1 cm−3 , p = 2.5tobs [days]Fνθobs=0θobs=0.21θobs=0.050.16θobs=0.1050.1,0.11a "thin ring"jet structureTdecFigure 4.6: Light curves from a jet a thin uniform ring structure [39]steepening that is typically seen in GRB afterglows. If the ring is thick (∆θ & θc), the light curveis practically indistinguishable from that of a uniform jet.Granot also calculated the light curves arising from a thin, fan-shaped jet [39]. He foundthat the steepening is about half what would be observed from a uniform jet and much smallerthan is typically observed in GRB afterglows. Depending on the viewing angle, the light curvewould peak a few hours to a few days after the burst (see Figure 4.7).High Redshift ModelIt has also been suggested that x-ray flashes may be gamma-ray bursts originating at highredshifts [43]. For instance, if a gamma-ray burst with a peak energy at the source Esrcpeak of 100keV were detected at a redshift of 5, its peak energy would be shifted down to 20 keV. As we haveseen, there are reasons to suspect that gamma-ray bursts follow star-formation rates, and if thisis the case, we would expect GRBs to be present out to redshifts of ∼ 10 or even higher.If most XRFs were actually high-redshift GRBs, we would also expect other properties tobe affected by their high redshift. For instance, XRF durations would tend to be longer than GRBdurations due to time dilation. Observations to date, however, show that the duration distributionof XRFs is similar to that of GRBs [43]. Furthermore, those XRF redshifts which have been5310−410−310−210−110010110210−1410−1310−1210−1110−1010−910−810−710−610−510−410−310−210−1100θc = π/2−0.05, ∆θ = 0.1, η = 300E=1051 erg, n0 = 1 , p = 2.5b = −p/2tobs (days)Fνθobs=π/2θobs=π/2−0.05θobs=π/2−0.1 π/2−0.15π/2−0.2θobs=π/2−0.3a "thin fan"jet structureTdecFigure 4.7: Light curves for a jet with an angular structure of a thin fan, with an openingangle of ∆θ = 0.1 centered on θ = π/2 [39]measured have been not been drastically different from GRB redshifts.4.4.2 Models Based on Intrinsic PropertiesThere are also several models that propose that XRFs are intrinsically different from GRBs.Here we review four such models.Photosphere-Dominated Fireball ModelIn section 4.1.1, we described the fireball model, which predicts a radius called the “photo-spheric radius” rph, beyond which the ultrarelativistic outflow becomes optically thin. Mészáros etal. have proposed that if the plasma becomes optically thin before it reaches the matter-dominatedphase, internal shocks may produce enough photons to create a second photosphere [57]. Thereexists a radius, called the “pair-shock” radius (r±), beyond which this second photosphere cannotform. If the internal shocks form at a radius r > r±, a regular gamma-ray burst is generated.Shocks that form at rph < r < r± could produce x-ray flashes. These x-ray flashes would, ingeneral, be less variable than GRBs, their variability being somewhat damped by the pair-shock54photosphere. Internal shocks that are generated at a radius r < rph would be x-ray rich.This photosphere-dominated fireball model, together with the observed anti-correlation be-tween the isotropic-equivalent luminosity Liso and θ, provides a geometric interpretation for thecorrelation between variability and isotropic-equivalent luminosity Liso that has been observed [66].It also makes the testable prediction that x-ray flashes are less variable than gamma-ray bursts.Low Γ (“Dirty”) Fireball ModelAs we described in section 4.1.1, one of the requirements of the standard fireball modelis a small density of baryons which would enable the internal energy to be efficiently convertedinto kinetic energy and carry the plasma beyond the photospheric radius before internal shocksbegan. A fireball with a low baryon density is sometimes referred to as a “clean” fireball. A“dirty” fireball, on the other hand, would have a somewhat larger baryon density and, hence,would achieve a smaller Lorentz factor Γ by the time it reached the matter-dominated phase. Inthe fireball model, Eobspeak ∝ Γ4, so even a slightly smaller Γ could result in a significantly softerspectrum [45, 21]. A fireball with 1 Γ 100 would emit mostly in x-rays instead of gamma-rays.The afterglow of the burst should depend very little on the baryon load, so the afterglowof an XRF in this model would be just as prominent as that of a GRB. Calculations indicate thatat very early times (up until about ∼ 1000s or so), the light curve of an XRF afterglow would bequite flat, whereas the afterglow of a GRB would increase slightly with time. At these early times,XRF afterglows would tend to be slightly weaker (∼ 5×) than those of GRBs. At later times (afterabout a day or so), the afterglows would be indistinguishable (see Figure 4.8).Small Contrast in Γ Fireball ModelBarraud et al. [7] set out to determine under what conditions a set of internal shocks mightproduce an x-ray flash. To do this, they investigated the behavior of a toy model in which twoultrarelativistic shells of the same mass m but slightly different values of Γ collide, producingsynchrotron emission. The parameters of the model were Γ1 (the Lorentz factor of the first, slowershell), Γ2 (the Lorentz factor of the second, faster shell), and τ (the delay between the generation55Figure 4.8: A light curve calculation for typical GRB afterglows in the “dirty fireball” model.The thick solid line is plotted for a usual isotropic GRB with Γ0 = 300. The dashed linerepresents an isotropic GRB with Γ0 = 30. The dash-dotted line corresponds to a jetted GRBviewed from within the jet with Γ0 = 300, and the dotted line is for a jetted GRB viewed fromoutside the jet. The thin solid line is for a beamed GRB with Γ0 = 30, θobs = 0. The Insetshows the evolution of the Lorentz factor correspondingly. See [45] for details of the modelparameters.56of the first shell and the second shell). The model predicts thatEobspeak ∝Ėψxy(κ)τ2xΓ6x−1 , (4.25)where Ė = (mc2/τ)(Γ1 + Γ2) is the energy loss rate of the inner engine, Γ = (Γ1 + Γ2)/2 is theaverage Lorentz factor, κ = Γ2/Γ1, x and y are variables that depend on how the initial energy ofthe shells is distributed between the magnetic field and the electrons, and ψxy(κ) is an increasingfunction of κ for all reasonable values of x and y. From this equation, we note that since κ is largerwhen there is a larger contrast between Γ1 and Γ2, Eobspeak is also larger in those circumstances. Wealso note that as long as x > 1/6, Eobspeak decreases as Γ increases. Statistical simulations with thismodel verify that XRFs are created when Γ is large and/or when κ is small. On a physical basis,the production of XRFs in clean fireballs may be explained by a reduction in shock efficiency. Inthe case of dirty fireballs (with a greater contrast in Γ), internal shocks take place closer to thesource, where the density is higher. The injected power Ė has very little bearing on whether agamma-ray burst or an x-ray flash is created.This “small contrast in Γ” model reproduces the observed correlation between duration andhardness as well as the correlation between hardness and intensity [52, 35]. It also predicts similarduration distributions for XRFs and GRBs, as has been observed. Furthermore, it predicts thatthe redshift distribution for XRFs should be nearly the same as that of GRBs.Amati et al. describe a procedure by which they were able to constrain Γ using the afterglowenergy, the circumburst density, and the duration of the emission [2]. They found that in the caseof XRF 020427, Γ > 195, which is consistent with this model.Uniform jet modelThe final model we shall examine was proposed by Lamb et al. [54]. Based on the resultsthat XRFs tend to have low isotropic-equivalent energies and using the observed anti-correlationbetween Eiso and jet opening angle θ, they proposed that gamma-ray bursts tend to have verynarrow jets and x-ray flashes tend to have much wider jets. Unlike the other models we haveconsidered, this model is based entirely on observation and offers no particular physical explanation57or basis. It therefore may well turn out to be consistent with one of the other intrinsic modelsdescribed in this section.58Chapter 5The Swift Observatory5.1 Description of the Swift MissionOne of the greatest difficulties that confront gamma-ray burst studies is the transient natureof the phenomenon. The bursts themselves only last for a few seconds, and although the afterglowslast much longer, they also fade very quickly, typically at a rate proportional to around t−1 ort−2. Swift is the first observatory with the ability to begin multi-wavelength observations within∼ 1 minute of the burst, at a time when the afterglow is orders of magnitude brighter than it iseven an hour later. This early follow-up permits us to detect many more afterglows than has beenpossible in the past. It also provides the key to constraining afterglow models and evaluating x-rayflash models.Launched in November of 2004, the Swift observatory (see Figure 5.1) is a medium-sizedexplorer (MIDEX) mission whose primary scientific objectives are to determine the origin of GRBsand to pioneer their use as probes of the early universe. Unlike most space missions, which areeither acronyms or are named after famous scientists, Swift was named after a bird. These birdshave the ability to scan the sky for insects and quickly change direction in mid-flight to catchthem. In a similar way, Swift scans the sky for gamma-ray bursts and quickly slews to the positionof a burst when one is detected. The Swift spacecraft, which was built by Spectrum Astro, housesa trio of instruments: the Burst Alert Telescope (BAT), the X-Ray Telescope (XRT), and theUltraViolet and Optical Telescope (UVOT). The Burst Alert Telescope searches the sky for newGRBs and, upon discovery, triggers an autonomous spacecraft slew to bring the burst into theXRT and UVOT fields of view. Such autonomy allows Swift to perform X-ray and UV/opticalobservations of ≈ 100 bursts per year within 20 - 70 seconds of a burst detection. Once the burstis within the fields of view of the XRT and UVOT, the pointing stability is better than 0.1 arcsec.59 Figure 5.1: The Swift Observatory. The BAT, consisting of the coded aperture mask aboveand the detector array below, is shown in the forefront. A fringe shield that surrounds theBAT is not pictured. The XRT and UVOT are shown behind the BAT.Table 5.1 summarizes the basic mission profile parameters. The orbital radius of 600 kmwas selected so that the instrument would be in a low enough orbit that the Van Allen radiationbelts not interfere very much, yet the orbit would still be high enough to guarantee a longer than2-year orbit lifetime. The worst-case solar activity scenarios predict a minimum orbit lifetime of5 years. The Delta 7320 launch vehicle had a 280 kg mass margin beyond what was required bythe observatory, which made possible a maneuver during launch to decrease the orbit inclinationfrom the 28.5◦ to 22◦. The lower inclination minimizes the amount of time the spacecraft spendsin the South Atlantic Anomaly (SAA), which is a region over the south Atlantic Ocean and SouthAmerica with a large number of high energy protons that bombard spacecraft passing throughit. When Swift passes through the SAA, the BAT is swamped with counts due to these protons,so BAT effectively “shuts down” during that time. Less time in the SAA means more time60Table 5.1: Swift Mission CharacteristicsMission Parameter ValueSlew Rate 50◦ in < 75 secOrbit Low Earth, 600 km altitudeInclination 22◦Launch Vehicle Delta 7320-10 with 3 meter fairingTotal Mass 1270 kgTotal Power 1640 WLaunch Date November 2004Mission Life 2 yrOrbital Life > 5 yrwatching for bursts—about 10% more than would be possible at a 28.5◦ inclination. It also meansa slower increase in background due to activation of the spacecraft materials. The high energyprotons of the SAA interact with the materials of the spacecraft and cause some of them tobecome radioactive, emitting hard x-rays that increase the background level detected by BAT.Over time, this background level increases and interferes with the ability of BAT to detect anddistinguish counts due to gamma-ray bursts. The high energy proton bombardment can alsodegrade the electronic components of the spacecraft over time. Less time spent in the SAA slowsthis degradation. The observatory was launched from Cape Canaveral and has a nominal lifetimeof 2 years with a goal of 5 years and an orbital lifetime of greater than 8 years.5.2 The Burst Alert Telescope5.2.1 PurposeAs its name suggests, the Burst Alert Telescope (BAT) provides the initial detection ofthe gamma-ray burst. It then calculates the position of the burst and sends the position to thespacecraft attitude control system. Swift does all this within 12 seconds of the initial detection ofthe burst. In order to monitor as much of the sky at possible at once, BAT has a large FOV—about2 sr, or roughly 1/6 of the sky. This is approximately the same field of view that a human being hasstaring straight ahead. One of the few ways to produce hard x-ray images with such a large FOV isto use the coded-aperture technique, which is described in section 5.2.3. For each detected burst,61event data is accumulated during an adjustable time window surrounding the time of the initialburst trigger. This event data consists of the time that each photon was detected (to an accuracyof ∼ 200µs), the detector in which it was detected, and the size of the electrical signal induced bythe detector (the “pulse height”). From these event files, spectra (the number of photons detectedas a function of energy) and light curves (plots of the number of photons detected as a functionof time) can be created. A more detailed description of the BAT instrument and its pre-launchperformance is given by Barthelmy et al. [9].5.2.2 Physical DescriptionThe basic parameters of the BAT instrument are listed in table 5.2. It can detect photons inthe energy range of 15-150 keV and contains 32,768 pieces of 4mm×4mm×2 mm CdZnTe (CZT)room temperature semiconductor material which form a 1.2× 0.6 m sensitive area in the detectorplane. Groups of 128 detector elements are assembled into 8 × 16 sub arrays, with the detectorsconnected to a 128-channel readout XA 1.23 Application Specific Integrated Circuits ASIC; whichis designed and produced by Integrated Detector and Electronics (IDE AS) of Norway. Detectormodules, each containing two such sub-arrays, are further grouped by eights into 2048-detectorblocks. This hierarchical structure, along with the forgiving nature of the coded-aperture technique,means that the BAT can tolerate the loss of individual pixels, individual detector modules, and evenwhole blocks without losing the ability to detect bursts and determine locations. The CZT arrayhas a nominal operating temperature of 20◦C, and its thermal gradients (temporal and spatial) arekept to within ±1◦C. The typical bias voltage applied to the detectors is 200 V, with a maximumof 300 V. The detectors are calibrated in flight with an electronic pulser which periodically sendsan electronic signal of a known voltage into the electronics, and two 241Am tagged sources that arelocated just inside the mask and constantly illuminate the array with photons of specific energies.The tagged sources are small pieces of scintillator doped with 241Am. The scintillators detect thelight flash accompanying each 241Am decay so that any photon detected at the same time as thescintillator signal is identified as coming from the tagged source.62Figure 5.2: A drawing of the Burst Alert Telescope and other components of the Swift observatory [32]The BAT has a D-shaped coded mask made of 54,000 lead tiles (5mm × 5mm × 1 mm)mounted on a 5-cm-thick composite honeycomb panel, that is supported by composite fiber strutsat 1 meter above the detector plane. The BAT coded mask has a completely random, but known50% open-50% closed pattern tile. The mask area is 2.7 m2, resulting in a half-coded FOV (thatis, a field of view in which every point in the sky illuminates at least half the area of the arraythrough the mask) of 100◦ × 60◦, or 1.4 steradians.A “graded-Z” fringe shield, located both under the detector plane and surrounding the maskand detector plane, reduces the number of photons from sources outside the field of view and othersfrom background radiation by ∼ 95%. The shield is composed of layers of Pb, Ta, Sn, and Cu,which are thicker toward the bottom nearest the detector plane and thinner near the mask.A “Figure of Merit” (FoM) algorithm decides if a burst detected by the BAT is worthrequesting a slew maneuver by the spacecraft. If the new burst has more “merit” than the pre-programmed observations, a slew request is sent to the spacecraft. The ground control team canalso upload target positions, which are processed exactly the same as targets discovered by theBAT.63Table 5.2: Burst Alert Telescope CharacteristicsBAT Parameter ValueEnergy Range 15-150 keVEnergy Resolution ∼ 7 keVAperture Coded mask, random pattern, 50% openDetection Area 5240 cm2Detection Material CdZnTe (CZT)Detection Operation Photon countingField of View (FOV) 1.4 sr (half-coded)Detector Elements 128 Detector Modules with 256 detectors eachDetector Element Size 4 × 4 × 2 mm3Coded-Mask Cell Size 5 × 5 × 1 mm3 Pb tilesTelescope PSF 17 arcminSource Position Accuracy 1 − 3 arcmin (depending on source position)Sensitivity ∼ 10−8 ergs/cm2/sNumber of Bursts Detected ∼ 100/yr5.2.3 Coded Aperture ImagingAs we described in section 5.2.1, BAT is required to generate a gamma-ray image of a largearea of the sky on a short time scale. Focusing optics are very difficult above about 10 keV, becausethe techniques that are typically applied at x-ray energies, in which x-rays graze the surface ofmirrors, require grazing angles that are too small to be practical, although these angles can beincreased using multi-layer mirrors like those used by the InFocus mission [61]. Another difficultyis that most imaging techniques are not feasible for large fields of view. Coded aperture imagingsolves both of these problems [17, 22].As photons coming from a particular direction in the sky pass through the mask, they casta shadow across the detector array. Those detectors that are illuminated through the mask willdetect a high number of photons, whereas those that are in shadow will detect very few. The resultis a pattern of detected photons (or “counts”) in the detector array. By comparing the pattern ofthese counts with the known pattern of the mask, it is possible to determine from which directionthe photons originated. This process is automated and is conducted for every point in the skyusing a Fourier transform algorithm. The result is an image of the sky, with bright points wheregamma-ray sources are present and dark background everywhere else.64 Figure 5.3: The mask weight values assigned to various detectors. Detectors that are fully-illuminated are assigned a value of +1 and detectors that are fully-masked are assigned a valueof -1.We can use a similar technique to produce a count rate for a particular source in the sky. Wedo this by assigning a weight to each detector, according to the degree to which it is illuminatedthrough the mask. Detectors that are fully-illuminated would receive a mask weight of +1 anddetectors that are fully-masked would receive a mask weight of -1. All other detectors would receivea mask weight somewhere between -1 and +1 (see Figure 5.3). If the count rates from eachindividual detector are all added together, weighted by these mask weight values, the result is avalue that is proportional to the number of counts received in a single fully-illuminated detectoras a result of photons arriving directly from the source in question, with all other counts removed.See Appendix A for a complete derivation of the mask-weighted count rate.5.2.4 Effective AreaThe “effective area” of an x-ray or gamma-ray instrument is typically defined as the ratioof the count rate due to photons of a given energy and the actual incident flux of those photons.Thus, effective area has units of area (as the name suggests) and depends upon the energy and65direction of the incident photons.For a typical x-ray or gamma-ray instrument, the effective area is defined in terms of everycount recorded by the instrument that results from the incident photon flux coming from thedirection of the source of interest. As such, it includes counts from photons that arrive directlyfrom the source as well as those that enter the detectors after first scattering off of some passivematerial. The mask-weighted count rate differs from the full count rate typically considered inseveral ways:• Counts resulting from scattered photons are removed as background.• Counts resulting from photons that pass through the various passive materials, including thelead tiles, are removed as background.• Counts that arrive at detectors that are half-illuminated are removed, because such detectorsreceive a weight of 0. Likewise, counts that arrive at detectors that are less than half-illuminated actually contribute negatively to the total rate. The overall result is that manycounts that really do result from photons coming directly from the source are cancelled out.When we define the effective area in terms of the mask-weighted count rate, it is likewise reducedfor these reasons. Figure 5.4 shows the mask-weighted effective area as a function of incident pho-ton energy at two different incident angles. Figure 5.5 shows the traditionally-defined effective areafor the BAT instrument. The latter curve was generated by Monte Carlo GEANT simulations ofincident photons interacting with the materials comprising the entire Swift observatory. The sim-ulations use a model called SwiMM (short for “Swift Mass Model”) which was developed by DavidWillis, formerly of the University of Southampton, and is currently maintained by collaboratorsat JAXA/ISAS in Japan.5.2.5 The Trigger AlgorithmThe Burst Alert Telescope constantly watches for new gamma-ray bursts. Since burstdurations can range from a few milliseconds to a number of minutes, BAT has many different66Figure 5.4: The “mask-weighted” effective area curve for the Burst Alert Telescope. Thiseffective area curve is defined as the ratio of the mask-weighted count rate to incident photonflux. The top curve is the effective area for photons that hit the array from directly above.The bottom curve is the effective area for photons incident on the array at a 45◦ angle.67Figure 5.5: The “traditional” effective area curve for the Burst Alert Telescope for photonsincident from directly above the array. This effective area curve is defined as the ratio of thetotal count rate to incident photon flux. This particular plot was generated using a MonteCarlo simulation, so there are small statistical fluctuations that give it a jagged appearancein places.68search criteria that it uses to identify a burst. When a potential burst is identified by a combinationof these criteria, We call that detection a “trigger”, because it sets in motion a set of automatedactions, including possibly slewing to the position of the burst. There are two different ways inwhich a potential burst may be identified: either as a result of an increase in the count rate inthe array (which we call a “rate trigger”) or as a result of the detection of a new source in animage generated by the onboard software (an “image trigger”) [23]. The rate triggers can furtherbe classified as “short rate triggers”, in which BAT searches for rate increases on time scales ofless than 64 ms, or as “long rate triggers”, in which BAT searches for rate increases on time scaleslonger than 64 ms.Short Rate TriggersBAT continuously watches for statistically significant increases in the count rate during 4ms, 8 ms, 16 ms, 32 ms, and 64 ms intervals. A burst is more likely to be detected in a timeinterval that is similar to its duration. During each of these time intervals, the BAT softwaremonitors the count rates in 9 different regions of the array: the 4 quadrants, the top half, thebottom half, the left half, the right half, and the entire array. A burst in a part of the sky thatis shielded from all but one quadrant of the array would result in a rate increase mostly confinedto that quadrant. Furthermore, 4 different energy ranges are continuously monitored: for countsthat are likely to originate from photons with energies between 15 and 25 keV, between 25 and 50keV, between 50 and 100 keV, and between 100 and 350 keV (although these limit values range arecommandable). A “hard” burst (that is, one with many high-energy photons and fewer low-energyphotons) would more likely be detected in a higher energy range. Considering all combinations ofthese criteria, there are 26,496 samples that must be checked every 1.024 seconds. The on-boardsoftware compares the maximum count rate in each sample to the average count rate in a 1.024second interval and calculates a detection “score”. If the score is greater than some commandablevalue, the on-board software searches backward in time until it finds the moment when the rateincrease began. Using data from that time interval, it generates an image and compares it to the69most recent 8-second image (which the software is constantly producing) to see if there are anynew sources. This step permits BAT to eliminate false triggers from rate increases due to knowngamma-ray sources and due to showers of charged particles that illuminate the array from timeto time. If a new source exists (to a certain commandable level of significance), the trigger isannounced to the world via the Gamma-ray burst Coordinates Network (GCN).Long Rate TriggersLong triggers are significant rate increases in time intervals longer than 64 ms. BAT monitorsthe same energy ranges and regions of the array for long triggers as for short triggers. However, onthese longer time scales, the background count rate may be changing, so it isn’t possible to simplycompare the maximum rates to an average rate in some longer time interval, as is done with shorttriggers. Instead, a linear or quadratic function is fit to a set of background regions to remove anytrends in the overall background rates. Then the count rate in some specified “foreground” intervalis calculated, with the background rate subtracted off. Up to 3 background time intervals can bespecified. Commandable parameters also specify the degrees of the polynomials that should beused to fit the background and the threshold for how high the trigger “score” must be before it isconsidered real (or likely to be real). As with the short triggers, an image is formed and searchedfor new sources before the trigger is announced.Image TriggersWe have already mentioned that the on-board software creates an image of the sky every 8seconds. These 8-second images can be combined into images comprising larger durations. Threesuch durations are possible at a time, and those durations are adjustable. These durations mustbe multiples of 8 seconds, and examples might be 64 seconds, 10 minutes, and 45 minutes (halfan orbit). The on-board software searches these combined images for new sources, ignoring thosethat are in positions where known gamma-ray sources exist. If the source detection score exceedssome commandable value, the trigger is announced.70The TDRSS and GCN SystemsTriggers are relayed to the ground by the Tracking and Data Relay Satellite System (TDRSS).Once on the ground, they are transferred to the Gamma-ray burst Coordinates Network (GCN)system, which distributes the information to any instrument, person, or institution wishing theinformation (to make follow-up observations or for general awareness about the current state ofGRB activity) [8]. GCN circulars are also distributed by the Swift instrument teams and byfollow-up instrument teams, providing additional information about the burst. Throughout thiswork, sources that are cited as “GCN” followed by a number refer to these circulars, and can befound at http://gcn.gsfc.nasa.gov/gcn3 archive.html. The position of the burst is available on theground within about 20 seconds of the initial detection. Light curve information on the burstfollows about 110 seconds later.5.2.6 GRB Data ProductsThe key aspect of the BAT instrument is to produce burst information quickly and have itdistributed to other spacecraft and to ground-based observatories as soon as possible. Table 5.3lists the burst-related data products, and how soon they become available. The burst trigger startsthe time-available delay clock listed in the table. The Trigger Alert message alerts ground-basedpersonnel that the BAT trigger algorithm has been satisfied. It contains the time of the triggerand which particular time, energy, and geometry trigger was satisfied. This alert message is sentto the ground through TDRSS and distributed to the world through the GCN. Meanwhile theBAT continues to process the event data to determine the location of the burst, which is alsotransmitted to the ground through TDRSS. BAT produces burst location error circles with a 4arcmin diameter, that are made available on the ground within 20 sec of the start of the burst.The results of the decision made by the spacecraft as to whether or not to slew to the burst isalso transmitted. After 130 sec, 30 sec of pre-trigger and 120 sec of post-trigger light curveinformation in 4 energy ranges is transmitted. Once the XRT and UVOT instruments are slewedinto position, they also produce position, spectra, and images of the burst afterglow, which also71Table 5.3: BAT burst-related data products and their time delays after the burst triggerData Product Time AvailableTrigger Alert 5 secBurst Position 12 secFOM Will/Won’t Observe 14 secSpacecraft Will/Won’t Slew 14 secBurst Light Curve 130 secBurst Afterglow Light Curve ∼ 20 minBurst Event-By-Event Data 2-4 hrsgo through TDRSS to the ground and through GCN. More detailed event-by-event informationis transmitted to the ground some time later when the Swift spacecraft passes over the Malindiground station in Kenya.5.3 The X-Ray Telescope5.3.1 PurposesSwift’s X-Ray Telescope (XRT) is designed to measure the 0.2 to 10 keV X ray fluxes,spectra, and light curves of GRBs and afterglows over a wide dynamic range covering more than7 orders of magnitude in flux. The XRT pinpoints GRBs to 5 arcsec accuracy within 10 secondsof target acquisition for a typical GRB and studies the X-ray counterparts of GRBs beginning atthe time the XRT has stabilized on-target (20 - 70 seconds after burst discovery) and continuingfor days to weeks. Figure 5.6 shows a schematic of the XRT, and Table 5.4 summarizes XRTparameters. Further information on the XRT is given by Burrows et al. [15].5.3.2 Physical Description and PerformanceThe XRT is a focusing X-ray telescope with a 110 cm2 effective area at 1.5 keV, 23 arcminFOV, 18 arcsec resolution (half-power diameter), and 0.2 - 10 keV energy range. The XRT uses agrazing incidence Wolter 1 telescope to focus X-rays onto a CCD. The complete mirror module forthe XRT consists of the X-ray mirrors, thermal baffle, a mirror collar, and an electron deflector.To prevent on-orbit degradation of the mirror module’s performance, the mirror is maintained at72Figure 5.6: Block diagram of the X-Ray Telescope [32]20 ± 5◦C with gradients of < 1◦C by an actively controlled thermal baffle. A composite telescopetube holds the focal plane camera, containing a single CCD detector. The CCD consists of animage area with 600 × 602 pixels (40 × 40 mm) and a storage region of 600 × 602 pixels (39 × 12mm). The FWHM energy resolution of the CCD decreases from ∼ 190 eV at 10 keV to ∼ 50eV at 0.1 keV, where below ∼ 0.5 keV the effects of charge trapping and loss to surface statesbecome significant. The detectors operate at ≈ −50◦C to ensure low dark current and to reducethe CCD’s sensitivity to irradiation by protons (which can create electron traps that ultimatelyaffect the detector’s spectroscopy).5.3.3 GRB Data ProductsThe XRT supports three readout modes to enable it to cover the large range of intensitiesexpected from GRB afterglows and autonomously determines which read-out mode to use. Inorder of bright flux capability (and the order that would normally be used following a GRB), themodes are:1. Imaging Mode, which produces an integrated image measuring the total energy depositedin each pixel and does not provide any spectroscopic information and only coarse timing73Table 5.4: X-Ray Telescope CharacteristicsXRT Parameter ValueEnergy Range 0.2-10 keVTelescope Wolter 1 (3.5 m focal length)Detector e2v CCD-22Detection Elements 600 × 600 pixelsPixel Size 40 µm × 40 µmPixel Scale 2.36 arcsec/pixelEffective Area ∼ 125 cm2 at 1.5 keV∼ 20 cm2 at 8.1 keVReadout Modes Imaging (IM) mode, Photodiode (PD) mode,Windowed Timing (WT) mode, Photon-Counting (PC) modeField of View (FOV) 23.6 × 23.6 arcminPoint Spread Function (PSF) 18 arcsec (HPD) at 1.5 keV22 arcsec (HPD) at 8.1 keVPosition Accuracy 3 arcsecSensitivity 2 × 10−14 ergs/cm2/s in 104 sinformation. Depending on the brightness of the source, the images are accumulated foreither 0.1 or 2.5 seconds. Images provide a position for the source and an estimate of theflux. They are useful for fluxes up to 7× 10−7 ergs/cm2/s.2. Photodiode Mode, which is useful for obtaining timing information for very bright sources(up to about 4 × 10−6 ergs/cm2/s). Photodiode Mode enables the XRT to produce a set oftime-resolved images on a time scale of about 0.14 ms. Unless the flux is too high (aboveabout 10−7 ergs/cm2/s), the energy and “grade” (or pattern of detected photons in nearbypixels) of each recorded photon can be extracted from these images. The grade providesinformation that can be used to screen valid events from those that most likely result fromcharged particles or piled-up events.3. Windowed Timing Mode uses a 200 column window covering the central 8 arcminutes ofthe XRT FOV. It provides position information in one dimension, and each image columnprovides a measured flux at a different moment in time. The time resolution is about 1.8 ms inthis mode. Useful fluxes can be measured up to about 3×10−7 ergs/cm2/s, and spectroscopicinformation can be obtained for fluxes up to about 6×10−8 ergs/cm2/s allowing bright source74Table 5.5: Summary of the XRT mode characteristics. 1 Crab is the flux emitted by the Crabnebula in the energy range of interest. In XRT’s energy range, 1 Crab ≈ 6×10−8 ergs/cm2/s.Mode Image Spectral Time On-Board Event Flux LevelCapability Capability Resolution Reconstruction Mode SwitchPU & LR No Yes 0.14 ms No, Done on ground 0.6-60 CrabWT 1D Yes 1.7 ms No, Done on ground 1-600 mCrabPC 2D Yes 2.5 s Yes < 1 mCrabIM 2D No 0.1 s (short) Not Applicable > 140 mCrab (short)2.5 s (long) < 5.6 mCrab (long)spectroscopy through rapid CCD readouts. For each photon, the arrival time, 1-D position,energy, and grade are provided in this mode.4. Photon-Counting Mode permits full spectral and spatial information to be obtained forsource fluxes ranging from 2 × 10−14 to 2 × 10−11 ergs/cm2/s. However, the CCD is onlyread out every 2.5 seconds, so for fluxes larger than ∼ 2 × 10−11 ergs/cm2/s, pile-up is aproblem. For each photon, this mode provides the 2-D position, arrival time, energy, 3 × 3pixel neighborhood centered on the event, and the grade of the event.With the exception of Imaging Mode, all data is collected and transmitted to the groundin the form of an event list. Each event list file consists of a time-ordered list of photon attributes(for example, time, position and pulse-height information). Images, spectra, and light curves canbe created from these event lists.5.4 The UltraViolet and Optical Telescope5.4.1 PurposesThe UltraViolet and Optical Telescope (UVOT) generates spectra, images, and light curvesin several different ultra-violet and optical wavebands. It is co-aligned with the XRT and carriesan 11-position filter wheel which allows low-resolution grism spectra of bright GRBs and broad-band UV/visible photometry. Photons register on the microchannel plate intensified CCD (MIC).Figure 5.7 shows a schematic of the UVOT, and Table 5.6 summarizes the UVOT parameters.Further information on the UVOT is given by Roming et al. [67].75Figure 5.7: Block diagram of the UltraViolet and Optical Telescope [32]5.4.2 Physical Description and PerformanceThe UVOT’s optical train consists of a 30 cm Ritchey-Chrétien telescope with a primaryf-ratio of f/2.0 increasing to f/12.72 after the secondary. An INVAR structure that is intrinsicallythermally stable is used between the mirrors and maintains the focus. Fine adjustment to the focusis achieved by activating heaters on the secondary mirror support structure and on the INVARmetering rods that separate the primary and secondary mirrors. The UVOT carries two redundantphoton-counting detectors that are selected by a steerable mirror mechanism. Each detector has afilter wheel mounted in front of it carrying the following elements: a blocked position for detectorsafety; a white light filter; a field expander; two grisms; U, B, and V filters; and three broadbandUV filters centered on 190, 220 and 260 nm. One grism on each wheel is optimized for the UV, theother for optical light, and both offer a spectral resolution of ∼ 1 nm/pixel. The UVOT operatesas a photon-counting instrument. The two detectors are MICs incorporating CCDs with 384×288pixels, 256 × 256 of which are usable for science observations. Each pixel corresponds to 4 × 4arcsec on the sky, providing a 17× 17 arcmin FOV. Photon detection is performed by reading outthe CCD at a high frame rate and determining the photon splash’s position using a centroidingalgorithm. The frame rate for the UVOT detectors is 10.8 ms.76Table 5.6: UltraViolet and Optical Telescope CharacteristicsUVOT Parameter ValueWavelength Range 170-600 nmTelescope Modified Ritchey-ChrétienAperture 30 cm diameterF-number 12.7Detector Intensified CCDDetector Operation Photon countingField of View (FOV) 17 × 17 arcminDetection Elements 2048× 2048 pixelsTelescope PSF 0.9 arcsec FWHM at 350 nmColors 6Sensitivity B = 24 in white light in 1000 sPixel Scale 0.5 arcsecFigure 5.8: The Effective Area of the UltraViolet and Optical Telescope with Different ColorFilters in Place [14]775.4.3 GRB Data ProductsWhen the spacecraft slews to a gamma-ray burst, the UVOT records a 100-second imageusing its V filter. This image is called a “finding chart” and contains an image of the burst afterglowand any stars that are in the field of view. This finding chart is to aid ground-based observers inlocalizing the afterglow. The positional accuracy of the finding chart is approximately 0.3 arcsecrelative to the background stars in the FOV. After recording this finding chart, the UVOT beginsproducing a set of exposures using an automated sequence of filters.There are two data collection modes for the UVOT: Event and Imaging, which can be run atthe same time if desired. In Event Mode, the UVOT stores time-tagged photon events in memoryas they arrive. The timing resolution is ∼ 11 ms. In Imaging Mode, photon events are summedinto an image for a time period known as the tracking frame time (≤ 20 s). The advantage ofImaging Mode is that it minimizes the telemetry requirements when the photon rate is high, butat the expense of timing information.78Chapter 6Calibration of the Burst Alert Telescope6.1 The Physics of Gamma Ray Detection6.1.1 Processes of Photon InteractionWhen a photon enters a detector (or any other material), it may interact with the detector(and in the process transfer a portion of its energy to the detector) by one of three main processes:photoelectric absorption, Compton (or incoherent) scattering, or pair production. Any of theseprocesses might take place when the photon comes in contact with an atom, but depending on theenergy of the photon and the type of atom, some processes are more likely than others.Photoelectric AbsorptionWhen a photon interacts with an atom by photoelectric absorption, its energy is absorbed bythe atom, freeing one of the electrons in the atom (usually one of the innermost “K shell” electrons),and leaving a vacancy in the electron structure (see Figure 6.1). The remaining electrons rearrangethemselves, filling the vacancy and emitting a series of photons with characteristic energies in theprocess. These characteristic photons may then be absorbed themselves by other atoms, or theymay manage to escape from the detector altogether.The likelihood of photoelectric absorption taking place is much greater for atoms with alarge number of electrons. For neutral atoms, the number of electrons is the same as the numberof protons in the nucleus (the “atomic number” Z). The probability of absorption is also muchgreater for photons with a small amount of energy E:P ∝ Z5E3.5. (6.1)Photoelectric absorption is the dominant interaction process for photon energies less than about200 keV. Hence, this is the dominant interaction mechanism for the BAT detectors.79Figure 6.1: Photoelectric absorption is the process whereby a photon of energy E impartsits energy to an electron, allowing it to escape from the atom. In the process, one ormore fluorescence photons (and sometimes an Auger electron) are emitted. Adapted fromwww.indyrad.iupui.edu/public/ radsci/R250/Assignment%20Chapter%2013.pdfCompton ScatteringCompton scattering occurs when an electron absorbs only a portion of the photon’s initialenergy, resulting in the creation of a new, low energy photon that takes off in a new direction. Theenergy of the new photon is given byE′ =E1 + Em0c2 (1 − cos θ), (6.2)where E is the energy of the initial incident photon, E′ is the energy of the final photon, and cos θis the angle between the direction of the initial photon and the direction of the final photon. Thegreater the scattering angle θ, the more energy is absorbed by the electron. Incident photons withgreater energies tend to result in smaller scattering angles. The probability that a photon willCompton-scatter is proportional to the electron density of the material and inversely proportionalto the energy of the photon:P ∝ ρeE. (6.3)This is the dominant process for gamma-rays with energies between about 200 keV and about 5MeV.80θFigure 6.2: Compton scattering, whereby a photon imparts a portion of its energyE to an electron, allowing it to escape from the atom. In the process, a newphoton is created with energy E′. Adapted from www.indyrad.iupui.edu/public/ rad-sci/R250/Assignment%20Chapter%2013.pdfPair ProductionDuring pair production, a photon interacts with the strong nuclear force of an atom andcreates an electron and positron in its place. Since electrons and positrons have masses equivalentto 511 keV, the photon must have at least twice that energy (or 1.022 MeV) for this process tobe possible. Any energy the photon has above and beyond 1.022 MeV becomes kinetic energyfor both the electron and positron. The positron doesn’t last very long and annihilates fairlyquickly with another electron, producing two “annihilation photons”, each with an energy of 511keV. The probability of pair production increases as the number of protons in the nucleus (Z)increases, because such atoms have a greater nuclear “strong force” which facilitates the process.Pair production is only significant for photons with energies greater than about 5 MeV.6.1.2 What Takes Place Inside a Semiconductor DetectorWhen a gamma-ray photon transfers a portion of its energy to an electron in a semiconduc-tor detector by one of the processes we’ve described, the electron usually carries with it a significant81Figure 6.3: Pair production, whereby a photon is converted into apositron and an electron. Adapted from www.indyrad.iupui.edu/public/ rad-sci/R250/Assignment%20Chapter%2013.pdfamount of kinetic energy. As this energetic electron collides with nearby atoms, it transfers itskinetic energy to the electrons in those atoms, liberating them as well. Soon, there is a “cloud”of free electrons and a corresponding cloud of atoms with electron vacancies (or “holes”). Duringnormal detector operation, a bias voltage is applied between the metallized top and bottom elec-trode surfaces of the detector, so that an electric field exists inside the detector and the cathode isthe side facing the photon source. Under the influence of this electric field, the free electrons mi-grate toward the bottom surface of the detector. The “holes” behave very much like free particlesthemselves, migrating toward the cathode (the top surface of the detector — see Figure 6.4).The migration of these electrons and holes results in a current that begins to flow throughthe detector and through the Application-Specific Integrated Circuit (ASIC) to which it is coupled.This current is integrated over time and converted into a voltage pulse with a pulse height amplitude(PHA) proportional to the time-integrated current (i.e., the total charge of the electrons and holesthat were created as a result of the initial photon interaction). This PHA value is then transformedinto a number between 0 and 4095, recorded in the instrument’s solid-state recorder, and eventuallytransmitted to the ground in a data stream. Usually when we refer to a PHA value, we mean the82 Figure 6.4: When a photon deposits energy within a detector, a cloud of electrons and ofholes is generated that migrates toward the electrodes on the surface of the detector, due tothe applied bias voltage.number between 0 and 4095 that the voltage pulse height amplitude has been converted into. Weoften refer to each of these numbers as a “channel”.6.2 Properties of CZT DetectorsCdZnTe (or CZT) detectors are a relatively new technology that has only been around forthe last 10-15 years. They have several advantages over other semiconductor materials that arecommonly used for gamma-ray detection. For one thing, they have a relatively large band gap(the amount of energy required to excite a single electron-hole pair): 1.6 eV. The size of the bandgap dictates the magnitude of the detector’s “leakage” current simply due to thermally excitedelectron-hole pairs. If the band gap is low and the temperature is high, then there is enoughthermal energy to excite electron-hole pairs, and it is difficult to distinguish detected photonsfrom the counts generated by thermal energy. Silicon and germanium, which are also commonsemiconductor detector materials, have band gaps of only 1.1 and 0.7 eV, respectively. Thus,germanium must be cooled to cryogenic temperatures (for example, with liquid nitrogen) in order83to operate effectively as a detector. CZT, on the other hand, operates well at room temperatures(20◦ – 25◦).Another advantage of CZT is that its constituent elements (Cadmium, Zinc, and Tellurium)have large atomic numbers (48, 30, and 52, respectively). As we have seen in the last section, theprobability of photon interaction is greater when the atoms involved have large atomic numbers.By comparison, silicon has an atomic number of only 14 and germanium of 32. Thus, CZT has asignificantly greater detection efficiency. These two advantages—the ability of CZT to operate atroom temperature and the high atomic numbers of its component atoms–are the primary reasonsthat CZT was selected as the material for the BAT detectors.CZT detectors have a drawback, though—one that is common among “composite” semi-conductor detectors (those made of more than one element). It relates to a phenomenon known ascharge trapping. Defects within the semiconductor material result in sites at which charge carriers(electrons or holes) are trapped for a period of time that is long compared to the time over whichthe detector circuitry measures the induced current. As the charge carriers migrate through thedetector, their numbers decrease exponentially as more and more carriers are trapped, reducingthe overall current flowing through the detector. The overall reduction in integrated current de-pends on the depth at which the photon interaction takes place. The ability of a charge carrier(an electron or a hole) to migrate successfully through a detector depends on carrier’s “mobility”µ and its “lifetime” τ . Mobility is a measure of how quickly a charge is able to move through thematerial when a given electric field is applied:µ =vdE, (6.4)where vd is the drift velocity of the charge carrier, and E is the electric field. The typical units ofmobility are cm2/(V · s). The lifetime of a charge carrier is defined as the average amount of timebetween the liberation of the charge carrier (either as a free electron or as a hole) and the chargecarrier’s entrapment. A useful combination of these quantities gives the average distance that acharge carrier is able to travel before being trapped:λ = µ τ E. (6.5)84Charge trapping is a phenomenon that occurs, to some extent, in all semiconductor detec-tors. As long as the average distance the charge carrier can travel is significantly larger than thethickness of the detector, little charge is lost. In Ge and Si detectors, this is generally the case,and λ for electrons and for holes are comparable to each other. In CZT, however, λ for holes isoften a factor of 10 or more smaller than λ for electrons. In the case of the BAT detectors, forwhich the applied bias voltage is nominally 200 V, the average path length of electrons in theBAT detectors is on the order of 2 cm. This is much longer than the distance between the topand bottom surfaces of the detectors (2 mm), so electrons have no trouble migrating through thedetectors with little charge lost. The average path length for holes, on the other hand, is muchsmaller—on the order of 0.2 mm—so hole trapping is significant. The voltage is applied in sucha way that holes migrate toward the top surface of the detectors, so that the deeper the photoninteraction occurs, the more holes are trapped before arriving at the top surface, and the smallerthe overall measured signal will be.There is an equation, known as the “Hecht relation”, which gives the “effective charge”measured by the detector electronics as a function of the depth of the photon interaction:QQ0=(λeD)[1 − exp(−D − zλe)]+(λhD)[1 − exp(− zλh)], (6.6)where Q is the “effective” charge that is measured by the detector electronics, Q0 is the chargethat would have been measured if all of the charge had been collected (that is, if there had beenno charge trapping), D is the thickness of the detector (in our case, 2 mm), and z is the depthof the photon interaction, where the top surface of the detector is taken to be z = 0 and thebottom surface is at z = 2 mm [42]. A plot of the charge collection efficiency (Q/Q0) as a functioninteraction depth is shown in Figure 6.5.Figure 6.5 illustrates the shape of the Hecht relation for typical values of BAT CZT detectors.One can see that photons that interact deeper in the detector result in a smaller charge collectionefficiency. In the case of photons with low energies (E < 50 keV or so), most of the photons do notpenetrate very deeply into the detector, so most interactions occur near the top surface (z = 0)where the charge collection efficiency is high and is nearly the same for all interactions. If we plot85 Figure 6.5: The charge collection efficiency (Q/Q0), as given by the Hecht relation [42], asa function of photon interaction depth, for λe = 2 cm and λh = 0.02 cm. These are typicalvalues for the BAT CZT detectors.86 Figure 6.6: An example of a PHA spectrum from 122 keV photons interacting with a CZTdetector. Peaks resulting from higher photon energies result in tails that are more pronounced.Note the two small “escape peaks” sitting on the tail of the main peak. These are discussedin section 6.3.1.the number of events in each PHA channel as a function of PHA channel, we see a fairly narrowpeak. Photons with high energies, however, are able to penetrate deeper into the detector andthus a significant number of photons produce voltage pulses that are much smaller than wouldhave been the case if the interaction had occurred near the top surface. Instead of a narrow peak,a monoenergetic beam of these photons results in a peak with a long “tail” which extends to lowerPHA values (see Figure 6.6). This “tailing effect” is the primary drawback of using CZT detectors.6.3 Measuring the µτ ProductsIn the Hecht relation given by equation 6.6, there are two quantities that describe the charge-trapping characteristics of a detector: λe and λh. These values depend on the electric field inside87the detector (which is known) and on the product of the mobility and the lifetime of electrons andof holes (µτe and µτh). In order to describe how charges are trapped within the detector, we needto know these values. Note that, for our purposes, it doesn’t matter what the mobility µ or thelifetime τ of a charge carrier is—all that we need to know is their product (µτ).Every one of the 32,768 detectors that make up the BAT array has its own value for µτe andfor µτh. Our first task in calibrating the BAT instrument was to determine what those values are.This was done by measuring the PHA spectrum resulting from a beam of monoenergetic photons.The photon energy had to be sufficient so that the photons were able to penetrate all the waythrough the detector. From this measured spectrum, we were able to determine what µτe and µτhvalues were required to reproduce the shape of the spectrum. We developed a “single-detector”model that reproduces the shape of the spectrum, given a particular set of values for µτe andµτh. The values of µτe and µτh were adjusted until the model spectrum matched the real one. Inparticular, we performed a least-squares fit of the single-detector spectral model to the measuredspectrum, with the µτe and µτh values as free parameters. This process was repeated for each ofthe 32,768 detectors to find µτ values for each detector.6.3.1 Energy Deposition DistributionsFor a particular depth within the detector, the Hecht relation tells us the size (or “am-plitude”) of the voltage pulse generated when photons deposit their energy at that depth. Buta spectrum tells us the number of voltage pulses that have a particular voltage amplitude. Soin order to use the Hecht relation to create a single-detector model spectrum, we not only needa pair of µτ values, we also need to know how many photons deposit their energy at differentdepths in the detector. To determine the fraction of the incident monoenergetic photons that areabsorbed at various depths in the detector, we ran a set of “Monte Carlo” simulations, in whichwe use a software tool called grmcflight to simulate a beam of monoenergetic photons interact-ing with a detector. This tool tracks the path of each photon as it enters the detector. As thephoton travels through the detector, the program calculates the probability that it will deposit a88portion of its energy (either by photoelectric absorption or by Compton scattering) at any givenposition along its path. It then chooses a position (based on the calculated probabilities) at whicha particular energy deposition takes place. It records the position and the amount of energy thatwas deposited and, if a new photon was generated in the process, it continues tracking the newphoton. It continues to do this until the photon is either entirely absorbed or exits the geometryof the simulation. This simulation tool repeats this whole process for many photons, building upa long list of locations and deposited energies. After a predetermined number of photons havebeen tracked through the simulation, the detector is divided into 1000 “slices”, and the numberof photons that deposited energy in each slice is calculated and divided by the total number ofphotons/cm2 that the simulation generated and by the thickness of the slice. The result is a listof 1000 numbers, each in units of cm, that represents the probability that a photon with the par-ticular energy we simulated will deposit its energy at a given depth (see Figure 6.7). We call thislist a “depth distribution” list. We repeat this whole process using 50 different photon energies,ranging from 10 keV all the way up to 9 MeV.If a photon enters the detector directly from above, it will have a different probability ofdepositing its energy in a given slice than if it enters the detector at an angle (see Figure 6.8).Because of this angular dependence, we must repeat the simulations using many different incidentangles. We chose 105 different incident photon angles and repeated all of the simulations describedabove using each of those angles. Because most of the detectors in the array are surrounded byneighboring detectors, we included a layer of CZT around the lone detector in the simulation, sothat photons that are absorbed before they even reach the detector are properly accounted for (seeFigure 6.9).Another complication in the process is that a photon rarely deposits all of its energy at onelocation within the detector. Usually (especially at high incident photon energies where Comptonscattering begins to dominate and pair production becomes energetically possible), a portion of itsenergy is deposited at one location and another portion at some other location. Because of this,89 Figure 6.7: Two examples of depth distributions generated by the Monte Carlo simulations.The pink line gives the distribution for 60 keV photons and the blue line for 122 keV photons.The y-axis gives the number of photons that deposit their full energy in the detector at agiven depth, divided by the slice thickness (in cm) and by the incident flux of photons (inphotons/cm2). The simulated photons all originate from above the detector in a downward-pointing, collimated beam, with a beam area sufficient to encompass the entire area of thedetector.90 Figure 6.8: Depth distributions generated using 129 keV photons incident on a detector fromtwo different incident angles: 0◦ (blue) and 61◦ (pink). The y-axis gives the number of photonsthat deposit their full energy in the detector at a given depth, divided by the slice thickness(in cm) and by the incident flux of photons (in photons/cm2). θFigure 6.9: The geometry of the simulations used to generate the depth distribution lists.A stream of photons is incident from a given angle on a single CZT detector surrounded byneighboring detectors.91 = + + Figure 6.10: The weighted average depth at which a photon deposits its energy is calculatedfrom the various depths at which portions of its energy are deposited.we calculated the weighted average depth at which each photon deposited its energy:zave =∑ziEi∑Ei, (6.7)where zi is the depth at which a particular amount Ei of the photon’s energy is deposited. Thisaverage depth is the depth we counted as the one at which the photon’s energy was deposited (seeFigure 6.10).Sometimes, one of the secondary photons leaves the detector entirely, so that not all ofthe energy of the incident photon is deposited in the detector. Compton scattering is less likelythan photoelectric absorption at the energies at which the BAT array is most sensitive, so weneglect those photons that Compton scatter out of the detector. However, it is also possible for asecondary photon generated during a photoelectric absorption event to leave the detector withoutbeing reabsorbed. By far the most likely energies of these secondary photons are 27 keV and 31keV, which result when the incident photon interacts with an electron in the innermost shell (theK shell) of either a Cd or a Te atom. We therefore keep track of all incident photons that depositall of their energy except 27 or 31 keV. These interactions produce small so-called “escape peaks”in the measured spectrum that appear to have energies equal to 27 keV and 31 keV less than theincident photon energy (see Figure 6.6). We record the events associated with these escape peaksin separate depth distribution lists. In the end, we have a set of 15,750 depth distribution lists—a92Figure 6.11: A full-energy depth distribution for 129 keV photon incident on a detector, fitto a 20th-order polynomial. The depth distributions corresponding to lower photon energies,especially at large incident angles, have fairly sharp features which are also well-fit by a 20th-order polynomial.full-energy list, a Cd K-escape list, and a Te K-escape list for each energy and angle.These depth distributions take up a fair amount of computer memory— 15,750 distributions× 1000 numbers per distribution × 8 bytes per number, which comes to about 120 MB. To reducethe size of the file that stores these numbers, we fit the distributions to a 20th-order polynomial.This enabled us to store 20 numbers (the coefficients of the polynomial) for each distributioninstead of 1000 numbers, reducing the memory required by a factor of 50. A 20th-order polynomialis able to fit the shape of the distribution sufficiently well to reproduce the essential features of thedistribution’s shape (see, for example, Figure 6.11).6.3.2 The Single-Detector Spectral ModelWith a depth distribution and a set of µτ values, we have everything we need to generate amodel spectrum for a given incident photon energy and a given detector. We begin by evaluatingthe Hecht relation at the center of each of the 1000 slices into which we have divided our detector.This tells us the charge collection efficiency resulting from photons that deposit their energy atthose depths. For a given incident photon energy, the charge collection efficiency is proportional93to the voltage pulse height amplitude, or PHA channel:PHA = K ·(QQ0). (6.8)For our single-detector model spectrum, we are free to choose the scale of the PHA channels, sowe can define the proportionality constant K any way we wish. For convenience, we chose K suchthat a full charge collection (Q = Q0) would place a count in a PHA channel with a value equalto 4 times the energy that is deposited:PHA = 4 · E ·(QQ0). (6.9)With this choice, a 122 keV photon that deposited all of its energy and for which the full chargewas collected would result in a count in PHA channel 488. Since the charge trapping properties ofour detectors never permit a full charge to be collected, the peak resulting from 122 keV photonswould actually lie at a PHA channel that is somewhat smaller than 488. With our PHA channelsdefined in this way, we simply place the values from the depth distribution into the PHA channelscorresponding to the charge collection efficiencies of those depths. Because of random fluctuationsin the electronics, a range of voltage pulse amplitudes may result from a set of interactions, evenif those interactions all occur at the same depth and all deposit the same amount of energy. Thisresults in peaks that are somewhat “smeared out”. This is known as energy resolution. We include,in our single-detector model, a parameter which dictates the width of the energy resolution. Wegenerate a Gaussian-shaped peak with this width and convolve this peak with the spectrum. Thisproduces the desired energy resolution effects.6.3.3 Correcting for Electronics NonlinearityAs we described in section 6.1.2, the voltage pulses that are produced when energy is de-posited in a detector are transformed into PHA values that range from 0 to 4095. Because ofthe properties of the ASIC circuitry that measures these voltage pulses, the resulting PHA valuesare not directly proportional to the amplitudes of the voltage pulses (we can’t simply say thatthe signal which was transformed into a PHA value of 2000 is twice as large as the signal which94was transformed into a PHA value of 1000). This is known as electronics nonlinearity. Before wecan directly compare a single-detector model spectrum with a real spectrum, we need to apply acorrection for this nonlinearity, so that the PHA values are directly proportional to the voltagepulses. This is done by measuring the positions of peaks resulting from a set of voltage pulseswith known amplitudes that are sent through the ASIC circuitry. For instance, in a particulardetector, we may find that a 0.5 V pulse results in a peak that is centered at a PHA channelof 1500. By repeating this for several voltage pulses, we can determine the relationship betweenPHA channel and voltage pulse, which turns out to be well approximated by an equation of theform V=a·(PHA)3 + b·(PHA)2 + c·(PHA) + d. We use this relationship to help define a new setof “corrected” PHA values that are directly proportional to the amplitude of the voltage pulses,and we “rebin” the spectra by redistributing the counts in the original PHA channels into theappropriate “corrected” PHA channels.As illustrated in Figure 6.12, the position of the model peak is quite sensitive to the valueof µτe, while the value of µτh strongly affects the prominence of the tail. Since λe and λh inequation 6.6 are both proportional to the electric field E within the detector, a change in E affectsboth λe and λh. Therefore, a change in the bias voltage applied to the detector influences thespectral shape. As the bias voltage increases, the peak channel approaches some limit (which welabel CHmax) which corresponds to the channel at which the efficiency given by the Hecht relationbecomes unity. Also, at higher bias voltages, the tail structure is reduced. By fitting our single-detector spectral model to multiple spectra measured with different bias voltages, we can obtainvalues for µτe and µτh.For our measurements, we illuminated each detector with photons from a 57Co source, whichemits 14.4, 122.1 and 136.5 keV photons (the most prominent being 122.1 keV photons). For eachdetector, we made three measurements: one with a bias voltage of 100 V, another of 200 V, anda third of 300 V. To minimize counts resulting from photons that Compton scattered off of othermaterials in the vicinity before entering the detector, we restricted our fitting region to the portionof the spectrum above the escape peaks. The temperature for all three measurements was held at95Figure 6.12: The dependence of the model peak shape on different values of µτe and µτh(given here in units of cm2/V). Different values of µτe (left panel) strongly affect the positionof the peak, while different values of µτh strongly affect the prominence of the tail.20◦C, the nominal operating temperature of the detectors in orbit.The results of our measurements are shown in Figure 6.13. Note that the µτe values rangefrom about 10−3 to 10−2 cm2/V, whereas the µτh values are much smaller, ranging from about10−5 to 10−4 cm2/V. We discovered that detectors grown from the same crystal ingot tended tohave similar values for µτe, but that µτh varied widely from detector to detector even within thesame ingot.6.4 Determining the Energy ScaleEven after the rebinning process described in section 6.3.3, the PHA channels at whichthe peaks of each particular photon energy are centered differ significantly from one detector toanother. In one detector, the 122 keV peak may lie at PHA channel number 500, whereas inanother detector, the 122 keV peak may lie at PHA channel number 450. The primary cause ofthese energy scale differences is variations in µτ values (particularly µτe) from one detector to thenext, as illustrated in Figure 6.12. As we’ve seen from the Hecht equation in Section 6.2, differentµτ values result in different maximum charge collection efficiencies.In Section 5.2.3, we described how a mask-weighted count rate could be generated by com-bining the count rates from each individual detector in such a way that background counts are96Figure 6.13: A plot of the µτe and µτh values measured for the 32,768 detectors that makeup the BAT array.removed. It is possible to use the same process to combine individual detector spectra to buildup a background-subtracted, mask-weighted spectrum. Before we can combine the spectra fromeach detector, however, we need to adjust each spectrum so that all of the peaks corresponding toa given incident photon energy appear at the same PHA channel. In section 6.3.3, we describeda process in which we defined a new set of PHA channels that are proportional to the voltageamplitude of the signal generated in the ASIC. This time we use a similar process to define a setof PHA channels that are proportional to the peak energy, so that for every detector, a photon ofa given energy always produces a peak centered at the same PHA channel. We do this by findinga relationship between peak energy and PHA channel:E = f(PHA), (6.10)defining a new set of channels (which we shall label “PHA′”) that are proportional to the peakenergy:PHA′ ∝ E, (6.11)and rebinning the spectra into these new PHA′ channels.Finding the specific relationship described by equation 6.10 is done in several steps. First,we find the PHA channels at which several calibration source energy peaks are centered. We use97photons from an 241Am source (which emits 59.5 keV photons, among others), a 133Ba source(which emits 30.9 keV and 81 keV photons), a 109Cd source (which emits 22.1 keV photons), and a57Co source (which emits 122.1 keV photons). We use these PHA channels along with the relation-ship between voltage amplitude and PHA channel found previously (as described in section 6.3.3)to find a relationship between voltage pulse amplitude and peak energy. This relationship iswell-approximated by a linear function:V = G ·E +O, (6.12)where G and O are constants, E is the photon energy, and V is the voltage pulse that is producedby that photon energy that results in a peak in the spectrum. This equation can be combined withthe relationship between V and PHA described in section 6.3.3 to derive a relationship betweenE and PHA that has the form:E = A · (PHA)3 +B · (PHA)2 + C · (PHA) +D, (6.13)where A, B, C, and D are constants. This process is repeated for each of the 32,768 detectors.We have described how some of the properties of the detectors (for example, the energyresolution) depend upon their temperature. The relationship between peak energy and PHAchannel (equation 6.13) is also temperature-dependent. By measuring the 122 keV peak positionat different temperatures, we have found that the constant G in equation 6.12 increases by between1% and 4% (depending on the detector) when the temperature is reduced from 20◦C to 4◦C. Thisis one reason that the temperature of the detectors is carefully maintained at 20◦C.6.5 The Angular Dependence of the Peak ShapeIn section 6.3.1, we described the need to find the depth distributions resulting from photonsilluminating a detector from several different angles, and Figure 6.8 gives an example, illustratingthe depth distributions for 129 keV photons when the photons are incident on the detector fromdirectly above the detector and also from angle of 61◦C. We expect fewer photons overall to interactin the detector when the photons are incident at an angle (because the area intersected by the98Figure 6.14: Simulated spectra from 122 keV photons incident on a detector from variousangles. As the angle increases, the tail becomes smaller.stream of photons is smaller). There are also good physical reasons to suspect that the overallshape of the peak will be different depending on the angle of the incident photons. A photonthat is traveling through a detector at an angle is able to travel the same average distance beforebeing absorbed as a photon that is incident on-axis, but because it travels at an angle, it doesnot penetrate as deeply into the detector. For this reason, when photons are incident at largerangles, a larger fraction deposit their energy near the top surface of the detector. As we note inFigure 6.5, photons that penetrate deeper into the detector result in a smaller charge collectionefficiency. These are the photons that contribute to the tail of the peak. As a result, the tail is notas prominent when photons are incident at a larger angle. Figure 6.14 shows a set of simulatedspectra resulting from 122 keV photons incident at various angles. As the angle increases, the tailbecomes smaller. Note that for small angles (less than about 40◦ or so), the effect is quite small.Most of the detectors in the array are surrounded on all sides by other CZT detectors, butdue to engineering considerations, the detectors are clustered in groups of 128 detectors, with gapsseparating each group from the others (see Figure 6.15). If photons are incident on the array at anangle, detectors on the “front” (or “leading”) edges of these groups will have their sides illuminatedmore than those that are surrounded by photons. This means that more will interact at greater99Figure 6.15: The array is divided into groups of 128 detectors that are separated from eachother by gaps.depths in those leading edge detectors, contributing more to the tail in their spectra. Differences inthe spectra of these leading edge detectors and those of other detectors do not become significantuntil the incident angle reaches about 60◦ or so.The spectra from all of the detectors are summed (weighting them by their mask weightfactors) to produce a composite spectrum. Leading edge detectors only make up 18% of the totalarray, so when they are combined with the rest of the detectors, the overall affect these differenceshave on the shape of the composite spectrum is negligible.6.6 Adjusting for Distance and Angle VariationsThe spectral model we have discussed so far is a single-detector model, able to reproducethe shape of the spectrum from a single detector. Since the spectra from all of the detectors are100combined in a mask-weighted sum, we need a full-array spectral model that can reproduce theshape of the composite spectrum. To assist us in creating this full-array spectral model, we usedBAT to measure the real gamma-ray spectra from radioactive sources that emit photons at specificenergies. We placed these sources in different positions relative to the detector array and measuredtheir spectra. Usually, the sources were placed at a distance of about 3 meters above the array.When a source is placed 3 meters directly over the center of the array, detectors at thefarthest corners of the array are slightly farther from the source (308 cm) than detectors at thecenter of the array (300 cm). This means that the number of photons/cm2/s that fall on thesedetectors will be smaller than those at the center of the array by a factor of 3002/3082 = 0.95.This isn’t a large difference, but it is significant, and when the source is not directly above thecenter of the array, the difference is larger.Furthermore, photons from the calibration source approach different detectors at differentangles. When photons are incident on the top surface of a detector at an angle, the projectionof the area perpendicular to the incident photons is smaller than when photons are incident fromdirectly above, by an amount cos θ (see Figure 6.16).Both of these effects result in differences in the total number of photons that hit detectorsat different positions on the array. In orbit, photons come from so far away that these differencesare undetectable. We must correct for these differences before combining the spectra so that ourground measurements will give us an accurate idea of how the spectra from distant sources willlook. We correct for differences in illumination angle and distance to the source done by multiplyingthe count rate observed in each detector by a correction factor:(r2dCangle,d)(Cangle,cr2c), (6.14)where the subscripts “d” and “c” refer to the detector to which the factor is being applied and tothe center of the array, respectively, r is the distance from the source to the detector (or to thecenter of the array), and Cangle is a constant that is equal to the perpendicular area visible to theincident photons at the detector (or at the center of the array) divided by the full area of the topsurface of the detector (0.16 cm2). (Cangle) is approximately equal to cos θ, but since photons at101 θFigure 6.16: Photons that are incident from directly above a detector intersect an area ofAon−axis (= 0.16 cm2), whereas photons that are incident from an angle θ intersect an areaof Aangle, where Aangle = Aon−axis · cos θ (plus a little bit, due to the edges of the detectorbeing visible).an angle will intersect the sides of the detectors to a certain extent, there are two terms that areincluded to account for this:Cangle = cos θz+ min[0.15, 0.05 ·∣∣∣zsxs∣∣∣]· cos θx+ min[0.15, 0.05 ·∣∣∣zsys∣∣∣]· cos θy, (6.15)The overall result of multiplying the spectrum from each detector by the factor given in equa-tion 6.14 is that it adjusts the overall height of each detector spectrum to what it would have beenif the detector were at the center of the array.6.7 The Full-Array Spectral ModelIn the beginning of the last section, we described the need for a full-array spectral modelthat can reproduce the shape of the mask-weighted summed spectrum. This is somewhat differentfrom the single-detector spectral model we’ve discussed up to this point. The single-detector modelreproduces the shape of a peak resulting from the photoelectric absorption of photons of a given102energy in a single detector, and it depends on the µτe and µτh values of the detector. As we sawin Figure 6.13, these values are very different from one detector to another.The full-array spectral model must be a combination of the single-detector spectral modelsof all of the detectors in the array. One way to generate this full-array model spectrum would beto compute the single-detector model spectrum for every detector, and then take the average ofthese spectra, but generating so many computing so many spectra would be very computationallyintensive. So instead, we divide Figure 6.13 into 35 regions and use the average µτ values in eachregion to calculate a model spectrum for that region. Then we combine the 35 spectra, weightingeach one according to the number of detectors within the corresponding region. This produces afull-array model spectrum that matches the real data reasonably well.Another matter we must consider for our full-array spectral model is the absorption dueto passive material in the BAT FOV that might reduce the number of photons/cm2/s hitting thedetector array. There are several materials that lie between the photon source and the array. Theseinclude a Multi-Layer Insulation (MLI) needed for thermal control, a wire grid that maintains thebias voltage, epoxy dots that hold the grid in place, the mask structure to which the lead tilesare fastened, and layers of thin conducting materials intended to protect the array from strayelectrical discharges. All of these materials absorb gamma-ray photons. Our full-array modeltherefore includes a calculation of the fraction of photons that will successfully arrive at the array.This fraction depends on what materials the photons pass through and how thick they are, andit is also a function of photon energy. When we are making measurements on the ground withradioactive calibration sources, the air between the source and the detectors also absorb photons,so we must also include this in our calculation. Rather than perform this calculation for eachindividual detector, we make the assumption that the absorption at different detector positions isabout the same, and we simply calculate the absorption for the center of the array. More detailedcalculations have shown that this assumption is reasonable.There are 7 parameters required for our full-array spectral model:• A parameter that specifies the energy resolution. This is defined as the standard deviation103(σ) of the best-fit Gaussian function that is convolved with the peaks to smear them out.• Two parameters that enable the model peak positions to be slightly adjusted by a factor ofC · EI , where we label C the “gain coefficient” and I the “gain index”. If no adjustmentwere needed, C would be 1 and I would be 0. The best fit parameters were indeed found tobe close to 1 and 0, respectively.• Three parameters that add an extra “tail” component to the peaks at lower energies. Itwas found during the fitting process that our model was unable to provide a good fit to the133Ba 31 keV peak. The real data seemed to show a large tail that our model didn’t predict.Consequently, we incorporated an empirical exponential tail into the model of the formF (PHA) = A · exp(PHA− PHApeakPHApeak · λ), (6.16)with A = C ·(max(E,E0)E0)Iwhere PHApeak is the PHA channel at which the photopeak generated by photons of energyE appears, E0 = 31 keV, C is a constant which we label the “exponential tail coefficient”, λis a constant which determines how steeply the exponential tail falls off, and I is a constantwhich we label the “exponential tail index”. The overall prominence of the exponential peak(as quantified by A) becomes smaller as the photon energy E increases, so the exponentialtail is less pronounced at higher energies. A portion of the counts in the main photopeak(equal to A) is removed, and then F (PHA) is added to the spectrum.• A parameter that adjusts the overall normalization of the entire spectrum, which we labelthe “normalization adjustment”. This is a value that is multiplied to the model spectrum,so that if no adjustment were necessary, it would be equal to 1.A 133Ba source was placed at 57 distinct positions, and measurements were taken with theBAT instrument (see Figure 6.17). 133Ba emits photons with four energies that are within therange in which the BAT is particularly sensitive: 31 keV, 35 keV, 53 keV, and 81 keV. The two104 Figure 6.17: The positions of the 57 133Ba spectra measured with the BAT instrument forcalibration purposes are shown here. The sources were placed at various distances from thedetector array, but most were placed about 300 cm above the array. This plot shows thepositions of these sources, in coordinates centered at the detector array. For reference, theposition of the detector array is outlined in red. The UVOT and XRT are located in the +ydirection from the detector array. Depending on the position of the source, a certain fractionof the detectors are illuminated through the mask. Those points inside the green contourilluminate the entire array. Such points are “100% coded”. Points lying on the blue contourwould illuminate exactly half of the array (that is, for half of the detectors, the graded-Z shieldwould lie between the source and the detector). These points would be “50% coded”. For anypoints outside the purple contour, the shield would lie between the source and every detectorin the array (that is, they would be “0% coded”).105 Figure 6.18: A typical composite spectrum obtained by measuring the photons incident onthe BAT array from a 133Ba source. 133Ba emits photons at 4 energies to which BAT isparticularly sensitive: 31 keV, 35 keV, 53 keV and 81 keV. the 31 keV and 81 keV peaks arethe most prominent (peaking at PHA channels of 62 and 163, respectively). The 53 keV peakshows up as a very small peak at a PHA channel of 107, and the 35 keV peak shows up at achannel of 70, right next to the 31 keV peak and almost unresolvable from it.most prominent lines are at 31 and 81 keV. Figure 6.18 shows a typical spectrum measured atone of these positions. The source emits about 6 × 107 photons/cm2/s in all directions at thesefour energies, resulting in about 59 photons/cm2/s at the center of the array when the sourcewas placed 300 cm above it. A duration of about 600 s was required to accumulate spectra withsufficient counts to perform a good fit. Once the spectra were measured, they were adjusted usingthe procedures described in sections 6.4 and 6.6 so that they could be combined, and then theywere added together, weighted by their mask-weighting factors. The full-array spectral model wascalculated and compared to the real combined spectrum, and the 7 parameters described abovewere adjusted until the model spectrum matched the real one. This process involved a least squaresfit of these 7 free parameters using the Levenberg-Marquardt algorithm. Of the 57 measurements106Table 6.1: The mean and standard deviations of the 7 full-array model fit parametersParameter Mean Standard Deviation % Standard Deviationsigma 1.345 keV 0.03550 keV 2.6%gain coefficient 1.014 0.002842 0.28%gain index -0.002140 0.0005344 25%exponential tail coefficient 0.2197 0.02975 14%exponential tail index 1.957 0.7318 37%exponential tail λ 0.2768 0.05422 20%normalization adjustment 1.069 0.05546 5.2%made, 39 resulted in good fits and 18 of the measurements were excluded from the sample forreasons such as improper measurement setups or unreasonable fit values. Table 6.1 gives themean values and the standard deviations for the 7 fit parameters. As you can see, in some cases,there was a significant spread in the measured parameters. Our original expectation was that asmooth variation with source position would be present in these parameters. In fact, when themeasurements were made, we found no clear correlation between the parameters and the positionsof the source. This led us to conclude that either (1) the parameters vary with source position ona very small scale—much smaller than the distance between measured source positions, or (2) thevariations seen in the fit parameters are smaller than the uncertainties in those fit parameters.After we derived these parameters using our 133Ba measurements, we checked the resultsagainst measurements with other calibration sources. We noticed that the model wasn’t able toreproduce the position or width of the peak resulting from the 122 keV photons produced by 57Co.This prompted us investigate whether we needed a better functional form for the factor used toadjust the peak positions than the power law function we had assumed (of which, the coefficientand index were among the 7 parameters in our fit). It also prompted us to investigate whether theenergy resolution was a function of photon energy. We measured the required position adjustmentfactor and energy resolution width σ for several calibration peaks ranging from 14 keV all the wayto 384 keV. We then found functions that reproduced these values (see Figures 6.19 and 6.20).These functions were incorporated into our model, and we found that it was able to reproducethe measurements at photon energies above 80 keV much more accurately.107Figure 6.19: The peak position adjustment factor, as a function of photon energy. We foundthat a quadratic function of the form (−4.66× 10−7)E2 + (2.19 × 10−4)E + 0.998 adequatelyfit the data points.Figure 6.20: The energy resolution width “sigma”, in units of keV, as a function of photonenergy. We found that a constant value of 1.35 keV up to a photon energy E of 77 keV and alinear function of the form 0.00918E +0.641 beyond E=77 keV adequately fit the data points.1086.8 On-Orbit Calibration ResultsAfter Swift was launched, we began making measurements of astrophysical sources. Onesource in particular with a well-known measured spectrum in the BAT energy range is the CrabNebula. It emits a spectrum of photons in which the number of photons N varies with photonenergy E according todNdE= A ·(E1 keV)I, (6.17)where A = 10.17 photons/cm2/s/keV and I = 2.15. We made several measurements of the Crabspectrum by orienting the BAT instrument at 40 different angles relative to the position of theCrab Nebula. We then compared the spectrum we measured with the spectrum we expected tomeasure (the spectrum predicted by our model).Figure 6.21 shows the measured spectrum vs. the model spectrum for one particular mea-surement. One thing we note right away is that the regions where the model is less accurate arealso just outside the range of the 133Ba photon energies where we primarily focused our calibrationefforts.To correct these problems, we developed an empirical correction model which consisted ofthree parts:1. a parameter that makes it possible to include extra absorption that may be present but whichwe didn’t originally account for,2. a parameter that dictates to what degree edges of the lead tiles affect the absorption,3. a six-degree polynomial function which, when multiplied to the model spectrum of the CrabNebula, “forces” it to match the measured spectrum.The first two components of this correction model are motivated by actual, physical processesthat we expect may be at play. It is conceivable that extra absorptive material may be present,and we know that the edges of the lead tiles play some roll in reducing the number of photonsarriving at the detector array. The last component is entirely empirical, without any attempt toexplain it in terms of known physical processes (though this only reflects our ignorance of what109Figure 6.21: A measurement of the Crab Nebula spectrum (black), along with our model ofthe Crab Nebula spectrum (red). Our model is a bit high for PHA channels correspondingto photon peak energies below about 25 keV and a bit low for PHA channels correspondingto photon peak energies above about 90 keV. To adjust for these differences, an empiricalcorrection model was applied.110processes are not being accounted for by our model). The empirical correction model involves 8free parameters which are derived by comparing the Crab spectrum from our full-array spectralmodel to the actual Crab spectrum measured at five different positions relative to the BAT array.Although this correction model enables us to reproduce the measured Crab Nebula spectrum,it is imperfect for astrophysical sources with spectra that are much different from that of theCrab. Ongoing efforts seek to understand the physical reasons behind the need for this empiricalcorrection model, so that they can be more properly accounted for.Using this empirical correction, we compared the model spectrum to the measured spectrumfor each of the different Crab measurements. By allowing the spectral index and the normalizationto vary in the model, we fit the model to the measured data to determine whether the best-fitspectral index and flux from the Crab Nebula matched the known canonical spectral index andflux. The results are shown in Figures 6.22 and 6.23. There is some spread in the best-fit values,and in some cases the best-fit value is farther from the true value than the size of the error inthe best-fit value would permit. There is also a slight systematic drift to higher photon indicesat greater angles. In general, though, the measured values are fairly consistent with the best-fitvalues.6.9 The Purpose of a Response MatrixUp to this point, we have described in detail the single-detector and full-array spectralmodels that we have developed which reproduce the spectra generated by the BAT instrument.But so far, we have said nothing of their purpose. We shall now describe why we need such aspectral model.If life were simple, photons of a given energy that come in contact with a detector would alldeposit all of their energy in the detector, and the electrical signal generated by the detector wouldalways have the same amplitude. Thus, photons of a given energy would all result in counts in aparticular PHA channel. It would be very easy to interpret the measured PHA spectrum becauseit would exactly resemble the spectrum of incident photons. As we have seen, though, life is not111Figure 6.22: θ verses best-fit photon index for the Crab spectrum measurements. The hori-zontal dashed line indicates the known “canonical” value of the Crab spectrum photon index.Ideally, all of the measurements would fall right on that line.Figure 6.23: θ verses best-fit 15-150 keV flux for the Crab spectrum measurements. Thehorizontal dashed line indicates the known “canonical” 15-150 keV flux from the Crab Nebula.Ideally all of the measurements would fall right on that line.112 Figure 6.24: An m × n response matrix transforms an m-element photon spectrum into ann-element count spectrum.so simple. A stream of monoenergetic photons results in a PHA spectrum with counts in manydifferent PHA channels. Therefore, a stream of photons of many different energies will depositcounts in overlapping PHA channels, and the result is a rather convoluted PHA spectrum that isnot obvious to interpret.We can describe the way in which an instrument transforms a photon spectrum into aPHA (or “count”) spectrum by describing it as a matrix (see Figure 6.24). We can describethe incident photon spectrum as a vector, in which each vector element represents the numberof photons/cm2/s that have energies within some energy range E1 − E2. For instance, perhapsthere are 20 photons/cm2/s incident on the instrument with energies ranging from 10-12 keV, 16photons/cm2/s with energies ranging from 12-14 keV, and so forth. We can describe the PHAspectrum in the same way: as a vector whose elements give the number of counts/s measuredin a particular PHA channel. The “response matrix” is then a matrix that deterministicallytransforms any photon spectrum into a corresponding measured count spectrum. If we choose thephoton spectrum to have N elements (or “bins”) and the count spectrum to have M bins, then theresponse matrix will be an M × N matrix. Note that because the elements in the count spectrumhave units of counts/s and the elements in the photon spectrum have units of photons/cm2/s, theelements in the response matrix must have units of “counts/(photons/cm2)”, or (since “counts”113and “photons” are technically dimensionless numbers) just cm2—the units of effective area, whichwe described in Section 5.2.4. In fact, if the matrix were “collapsed” by adding together all ofthe elements in each column, the result would be a vector containing the effective area of theinstrument for each photon energy range.If only a single element of the photon spectrum were non-zero —say, for example, thethird element—then only the third column in the response matrix would contribute to the countspectrum. One can see, then, that each column of the matrix is simply the count spectrum thatwould result if 1 photon/cm2/s in the corresponding photon energy range were incident on theinstrument. So in order to build up and generate a response matrix, all we need to do is find eachof the count spectra that would result from photons in each of the incident photon energy ranges.This, then, is the purpose of our spectral model. Because the depth distributions from which theresponse matrix is created depend on the direction of the incident photons, the response matrixis likewise a function of incident direction. In fact, for any given incident direction relative to theBAT instrument, there exists a unique response matrix.How, then, do we use the response matrix? When the BAT measures a count spectrum, andthe position of the source is determined (by the processes described in section 5.2.3), we would liketo be able to insert this count spectrum into some mathematical tool and have it return the photonspectrum (which is, after all, what we are really interested in). This would require a matrix thattransforms a count spectrum into a photon spectrum, which is the inverse of what the responsematrix does. In fact, we would love to have the inverse of the response matrix, which woulddo just that. Unfortunately, deriving the inverse of the response matrix from first principles isnot possible, and finding it by inverting the response matrix is not generally possible because suchmatrices tend to be singular (or nearly singular) and therefore do not have inverses. So we are stuckwith finding a way to do the best we can with what we have: the response matrix, which transformssomething we don’t have (the photon spectrum) into something we do (the count spectrum). Acommon method is to choose a model of some kind for the photon spectrum—a model that canbe described in terms of a few parameters. A good example is the Band function we described114in section 2.4.4, which is characterized by four parameters: a low-energy index α, a high-energyindex β, a peak energy Eobspeak, and a normalization constant A. We begin with our best guesses asto what four values would generate a photon spectrum that would result in the count spectrum wehave measured. Then, we iteratively make small changes to these parameters (usually using someautomated fitting software tool) until they result in a count spectrum that matches the one wehave measured. We can even assign a confidence to each parameter, stating for example, a “90%confidence interval”, which, roughly speaking, gives us a range of values that the “true” value hasa 90% probability of falling within.The size and number of photon bins are entirely at our discretion, and since we rebin thePHA spectra prior to using the response matrix, we can also assign PHA bins that cover any rangeand have whatever widths we desire. Because we are not trying to invert the matrix, we don’tneed to worry about making sure that it is square (with the number of photon bins equaling thenumber of count bins). In order to not lose any resolution in the PHA spectrum, it is wise tomake the PHA bins narrower than the widths of the peaks. The standard choice for BAT spectrais to have 80 count bins, with widths such that 200 keV photons would produce peaks centered inthe 79th bin. In order to guarantee the optimum conditions for fitting parameters, it is wise tochoose the photon bin widths to be somewhat finer than the count bin widths, by at least a factorof two. The standard choice for defining such photon bins for BAT spectra is to choose about200 bins that range from 10 keV up to 500 keV, with exponentially increasing bin widths suchthat the first bin has a range from 10 to 10.1975 keV and the 200th bin has a range from 490.314keV to 500 keV. Photons with energies less than 10 keV are not detected by BAT, and BAT hasvery little sensitivity to photons with energies greater than about 300 keV (let alone 500 keV), asdemonstrated by the effective area curve in Figure 5.4.With this response matrix, we now have the tool we need for investigating the spectralproperties of gamma-ray bursts and x-ray flashes detected by BAT.115Chapter 7Analysis Procedures7.1 Data Reduction7.1.1 BAT DataAll data that comes from the Swift spacecraft and from its three instruments is availablethrough the High Energy Astrophysics Science Archive Research Center (HEASARC) which islocated at Goddard Space Flight Center. The HEASARC provides:• “raw” data files—these are files that have not yet been processed (at least, very much) bydata analysis tools. One example is event files, which give the time, PHA channel, anddetector of each count recorded by the BAT. Other raw data files give descriptions of whichdetectors are enabled at a given time, what the bias voltage is, where the spacecraft is locatedand where it is pointing at any given moment, etc.• data analysis tools—these are software tools that can be used to create (for example) lightcurves, spectra, and response matrices from the raw data files.• some processed files—some files that are generated automatically by the HEASARC, usingthe data analysis tools. These include light curves and spectra. Often, scientists want toobtain files that are processed in a specific way and will therefore run the data analysis toolsthemselves. Some files (like response matrices) are not provided by the HEASARC and mustbe generated by the user.The data files are provided in FITS format, which is a standard form for many astrophysical datafiles supported by the HEASARC. Likewise, the analysis tools are designed to process FITS files.All of these tools and data files can be obtained from the ’Archive’ or ’Quicklook Data’ pagesaccessible from the main Swift web page at http://swift.gsfc.nasa.gov.116The process by which one generates a spectrum and a response matrix from BAT event datais as follows:1. Assign an energy to each PHA channel for each detector. This is done using the relationshipbetween peak energy and PHA channel described in section 6.4. The tool that assigns thisenergy is bateconvert.2. Create a “detector plane image” from the event file. This is just a map showing the numberof counts recorded in each detector. The tool that generates this is called batbinevt.3. Determine which detectors are “noisy”. For reasons not well understood, some detectorssometimes begin recording a large flow of counts that aren’t the result of detected photons.These detectors can be singled out so that they can be excluded when an image of the sky,a light curve, or a spectrum is generated. The tool that finds these noisy detectors is calledbathotpix.4. Mask out noisy detectors, then convolve the detector plane image with the pattern of themask to create an image of the sky. The tool that does this is batfftimage.5. Determine the position of the gamma-ray burst. For bright bursts, this is the most prominentobject in the sky image. For weaker bursts, sometimes there are other gamma-ray sourceswith known positions that can be ignored. The tool that determines the positions of sourcesin the image is batcelldetect.6. Calculate the mask weight for each detector. This requires a knowledge of the position ofthe burst, which we found using batcelldetect. The tool that calculates these mask weights isbatmaskwtevt.7. With the energy scale and the mask weights, we can now combine the individual spectra intoa single composite mask-weighted spectrum. This is done by batbinevt.8. Finally, generate a response matrix appropriate for the spectrum we have just produced.This is done with batdrmgen.117For my analysis, I have used the following versions of these tools:Tool Versionbatbinevt 1.23bathotpix 1.7batfftimage 1.15batcelldetect 1.28bateconvert 4.1batmaskwtevt 1.11batdrmgen 3.1Because we introduced a set of empirical corrections to the batdrmgen model, we recognizedthat there are still, at this point, some limitations in our understanding of the response. Weevaluated the size of the empirical corrections at each channel and derived a set of “systematicerrors”. The overall magnitudes of these errors were adjusted until the errors in the best-fitparameters from the various Crab measurements were consistent with each other. This systematicerror vector is found in a calibration file distributed with the BAT tools and was applied to eachBAT spectrum before fitting it with the response matrix. Channels corresponding to photonenergies of 14 to 200 keV were included in each fit.7.1.2 Wind Konus DataThe Wind satellite was launched on 1 November 1994 and currently resides in a halo orbitabout the L1 Lagrangian point between the Earth and the sun. The L1 Lagrangian point is oneof 5 points at which a spacecraft can be positioned where it will remain “fixed” with respect tothe sun and the Earth. The L1 point lies directly between the Earth and the sun at a distanceof about 900,000 miles from the Earth (which is about 1/100th of the distance between the Earthand the sun). One of Wind’s primary missions is to measure the solar wind, but there are alsogamma-ray burst instruments aboard such as Konus.Konus consists of two identical detectors with nearly omnidirectional sensitivity that are118placed on opposite sides of the spacecraft so that together they can observe almost the entire skyat once [3]. Each detector consists of a NaI(Tl) scintillator crystal that is 5 inches in diameterand 3 inches high that is housed in an aluminum container with a beryllium entrance window.When a gamma-ray photon interacts with the crystal, it produces visible light that is detected andconverted into an electrical signal by a photomultiplier tube.The Konus detectors can measure incident photons with energies ranging from 10 keV to 10MeV. Konus has no independent imaging ability and can only determine the origin of a photon bywhich detector is hit. Konus records 64 individual spectra each time it detects a gamma-ray burst.The first four are accumulated for 64 ms each. The accumulation times of the next 52 spectra aredetermined by an adaptation system and depend on the time structure of the burst. They rangefrom 0.256 s to 8.192 s. The last 8 spectra are accumulated for 8.192 s each.Because the mask-weighting technique removes most counts resulting from photons above200 keV or so, it is difficult to measure Eobspeak if Eobspeak happens to be above 200 keV. Konus, on theother hand, can measure photons with energies up to 10 MeV. For this reason, it is very helpfulto fit the spectrum measured by BAT and the one measured by Konus simultaneously. Of course,each spectrum requires its own response matrix.One difficulty in doing this simultaneous fit arises from the delay time between the twospacecraft. Because Konus is so far from the Earth (and hence, from Swift), there can be asignificant delay between when the photons of a given gamma-ray burst are first measured by onespacecraft and when they are measured by the other. In the most extreme cases, this delay timecan be as large as 5 seconds. A short burst could be over and finished in the Konus detectorsbefore BAT even begins to detect it, or vice versa.In order to simultaneously fit spectra from both instruments, it is important that the twospectra include the same time intervals, relative to when the burst photons arrived at the twoinstruments. This requires us to know the time delay, for a given incident direction, between whena plane wave would intersect Swift and when it would intersect the Wind satellite. In order todetermine this delay time, we need 3 pieces of information:1191. The position of the Swift spacecraft at the time it detected the burst. This information isavailable in an “attitude” file made available through HEASARC.2. The position of the Konus spacecraft at the time the burst triggered it. This information iscontained in the Konus spectral files that are provided to us by the Konus team.3. The direction from which photons came for the burst in question. Typically this directionis measured in celestial coordinates of Right Ascension (RA) and Declination (Dec). Thisinformation comes through the TDRSS message but can also be found independently usingthe steps listed in Section 7.1.1.Once we have these 3 pieces of information, the delay time between when Konus would haveobserved the burst and when Swift would have observed the burst is given by(kx − sx)ex + (ky − sy)ey + (kz − sz)ezc, (7.1)where (kx, ky, kz) are the x, y, and z coordinates of the position of the Konus spacecraft in geocentricinertial coordinates (coordinates in which the center of the Earth is the origin, the +z-axis extendsthrough the north pole, and the +x-axis points in the direction of the Vernal Equinox), and(sx, sy, sz) are the x, y, and z coordinates of the position of the Swift spacecraft in geocentricinertial coordinates, and e is the unit vector in the direction of the burst, which is given in termsof RA and Dec:ex = cos(Dec) cos(RA)ey = cos(Dec) sin(RA) (7.2)ez = sin(Dec)With this delay time, it is a simple matter to specify time intervals that are the same forboth instruments. Since the time intervals of the Konus spectra are rigidly set by the instrument,it is necessary to find the corresponding times for Swift and specify them when producing the BATspectra.1207.1.3 HETE-2 FREGATE DataIn the case of XRF 050219B, the HETE-2 French Gamma-Ray Telescope (FREGATE)observed the burst and obtained spectral measurements. The energy range of the FREGATEextends from 6 – 400 keV, which is significantly larger than BAT’s nominal range of 14 – 200 keV.The HETE-2 science team provided me with spectral and response matrix FITS files, so that Iwas able to fit the FREGATE data jointly with the BAT data and better constrain the spectralparameters of this burst. For more information on the HETE-2 mission, see Section 3.4.7.1.4 XRT DataAs with the BAT data files and software, the XRT files and software are also distributedthrough the HEASARC. An xrtpipeline script is available that simplifies the data analysis processconsiderably, automating the basic data reduction tasks and producing (among other files) a setof event files that is “clean”—that is, in which counts that have been deemed unreliable or invalidare removed. Examples of such counts would be those that occur in bad pixels, those that arejudged invalid according to their “grade” (or the pattern of counts in surrounding pixels), or thosethat are due to photons coming from a calibration source. The XRT can sometimes overheat dueto the failure of the Thermal Electric Cooler (TEC) early in the mission, and this invalidates thedata taken during such time periods, so the event files also must be screened to include only thosetime intervals during which the CCD temperature was below −40◦. xrtpipeline was run for eachdata set to produce clean event files, and then each event file was screened for temperature.Extracting a spectrum from a cleaned event file can be done using a tool developed byHEASARC called “xselect”, which allows the user to select a portion of the CCD array fromwhich to extract the spectrum. Since each position on the CCD array corresponds to a point inthe sky, this makes it possible to extract a “foreground” spectrum (one that includes the positionof the afterglow) as well as a “background” spectrum (one that does not include the afterglow).By subtracting these two from each other, we obtain a spectrum due entirely to the afterglow.Xselect allows the user to specify the radius of the region within the image (in pixel coordinates)121to use for extracting each spectrum. For the analysis described herein, a radius of 20 pixels wasused for foreground spectra and a radius of 70 pixels was used for background spectra. Xselectversion 2.3 was used for this analysis. When performing the spectral fits, channels correspondingto photon energies of 0.3 to 10 keV were included.It is also possible to find the number of counts/s as a function of time from the cleaned eventfile using xselect, but it is a rather cumbersome and time-consuming process, especially for GRBafterglow data in which the count rate diminishes with time (as opposed to the count rates frommany astrophysical phenomenon, which remains constant or varies much more slowly). For mostof the analysis described in this work, I have used a tool called plot lc, which was developed by amember of the XRT instrument team named David Morris. This tool produces a light curve forwhich each data point contains a specified number of counts each. In this way, the relative errorfor each data point can be precisely controlled. Including many counts in a data point results ina small measurement error but fewer overall data points, while including fewer counts results in alarger error but more data points.plot lc also corrects for an effect known as “pile-up”. Pile-up refers those counts that arriveat the instrument nearly simultaneously and are recorded as a single event rather than as multipleevents. This reduces the number of counts recorded, the flux inferred from the measured countrate is lower than the true flux. By observing the shape of the point spread function, a “flattening”can be detected near the center of the point spread function when the count rate exceeds a certainlevel. To correct for pile-up, one must exclude all counts from pixels near the center of the pointspread function and multiply the remaining counts by a correction factor. If the count rate exceeds1 count/s, plot lc excludes all counts recorded in pixels that are within a radius of 4 pixels of thecenter. These are known to be the pixels affected by pile-up and high count rates. The remainingcounts within the region of interest are multiplied by a factor of 2.08, which corrects for the lossof the pixels that were excluded. If the count rate exceeds 10 counts/s (which occurred in the caseof XRR 050315), the radius is expanded to 8 pixels and a correction factor of 4.01 is used.1227.2 Spectral ModelsWe have count spectra and response matrices available to us for both the prompt gamma-ray emission and the longer-lasting afterglow emission. In order to find photon spectra, we mustselect some suitable models that we suspect may (with the proper parameters) match those photonspectra.7.2.1 Prompt EmissionBand ModelAs we described in section 2.4.4, the photon spectrum of the prompt gamma-ray emissionis, in general, well-fit by what has come to be known as the Band function:N(E) = A(E50 keV)αexp(−E(2 + α)Eobspeak)E <((α− β)Eobspeak2 + α)= A[(α− β)Eobspeak(50 keV)(2 + α)]α−βexp (β − α)(E50 keV)βE >((α− β)Eobspeak2 + α)(7.3)whereN(E) is the number of photons/cm2/s/keV,α is the spectral index of the low-photon-energy portion of the spectrum,β is the spectral index of the high-photon-energy portion of the spectrum,Eobspeak is the photon energy at which the greatest amount of total energy is detected, andA is a normalization constant with units of photons/cm2/s/keV.The form of the Band function presented here is slightly different from the form given inEquation 2.2. We have substituted a value of 50 keV in place of 100 keV as the denominator inseveral of the fractions. BAT is most sensitive to photons with energies near 50 keV, so this choiceenables us to find normalization constants A that are better constrained. The Band function willbe the primary model of choice for studying the prompt emission.For some bursts, we expect that Eobspeak will be above the energy range of the instrumentsfrom which we have data (BAT, Konus, and FREGATE). In these instances, we will only detect123the low-energy portion of the Band model spectrum, and the data will be well fit by either thesimpler power law model or the so-called “cut-off” power law model (which is a power law functionmultiplied by an exponential “cut-off” term).Constrained Band ModelWe also recognize that for some bursts, Eobspeak will fall below the energy range of our instru-ments. For these bursts, a simple power law model is often sufficient to fit the spectrum. However,since we are often interested in obtaining a value for Eobspeak, we may be tempted to use the Bandfunction as our model in all cases and see what constraints it provides for Eobspeak. Unfortunately,the constraints it provides may be inaccurate, since the high energy AND the low energy portionsof the Band function can be made to fit the data equally well, giving us, in one instance, a valueof Eobspeak below our energy range and in the other, a value above our energy range.An alternative approach is to use a spectral model called the “constrained” Band function,which was developed by Sakamoto et al. [69] This model has three spectral parameters instead offour: β, Eobspeak, and a normalization constant A, along with a fixed parameter Epivot. If the data iswell fit by a power law with some small curvature at the lower energy end, this model is able to fitthat data well. If, on the other hand, the data is well fit by a power law alone, this model forcesthat power law to coincide with the high-energy portion of the Band function, so that Eobspeak willbe constrained to some value lower than the energy range of our instrument.The results of the constrained Band function are only meaningful if β < −2. For highervalues of β, the constrained Band function may produce a value of Eobspeak that is higher than thetrue value. For a more detailed explanation of this model, see the description given by Sakamotoet al. [69].1247.2.2 Afterglow EmissionIn section 2.5, we described the photon spectra of x-ray afterglow emission as typically apower law function with some absorption by intervening hydrogen atoms:N(E) = A · E−I · exp[nH · σ(E)], (7.4)where I is the photon index, nH is the number of hydrogen atoms per cm2 between the detectorand the source, σ(E) is the photoelectric cross-section of hydrogen (which is just a known functionof photon energy), and A is a normalization constant. This will be our model of choice for thex-ray afterglow emission.We note that in some cases, nH will be consistent with values measured for our own galaxy,indicating that hydrogen concentrations within our own galaxy are sufficient to explain what wesee. In other cases nH will be higher than can be explained by absorption due to hydrogen in ourown galaxy, indicating the presence of absorbing material beyond our galaxy. Galactic hydrogenconcentrations are obtained using the HEASARC galactic hydrogen column density calculator(http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl).7.2.3 Error BarsThe errors cited throughout the following chapter are, for the most part, for the 90% con-fidence interval (meaning, loosely speaking, that there is a 90% probability that the true valuelies within the interval specified by the error values). The exceptions are the light curve temporaldecay indices and break times. In those instances, the errors cited are for the 68% confidenceinterval.7.3 Estimating RedshiftsMany times, despite the best efforts of observers, it isn’t possible to measure the redshiftof a gamma-ray burst. A redshift measurement typically requires the observation of an opticalafterglow, and sometimes the optical afterglow either isn’t observed or is too faint to provide a125redshift measurement. Fortunately, there are ways to estimate the redshift. One of these is to usethe correlation between peak Liso and Esrcpeak that was discussed in section 2.4.6.For a given redshift, the peak isotropic luminosity resulting from photons with energiesbetween 30 keV and 10 MeV in the source frame is given byLiso(30 keV, 10 MeV = 4πd2LF(30 keV1 + z,10 MeV1 + z), (7.5)where F (30 keV1+z ,10 MeV1+z ) is the measured peak flux that results from photons with energies rangingfrom (30 keV)/(1 + z) to (10 MeV)/(1 + z), and the “luminosity distance” dL is a function ofredshift z. Likewise, Esrcpeak is given byEsrcpeak = Eobspeak · (1 + z). (7.6)Both quantities are functions of redshift z.If we fit an appropriate model (in our case the Band function) to the peak 1 second of theprompt emission count spectrum, we can derive spectral parameters that allow us to calculateF (30 keV1+z ,10 MeV1+z ) for some value of z. We can then see if this z leads to values of Liso and Esrcpeakthat agree with equation 2.8. Since there is a range of possible values for the coefficient and thepower law index in equation 2.8, there will also be a range of redshifts that result in values thatmatch the correlation. We can find this range of values by repeating this process for multipleredshift values. Because the spectral parameters are poorly constrained when only one second ofdata is used, we use α, β, and Eobspeak from a fit to the full time interval and the normalizationparameter A from a fit to the peak 1 second of the burst.We find that this method works much better for bursts with redshifts less than about 1. Forbursts with higher redshifts, a much larger range of redshifts result in values that are consistentwith equation 2.8, so the uncertainty in a given estimated redshift is larger.7.4 The Kolmogorov-Smirnov TestThere is a statistical test known as the Kolmogorov-Smirnov (or K-S) test, which allows oneto determine the likelihood that two distributions are drawn from the same parent distribution.126This test gives a probability that the two distributions in question are consistent. This test is usedin Chapter 9 to compare distributions of burst properties observed by Swift with those of othermissions, as well as to compare properties of XRFs with those of GRBs. For our purposes, wewill consider a probability below 0.1 as an indication that the two distributions are inconsistent, aprobability between 0.1 and 0.3 as an indication of marginal consistency, and a probability greaterthan 0.3 as an indication that the two distributions are consistent.7.5 Definition of GRBs, XRRs, and XRFsThere hasn’t been a universally agreed upon definition of an “x-ray flash”, other than thequantitatively imprecise definition that it emits most of its energy in x-rays and has a shortduration. The definitions adopted by those whose work we reviewed in chapter 3 tended (under-standably) to be based on the characteristics and energy ranges of the instruments that collectedthe data, and none of those definitions are quite suitable for BAT. And yet, we desire a definitionthat will be close enough to the others that we may reliably compare the characteristics of theBAT-detected XRF population with those from other missions.As we discussed in section 3.4, Sakamoto et al. defined x-ray flashes in terms of the fluenceratio SX(2-30 keV)/Sγ(30-400 keV) [70]. GRBs, XRRs, and XRFs were classified according totheir value of this fluence ratio. They also found a strong correlation between Eobspeak and fluenceratio (see Figure 7.1). The borderline Eobspeak between XRFs and XRRs is ≈ 30 keV, and theborderline Eobspeak between XRRs and GRBs is ≈ 100 keV.In the BAT energy range, a fluence ratio of S(25 – 50 keV)/S(50 – 100 keV) is more naturaland easier to measure with confidence. I therefore chose my working definition in terms of thisratio. In order to ensure that my definition was close to that adopted by Sakamoto et al., Icalculated the fluence ratio for a burst for which the parameters of the Band model were α = −1,β = −2.5, and Eobspeak = 30 keV. These values of α and of β are typical of the distributions foundby BATSE. The ratio thus found was 1.3217. I likewise calculated the fluence ratio for a burst forwhich α = −1, β = −2.5, and Eobspeak = 100 keV, which I found to be 0.7218. My working definition127Figure 7.1: Distribution of bursts in the Eobspeak–S(2–30 keV)/S(30–400 keV) plane. Overlaidare curves corresponding to the X-ray to γ-ray fluence ratio as a function of Eobspeak, assumingthe Band function with α = −1 and β = −2.5 (red), −3.0 (blue), and −20.0 (green) [70].for XRFs, XRRs, and GRBs thus became:S(25 – 50 keV)/S(50 – 100 keV) ≤ 0.7218 GRB0.7218 < S(25 – 50 keV)/S(50 – 100 keV) ≤ 1.3217 XRR (7.7)S(25 – 50 keV)/S(50 – 100 keV) > 1.3217 XRFThe BAT spectrum for each burst detected through 31 July 2005 was extracted and acorresponding response matrix was generated. The “xspec” spectral fitting tool (version 11.3.1)was used to fit each spectrum to a Band model, a power law model, and a cut-off power law model.The latter two models were chosen so that good fits would be possible when Eobspeak extendedto energies below or above the BAT energy range, so that only one branch of the Band modelwas visible. From the best fit model in each case, a fluence ratio was calculated. The fluenceratios found for each burst are shown in Table 7.1. A histogram of these fluence ratios is shownin Figure 7.2. Of the 49 bursts for which BAT obtained spectra between December 2004 when128Figure 7.2: Histogram of the fluence ratios of BAT bursts observed between launch and 31July 2005. The fluence ratios defining the boundaries between XRFs, XRRs, and GRBs arealso shown.the instrument went on line and 31 July 2005, 5 are classified as x-ray flashes, 24 as x-ray richgamma-ray bursts, and 20 as “regular” gamma-ray bursts.In some cases, the spacecraft began to slew while the burst was still in progress. Thiscomplicates the data analysis because, while some geometric corrections are applied to the dataas the slew occurs to help normalize the count rates, the response matrix strictly only applies toa particular position in the BAT’s field of view, and while the spacecraft is slewing, this positionchanges. A response matrix generated for the pre-slew source position should only be used toanalyze spectra accumulated before the spacecraft began to slew. In determining the fluence ratiosgiven in Table 7.1, the spectrum from the entire duration of the burst was fit using the responsematrix for the pre-slew position of the burst. Therefore, we expect that for those bursts in whichthe slew began during the burst, the fluence ratios we calculated would not be accurate. Anextreme example is GRB 050128, for which the spacecraft slewed 51 degrees. Fitting the pre-slewdata with the pre-slew response matrix, we obtain a hardness ratio of 1.6205. When we use thepost-slew response matrix, we obtain a hardness ratio of 1.9949, which is 23% larger. This is thelargest error we might possibly see in any of the values in the table. Swift did not slew during anyof the XRFs detected by BAT, so none of the values for those bursts are affected by this problem.129Table 7.1: The fluence ratio S(25 – 50 keV)/S(50 – 100 keV) for all bursts detected by BATthrough 31 July 2005, in descending order of fluence ratio. Those in the first group qualify asXRFs, those in the second as XRRs, and those in the third as GRBs.XRFs XRRs GRBsName Ratio Name Ratio Name Ratio050416A 4.0351 050315 1.1429 050509B 0.7204050406 3.1217 050318 1.0742 050506 0.7143050714B 2.6441 050509A 1.0693 050117 0.7139050726B 1.9190 050507 0.9750 050401 0.6962050215B 1.4115 050319 0.9551 050416B 0.6944050607 0.9541 041226 0.6828050421 0.9434 050422 0.6781050223 0.9254 050215A 0.6775041224 0.8942 050717 0.6653050721A 0.8825 050716 0.6640050724 0.8231 050124 0.6624050410 0.8121 050128 0.6497041228 0.8033 050126 0.6353050418 0.7947 050202 0.6235050525A 0.7930 050219A 0.6000041220 0.7874 050326 0.5953050505 0.7695 050603 0.5920050502B 0.7665 041223 0.5428050715 0.7653 050726A 0.4894050713A 0.7500 050412 0.4388050219B 0.7444050713B 0.7391050730 0.7323050712 0.73201307.6 Selection CriteriaI conducted a detailed analysis of the five x-ray flashes listed in Table 7.1. An additionalx-ray flash that was observed on 24 August 2005, for which a redshift was obtained, was added tothe set, making a total of 6 XRFs.The properties of these x-ray flashes were compared to those of a set of XRRs and GRBsthat were detected by BAT. For extensive analysis, I selected two XRRs and 4 GRBs that were alsoobserved by Konus. Using Konus data in addition to BAT data made possible better constraintson the model parameters and permitted the measurement of Eobspeak above the range in which BATis sensitive. Two additional XRRs were selected for which redshifts had been measured, bringingthe total number of XRRs studied to 4.131Chapter 8BAT XRFs, XRRs, and GRBs8.1 XRFsThis section describes the prompt emission and x-ray afterglow properties of the 5 XRFsobserved by BAT prior to 31 July 2005, as well as the XRF observed on 24 August 2005.8.1.1 XRF 050416AThis was the softest of the x-ray flashes observed by Swift as of 31 July 2005, with a S(25 –50 keV)/S(50 – 100 keV) fluence ratio of only 4.0355. BAT triggered on this burst at 11:04:44.5 UT(MET=135342283.968), and the spacecraft slewed immediately so that the XRT was able to beginaccumulating data within 92.5 s (GCN 3264). The position determined by BAT was RA=12h33m 57.6s and Dec=+21d 03’ 10.8” with an estimated uncertainty radius of about 3 arcmin(95% confidence) (GCN 3273). The XRT-determined position was found to be RA=12h 33m54.8s, Dec=+21d 03’ 25.1”, with a 5 arcsec uncertainty (90% confidence) (GCN 3275). Afterglowemission in the optical, infrared, and radio wavelengths was also detected (GCN 3265, 3269, 3318).The optical observations made possible a redshift determination of z = 0.6535 ± 0.0002 (GCN3542).Prompt EmissionThe duration of the burst was about 2.4 seconds (±0.2 seconds). The light curve shows asingle prominent peak with a slightly longer rise time than fall time, followed by two smaller peaks.The emission comes entirely from 15-50 keV photons (see Figure 8.1).The Band model failed to constrain Eobspeak, so the constrained Band model was used instead.Eobspeak was found to be 16.7+5.3−9.5 keV (90% confidence). The time interval from which the spectrumwas extracted spans from 0.5 s before the trigger to 3 s after the trigger. Figure 8.2 shows the132Figure 8.1: The light curve of the XRF 050416a prompt gamma-ray emission. The dashedlines mark the time interval used for the spectral analysis. The solid line marks the beginningof the spacecraft slew.133Figure 8.2: The count spectrum of the XRF 050416a prompt gamma-ray emission, alongwith the best fit model. See Table 8.1 for the best fit parameter values.count spectrum and the best-fit model for this spectrum. The probability density distribution forEobspeak is shown in Figure 8.3.X-ray AfterglowThe x-ray afterglow, as measured by the XRT, is shown in Figure 8.4. The best fit tothe data prior to 106 s results from a broken power law, which gives an early decay index of−0.672± 0.026, a late decay index of −0.905± 0.034, and a break at 8400± 3500 s. There is alsoevidence of some late time flaring about 6 × 106 seconds (68 days) after the trigger.The count spectra measured by the XRT is shown in Figure 8.5. Two different time intervalswere chosen, one before the break in the light curve and one after it. The best fit spectral indicesin both cases are consistent with each other (2.2±0.3 and 2.2±0.2). The best fit hydrogen columndensity is (2.8 ± 1.2) × 1021 atoms/cm2, which is considerably higher than the hydrogen columndensity due to our galaxy alone in the direction of the burst, which is only 2.06× 1020 atoms/cm2.134Figure 8.3: The Eobspeak probability density distribution for XRF 050416a. The y-axis gives thedifferential probability dPdE, such that the probability that Eobspeak lies between two values E1and E2 is given by the area under the curve between E1 and E2. There is a 90% probabilitythat Eobspeak lies within the region bounded by the two vertical lines.Table 8.1: The best fit spectral parameters for the XRF 050416a prompt emission, usinga power law model, a cutoff power law model, and the constrained Band model. Eobspeak ismeasured in keV, A in ergs/cm2/s/keV, and average flux is given in ergs/cm2/s for 15 – 150keV photons. The exposure was 2.728 s.power law cutoff power law BandParameter Value Parameter Value Parameter Valueα −3.09+0.21−0.22 α −1.31+1.43−1.24 β −3.86+0.67−1.88A (9.0 ± 1.6) × 10−3 Eobspeak < 20.4 Eobspeak 16.7+5.3−9.5A 0.128+1.105−0.080 Epivot 20A < 3.19flux 3.78 × 10−8χ2ν/ν 1.046/73 χ2ν/ν 0.966/72 χ2ν/ν 0.983/72135Figure 8.4: The light curve of the XRF 050416A x-ray afterglow, along with the best-fitbroken power law model. Also shown with a circle is the average flux of the prompt emissionextrapolated into the 0.6 – 10 keV energy range. Note that at very late times, the light curveflares up for a short period of time.Table 8.2: The best fit spectral parameters for the XRF 050416a x-ray afterglow, beforeand after the break in the light curve. nH is measured in atoms/cm2, A in units of 10−3ergs/cm2/s/keV, and flux in units of ergs/cm2/s. Start and stop times are in seconds sincethe BAT trigger.Parameter Before Break After BreakI 2.2 ± 0.3 2.2 ± 0.2nH (2.8 ± 1.2)× 1021 (3.1 ± 0.8) × 1021A 5.01.9−1.3 0.27+0.06−0.05flux 2.540.770.45 × 10−11 1.38+0.26−0.18 × 10−12start time 600 s 11510.6576 sstop time 2158.6576 s 156961.70812 sexposure 1543.5576 s 45292.556 sχ2ν/ν 0.775/17 0.530/70136Figure 8.5: The count spectrum of the XRF 050416a x-ray afterglow, before (top) and after(bottom) the break in the light curve. Count bins corresponding to photon energies of 0.3 –10 keV were included. See Table 8.2 for the spectral parameters.1378.1.2 XRF 050406BAT triggered on this burst at 15:58:48.4 UT (MET=134495927.872), and the spacecraftslewed immediately so that the XRT was able to begin accumulating data within 87 s (GCN3180, 3181). The position determined by BAT was RA=2:17:53.04 and Dec=-50:10:51.6 withan estimated uncertainty radius of about 3 arcmin (95% confidence) (GCN 3183). The XRT-determined position was found to be RA=2h 17m 52.64s, Dec=-50d 11’ 18.8”, with a 5 arcsecuncertainty (90% confidence) (GCN 3184). Optical afterglow emission was also detected (GCN3185), but a redshift was not measured. Using the technique described in Section 7.3 we are ableto constrain the redshift to z > 3.Prompt EmissionThe T90 duration of the burst was 5.78±0.23 seconds. The light curve shows a single peak,but the burst was not bright enough to enable us to discern finer structure. Like XRF 050416a,the emission comes entirely from 15-50 keV photons (see Figure 8.6).The Band model failed to adequately constrain Eobspeak, so the constrained Band model wasused instead. Eobspeak was found to be 27.9+7.9−18.1 keV (90% confidence). The time interval fromwhich the spectrum was extracted spans from 2.504 s before the trigger to 3.884 s after the trigger.Figure 8.7 shows the count spectrum and the best-fit model for this spectrum. The probabilitydensity distribution for Eobspeak is shown in Figure 8.8.X-ray AfterglowThe x-ray afterglow, as measured by the XRT, is shown in Figure 8.9. The afterglow israther faint, even though it was observed immediately. While the light curve is poorly fit with asingle power law it is consistent with an offset power law or with a broken power law. The offsetpower law results in a t0 of 138± 21 s, which is long after the prompt emission. The broken powerlaw, on the other hand, results in a best fit early index of −2.01±0.22, a late index of −0.50±0.29,and a break time of 3900 ± 2800 s. There is a hint of a possible second break in the afterglow at138Figure 8.6: The light curve of the XRF 050406 prompt gamma-ray emission. The two dashedlines indicate the time interval used to measure the spectrum. The solid line indicates thetime at which Swift began to slew.139Figure 8.7: The count spectrum of the XRF 050406 prompt gamma-ray emission, along withthe best fit model See Table 8.3 for the spectral parameters.Figure 8.8: The Eobspeak probability density distribution for XRF 050406. The y-axis gives thedifferential probability dPdE, such that the probability that Eobspeak lies between two values E1and E2 is given by the area under the curve between E1 and E2. There is a 90% probabilitythat Eobspeak lies within the region bounded by the two vertical lines.140Table 8.3: The best fit spectral parameters for the XRF 050406 prompt emission, usinga power law model, a cutoff power law model, and the constrained Band model. Eobspeak ismeasured in keV, A in ergs/cm2/s/keV, and average flux in ergs/cm2/s for 15 – 150 keVphotons. The exposure was 6.388 s.power law cutoff power law BandParameter Value Parameter Value Parameter Valueα −2.59+0.44−0.37 α −0.541+3.54−1.70 β < −2.78A 1.23+0.38−0.37 × 10−3 Eobspeak 25.4+9.1−23.5 Eobspeak 27.97.9−18.1A < 6.23 × 10−2 Epivot 20A < 0.32flux 1.11 × 10−8χ2ν/ν 1.046/73 χ2ν/ν 1.196/72 χ2ν/ν 1.186/72around 300 ks, but the data isn’t really sufficient to draw any conclusions.An upper limit on the average 0.6 – 10 keV flux of the prompt emission is also shown,as extrapolated from the BAT data. The trend of the early afterglow power law overshoots theprompt emission data.The count spectrum measured by the XRT is shown in Figure 8.10. Due to the faintnessof the afterglow, it was not possible to produce a spectrum after the light curve break, and thelocation of the break itself is poorly constrained. The hydrogen column density was not wellconstrained, but was consistent with the hydrogen column density due to our galaxy alone in thedirection of the burst (which is 2.77 × 1020 atoms/cm2), so we set this parameter to 2.77 × 1020atoms/cm2 in the model and froze it at that value. With nH thus fixed, the best fit spectral indexwas 1.69 ± 0.32.8.1.3 XRF 050714BOn 14 July 2005, BAT detected two bursts the second was labeled 050714B. This bursttriggered BAT at 22:40:32 UT (MET=143073632s), and the spacecraft slewed immediately so thatthe XRT was able to begin accumulating data within 151 s (GCN 3613). The position determined141Figure 8.9: The light curve of the XRF 050406 x-ray afterglow, along with the best-fit brokenpower law model. The two power laws intersect at about 4 ks. Also shown with a circle is the90% upper limit of the average flux of the prompt emission, as extrapolated from the BATdata.Table 8.4: The best fit spectral parameters for the XRF 050604 x-ray afterglow. A is measuredin units of photons/cm2/s/keV and flux in units of ergs/cm2/s. Start and stop times are inseconds since the BAT trigger.Parameter ValueI 1.69 ± 0.32A 1.51+0.66−0.23 × 10−50.6 – 10 keV flux 1.37+0.42−0.34 × 10−13start time 98.769023 sstop time 150615.92636 sexposure 49343.6 sχ2ν/ν 0.775/17142Figure 8.10: The count spectrum of the XRF 050406 x-ray afterglow. Counts bins cor-responding to photon energies of 0.3 – 10 keV are shown. See Table 8.4 for the spectralparameters.by BAT was RA=11h 18m 47.3s, Dec=-15d 32m 6s, with an estimated uncertainty radius of about3 arcmin (95% confidence) (GCN 3615). The XRT-determined position was found to be RA=11h18m 48.0s, Dec=-15d 32m 49.9s, with a 6 arcsec uncertainty (90% confidence) (GCN 3184). Noafterglow was detected in any other waveband, and a redshift measurement could not be made.Using the technique described in Section 7.3, we constrain the redshift to z > 1.8. A seriouslimitation in our ability to constrain z better than this comes from the uncertainty in the spectralparameters.Prompt EmissionThe T90 duration of the burst was 45.9 ± 0.7 seconds. The light curve may show threeseparate peaks, but the burst was too dim to enable us to discern structure with much confidence.The emission comes mostly, if not entirely, from 15-50 keV photons (see Figure 8.11).The Band model failed to adequately constrain Eobspeak, so the constrained Band model was143Figure 8.11: The light curve of the XRF 050714b prompt gamma-ray emission. The twodashed lines indicate the time interval used to measure the spectrum. The solid line indicatesthe time at which Swift began to slew.144Figure 8.12: The count spectrum of the XRF 050714b prompt gamma-ray emission, alongwith the best fit model. See Table 8.5 for the spectral parameters.used instead. Eobspeak was found to be 23.1+6.4−20.5 keV (90% confidence). The time interval fromwhich the spectrum was extracted spans from 19.492 s after the trigger to 68.248 s after thetrigger. Figure 8.12 shows the counts spectrum and its best-fit model. The probability densitydistribution for Eobspeak is shown in Figure 8.13.X-ray AfterglowThe x-ray afterglow light curve, as measured by the XRT, is shown in Figure 8.14. Thebest fit is a broken power law with an early decay index of −6.605 ± 0.006, a late decay index of−0.89 ± 0.06, and a break time of 225 ± 11 s. The early decay index is considerably steeper thanthe decay observed in most bursts, even when very early afterglow data is available.The average 0.6 – 10 keV flux of the prompt emission is also shown, as extrapolated fromthe BAT data. The trend of the early afterglow overshoots the prompt emission data.The count spectra for intervals before and after the break in the light curve are shown inFigure 8.15. Before the break, the spectrum fits a power law and a black body model equally well,145Figure 8.13: The Eobspeak probability density distribution for XRF 050714b. The y-axis givesthe differential probability dPdE, such that the probability that Eobspeak lies between two values E1and E2 is given by the area under the curve between E1 and E2. There is a 90% probabilitythat Eobspeak lies within the region bounded by the two vertical lines.Table 8.5: The best fit spectral parameters for the XRF 050714B prompt emission, usinga power law model, a cutoff power law model, and the constrained Band model. Eobspeak ismeasured in keV, A in ergs/cm2/s/keV, and average flux in ergs/cm2/s for 15 – 150 keVphotons. The exposure was 48.756 s.power law cutoff power law BandParameter Value Parameter Value Parameter Valueα −2.66+0.32−0.37 α −1.04+0.65−1.56 β −3.1+0.7−2.1A (1.14 ± 0.28) × 10−3 Eobspeak < 33.2 Eobspeak 23.1+6.4−20.5A < 5.51 × 10−2 Epivot 20A < 0.68flux 1.12 × 10−8χ2ν/ν 1.218/73 χ2ν/ν 1.192/72 χ2ν/ν 1.210/72146Figure 8.14: The light curve of the XRF 050714B x-ray afterglow, along with the best-fitmodel of two intersecting power laws. The circled data point is the average flux of the promptemission, extrapolated into the 0.6 – 10 keV energy range. When extrapolated backward, theearly power law overshoots the average 0.6 – 10 keV flux of the prompt emission.147Table 8.6: The best fit spectral parameters for the XRF 050714B x-ray afterglow. nH ismeasured in 1021 atoms/cm2, A in units of ergs/cm2/s/keV, kT in units of keV, and 0.6 –10 keV flux in units of 10−8 ergs/cm2/s. Start and stop times are in seconds since the BATtrigger.Before Break Before Break After Break(Power Law) (Black Body)nH 8.2+1.6−1.0 nH 2.5+0.9−0.8 nH 1.4+0.9−0.8I 6.09+0.71−0.61 kT 0.23 ± 0.02 I 2.53+0.45−0.38A 1.11+0.55−0.34 A 6.6+2.6−1.6 × 10−3 A 2.36+0.85−0.60 × 10−4flux 6.39+11.47−4.01 flux 214.8 flux 1.15+0.50−0.25start time 157.4917 s start time 157.4917 s start time 414.47 sstop time 207.8529 s stop time 207.8529 s stop time 58268.59464 sexposure 50.299 s exposure 50.299 s exposure 22542.0 sχ2ν/ν 0.998/24 χ2ν/ν 1.059/24 χ2ν/ν 1.500/14but the best fit nH is not consistent with the spectrum after the break when a power law modelis used. The best fit hydrogen column density for the spectrum after the break is 1.4+0.9−0.8 × 1021atoms/cm2, which is slightly higher than the hydrogen column density due to our galaxy alone inthe direction of the burst, which is 5.3 × 1020 atoms/cm2.8.1.4 XRF 050824This burst triggered BAT at 23:12:16 UT (MET=146617936s) (GCN 3866). The proximityof the moon prevented Swift from slewing immediately, but the XRT was able to begin accumu-lating data beginning 6089 s after the trigger (GCN 3872). The position determined by BAT wasRA=0h 49m 1.44s, Dec=+22d 37m 4.8s, with an estimated uncertainty radius of about 3 arcmin(90% confidence) (GCN 3871). The XRT-determined position was found to be RA=0h 48m 56.0s,Dec=+22d 36m 28.5s, with a 6.8 arcsec uncertainty (90% confidence) (GCN 3872). An opticalafterglow was also detected (GCN 3865), and a redshift of 0.83 was measured (GCN 3874).148Figure 8.15: The count spectrum of the XRF 050714B x-ray afterglow before (top) and after(bottom) the break in the light curve. The count spectrum before the break is fit with a blackbody model, while the count spectrum after the break is fit with a power law model. Countsbins corresponding to photon energies of 0.3 – 10 keV are shown. See Table 8.6 for the spectralparameters.149Table 8.7: The best fit spectral parameters for the XRF 050824 prompt emission, usinga power law model, a cutoff power law model, and the constrained Band model. Eobspeak ismeasured in keV, A is in ergs/cm2/s/keV, and average flux is in ergs/cm2/s for 15 – 150 keVphotons. The exposure was 29.092 s.power law cutoff power law constrained BandParameter Value Parameter Value Parameter Valueα −2.78+0.37−0.44 α −2.61+2107−0.47 β −2.86+0.42−0.68A 9.1+2.8−2.7 × 10−4 Eobspeak 0.41+23.17−0.18 Eobspeak < 19.5A 8.8+3.1−4.3 × 10−4 Epivot 10A 102.2flux 9.81 × 10−9χ2ν/ν 0.935/73 χ2ν/ν 0.950/72 χ2ν/ν 0.944/72Prompt EmissionThe T90 duration of the burst was 25.5 ± 0.1 seconds. A single broad peak can be distin-guished in the light curve, although the signal is weak. The emission comes mostly from 15-50 keVphotons, although there may be a hint of emission in the 50-100 keV band (see Figure 8.16).Using the constrained Band model, we were able to constrain Eobspeak to < 19.5 keV. The timeinterval from which the spectrum was extracted spans from 32.2206 s after the trigger to 61.3126s after the trigger. Figure 8.17 shows the count spectrum from BAT and the best-fit constrainedBand model. The probability density distribution for Eobspeak is shown in Figure 8.18.X-ray AfterglowThe x-ray afterglow, as measured by the XRT, is shown in Figure 8.19. The data is best fit(with a χ2/nu of 1.37) by a broken power law with an early index of −0.49 ± 0.05, a late index of−1.1 ± 0.3, and a break time of 300000 ± 150000 s. A single power law fit results in an index of−0.60 ± 0.27 and a χ2ν of 1.85.The average 0.6 – 10 keV flux extrapolated from the fit to the prompt emission is also shownin Figure 8.19. When extrapolated backward, the early power law undershoots the average 0.6 –150Figure 8.16: The light curve of the XRF 050824 prompt gamma-ray emission. The twodashed lines indicate the time interval used to measure the spectrum. Due to Viewing con-straints posed by the Moon, Swift did not slew to this burst until long after this time interval(To+6089s).151Figure 8.17: The count spectrum from BAT of the XRF 050824 prompt gamma-ray emission,along with the best fit constrained Band model. See Table 8.7 for the spectral parameters.Figure 8.18: The Eobspeak probability density distribution for XRF 050824. The y-axis givesthe differential probability dPdE, so that the probability that Eobspeak lies between two values E1and E2 is given by the area under the curve between E1 and E2. There is a 90% probabilitythat Eobspeak lies below 19.5 keV.152Figure 8.19: The light curve of the XRF 050824 x-ray afterglow, along with the best-fit brokenpower law fits to the light curve. The circled data point is the average flux of the promptemission, extrapolated into the 0.6 – 10 keV energy range. When extrapolated backward, theearly power law undershoots the average 0.6 – 10 keV flux of the prompt emission.10 keV flux of the prompt emission.The counts spectrum measured by the XRT is shown in Figure 8.20. The best fit spectralindex is 1.9 ± 0.2. The best fit hydrogen column density is 8.1+6.6−6.0 × 1020 atoms/cm2, which isconsistent with the hydrogen column density due to our galaxy alone in the direction of the burst,which is 3.6× 1020 atoms/cm2. The spectrum used here consists of counts that were accumulatedbetween 6092s after the trigger and 87622 s after the trigger. The afterglow had become too faintafter the break for spectral analysis.8.1.5 XRF 050528This burst triggered BAT at 04:06:45 UT (MET=138946004.672s) (GCN 3496). While Swiftimmediately slewed, afterglow measurements had to be delayed because XRT was in an engineeringmode at the time of the burst. XRT began collecting data 14.15 hours after the BAT trigger (GCN3505). The position determined by BAT was RA=23h 34m 8.8s, Dec=+45d 56m 37.2s, with an153Figure 8.20: The count spectrum of the XRF 050824 x-ray afterglow. Count bins corre-sponding to photon energies of 0.3 – 10 keV were included. See Table 8.8 for the spectralparameters.Table 8.8: The best fit spectral parameters for the XRF 050824 x-ray afterglow. nH ismeasured in atoms/cm2, A in units of photons/cm2/s/keV, and flux in units of ergs/cm2/s.Start and stop times are in seconds since the BAT trigger.Parameter ValueI 1.9 ± 0.2nH 8.1+6.6−6.0 × 1020A 8.21.6−1.3 × 10−50.6 – 10 keV flux 1.24+0.16−0.14 × 10−12start time 6091.773543 sstop time 87622.19248 sexposure 22773.8 sχ2ν/ν 0.992/17154Table 8.9: The best fit spectral parameters for the XRF 050528 prompt emission, usinga power law model, a cutoff power law model, and the constrained Band model. Eobspeak ismeasured in keV, A is in ergs/cm2/s/keV, and average flux is in ergs/cm2/s for 15 – 150 keVphotons. The exposure was 12.248 s.power law cutoff power law constrained BandParameter Value Parameter Value Parameter Valueα −2.34+0.26−0.29 α −0.78+1.85−1.32 α −0.96+1.75−1.73A 3.73+0.69−7.21 × 10−3 Eobspeak 30.1+10.6−30.1 β < −2.45A < 9.86 × 10−2 Eobspeak 30.5+9.9−9.9A 2.42+38.36−1.73 × 10−2flux 9.78 × 10−9χ2ν/ν 0.905/73 χ2ν/ν 0.861/72 χ2ν/ν 0.873/71estimated uncertainty radius of about 3 arcmin (90% confidence). Three sources were detectedinside the BAT error circle (GCN 3505) but they were too faint for my analysis to determine which,if any, was the afterglow. No other afterglow emission was detected, and no redshift was measured.The inferred redshift, using the technique described in Section 7.3, is 0.3+1.7−0.2.Prompt EmissionThe T90 duration of the burst was 11.168 ± 0.036 seconds. There are possibly two peaks,one beginning at about 5 seconds before the trigger, and a second larger one centered about 3.5seconds after the trigger. The most prominent emission results in counts in channels correspondingto photon energies of 15 - 50 keV, with a hint of counts in the 50 - 100 keV band. No counts areseen in the 100 - 350 keV band (see Figure 8.21).For this burst, the Band model provided a good fit to the spectrum, and was able to constrainEobspeak reasonably well to 30.5+9.9−9.9 keV. The time interval from which the spectrum was extractedspans from 7.264 s before the trigger to 4.984 s after it. Figure 8.22 shows the counts spectra andthe best-fit model.155Figure 8.21: The light curve of the XRF 050528 prompt gamma-ray emission. The twodashed lines indicate the time interval used to measure the spectrum. Swift slewed after thetime interval shown on this plot.156Figure 8.22: The count spectra of the XRF 050528 prompt gamma-ray emission, along withthe best fit model. See Table 8.9 for the spectral parameters.8.1.6 XRF 050215BThis burst triggered BAT at 02:35:00 UT (MET=130127622.592s) (GCN 3027). While Swiftimmediately slewed, afterglow measurements had to be delayed because XRT was in a manual statecollecting calibration data, and Swift entered the SAA shortly after the slew. XRT began collectingdata about 96 minutes after the BAT trigger. The position determined by BAT was RA=11h 37m48s, Dec=+40d 48m 16s, with an estimated uncertainty radius of about 4 arcmin (90% confidence)(GCN 3024). The XRT-determined position was found to be RA=11h 37m 47.7s, Dec=+40d 47m44.0s, with a 6 arcsec uncertainty (90% confidence) (GCN 3872). In addition to the x-ray afterglow,optical and infrared afterglow emission was detected (GCN 3031, 3037). No redshift was measuredfor this burst, but the inferred redshift, based on the Yonetoku relation, is 0.8+1.4−0.3.157Table 8.10: The best fit spectral parameters for the XRF 050215B prompt emission, using apower law model, a cutoff power law model, and the Band model. Eobspeak is measured in keV,A in ergs/cm2/s/keV, and average flux in 10−8 ergs/cm2/s for 15 – 150 keV photons. Theexposure was 11.776 s.power law cutoff power law BandParameter Value Parameter Value Parameter Valueα −2.22+0.19−0.21 α −1.39+0.74−0.71 α −1.48+0.68−0.40A 2.28+0.30−0.32 × 10−3 Eobspeak > 38.5 β < 2.22A < 2.10 × 10−2 Eobspeak 26.7+10.7−12.5A < 8.60 × 10−3flux 2.04 ± 0.30χ2ν/ν 0.790/192 χ2ν/ν 0.771/191 χ2ν/ν 0.775/190Prompt EmissionThe T90 duration of the burst was 10.43± 0.06 seconds. There is a strong peak at the timeof the trigger, followed by two smaller peaks at 5 s and 9 s after the trigger. The most prominentemission results in channels corresponding to photon energies of 15-50 keV, but this time we alsosee significant emission in the 50-100 keV band. No emission is apparent in the 100-350 keV band(see Figure 8.23).HETE-2 also observed this burst, and we were able to use both BAT and HETE-2 data tofind the photon spectrum. The Band model provided a good fit to the data and constrained Eobspeakto 26.7+10.7−12.5 keV. The time interval from which the spectrum was extracted spans from 1.776 sbefore the trigger to 10 s after it. Figure 8.24 shows the counts spectra and the best-fit model.X-ray AfterglowThe x-ray afterglow, as measured by the XRT, is shown in Figure 8.25. As we noted before,afterglow data was recorded beginning about 96 minutes after the burst. A power law with anindex of −1.00 ± 0.08 fits the data quite well. An offset power law provides a better fit, but theoffset time is very poorly constrained. There are too few data points to provide well-constrained158Figure 8.23: The light curve of the XRF 050215B prompt gamma-ray emission. The twodashed lines indicate the time interval used to measure the spectrum. The solid line indicatesthe time at which Swift began to slew.159Figure 8.24: The count spectra of the XRF 050215B prompt gamma-ray emission, as recordedby BAT and HETE-2, along with the best fit Band model. The BAT spectrum is shown inblack and the HETE-2 spectrum is shown in red. See Table 8.10 for the spectral parameters.parameters from a broken power law fit. The power law is consistent with the upper limit of theextrapolated 0.6 – 10 keV flux from the prompt emission.The count spectrum measured by the XRT is shown in Figure 8.26. The hydrogen columndensity was constrained to < 3.3 × 1021 atoms/cm2. Since this was consistent with the hydrogencolumn density due to our galaxy alone in the direction of the burst (which is 2.07×1020 atoms/cm2)and was not very well constrained, we set this parameter to 2.07 × 1020 atoms/cm2 in the modeland froze it at that value. With nH thus fixed, the best fit spectral index was 1.10+0.55−0.58. Thespectrum used here consists of counts accumulated between 5770 s after the trigger and 35999.7 safter the trigger.8.2 XRRsFor our sample of X-ray rich gamma-ray bursts (XRRs), which are defined as those with afluence ratio S(50−100 keV)/S(25−50 keV) between 0.7566 and 1.3855, we selected 4 bursts. Two160Figure 8.25: The light curve of the XRF 050215B x-ray afterglow, along with the best-fitpower law model. Also shown, with a circle, is the upper limit of the average flux of theprompt emission extrapolated into the 0.6 – 10 keV energy range.Figure 8.26: The count spectrum of the XRF 050215B x-ray afterglow. Count bins corre-sponding to photon energies of 0.3 – 10 keV were included. See Table 8.11 for the spectralparameters.161Table 8.11: The best fit spectral parameters for the XRF 050215B x-ray afterglow. A ismeasured in units of photons/cm2/s/keV and flux in units of ergs/cm2/s. Start and stoptimes are in seconds since the BAT trigger.Parameter ValueI 1.10+0.55−0.58A 1.09+0.32−0.28 × 10−40.6 – 10 keV flux 1.52+1.51−0.72 × 10−12start time 5770.0086 sstop time 35999.67288 sexposure 5131.1 sχ2ν/ν 0.120/3of these have a fairly low fluence ratio, making them “borderline XRRs” and were selected becausewe have measured redshifts for them. The other two were selected because they were simultaneouslyobserved by Konus, enabling us to determine their spectral parameters with greater precision.8.2.1 XRR 050315This burst triggered BAT at 20:59:42 UT (MET=132613181.952) (GCN 3094). Swift slewedimmediately and the XRT began accumulating data at 21:01:05.5 UT, 83.5 s after the trigger(GCN 3097). The position determined by BAT was RA=20h 25m 57.2s, Dec=-42d 35m 30.6s,with an estimated uncertainty radius of about 3 arcmin (90% confidence). The XRT-determinedposition was found to be RA=20h 25m 53.9s, Dec=-42d 36m 01.4s, with a 6 arcsec uncertainty(90% confidence) (GCN 3133). Afterglow emission in optical (GCN 3100) and radio (GCN 3102)wavebands was detected, and a redshift of 1.949 was measured (GCN 3101).Prompt EmissionXRR 050315 was rather long, with a T90 duration of 95.46± 0.14 seconds. There is a smallpeak beginning about 56 seconds before the trigger, followed by three other small peaks centered atabout 35, 25, and 15 seconds before the trigger. A large broad peak extends from 10 seconds before162the trigger to about 20 seconds after the trigger, followed by another large peak that begins about25 seconds after the trigger and gradually decays to zero at about 8 seconds after the trigger. Mostof the flux comes from channels corresponding to photon energies of 15 - 50 keV, with significantemission also coming from 50 - 100 keV channels. There is a hint of possible emission in the 100 -350 keV band, too (see Figure 8.27).A power law fit to this count spectra indicated that the power law index was −1.89 ± 0.10which is less than -2, so the constrained Band model cannot be used. The Band model, with allof the parameters free to vary, was not able to constrain any of the parameters, and a confidencecontour plot of α verses Eobspeak shows Eobspeak to be smaller than 100 keV if α is greater than -1.5,which is admittedly a very weak constraint (see Figure 8.28).The time interval from which the spectrum was extracted spans from 56.18 s before thetrigger to 14.14859 s after it. Figure 8.29 shows the count spectra and the best-fit power lawmodel.X-ray AfterglowThe x-ray afterglow, as measured by the XRT, is shown in Figure 8.30. The best fit powerlaw index for the early phase of the afterglow (between about 90 s and about 500 s after the BATtrigger) is −3.21±0.04. The data was heavily piled up over much of this range, and the difficulty incorrecting for pile up may have contributed to the poor fit. The late phase of the afterglow (from5 ks to the end of the observation) is well fit by a broken power law with an early decay index of−0.65± 0.02, a late decay index of −1.31 ± 0.07, and a break time of 125000± 13000 s. Vaughanet al. have published a paper describing this burst [79], and the x-ray light curve they derived isessentially the same as the one presented here. Their light curve does show some evidence of theearly phase and the late phase smoothly joining each other.The 0.6 – 10 keV flux of the prompt emission is also shown in Figure 8.30, as extrapolatedfrom the BAT data.The XRT count spectra during the early phase of the light curve, and during the late phase163Figure 8.27: The light curve of the XRR 050315 prompt gamma-ray emission. The dashedvertical line indicates the beginning of the time interval used to measure the spectrum. Swiftbegan to slew at the time indicated by the solid vertical line. Because of difficulties associatedwith fitting data that has been taken during a slew, the spectrum was extracted from a timeinterval ending at the slew.164Figure 8.28: The confidence contour between α and Eobspeak for the XRF 050315 promptgamma-ray emission. The probability that α and Eobspeak have values between the black lines is68%. The probability that their values lie between the red lines is 90%. The probability thattheir values lie below the green line is 99%. Based on this plot, there is a 90% probabilitythat Eobspeak is less than 100 keV if α is greater than -1.5.165Figure 8.29: The count spectrum of the XRR 050315 prompt gamma-ray emission, alongwith the best fit power law model. See Table 8.12 for the spectral parameters.Figure 8.30: The light curve of the XRR 050315 x-ray afterglow. The early phase of theafterglow (between 90 s and 500 s) is fit with a power law, and the late phase (after 5000 s) isfit with a broken power law. The circled data point is the average flux of the prompt emission,extrapolated into the 0.6 – 10 keV energy range.166Table 8.12: The best fit spectral parameters for the XRR 050315 prompt emission, using theBand model with α frozen at -1.5, -1 and at -0.5 as well as a simple power law model. Eobspeakis measured in keV, A in photons/cm2/s/keV, and average 15 – 150 keV flux in ergs/cm2/s.It is important to note that since only the interval prior to the slew was used to extract thespectrum, the average flux given below multiplied by the exposure gives the total fluence onlyof that portion of the burst, not for the full burst.Model Parameter ValueBand (α = −1.5) β < −1.84Eobspeak < 90.9A 5.01+37.66−0.57 × 10−3χ2ν/ν 0.930/72Band (α = −1) β −2.07+0.24−0.54Eobspeak < 59.2A 1.02+23.91−0.21 × 10−2χ2ν/ν 0.936/72Band (α = −0.5) β −1.95+0.14−0.16Eobspeak < 46.4A 2.62+149.88−0.95 × 10−2χ2ν/ν 0.942/72Cutoff Power Law α −1.54+0.44−0.40Eobspeak > 50.5A 4.8+3.0−1.7 × 10−3χ2ν/ν 0.932/72Power Law α −1.89 ± 0.1A 3.20+0.19−0.18 × 10−3χ2ν/ν 0.951/73flux 2.94 × 10−8exposure 70.32859 s167Figure 8.31: The count spectra of the XRR 050315 x-ray afterglow. The black line shows thespectrum and best fit during the early phase of the afterglow, the red line is the late phasebefore the break in the light curve, and the green line represents the late phase after the breakin the light curve. Count bins corresponding to photon energies of 0.3 – 10 keV were included.See Table 8.13 for the spectral parameters.before and after the break in the light curve are shown in Figure 8.31. The best fit hydrogencolumn density from the spectrum with the highest statistics (the portion of the late phase of theafterglow before the break in the light curve) was 2.0+0.4−0.3 × 1021 atoms/cm2. This is a bit higherthan the hydrogen column density due to our galaxy alone in the direction of the burst (which is4.34 × 1020 atoms/cm2). The spectral index after the break is a bit higher than the index beforethe break, indicating some spectral softening.8.2.2 XRR 050318This burst triggered BAT at 15:44:37 UT (MET=132853476.608) (GCN 3111). Swift didnot slew immediately due to an observing constraint. The XRT began accumulating data at16:39:14.5 UT, about 55 minutes after the trigger (GCN 3113). The position determined by BATwas RA=3h 18m 45.4s, Dec=-46d 23’ 14.8”, with an estimated uncertainty radius of about 3 arcmin168Table 8.13: The best fit spectral parameters for the XRR 050315 x-ray afterglow during theearly phase and before and after the break in the light curve during the late phase. nH ismeasured in 1021 atoms/cm2, A in units of 10−3 ergs/cm2/s/keV, and 0.6 – 10 keV flux inunits of 10−12 ergs/cm2/s. Start and stop times are in seconds since the BAT trigger.Parameter Early Phase Late Phase Late PhaseBefore Break After BreaknH < 0.88 2.0+0.4−0.3 3.2+1.9−1.6I 1.75+0.46−0.27 2.12+0.12−0.11 2.89+0.780.65A 3.33+0.87−0.36 1.87+0.23−0.20 0.21+0.15−0.09flux 13.4+2.1−3.6 7.59+0.45−0.43 0.55+0.27−0.13start time 300 s 5057.663403 s 271409.262923 sstop time 778.062 s 57387.17494 s 528072.4431 sexposure 476.3 s 11587.1 s 18724.7 sχ2ν/ν 0.374/3 1.253/65 0.818/4(90% confidence). The XRT-determined position was found to be RA=3h 18m 51.1s, Dec=-46d23’ 44.7”, with a 6 arcsec uncertainty (90% confidence) (GCN 3133). Optical afterglow emissionwas also detected (GCN 3114), and a redshift of 1.44 was measured (GCN 3122).Prompt EmissionThe T90 duration of XRR 050318 was 31.260 ± 0.004 seconds. There were two distinctepisodes of emission: one lasting from about the time of the trigger until about 3 seconds later,and the second lasting from about 23 seconds after the trigger until about 35 seconds after thetrigger. The first episode has a single discernible peak centered at about 1 second. The secondepisode has three distinct peaks: one at about 26 s, a larger peak at about 29 s, and a smallerpeak at about 32 s. Most of the counts are detected in the band corresponding to photon energiesof 25 - 50 keV, with significant counts also appearing in the 15 - 25 and 50 - 100 keV band. Nodiscernible counts above background are present in the 100 - 350 keV band (see Figure 8.32).For this burst, the Band model provided a good fit to the spectrum, and it was able to con-169Figure 8.32: The light curve of the XRR 050318 prompt gamma-ray emission. The dashedvertical line indicates the beginning of the time interval used to measure the spectrum. In-dividual detector count rates were not available for this burst after 32.4 s, so the light curveshown here is the total number of counts above background in each 64 ms time interval.Also for this reason, the interval used to measure the spectrum extends only to 32.4 s. Thespacecraft slewed at a time later than the time interval shown here.170Figure 8.33: The count spectrum of the XRR 050318 prompt gamma-ray emission, alongwith the best fit power law model. See Table 8.14 for the spectral parameters.strain Eobspeak quite well to 44.8+7.0−10.6 keV. The time interval from which the spectrum was extractedspans from 0.576 s before the trigger to 32.404 s after it. Figure 8.33 shows the count spectra andthe best-fit model.X-ray AfterglowThe x-ray afterglow, as measured by the XRT, is shown in Figure 8.34. As we mentionedbefore, Swift was unable to slew immediately due to an observing constraint, so XRT data is notavailable earlier than about 3 ks after the initial trigger. The light curve is well fit by a brokenpower law with an early decay index of −1.17 ± 0.05, a late decay index of −2.08 ± 0.15, and abreak time of 17700± 2200 s.The 0.6 – 10 keV flux of the prompt emission is also shown, as extrapolated from the BATdata. The value is consistent with the decay of the x-ray afterglow emission.The XRT count spectrum before and after the break in the light curve is shown in Fig-ure 8.35. The hydrogen column density was constrained to 5.9+2.9−2.5 × 1020 atoms/cm2. This is a171Table 8.14: The best fit spectral parameters for the XRR 050318 prompt emission, using apower law model, a cutoff power law model, and the Band model. Eobspeak is measured in keV,A in ergs/cm2/s/keV, and average flux in 10−8 ergs/cm2/s for 15 – 150 keV photons. Theexposure was 32.980 s.power law cutoff power law BandParameter Value Parameter Value Parameter Valueα −2.03 ± 0.09 α −1.23+0.44−0.40 α −1.08+1.650.58A 4.79+0.23−0.24 × 10−3 Eobspeak 44.8+6.9−6.3 β < 2.19A 1.24+0.87−0.47 × 10−2 Eobspeak 26.7+10.7−12.5A 1.44+19.94−0.67 × 10−2flux 2.04 ± 0.30χ2ν/ν 0.909/73 χ2ν/ν 0.748/72 χ2ν/ν 0.762/71Figure 8.34: The light curve of the XRR 050318 x-ray afterglow, along with the best-fit powerlaw model. The circled data point is the average flux of the prompt emission, extrapolated intothe 0.6 – 10 keV energy range. The data is well fit by a broken power law. When extrapolatedbackward, the early power law is consistent with the 0.6 – 10 keV flux of the prompt emission.172Table 8.15: The best fit spectral parameters for the XRF 050318 x-ray afterglow. nH ismeasured in 1020 atoms/cm2, A in units of 10−3 ergs/cm2/s/keV, and 0.6 – 10 keV flux inunits of 10−12 ergs/cm2/s. Start and stop times are in seconds since the BAT trigger.Parameter Interval 1 Interval 2nH 5.9+2.9−2.5 < 10.5I 1.98+0.14−0.13 1.97+0.29−0.25A 2.27+0.30−0.25 0.226+0.058−0.045flux 10.4+0.8−0.7 1.05+0.14−0.13start time 3284.760163 s 20654.933463 sstop time 17387.4133 s 63671.4133 sexposure 5225.0 s 18206.5 sχ2ν/ν 0.980/48 0.800/16little higher than the hydrogen column density due to our galaxy alone in the direction of the burst(which is 2.77 × 1020 atoms/cm2). The spectral indices in both spectra are consistent with eachother, indicating little spectral evolution.8.2.3 XRR 050525AOn 25 May 2005, a fairly bright burst triggered BAT at 00:02:53 UT (MET=138672172.8)(GCN 3466). Swift immediately slewed, and the XRT began accumulating data at 00:04:58 UT,125 s after the trigger. The position determined by BAT was RA=18h 32m 34s, Dec=+26d 20’ 38”,with an estimated uncertainty radius of about 5 arcmin (90% confidence). The XRT-determinedposition was found to be RA=18h 32m 32.3s, Dec=+26d 20’ 17.5”, with a 6 arcsec uncertainty(90% confidence) (GCN 3466). Afterglow emission in the optical, infrared, and radio bands wasalso detected (GCN 3465, 3471, 3495), and a redshift of 0.606 was measured (GCN 3483).Prompt EmissionThis burst has a T90 duration of 8.848 ± 0.004 seconds, but emission is detected out toabout 12 seconds following the trigger. It consists of several peaks, with the highest centered at173Figure 8.35: The count spectrum of the XRF 050318 x-ray afterglow before (top) and after(bottom) the break in the light curve. Both count spectra are fit with a power law model,with intervening absorption. Count bins corresponding to photon energies of 0.3 – 10 keVwere included. See Table 8.15 for the spectral parameters.174Table 8.16: The best fit spectral parameters for the XRR 050525A prompt emission, using apower law model, a cutoff power law model, and the Band model. Eobspeak is measured in keV,A in ergs/cm2/s/keV, and average flux in 10−8 ergs/cm2/s for 15 – 150 keV photons. Theexposure was 12.8463 s.power law cutoff power law BandParameter Value Parameter Value Parameter Valueα −2.06± 0.02 α −1.14 ± 0.07 α −1.057+0.099−0.092A 0.112± 0.002 Eobspeak 84.0+2.7−2.6 −3.29+0.26−0.50A 0.233+0.015−0.013 Eobspeak 81.00+3.32−3.26A 0.253+0.026−0.021flux 115.4 ± 1.8χ2ν/ν 10.036/99 χ2ν/ν 1.399/98 χ2ν/ν 1.325/97about 1.5 s after the trigger, preceded by a slightly lower peak at about 0.3 s after the trigger.Two more peaks follow at about 5.5 s and about 7 s. The largest count rates appear in channelscorresponding to photon energies of 25 - 50 keV, with slightly smaller rates appearing in the 15 - 25and 50 - 100 keV bands. Some emission is also evident in the 100 - 350 keV band (see Figure 8.36).As with most of the other XRRs we selected for study, this burst was also observed byKonus. Since Swift began to slew during the burst, and due to the difficulty in fitting data takenduring a slew, it was only practical to use BAT data from before the slew. Because of the timeintervals into which the Konus data was divided, doing a joint fit with BAT and Konus datawould have eliminated a large portion of the burst, so instead Konus data alone was used for thespectral analysis. We were able to constrain Eobspeak to 81.00+3.32−3.26 keV, using the Band model. Thetime interval from which the spectrum was extracted spans from 0.26502 s after the BAT triggerto 14.85702 s after it. Figure 8.37 shows the count spectrum from Konus as well as the best-fitspectral model.175Figure 8.36: The light curve of the XRR 050525A prompt gamma-ray emission. The dashedvertical lines bound the time interval used to measure the spectrum. Swift began to slewat the time indicated by the solid vertical line. Since the slew began during the burst, onlyKonus data were used for the spectral fitting.176Figure 8.37: The count spectra recorded by Konus for the XRR 050525A prompt gamma-rayemission, along with the best fit Band model. See Table 8.16 for the spectral parameters.X-ray AfterglowThe x-ray afterglow, as measured by the XRT, is shown in Figure 8.38. Although Swiftslewed immediately, photon counting and window timing data were only available beginning about6 ks after the burst. The light curve is best fit by a broken power law, with an early decay indexof −1.21± 0.14, a late decay index of −1.68 ± 0.10, and a break time of 14700± 5000 s.The 0.6 – 10 keV flux of the prompt emission is also shown, as extrapolated from the BATdata. The value is consistent with the decay of the x-ray afterglow emission.The XRT count spectra before and after the break are shown in Figure 8.39. The hydrogencolumn density was constrained to (2.2 ± 0.8) × 1021 atoms/cm2. This is slightly higher thanthe hydrogen column density due to our galaxy alone in the direction of the burst (9.1 × 1020atoms/cm2). The spectral index in the second spectrum, though not well constrained, is consistentwith that of the first spectrum.177Figure 8.38: The light curve of the XRR 050525A x-ray afterglow, along with the best-fitbroken power law model. The circled data point is the average flux of the prompt emission,extrapolated into the 0.6 – 10 keV energy range. When extrapolated backward, the earlypower law is consistent with the 0.6 – 10 keV flux of the prompt emission, as measured byBAT.Table 8.17: The best fit spectral parameters for the XRR 050525a x-ray afterglow. nH ismeasured in 1021 atoms/cm2, A in units of 10−3 ergs/cm2/s/keV, and 0.6 – 10 keV flux inunits of 10−12 ergs/cm2/s. Start and stop times are in seconds since the BAT trigger.Parameter Interval 1 Interval 2nH 2.2+0.9−0.8 < 6.7I 2.11+0.26−0.23 1.96+2.08−0.80A 4.00+1.15−0.89 0.226+0.058−0.045flux 16.3+1.9−1.8 1.82+0.59−0.60start time 5858.924463 s 16087.577083 sstop time 12842.995612 s 87212.735492 sexposure 1835.0 s 3915.3 sχ2ν/ν 0.585/17 0.770/1178Figure 8.39: The count spectrum of the XRR 050525a x-ray afterglow before (top) and after(bottom) the break in the light curve. Both count spectra are fit with a power law model,with intervening absorption. Count bins corresponding to photon energies of 0.3 – 10 keVwere included. See Table 8.17 for the spectral parameters.1798.2.4 XRR 050219BOn 19 Feb 2005, two bursts triggered BAT. The second of these (050219B) was detected at21:05:51 UT (MET=130539950.656) (GCN 3043). Swift wasn’t able to slew immediately due toan Earth limb observing constraint, but a slew took place about 50 minutes later, and 52 minuteslater, the XRT began accumulating data. The position determined by BAT was RA=5h 25m 9.8s,Dec=-57d 45’ 48.5”, with an estimated uncertainty radius of about 3 arcmin (90% confidence).The XRT-determined position was found to be RA=5h 25m 16.31, Dec=-57d 45’ 27.31”, with a6.3 arcsec uncertainty (90% confidence) (GCN 3049). In addition to the x ray, afterglow emissionwas detected in the infrared band, but not in any other bands (GCN 3064). A redshift was notdirectly measured, but the technique discussed in section 7.3 gives an inferred redshift of 1.0+0.5−0.2.Prompt EmissionThis burst has a T90 duration of 30.592 ± 0.064 seconds. It consists of at least threediscernible peaks, with the highest centered at about 3 s after the trigger, followed by a slightlylower peak at about 8 s after the trigger. Another much smaller peak is visible at about 38 s afterthe trigger. The largest count rates appear in channels corresponding to photon energies of 25- 100 keV, with slightly smaller rates appearing in the 15 - 25 keV band. Some emission is alsoevident in the 100 - 350 keV band (see Figure 8.40).This burst was also observed by Konus, and we were able to use both data sets together tofind the best spectral model. We were able to constrain Eobspeak to 161.5+13.1−11.1 keV, using the Bandmodel. The time interval from which the spectrum was extracted spans from 1.4515 s before thetrigger to 53.0765 s after the trigger. Figure 8.41 shows the count spectra from both BAT andKonus as well as the best-fit model for these spectra.X-ray AfterglowThe x-ray afterglow, as measured by the XRT, is shown in Figure 8.42. The light curve isconsistent with a power law with a decay index of −1.03± 0.09. The average 0.6 – 10 keV flux of180Figure 8.40: The light curve of the XRR 050219B prompt gamma-ray emission. The dashedvertical lines bound the time interval used to measure the count spectrum. Swift slewed longafter the interval shown in this plot.181Figure 8.41: The count spectra recorded by BAT and Konus for the XRR 050219B promptgamma-ray emission, along with the best fit power law model. Konus spectra are shown inred and green, and the BAT spectrum in black. See Table 8.18 for the spectral parameters.Table 8.18: The best fit spectral parameters for the XRR 050219B prompt emission, using apower law model, a cutoff power law model, and the Band model. Eobspeak is measured in keV,A in ergs/cm2/s/keV, and average flux in 10−8 ergs/cm2/s for 15 – 150 keV photons. Theexposure was 54.528 s.power law cutoff power law BandParameter Value Parameter Value Parameter Valueα −1.68± 0.02 α −1.05 ± 0.07 α −1.05 ± 0.07A (2.41 ± 0.06) × 10−2 Eobspeak 162.8+13.1−11.3 β < −2.86A 3.46+0.18−0.16 × 10−2 Eobspeak 161.5+13.1−11.1A 3.46+0.19−0.16 × 10−2flux 23.3 ± 0.6χ2ν/ν 3.860/175 χ2ν/ν 1.633/174 χ2ν/ν 1.637/173182Figure 8.42: The light curve of the XRR 050219B x-ray afterglow, along with the best-fitpower law model. The circled data point is the average flux of the prompt emission from BAT,extrapolated into the 0.6 – 10 keV energy range. When extrapolated backward, the power lawis consistent with the average 0.6 – 10 keV flux of the prompt emission.the prompt emission is also shown, as extrapolated from the BAT data. The trend in the afterglowis consistent with the prompt emission.The count spectrum measured by the XRT is shown in Figure 8.43. The best fit hydrogencolumn density is 1.09+0.92−0.80 × 1021 atoms/cm2. This value (with its rather large error bars) isconsistent with the hydrogen column density due to our galaxy alone in the direction of the burst(3.83×1020 atoms/cm2). The best fit spectral index is 1.80+0.27−0.25. The spectrum used here consistsof counts accumulated between 5394.5 s after the trigger and 120995.4 s after the trigger.8.3 GRBsWe now move on to a collection of those bursts for which the fluence ratio S(50−100 keV)/S(25−50 keV) ≥ 1.3855. There are 5 bursts in our sample, selected because they were also observed byKonus.183Figure 8.43: The count spectrum of the XRR 050219B x-ray afterglow. Count bins corre-sponding to photon energies of 0.3 – 10 keV were included. See Table 8.19 for the spectralparameters.Table 8.19: The best fit spectral parameters for the XRR 050219B x-ray afterglow. nH ismeasured in atoms/cm2, A in units of photons/cm2/s/keV, and flux in units of ergs/cm2/s.Start and stop times are in seconds since the BAT trigger.Parameter ValuenH 1.09+0.92−0.80 × 1021I 1.80+0.27−0.25A 8.14+2.52−1.87 × 10−40.6 – 10 keV flux 5.52+0.71−0.59 × 10−12start time 5395.353623 sstop time 120995.399052 sexposure 6603.8 sχ2ν/ν 0.589/171848.3.1 GRB 050401With a fluence ratio of 1.4363, this is the softest in our sample of bursts with a fluenceratio large enough to be designated actual “gamma-ray bursts”. This burst triggered BAT at14:20:15 UT (MET=134058014.784) (GCN 3161). There was a 9 second delay in the slew dueto an Earth observing constraint, after which the XRT began immediately to collect data. Theposition determined by BAT was RA=16h 31m 32.3s, Dec=2d 11’ 42.5”. The XRT-determinedposition was found to be RA=16h 31m 29s, Dec=2d 11’ 14”, with a 6 arcsec uncertainty (90%confidence) (GCN 3161). Afterglow emission was also detected in the optical and radio bands, anda redshift of 2.90 was measured (GCN 3163, 3187, 3176).Prompt EmissionThis burst has a T90 duration of 33.640 ± 0.004 seconds. There are two distinct episodesof emission—one beginning about 10 s before the trigger and ending about 10 s after the trigger,and the other beginning about 20 s after the trigger and ending about 30 s after the trigger. Thefirst episode consists of at least three distinct peaks: once centered at -6 s, one at 0 s, and one at 3s. The second episode consists of a single peak centered at 24 s. The largest count rates appear inchannels corresponding to photon energies of 50 - 100 keV, with slightly smaller rates appearing inthe 25 - 50 keV band. Prominent emission is also evident in the other two bands (see Figure 8.44).As with all of the bursts selected for this sample, this burst was also observed by Konus.Swift began to slew during the burst, and due to the difficulty in fitting data taken during a slew,it is not practical to use BAT data taken during the slew. So that emission from the full intervalof the burst could be analyzed, Konus data alone was used for the spectral analysis. We were ableto constrain Eobspeak to 117.5+20.1−16.4 keV, using the Band model.The time interval from which the spectrum was extracted spans from 5.24101 s before theBAT trigger to 35.97499 s after the trigger. Figure 8.45 shows the count spectrum from Konus aswell as the best-fit model for the spectrum.185Figure 8.44: The light curve of the GRB 050401 prompt gamma-ray emission. The dashedvertical lines bound the time interval used to measure the spectrum. Swift began to slewat the time indicated by the solid vertical line. Since the slew began during the burst, onlyKonus data was used for the spectral fitting.186Figure 8.45: The count spectra recorded by Konus for the GRB 050401 prompt gamma-rayemission, along with the best fit Band model. See Table 8.20 for the spectral parameters.Table 8.20: The best fit spectral parameters for the GRB 050401 prompt emission, using apower law model, a cutoff power law model, and the Band model. Eobspeak is measured in keV,A in ergs/cm2/s/keV, and average flux in 10−8 ergs/cm2/s for 15 – 150 keV photons. Theexposure was 38.3517 s.power law cutoff power law BandParameter Value Parameter Value Parameter Valueα −1.89± 0.04 α −1.18+0.18−0.17 α −0.90+0.34−0.26A (2.53 ± 0.12) × 10−2 Eobspeak 142.5+19.3−14.6 β −2.55+0.22−0.44A 3.50+0.41−0.33 × 10−2 Eobspeak 117.5+20.1−16.4A 4.30+1.25−0.71 × 10−2flux 23.3 ± 1.1χ2ν/ν 2.360/100 χ2ν/ν 1.402/99 χ2ν/ν 1.358/98187Figure 8.46: The light curve of the GRB 050401 x-ray afterglow, along with the best-fitbroken power law model. The offset of the early power law has a best fit value of 90 ± 16s. The existence of this offset causes the early power law to be plotted as a curved line withan asymptote at 90 s. The circled data point is the average flux of the prompt emission,extrapolated into the 0.6 – 10 keV energy range.X-ray AfterglowThe x-ray afterglow, as measured by the XRT, is shown in Figure 8.46. The light curve canbe fit with a broken power law, with a χ2ν of 1.689 (23 degrees of freedom). However, a betterfit is obtained if we allow the offset of the early power law to be different from the trigger time,in which case we obtain a χ2ν of 1.226 (with 22 degrees of freedom). An F -test reveals that thelatter model fits the data better to a confidence of 99.5%. Allowing the late power law offset to bedifferent from the trigger time doesn’t improve the fit, and requiring the two offsets vary togetherresults in the same χ2ν as allowing only the early power law offset to vary. The best fit early powerlaw index is −0.51±0.30 with a best fit offset of 90±16 s after the trigger. The best fit late powerlaw index is −1.33 ± 0.05. The break time is 9200 ± 1000 s.The XRT count rate before and after the break in the light curve is shown in Figure 8.47. Thebest fit hydrogen column density is 2.17+0.98−0.83 × 1021 atoms/cm2. This is higher than the hydrogen188Table 8.21: The best fit spectral parameters for the GRB 050401 x-ray afterglow. nH ismeasured in 1021 atoms/cm2, A in units of 10−3 ergs/cm2/s/keV, and 0.6 – 10 keV flux inunits of 10−12 ergs/cm2/s. Start and stop times are in seconds since the BAT trigger.Parameter Interval 1 Interval 2nH 2.17+0.98−0.83 2.6+1.2−0.9I 1.96+0.25−0.23 2.15+0.30−0.27A 13.3+4.0−3.0 3.66+1.31−0.90flux 62.3+7.2−6.8 14.5+2.1−1.7start time 1735.207623 s 13089.957232 sstop time 8526.682692 s 20104.5328 sexposure 458.3 s 1786.9 sχ2ν/ν 0.902/18 1.144/18column density due to our galaxy alone in the direction of the burst (4.84×1020 atoms/cm2). Thespectral indices are consistent with each other.8.3.2 GRB 050326This burst triggered BAT at 09:53:55 UT (MET=133523635.584) (GCN 3143). The space-craft did not slew immediately due to an Earth observing constraint. After the slew, the XRTbegan to collect data, beginning at about 10:48:13 UT, about 55 minutes after the burst (GCN3143). The position determined by BAT was RA=0h 27m 30.2s, Dec=-71d 22’ 40.0”. The XRT-determined position was found to be RA=0h 27m 48.7s, Dec=-71d 22’ 17.2”, with a 6 arcsecuncertainty (90% confidence) (GCN 3147). No afterglow emission in any other wave band wasdetected, and no redshift was measured. In this case, the Yonetoku relation doesn’t provide anextremely good constraint on z. It predicts a redshift of 2.4+3.4−0.9.Prompt EmissionThis is a very complex burst with many peaks. There are two distinct episodes of emission:one extending from 5 seconds before the trigger until 12 seconds after the trigger, and the other189Figure 8.47: The count spectrum of the GRB 050401 x-ray afterglow before (top) and after(bottom) the break in the light curve. Both count spectra are fit with a power law model,with intervening absorption. Count bins corresponding to photon energies of 0.3 – 10 keVwere included. See Table 8.21 for the spectral parameters.190Table 8.22: The best fit spectral parameters for the GRB 050326 prompt emission, using apower law model, a cutoff power law model, and the Band model. Eobspeak is measured in keV,A in ergs/cm2/s/keV, and average flux in 10−8 ergs/cm2/s for 15 – 150 keV photons. Theexposure was 30.6496 s.power law cutoff power law BandParameter Value Parameter Value Parameter Valueα −1.63± 0.02 α −0.910± 0.11 α −0.78+0.17−0.15A (2.99 ± 0.09) × 10−2 Eobspeak 240.8+2.4−2.3 β −2.48+0.24−0.63A 3.71+0.22−0.21 × 10−2 Eobspeak 206.4+32.8−27.1A 4.44+0.42−0.32 × 10−2flux 28.9+1.0−0.8χ2ν/ν 4.052/100 χ2ν/ν 1.070/99 χ2ν/ν 1.028/98extending from 16 to 26 seconds after the trigger. The first episode has at least five distinct peaks,with the largest centered at roughly the time of the trigger. The second episode has consists ofthree or four peaks with the most prominent centered at 20 and 22 seconds. The largest count ratesappear in bands corresponding to photon energies of 25 - 100 keV. Prominent rates also appear inthe other two bands (see Figure 8.48). The T90 duration for this burst is 29.328± 0.036 s.As with all of the other bursts selected for this sample, this burst was also observed byKonus. Swift began to slew early in the burst and due to the difficulty in fitting data taken duringa slew, it is not practical to use BAT data taken during the slew. Therefore, Konus data alone wasused for the spectral analysis. With the Band model, Eobspeak is found to be 206.4+32.8−27.1. Figure 8.49shows the count spectra from Konus and the best-fit model for these spectra.X-ray AfterglowThe x-ray afterglow, as measured by the XRT, is shown in Figure 8.50. The light curve isconsistent with a single power law with a decay index of −1.75± 0.04, with a reduced χ2 of 1.047.However, a broken power law gives a slightly better fit, with χ2/nu of 0.989, giving an early decayindex of −1.55 ± 0.21, a late decay index of −1.92 ± 0.11, and a break time of 14000 ± 10000 s.191Figure 8.48: The light curve of the GRB 050326 prompt gamma-ray emission. The dashedvertical lines bound the time interval used to measure the spectrum. Swift began to slewat the time indicated by the solid vertical line. Since the slew began during the burst, onlyKonus data was used for the spectral fitting.192Figure 8.49: The count spectra recorded by Konus for the GRB 050326 prompt gamma-rayemission, along with the best fit Band model. See Table 8.22 for the spectral parameters.An F -test favors the broken power law model at a 76% confidence level over that of a single powerlaw, which is somewhat marginal. The average 0.6 – 10 keV flux for of the prompt emission is alsoshown, as extrapolated from the BAT data. The value is a bit lower than the extrapolated lightcurve of the afterglow.The XRT count spectra both before and after the break in the light curve is shown inFigure 8.51. The best fit hydrogen column density is 4.2+2.8−2.4 × 1021 atoms/cm2. This is higherthan the hydrogen column density due to our galaxy alone in the direction of the burst (4.55×1020atoms/cm2). The best fit spectral indices of the two spectra are consistent with each other.8.3.3 GRB 050603This burst triggered BAT at 06:29:05 UT (MET=139472944.768) (GCN 3509). Swift didnot slew automatically because slewing had been disabled for some engineering tests. XRT beganto collect data beginning at 17:19:27 UT, about 11 hours after the burst (GCN 3514). The193Figure 8.50: The light curve of the XRR 050326 x-ray afterglow, along with the best-fitbroken power law model. The circled data point is the average flux of the prompt emission,extrapolated into the 0.6 – 10 keV energy range. When extrapolated backward, the earlypower law overshoots the 0.6 – 10 keV flux of the prompt emission.Table 8.23: The best fit spectral parameters for the GRB 050326 x-ray afterglow. nH ismeasured in 1021 atoms/cm2, A in units of 10−3 ergs/cm2/s/keV, and 0.6 – 10 keV flux inunits of 10−12 ergs/cm2/s. Start and stop times are in seconds since the BAT trigger.Parameter Interval 1 Interval 2nH 4.6+2.8−2.4 < 2.1I 2.23+0.56−0.51 2.05+0.60−0.34A 8.25+6.23−3.52 0.053+0.034−0.014flux 41.9+28.9−11.4 0.229+0.047−0.045start time 3272.194983 s 37954.304543 sstop time 11002.095892 s 196277.282632 sexposure 589.4 s 35534.9 sχ2ν/ν 0.866/6 1.201/5194Figure 8.51: The count spectrum of the GRB 050326 x-ray afterglow before (top) and after(bottom) the break in the light curve. Both count spectra are fit with a power law model,with intervening absorption. Count bins corresponding to photon energies of 0.3 – 10 keVwere included. See Table 8.23 for the spectral parameters.195position determined by BAT was RA=2h 39m 54.8s, Dec=-25d 10’ 57.8”. The XRT-determinedposition was found to be RA=2h 39m 56.8s, Dec=-25d 10’ 59.8”, with a 6 arcsec uncertainty (90%confidence) (GCN 3519). Afterglow emission was also detected in optical, submillimeter, and radiobands (GCN 3511, 3515, 3513). A redshift of 2.821 was measured.Prompt EmissionThis burst has a T90 duration of 12.608 ± 0.064 seconds. There are three distinct peaks.The first peak, which is shorter and broader than the others, reaches its maximum about 3 secondsbefore the trigger. The next peak begins about 1 s before the trigger, and the final peak, which isthe highest, reaches its maximum 0.2 s after the trigger. The largest count rates appear in channelscorresponding to photon energies of 50 - 100 keV, with slightly smaller rates appearing in the 25- 50 keV band. Prominent emission is also evident in the other two bands (see Figure 8.52).As with all of the other bursts selected for this sample, this burst was also observed byKonus. Because the Swift slew occurred long after the burst, we were able to use both data setstogether to find the best spectral model. Eobspeak was found to be 324.4+46.5−43.9 keV, using the Bandmodel. The time interval from which the spectrum was extracted spans from 3.05323 s before thetrigger to 0.78677 s after the trigger. Figure 8.53 shows the count spectra from both BAT andKonus and the best-fit model for these spectra.X-ray AfterglowThe x-ray afterglow, as measured by the XRT, is shown in Figure 8.54. The light curveis consistent with a power law with a decay index of −1.60 ± 0.08. The 0.6 – 10 keV flux of theprompt emission is also shown, as extrapolated from the BAT data. The trend of the afterglowovershoots the prompt emission data. We have often seen breaks in the afterglow light curves attimes earlier than 11 hours, however, and it is distinctly possible that one exists here.The count spectrum measured by the XRT is shown in Figure 8.55. The fit provides anupper limit for the hydrogen column density of 7.8 × 1020 atoms/cm2. This is consistent withthe hydrogen column density due to our galaxy alone in the direction of the burst (1.92 × 1020196Figure 8.52: The light curve of the GRB 050603 prompt gamma-ray emission. The dashedvertical lines bound the time interval used to measure the spectrum. Swift slewed long afterthe time interval shown here.197Figure 8.53: The count spectra recorded by BAT and Konus for the GRB 050603 promptgamma-ray emission, along with the best fit Band model. The Konus data is shown in redand green and the BAT data in black. See Table 8.24 for the spectral parameters.Table 8.24: The best fit spectral parameters for the GRB 050603 prompt emission, using apower law model, a cutoff power law model, and the Band model. Eobspeak is measured in keV,A in ergs/cm2/s/keV, and average flux in 10−8 ergs/cm2/s for 15 – 150 keV photons. Theexposure was 3.84 s.power law cutoff power law BandParameter Value Parameter Value Parameter Valueα −2.03 ± 0.01 α −0.902+0.052−0.054 α −0.74+0.10−0.08A (7.21 ± 0.32)× 10−2 Eobspeak 462.6+41.8−36.7 β −2.09+0.11−0.14A 0.1201± 0.0047 Eobspeak 324.4+46.5−43.9A 0.130+0.008−0.007flux 114.3± 6.5χ2ν/ν 18.569/156 χ2ν/ν 1.188/155 χ2ν/ν 0.915/154198Figure 8.54: The light curve of the GRB 050603 x-ray afterglow, along with the best-fit powerlaw model. The circled data point is the average flux of the prompt emission, extrapolatedfrom the BAT data into the 0.6 – 10 keV energy range. When extrapolated backward, thepower law overshoots the 0.6 – 10 keV flux of the prompt emission. It is distinctly possiblethat a break occurs before 11 hours.199Figure 8.55: The count spectrum of the GRB 050603 x-ray afterglow. Count bins corre-sponding to photon energies of 0.3 – 10 keV were included. See Table 8.25 for the spectralparameters.atoms/cm2). The best fit spectral index is 1.75+0.35−0.17. The spectrum used here consists of countsaccumulated between 34046.8 s after the trigger and 57789.9 s after the trigger.8.3.4 GRB 041223This was one of the first bursts detected by BAT. It triggered the BAT at 14:06:18 UT(MET=125503577.28) (GCN 2898). Swift did not slew automatically because autonomous slewinghad not yet been enabled at this early stage in the mission. XRT began to collect data beginningat 18:43:59 UT, about 4.6 hours after the burst (GCN 2901). The position determined by BATwas RA=6h 40m 48.8s, Dec=-37d 4’ 3.0”. The XRT-determined position was found to be RA=6h40m 47.4s, Dec=-37d 4’ 22.3”, with an estimated uncertainty of 15 arcsec (90% confidence) (GCN2910). Optical afterglow emission was also detected (GCN 2902), but a redshift was not measured.Any redshift higher than 3.2 would permit this burst to be consistent with the Yonetoku relation.200Table 8.25: The best fit spectral parameters for the GRB 050603 x-ray afterglow. nH ismeasured in atoms/cm2, A in units of photons/cm2/s/keV, and flux in units of ergs/cm2/s.Start and stop times are in seconds since the BAT trigger.Parameter ValuenH < 7.78 × 1021I 1.75+0.35−0.17A 3.47+1.11−0.39 × 10−40.6 – 10 keV flux 2.31+0.32−0.30 × 10−12start time 34046.753303 sstop time 57789.9225 sexposure 7889.3 sχ2ν/ν 1.263/13Prompt EmissionThis was a rather long burst with a T90 duration of 108.932± 0.018 seconds. It has a verycomplex structure with 15 or more peaks, the highest of which is centered about 35 seconds afterthe trigger. The largest count rates appear in channels corresponding to photon energies of 50 -100 keV, with significant rates also appearing in the other two bands (see Figure 8.56).As with the other bursts selected for this sample, this burst was also observed by Konus,and we were able to use both data sets together to find the best spectral model. Eobspeak was foundto be 344.7+32.8−28.7 keV, using the Band model. The time interval from which the spectrum wasextracted spans from 1.182 s before the trigger to 137.058 s after the trigger. Figure 8.57 showsthe count spectra from both BAT and Konus as well as the best-fit model for these spectra.X-ray AfterglowThe x-ray afterglow, as measured by the XRT, is shown in Figure 8.58. The light curve isconsistent with a power law with a decay index of −1.92± 0.27. The average 0.6 – 10 keV flux ofthe prompt emission is also shown, as extrapolated from the BAT data. The trend of the afterglowovershoots the prompt emission data. It is possible that a break in the afterglow takes place prior201Figure 8.56: The light curve of the GRB 041223 prompt gamma-ray emission. The dashedvertical lines bound the time interval used to measure the spectrum. Swift slewed long afterthe time interval shown here.202Figure 8.57: The count spectra recorded by BAT and Konus for the GRB 041223 promptgamma-ray emission, along with the best fit Band model. The BAT data is shown in blackand the Konus data in red. See Table 8.26 for the spectral parameters.Table 8.26: The best fit spectral parameters for the GRB 041223 prompt emission, using apower law model, a cutoff power law model, and the Band model. Eobspeak is measured in keV,A in ergs/cm2/s/keV, and average flux in 10−8 ergs/cm2/s for 15 – 150 keV photons. Theexposure was 138.24 s.power law cutoff power law BandParameter Value Parameter Value Parameter Valueα −1.34± 0.02 α −0.831+0.048−0.046 α −0.83 ± 0.05A (1.19 ± 0.02) × 10−2 Eobspeak 337.6+27.7−24.2 β < 2.63A 1.383+0.034−0.032 × 10−2 Eobspeak 335.9+28.8−26.5A 1.385+0.036−0.033 × 10−2flux 11.9 ± 0.2χ2ν/ν 5.425/135 χ2ν/ν 0.826/134 χ2ν/ν 0.832/133203Figure 8.58: The light curve of the GRB 041223 x-ray afterglow, along with the best-fit powerlaw model. The circled data point is the average flux of the prompt emission, extrapolatedinto the 0.6 – 10 keV energy range. When extrapolated backward, the power law overshootsthe 0.6 – 10 keV flux of the prompt emission.to XRT’s observations.The count spectrum measured by the XRT is shown in Figure 8.59. The best fit hydrogencolumn density is 1.71+0.71−0.64×1021 atoms/cm2. This is consistent with the hydrogen column densitydue to our galaxy alone in the direction of the burst (1.09×1021 atoms/cm2). The best fit spectralindex is 2.17+0.25−0.23. The spectrum used here consists of counts accumulated throughout the entireXRT observation, beginning 16658.4 s after the trigger and ending 28854.3 s after the trigger.204Figure 8.59: The count spectrum of the GRB 041223 x-ray afterglow. Count bins corre-sponding to photon energies of 0.3 – 10 keV were included. See Table 8.27 for the spectralparameters.Table 8.27: The best fit spectral parameters for the GRB 041223 x-ray afterglow. nH ismeasured in atoms/cm2, A in units of photons/cm2/s/keV, and flux in units of ergs/cm2/s.Start and stop times are in seconds since the BAT trigger.Parameter ValuenH 1.71+0.71−0.64 × 1021I 2.17+0.25−0.23A 1.96+0.49−0.39 × 10−30.6 – 10 keV flux 1.02+0.18−0.13 × 10−11start time 16658.405592 sstop time 28854.3362 sexposure 4016.5 sχ2ν/ν 1.088/24205Chapter 9Global Characteristics of BAT XRFs, XRRs, and GRBsIn this chapter, we will compare the prompt gamma-ray characteristics of bursts detected byBAT with those detected by other instruments. We will also compare the x-ray afterglow emissionof the sample of bursts described in Chapter 8 with bursts observed by other missions. Along theway, we will also note differences and similarities between the XRFs, XRRs, and GRBs observedby Swift.9.1 Prompt Emission9.1.1 FluenceFigure 9.1 shows the distribution of 15 – 150 keV fluences of all BAT bursts detected prior to31 July 2005. XRF 050826 is also included. Aside from two GRBs with very low fluences (which,incidentally, both have durations less than 200 ms, placing them in the class of short GRBs), theXRFs detected by BAT tend to have smaller fluences than the XRRs or GRBs detected by BAT.Sakamoto et al. also noted this general trend in bursts detected by HETE-2 [70].9.1.2 Spectral Peak Energy EobspeakFigure 9.2 shows the spectral peak energies Eobspeak for all BAT bursts detected prior to 31July 2005 for which Eobspeak was reasonably constrained. XRF 050826 is also included in the XRFhistogram. X-ray flashes tend to have lower values of Eobspeak than x-ray rich GRBs, which in turnhave lower Eobspeak values than GRBs. This is the same general trend observed in BeppoSAX [50]and HETE-2 bursts [70].206Figure 9.1: Fluence histograms of XRFs, XRRs, and GRBs detected by BAT. The x-axisshows the base-10 logarithm of the 15 – 150 keV fluence, in units of ergs/cm2. The arrow inone of the boxes in the XRF histogram denotes that only an upper limit on the fluence wasobtained for that burst.Figure 9.2: Eobspeak histograms of XRFs, XRRs, and GRBs detected by BAT. The x-axis showsthe logarithm of Eobspeak, in units of keV.207Table 9.1: K-S test results for Swift GRBs, XRRs, and XRFsDistributions ProbabilitySwift XRFs and Swift GRBs 0.17Swift XRFs and Swift XRRs 0.053Swift XRRs and Swift GRBs 0.61Swift (all) and BATSE (all) 0.001Swift (> 3 s) and BATSE (> 3 s) 0.389.1.3 DurationFigure 9.3 shows the T90 durations for all BAT bursts detected prior to 31 July 2005, as wellas for XRF 050826. Also shown are the durations of GRBs detected by BATSE. The KS-test givesa probability of 0.001 that there is no difference between the distribution of Swift burst durationsand that of the BATSE burst durations. If we only include bursts for which T90 is greater than3 seconds, the probability is 0.379. We can conclude that the distribution of bursts with a T90greater than 3 seconds detected by BAT is consistent with those detected by BATSE, but that ifwe include bursts with T90 durations less than 3 seconds (of which BAT has detected 2 as of 31July 2005), the two distributions are not consistent. We note that the two short-duration burstsdetected by BAT fall in the “GRB” category.Using the K-S test to compare the duration distribution of BAT-detected XRFs, XRRs,and GRBs, we obtain the values noted in Table 9.1. The T90 durations of the XRFs in our sampleare not consistent with those of the BAT-detected XRRs and are only marginally consistent withthe BAT-detected GRBs. Sakamoto et al. [70] found that the T90 durations of XRFs, XRRs, andGRBs detected by HETE-2 were consistent with one another. We note that both XRF samplesare small, with only 6 in our sample and 16 in the sample analyzed by Sakamoto et al., so thatthe results may be influenced by the small statistics of the samples.208Figure 9.3: Duration histograms of XRFs, XRRs, and GRBs detected by BAT. The x-axisshows the base-10 logarithm of T90, in units of seconds.9.1.4 Correlation of Eobspeak with Other QuantitiesSeveral studies examining potential correlations of Eobspeak with other quantities have beenconducted using observations from other missions. In this section, we compare the results of someof these studies with similar results from BAT.Eobspeak and DurationFigure 9.4 shows the distribution of burst durations and Eobspeak. In this case, the durationshown is T50, the interval between the time at which the fluence reaches 25% of its full finalamount and the time at which the fluence reaches 75% of its full final amount. Also shown arevalues taken from analysis done by Kaneko et al. on a sample of bursts detected by BATSE [47].This plot illustrates the differences in the sensitivity of the two instruments to bursts of differentdurations and peak energies. While the durations of the BAT bursts are consistent with those ofthe BATSE bursts, as we have already noted, BAT is less sensitive to the short-duration class ofbursts. Furthermore, the peak energies Eobspeak determined from BAT data alone tend to be lowerthan those determined from BATSE data. A similar study was done by Kippen et al. using XRFdata from BeppoSAX [50]. We note that the sample of XRFs presented here have a similar Eobspeak209Figure 9.4: A plot of T50 duration and peak spectral energy Eobspeak of XRFs, XRRs, and GRBsdetected by BAT. Also shown are the T50 durations and Eobspeak values for a sample of burstsdetected by BATSE.and T50 distribution as those presented by Kippen et al.Eobspeak and FluenceFigure 9.5 is a scatter plot showing Eobspeak verses 15 – 150 keV fluence. As before, allbursts detected prior to 31 July 2005 for which Eobspeak was reasonably constrained, as well as XRF050826, are included. We note a mild correlation between fluence and Eobspeak. The linear correlationcoefficient for this sample is 0.70903. The probability of such a correlation occurring by chancein a sample of this size (27 bursts) is 1.2 × 10−4. Sakamoto et al. [70] observed a similar thoughsomewhat smaller correlation between Eobspeak and the 2 – 400 keV fluence for bursts detected byHETE-2.210Figure 9.5: A plot of the 15 – 150 keV fluence and peak spectral energy Eobspeak of XRFs,XRRs, and GRBs detected by BAT. The solid line is the best linear fit to the data and isgiven by Eobspeak = 2158+1449−801 × [SE(15–150 keV)]0.29±0.04, with SE(15–150 keV) in ergs/cm2.211Figure 9.6: A plot of peak spectral energy in the source frame Esrcpeak verses the isotropic-equivalent energy Eiso of bursts detected by BeppoSAX, HETE-2, and Swift. The circleddata points are bursts for which a redshift was not measured and consequently the Yonetokurelation was used to derive an estimated redshift, using the technique described in Section 7.3.Since the Yonetoku relation is itself very similar to the Amati relation, it is not surprisingthat these points lie along the correlation.Eobspeak and EisoIn section 2.4.5, we described a correlation between Esrcpeak (the peak spectral energy in thesource frame) and Eiso (the total isotropic-equivalent gamma-ray energy emitted). Figure 9.6shows the data presented in Figure 2.7 along with data from bursts observed by BAT. The BATdata points lie right along this correlation and fill in a region of the Esrcpeak – Eiso plane not observedbefore. Table 9.2 gives the redshift, Esrcpeak, and Eiso values for Swift bursts.9.2 Afterglow Emission9.2.1 Light CurvesFigure 9.7 is a composite plot of all of the x-ray afterglow light curves presented in Chapter8. Figures 9.8, 9.9, and 9.10 show the light curves in each class. The 0.6 – 10 keV average flux212Table 9.2: Redshifts, Esrcpeak values and Eiso values for Swift bursts. Bursts in italics resultfrom inferred redshifts derived using the technique described in Section 7.3.Burst z Esrcpeak (keV) Eiso (1052 ergs)XRF 050416A 0.6535 27.6+8.8−16.7 0.076+0.017−0.011XRF 050406 > 3 > 39.2 > 0.214XRF 050714B > 1.8 > 7.28 > 0.641XRF 050824 0.83 < 35.7 0.179+6.877−0.083XRF 050528 0.3+1.7−0.2 39.7+53.4−14.2 < 0.72XRF 050215B 0.8+1.4−0.3 48.1+42.1−23.9 0.070+0.411−0.046XRR 050315 1.949 < 294.9 9.5+7.0−3.4XRR 050318 1.44 109.3+17.1−25.9 1.59+0.82−0.39XRR 050319 3.24 78.3+93.2−77.9 4.5+8.2−2.0XRR 050525A 0.606 130.1+4.0−3.8 2.663+0.100−0.094XRR 050505 4.27 336+466−147 38+21−25XRR 050219B 1+0.5−0.2 323+85−39 9.4+11.2−3.5GRB 050401 2.9 458+78−64 46.2+4.2−3.9GRB 050126 1.29 109+84−34 3.2+2.9−2.2GRB 050326 2.4+3.4−0.9 702+711−207 44+108−25GRB 050603 2.821 1240+178−168 61.8+3.1−3.1GRB 041223 > 3.2 > 1327.2 > 131.5213Figure 9.7: A composite plot of the GRB, XRR, and XRF x-ray afterglow light curvespresented in Chapter 8. The 0.6 – 10 keV average fluxes extrapolated from the promptemission spectra are designated by circles.extrapolated from the prompt emission varies over three orders of magnitude. There is a hintthat XRFs have a smaller extrapolated prompt emission on average than XRRs and GRBs. Theafterglow fluxes of all three classes are spread throughout a wide range of intensities at varioustimes along their evolution.9.2.2 Photon Index and nHFigure 9.11 shows the distribution of best-fit neutral hydrogen column densities nH andphoton indices I for the sample of bursts described in Chapter 8. Also shown are values gatheredand cited by Frontera [27]. No apparent correlation exists between these quantities. Using the K-Stest to compare nH values from Frontera’s sample with the nH values from our sample, we obtaina probability of 0.064 that the two samples are drawn from the same distribution. Thus, the twosets differ significantly. The probability for the photon indices of the two samples is 0.89. Thus,the photon indices of the two samples are consistent with one another. There is a hint that the214Figure 9.8: A composite plot of the afterglow light curves of x-ray flashes presented inChapter 8. The 0.6 – 10 keV average fluxes extrapolated from the prompt emission spectraare designated by circles. Eobspeak is also given for each burst.215Figure 9.9: A composite plot of the afterglow light curves of x-ray-rich gamma-ray burstspresented in Chapter 8. The 0.6 – 10 keV average fluxes extrapolated from the prompt emissionspectra are designated by circles. Eobspeak is also given for each burst.216Figure 9.10: A composite plot of the afterglow light curves of gamma-ray bursts presentedin Chapter 8. The 0.6 – 10 keV average fluxes extrapolated from the prompt emission spectraare designated by circles. Eobspeak is also given for each burst.217Figure 9.11: A plot of the best-fit neutral hydrogen column densities nH and photon indicesI of afterglows described in Chapter 8, along with values taken from Frontera [27]. The valuesplotted here are taken from spectra measured 10 hours after the burst trigger.GRBs in the sample tend to manifest systematically higher column densities and photon indicesthan XRFs or XRRs.9.2.3 Photon Index and Temporal IndexFigure 9.12 shows the distribution of best-fit temporal decay indices β and photon indicesI for the sample of bursts described in Chapter 8. Also shown are values gathered and cited byFrontera [27]. The values plotted are those measured 10 hours after the burst. We have alreadyseen that the photon indices from the two samples are consistent. The probability that the twodistributions of decay indices are drawn from the same parent distribution is 0.31. Thus, the decayindices of the two samples are also consistent. We note that the GRBs in our sample tend tohave higher (that is, steeper) temporal indices than XRRs, which in turn have higher indices thanXRFs.Figure 9.13 shows the temporal decay indices β and photon indices I for the Swift bursts218Figure 9.12: A plot of the best-fit temporal decay indices β and photon indices I of afterglowsdescribed in Chapter 8, along with values taken from Frontera [27]. The values plotted hereare taken from spectra measured 10 hours after the burst trigger.219Figure 9.13: A plot of the best-fit temporal decay indices β and photon indices I of afterglowsdescribed in Chapter 8, as measured 8000 seconds after the burst trigger. Two of the fourGRB afterglows in our sample were not observed until a later time, so they are not included.8000 seconds (2.2 hours) after the burst. We note that at 8000 s, as at 10 hours, the XRF afterglowstend to have smaller decay indices than XRRs and GRBs.9.2.4 2 – 10 keV FluxFigure 9.14 shows the distribution of 2 – 10 keV x-ray afterglow flux values measured 105seconds (a little over a day) after the burst for GRBs, XRRs, and XRFs. Also shown are valuesfrom BeppoSAX presented by Frontera [27]. The K-S test gives a probability of 0.30 that the Swiftvalues and the BeppoSAX values are drawn from the same distribution. There is a hint that theXRFs in our sample may tend to have slightly lower fluxes than the GRBs and XRRs.9.2.5 2 – 10 keV Isotropic LuminosityFigure 9.15 shows the distribution of 2 – 10 keV isotropic-equivalent luminosity Liso at atime 10 hours after the burst trigger for GRBs, XRRs, and XRFs. Also shown are values calculated220Figure 9.14: The distribution of 2 – 10 keV flux values from Swift GRB, XRR, and XRF after-glows measured 105 seconds after the burst, along with corresponding values from BeppoSAXbursts [27]. In two cases (namely, XRR 050219B and GRB 041223), the XRT observationsdid not extend out to 105 seconds, so those bursts were excluded from the plot.and presented by Berger and Kulkarni [10]. Using the K-S test to compare values from Berger’spaper with the those from our sample we obtain a probability of 0.058 that the two are consistent.Thus, the two distributions differ significantly. There is a very definite trend evident in this plot,with XRFs tending to have distinctly lower Liso values than XRRs, which in turn have lower valuesthan GRBs.Figure 9.16 shows a similar distribution to that of Figure 9.15. In this case, the luminosityis calculated for a time of 5 hours× (z+ 1) after the trigger for each burst. These luminosities aretherefore all measured at the same time in the source frame—namely, 5 hours. This may be a morevalid comparison of isotropic luminosities. The distinct trend of lower luminosities for XRFs andhigher luminosities for GRBs seen in Figure 9.15 is less prominent here, but it is still somewhatevident.9.2.6 Jet Breaks and Opening AnglesIn Section 2.5.3, we described a type of light curve break that is caused when the relativisticoutflow of the burst slows down enough that the edge of the jet becomes visible. Although distin-guishing between these so-called “jet breaks” and other breaks that result from spectral evolution221Figure 9.15: The distribution of 2 – 10 keV isotropic luminosities Liso from Swift GRB, XRR,and XRF afterglows at a time of 10 hours after the burst, along with corresponding valuesfrom bursts cited by Berger and Kulkarni [10]. In some cases, redshifts were not measuredfor the burst. In these instances, the Yonetoku relation was used to derive an approximateredshift. Such values are designated in the plot with a cross through the box. In some cases,only a lower limit for the redshift could be obtained, in which cases an arrow is placed in thecorresponding box.Figure 9.16: The distribution of 2 – 10 keV isotropic luminosities Liso from Swift GRB, XRR,and XRF afterglows at a time of 5 hours × (z + 1) after the burst. In some cases, redshiftswere not measured for the burst. In these instances, the Yonetoku relation was used to derivean approximate redshift. Such values are designated in the plot with a cross through the box.Also, in some cases, only a lower limit could be obtained, in which cases an arrow is placed inthe corresponding box.222Table 9.3: Constraints on jet break times and opening angles for the XRFs in our sample. Infinding the opening angles, we have followed the example of Ghirlanda et al. [34] and assumedn = 1 cm−3 and ηγ is 0.2. We note that for those bursts in italics, the redshifts used are theinferred redshifts, based on the Yonetoku relation, as described in Section 7.3. Jet breaks seenin past afterglows typically occur at or after about 1 day (105 s), and the decay index typicallychanges from about -1 to about -2 at the break.Burst z Jet Break Opening angle ∆Θ Comments050416A 0.6536 > 3.73 × 106 s > 40.6◦050406 > 3 > 2.81 × 105 s > 8.7◦ later data shows signs of a break,but the error bar is too large tobe sure050714B > 1.8 > 4.3 × 105 s > 10.0◦050824 0.83 3 × 105 s 13.6◦ if this break is a jet break—thedecay index after the break (-1.1)is shallower than is typically ob-served after a jet break, whichargues against this being the jetbreak> 1.22 × 106 s > 23.1◦ otherwise050215B 0.8 > 3.92 × 105 s > 16.7◦is difficult without multi-wavelength measurements, we can place some constraints on the jet breaktimes of the bursts in our sample. We can also calculate the corresponding jet opening angle usingEquation 2.6, if we assume that the bursts are viewed from within the jet. Tables 9.3, 9.4, and 9.5list the constraints we have on the jet break times and corresponding jet angles.9.3 Prompt Emission and Afterglow Emission9.3.1 Ratio of Prompt Emission Fluence and Afterglow Emission FluenceThe ratio of energy emitted during the afterglow to energy emitted during the promptemission is of interest with reference to many burst models. It has typically been presumed thatthe energy of the afterglow is orders of magnitude smaller than the energy of the burst. Swift’searly afterglow observations enable us to test whether this is the case. Table 9.6 gives the ratio ofafterglow fluence from 0.6 – 10 keV photons for the interval over which XRT measurements weremade to prompt emission fluence from photons of all energies. Because we are only able to includephotons in a limited energy band and because the measurements cover a finite time interval, theratios we obtain are only lower limits—the actual ratio could be much higher. Prompt emission223Table 9.4: Constraints on jet break times and opening angles for the XRRs in our sample.For more details, See the description in Table 9.3.Burst z Jet Break Opening angle ∆Θ Comments050315 1.949 1.25 × 105 s 5.0◦ if this is a jet break—the de-cay index after the break (-1.3)is shallower than is typically ob-served after a jet break, whichargues against this being the jetbreak> 8.92 × 105 s > 10.5◦ otherwise050318 1.44 17700 s 3.2◦ if this is a jet break—the decayindex changes from -1.2 to -2.1at the break, which is consistentwith most jet breaks, but it oc-curs at an earlier time than istypical> 4.47 × 105 s > 10.8◦ otherwise050525A 0.606 14700 s 3.3◦ if this is the jet break—the de-cay index changes from -1.2 to-1.7 at the break, which is asmaller change than occurs inmost jet breaks, and the breakoccurs much earlier than is typi-cal, both of which argue againstthis being the jet break> 97800 s > 6.7◦ otherwise050219B 1.0 < 10500 s < 2.3◦ if the jet break is earlier than theobservation—the decay index is -1.0, so this is not likely> 52600 s > 4.2◦ if the jet break is later than theobservation224Table 9.5: Constraints on jet break times and opening angles for the GRBs in our sample.For more details, See the description in Table 9.3.Burst z Jet Break Opening angle ∆Θ Comments050401 2.9 9200 s 1.4◦ if this is the jet break—the de-cay after the break is -1.3, whichis shallower than typical, and thebreak occurs much earlier thanis typical, both of which argueagainst this being the jet break> 7.89 × 105 s > 7.4◦ otherwise050603 2.821 < 37400 s < 2.3◦ if the jet break is earlier than theobservation—the break would beearlier than is typical for jetbreaks, so this is not likely. Thedecay index is -1.6, which doesn’ttell us much.> 1.01 × 106 s > 7.9◦ otherwise050326 2.4 14000 s 1.7◦ if this is the jet break—the de-cay index changes from -1.5 to-1.9 at the break, which is asmaller change than occurs inmost jet breaks, and the breakoccurs much earlier than is typi-cal, both of which argue againstthis being the jet break> 5.08 × 105 s > 6.6◦ otherwise041223 4 < 16800 s < 1.3◦ if the jet break is earlier than theobservation—the decay index is -1.9, which argues for this, but itwould mean a jet break that isearlier than is typical> 28200 s > 1.6◦ if the jet break is later than theobservation225Table 9.6: Ratio of 0.6 – 10 keV afterglow fluence to prompt emission fluence. The beginningand end of the time interval over which the fluence was calculated are given in seconds afterthe BAT trigger. Since only a limited energy range and time interval is available, the ratiospresented here are lower limits—the actual afterglow-to-prompt-emission fluence ratio may bemuch largerBurst Start of Interval (s) End of Interval (s) Ratio lower limitXRF 050416A 90 3.73 × 106 0.445XRF 050406 107 1.4 × 106 0.479XRF 050715B 158 9.42 × 105 0.0336XRF 050824 6092 1.23 × 106 0.349XRF 050215B 5796 5.74 × 105 0.200XRR 050315 87 8.92 × 105 0.0659XRR 050318 3367 4.47 × 105 0.100XRR 050525A 6011 3.10 × 105 0.0115XRR 050219A 10479 1.18 × 106 0.0417GRB 050401 147 7.89 × 105 0.0481GRB 050326 3323 1.61 × 105 0.00882GRB 050603 37430 1.01 × 106 0.00432GRB 041223 16770 28223 0.000241fluence covering all photon energies is calculated using the best fit Band parameters and assumingthe lower and upper spectral indices continue without breaks beyond the measured energy range.It is interesting to note that in the cases of the five bursts for which very early observationswere possible, two of them (XRF 050416A and XRF 050406) had very high afterglow-to-prompt-emission ratios (≈ 50%), while the other three (XRF 050715B, XRR 050315, and GRB 050401)had considerably lower ratios (3% – 7%). It appears that some physical process must be at workthat sometimes enhances the energy of the afterglow relative to the prompt emission and at othertimes suppresses it.9.3.2 Prompt Emission Eiso and Afterglow Emission LisoWe observed in the last section that XRFs tend to have lower isotropic luminosities thanXRRs and GRBs. Since this is also true of the prompt emission’s isotropic energy, we examinehere whether a correlation exists between Eiso and Liso. Figure 9.17 shows a scatter plot of theisotropic energy of the prompt emission Eiso and the isotropic luminosity Liso 10 hours after the226Figure 9.17: A scatter plot showing the correlation of the isotropic energy of the promptemission Eiso and the isotropic luminosity of the afterglow emission 10 hours after the bursttrigger. Bursts for which approximate redshifts were derived using the Yonetoku relation aremarked with circles. A definite correlation is evident in the plot. The solid line is the best linearfit to the distribution and is given by Eiso/(1052 ergs) = (3.019±0.017)× [Liso/(1045 ergs/s)]p,where p = 0.4672 ± 0.0061.trigger for the 13 bursts described in Chapter 8 for which afterglow measurements were made. Acorrelation is evident, with a linear correlation coefficient between log(Eiso) and log(Liso) of 0.722.The probability of such a correlation occurring by chance for this sample size (10 bursts) is 3.3%.If we exclude the three bursts for which an estimated redshift was used, the correlation coefficientbecomes 0.756, with a 6.5% probability of a chance occurrence.Figure 9.18 shows a similar scatter plot, but this time Eiso is plotted against the isotropicluminosity Liso at 5 hours× (z + 1) after the burst. The correlation is much weaker with a linearcorrelation coefficient between log(Eiso) and log(Liso) of 0.533. The probability of such a correlationoccurring by chance for this sample size (10 bursts) is 20.4%—relatively high. If we exclude thethree bursts for which an estimated redshift was used, the correlation actually improves slightly,with a coefficient of 0.627 and a probability of chance coincidence of 17.5%.As we noted in section 2.5.3, Eiso (and Liso) are functions of the jet angle, with the actual227Figure 9.18: A scatter plot showing the correlation of the isotropic energy of the promptemission Eiso and the isotropic luminosity of the afterglow emission 5 hours after the bursttrigger in the source frame (5 hours× (z +1) in the observer frame). Bursts for which approx-imate redshifts were derived using the Yonetoku relation are marked with circles. A weakercorrelation is evident in this plot than is seen in Figure 9.17. The solid line is the best linearfit to the distribution and is given by Eiso/(1052 ergs) = (2.473±0.017)× [Liso/(1045 ergs/s)]p,where p = 0.2993 ± 0.0061.228energy and luminosity given byEγ = (1 − cos(θ))Eiso (9.1)Lx = (1 − cos(θ))Liso. (9.2)Nardini et al. [59] noted a correlation similar to that shown in Figure 9.17 by studying a set ofbursts detected by other missions. He found that when Eiso and Liso were corrected for jet angle,the correlation disappeared. This indicates that the correlation is due entirely to the dependenceof the two quantities on jet angle.229Chapter 10Discussion10.1 The Continuity of XRFs, XRRs, and GRBsThe results of the analysis described in the last chapter strengthen the case that x-rayflashes and long-duration gamma-ray bursts are not separate and distinct phenomena, but insteadare simply ranges along a single continuum describing some sort of broader phenomenon. AsFigure 9.2 illustrates, XRFs, XRRs and GRBs form a continuum in peak energies Eobspeak, withXRF Eobspeak values tending to be lower than those of XRRs, which in turn are lower than thoseof GRBs. We have also seen that this same spread is evident in the peak energy in the frameof the burst progenitor Esrcpeak and that XRFs, XRRs, and GRBs all obey the same Esrcpeak – Eisocorrelation (see Figure 9.6). Further evidence of the continuous nature of these phenomena comesfrom the continuity in the fluences of XRFs, XRRs, and GRBs, with XRFs tending to manifestlower fluences than XRRs, which tend to have lower fluences than GRBs. This is illustrated bythe correlation between fluences and Eobspeak shown in Figure 9.5. Sakamoto et al. noted a similartrend in XRFs and GRBs detected by HETE-2 [70].10.2 Differences and similarities between the afterglows of XRFs and GRBsAs we examined the x-ray afterglow properties of XRFs, XRRs and GRBs in the last chapter,we noted that their spectral indices showed no strong correlation to indicate that the spectra ofXRF afterglows are distinctly different from those of XRRs or GRBs. We did, however, note adefinite distinction in the shape of the afterglow light curves. Table 10.1 gives a summary of thebreak times and decay indices of the light curves we have analyzed.We found that the gamma-ray bursts in our sample tend to have afterglows with shallowdecay indices (ranging in our sample from −0.5 to −1.6) at early times followed by steeper indices230Table 10.1: The decay indices and break times of the x-ray afterglows described in Chapter8, as well as the times of the earliest and latest observations by the XRT. Values for the 5XRFs in our sample are included first, followed by those of the XRRs in our sample. Lastly,we list the values for the 4 GRBs in our sample.Earliest 1st 1st 2ndObservation Decay Index Break Decay Index050416A 90 s −0.672± 0.026 8400 ± 3500 s −0.905 ± 0.034050406 107 s −2.01± 0.22 3900 ± 2800 s −0.50 ± 0.29050714B 158 s −6.6064± 0.0056 225 ± 11 s −0.886 ± 0.060050824 6092 s −0.491± 0.046 (3.1 ± 1.6) × 105 s −1.11 ± 0.32050215B 5796 s −0.998± 0.084 - -050315 84 s −3.210± 0.039 (758 – 5064 s) −0.653 ± 0.024050318 3281 s −1.172± 0.048 (1.77 ± 0.22)× 104 s −2.08 ± 0.15050525A 5858 s −1.21± 0.14 (1.47 ± 0.50)× 104 s −1.678 ± 0.096050219B 5394 s −1.027± 0.092 - -050401 132 s −0.507± 0.030 (9.1 ± 1.0) × 103 s −1.336 ± 0.050050326 3276 s −1.55± 0.21 (1.41 ± 0.99)× 104 s −1.92 ± 0.11050603 34048 s −1.599± 0.082 - -041223 16665 s −1.93± 0.28 - -2nd 3rd LatestBreak Decay Index Observation050416A - - 3.73 × 106 s050406 - - 1.4 × 106 s050714B - - 9.42 × 105 s050824 - - 1.23 × 106 s050215B - - 5.74 × 105 s050315 1.25 ± 0.13 × 105 s −1.309 ± 0.067 8.98 × 105 s050318 - - 8.32 × 105 s050525A - - 4.52 × 105 s050219B - - 2.30 × 106 s050401 - - 1.01 × 106 s050326 - - 5.31 × 105 s050603 - - 1.79 × 106 s041223 - - 28582 s231(ranging in our sample from −1.3 to −1.9) at later times, and that the breaks between these twoindices occur at about 104 seconds (see Figure 10.1). This pattern is evident in the compositelight curve plot in Figure 9.10. For example, the afterglow of GRB 050401 had a decay index of−0.5 until about 9000 s after the burst, after which its decay index changed to about −1.3, whichcontinued until it faded below the XRT’s detection threshold. The GRB 050326 x-ray afterglowhad an early index of −1.5, which steepened to −1.9 about 14 ks after the burst. The other twoGRBs in our sample (GRB 050603 and GRB 041223) likewise showed steep decay indices at latetimes. No breaks were observed for 050603 or 041223, but observations began rather late, so it isvery plausible that the early shallow decay index was missed.X-ray flash afterglows, on the other hand, seem to follow a different pattern. They oftenhave a very steep index early on, followed by a fairly early break, after which the index becomesmuch more shallow for the duration of the observation (see Figure 10.1). For two of the bursts inour sample (XRF 050824 and XRF 050215B), the early steep index was not observed, but in bothcases, the observation began relatively late, so the early steep indices may have been missed. Inone case (XRF 050824), the late shallow index is followed by a later steeper decay index, as wasobserved in the GRBs in our sample. However, the break preceding this steeper index occurredmuch later than the breaks observed in the GRB sample— 3.1×105 s after the burst. It is possiblethat such late breaks occurred in the other XRFs in our sample as well, but that the light curvehad faded below the sensitivity threshold of the XRT by that time. The XRF 050416A afterglowlight curve is the notable exception to this pattern. No steep index was observed at an early time,the break between shallow index and steep index was very early (at 8500 s), and the later indexwas relatively shallow (−0.9).The afterglows of the X-ray-rich gamma-ray bursts (XRRs) in our sample were split betweenthese two behaviors, with some manifesting a pattern like the XRF sample and others a patternlike the GRB sample. The afterglow of XRR 050315 looks like an x-ray flash afterglow, with asteep decay of index −3.2 followed by a shallow index of −0.6, followed by a steep index of −1.3.The break between the latter two indices occurred at 1.2×105 s, much later than the breaks in the232 Figure 10.1: The pattern of XRF and GRB afterglow light curves. GRB afterglow lightcurves tend to have a shallow index followed by a steep index, with a break between the twooccurring at about 104 to 2×104 seconds after the burst. XRF afterglows, on the other hand,tend to have a very steep early index followed by a shallower index. The shallow index, inthe case of one burst in our sample, is followed by another steeper index at a very late time(3 × 105 seconds). It may be that this steepening is present in all XRF afterglows at latetimes, but that the afterglows in our sample had become too dim by that point to allow us todetect the steepening.GRB sample. The XRR 050219B afterglow was a single power law with an index of −1.0 extendingfrom the beginning of the observation at 5394 s to the end of the observation at 2.3 × 106 s. Thisappears to fit the pattern of the XRF afterglows, with the observation beginning after the earlysteep index. The other two bursts in our sample (XRR 050318 and XRR 050525A) have afterglowsresembling those of GRBs, with an early shallow index (−1.2 in both cases) followed by a steeperindex (−2.1 and −1.7 respectively). The breaks occur at 1.8 × 104 s and 1.5× 104 s, respectively.It is possible that these two patterns form a continuum, with the break between shallowindex and steep index occurring at later times for XRFs (sometimes after the afterglow has faded233below our detection threshold) and at earlier times for GRBs. The very early steep index observedin two of the XRFs and one of the XRRs in our sample may be a special phenomenon that isunique to some bursts and which may be preferential to x-ray flashes.Another distinction we observe is that XRF afterglows have systematically lower fluxes thanGRB afterglows, particularly from about 100 seconds to about 3 × 104 seconds after the burst.This can be seen in the composite plot of the GRB, XRR, and XRF afterglow light curves inFigure 9.7. The plot of 2 – 10 keV flux at 105 seconds after the burst shown in Figure 9.14 alsoshows a slightly systematic tendency toward lower fluxes for XRFs.10.3 Implications for the off-jet modelIn Section 4.4.1, we described a model that suggests that x-ray flashes are intrinsically thesame as gamma-ray bursts, except that they are viewed from outside the relativistic outflow jet(see Figure 4.4). This model predicts an afterglow light curve that rises until it reaches a peakbefore diminishing again. Swift is the first mission capable of observing afterglows quickly enoughto fully test this prediction. The XRF afterglows described in Chapter 8 show a diminishing lightcurve throughout the observation, which in some cases began as early as 90 seconds after the burstand in all cases began by 6100 seconds after the burst. This means that if a peak occurred, it musthave appeared before the beginning of the observation.Equation 4.21 gives a relationship between the peak in the light curve tp, the time tj,0 ofthe jet break that would have been observed if the afterglow were viewed from within the jet, theviewing angle θv, and the half-opening angle of the jet ∆θ:tp =[5 + 2 ln(θv∆θ− 1)](θv∆θ− 1)2tj,0. (10.1)This equation is valid when θv & 2∆θ. If we divide both sides of Equation 10.1 by tj,0, it takesthe form:tptj,0=[5 + 2 ln(θv∆θ− 1)](θv∆θ− 1)2. (10.2)The GRB and XRF afterglow observations described in Chapter 8 allow us to place constraints on234the quantities in this equation. We note that breaks were observed in two of the GRB afterglowsin our sample: at 9000 seconds in the case of GRB 050401 and at 14000 seconds in the case ofGRB 050326. These may be jet breaks resulting from the relativistic outflow slowing to a pointthat the edge of the jet becomes observable, or they may be breaks due to spectral evolution as theafterglow progresses. If they are not jet breaks, then the jet breaks must occur at some time later,after the light curve faded below the XRT sensitivity threshold. In either case, we have an upperlimit on typical GRB jet breaks: tj > 9000 seconds. Jet breaks noted by others (for instance,Frail et al. [26]) are typically on the order of a day (105 s) or later after the burst. We also havea lower limit on tp from our XRF afterglow observations. In all cases, observations began earlierthan 6100 seconds after the burst, so we may say that tp < 6100 seconds for all of the XRFs inour sample (in some cases, much less). Consequently, the quantity tp/tj,0 in Equation 10.2 mustbe less than 6100/9000 = 0.68. Now,[5 + 2 ln(θv∆θ− 1)](θv∆θ− 1)2≥ 5 (10.3)for all possible choices of θv & 2∆θ. Thus, based on the x-ray flash afterglow observations presentedhere, Equation 10.1 cannot be satisfied.There are indications that Equation 10.1 may not be valid in cases of moderately lowLorentz factors (Γ . 100) and relative viewing angles (θv − ∆θ . 0.1 rad (6◦)). Yamazaki etal. [82] investigated a situation in which Γ = 100 and θv −∆θ = 0.05 rad and noted that the off-jetmodel predicts a peak before the jet break time—at around 40000 seconds—in the UV photonenergy range (see Figure 10.2). The position of the peak in the light curve should be independentof photon energy, so we would expect the same peak position at x-ray energies. We note that theseconditions (moderately low Γ and low θv −∆θ) may hold for at least some of the x-ray flashes weobserved. Equation 4.19 gives us a constraint on Γ and θv − ∆θ:Esrcpeak ≈Esrcpeak,in jet1 + Γ2(θv − ∆θ)2. (10.4)In the case of XRF 050416A for which Esrcpeak = 27.6 keV, if we assume Esrcpeak,in jet = 300 keV andΓ = 100, then θv − ∆θ = 0.03 rad (1.8◦), in which case we might expect a peak for XRF 050416A235Figure 10.2: The light curves of an x-ray flash afterglow in the UV band are shown at variousviewing angles, as predicted by the off-jet model. For these plots, the parameters are fixed as∆θ = 0.1 rad, Γ = 100, n = 1 cm−3, p = 2.25, E = 2 × 1054 ergs, εe = 0.1, εB = 0.01, andD = 1 Gpc [82].at a relatively early time—perhaps at around 104 seconds. Such a peak is not observed in any ofour measurements.Another indication of the prominence of the early afterglow emission is the afterglow-to-prompt-emission fluence ratios given in Table 9.6. These ratios are at least on the order of 50%in some instances. This is hard to account for in the off-jet model. This, along with the lack ofobservation of a peak in the afterglow, suggests that the off-jet model may need to be re-examinedor re-evaluated.10.4 ConclusionWe have seen that the XRFs observed by Swift form a continuum with the GRBs observedby Swift and by other missions, having systematically lower fluences and lower Eobspeak and Esrcpeak236values than GRBs. The bursts observed by Swift also confirm the Eiso – Esrcpeak correlation firstnoted by Amati et al. [1]We have also noted that the light curves of x-ray flashes follow a different “template” thanthose of gamma-ray bursts, with a steep decay index early in the afterglow, followed by a shallowindex that extends to very late times. Gamma-ray burst afterglows, on the other hand, do notshow this early steep decay and show a break to steeper indices at much earlier times than do XRFafterglows. XRF afterglows also have significantly lower fluxes than GRB afterglows.Perhaps the most significant finding in these studies is the lack any observed peak in theafterglow light curves of x-ray flashes. This contradicts predictions made by the off-jet and struc-tured jet models of x-ray flashes, indicating that the differences between gamma-ray bursts andx-ray flashes may be the result of intrinsic properties rather than solely the result of viewing ge-ometry. The sample of x-ray flashes observed to date is still very small, and as Swift continues itsmission, this data set will grow, enabling us to confirm and study these aspects more fully.237Appendix AMask-Weighted Count RatesThis appendix describes a procedure that is somewhat analogous to the one used to findbright sources in the sky. It is a technique called “mask-weighting” that enables us to find thecount rate resulting from photons from a given source, with counts from all other sources removed.Consider a stream of monochromatic photons incident on the array. Imagine that there are Sof these photons passing through each cm2 of area each second. Whenever a stream of gamma-rayphotons passes through some material, some of them are absorbed or scattered in other directions.Typically, the less energy the photon has, the bigger the chance it will be absorbed or scattered.This means that of these S photons/cm2/s, a certain fraction ftrans will be transmitted throughthe structure that holds the lead tiles in place, the grid of wire that maintains the high voltagein the detectors, and the epoxy holding the grid to the detectors. The number of photons/cm2/sarriving at a detector that is fully illuminated through the mask is thenS · ftrans. (A.1)Now, suppose a lead tile is partially obstructing the source, so that a fraction fillum of the detectoris illuminated and a fraction 1−fillum is obstructed by the lead tile. The number of photons/cm2/sarriving at the detector is thenS · ftrans · {fillum + [1 − fillum] · fPb} , (A.2)where fPb is the fraction of photons transmitted through lead (fPb = 0 for low energy photonsand fPb → 1 for high energy photons.We will not introduce a new quantity called the “mask-weighting factor”, w, defined asw ≡ 2fillum − CK, (A.3)238where C is chosen so that w satisfies the equation:∑detectorsw =∑detectors(2fillum − CK)⇒ 2∑detectors−C ·Ndetectors = 0 (A.4)⇒ C = 2Ndetectors∑detectorsfillumIt turns out that for a situation like ours, in which equal numbers of detectors are masked andilluminated, C ≈ 1. K is a normalization factor that may be chosen in a number of differentways, depending on how we desire the end result to be normalized. For now, we’ll hold off onchoosing what value to give it. Note that, if K were chosen to be 1, w would equal 2 fillum − 1, sothat detectors that are fully-illuminated would receive a mask weight of +1 and detectors that arefully-masked would receive a mask weight of -1. All other detectors would receive a mask weightsomewhere between -1 and +1, depending on the degree of illumination through the mask (seeFigure 5.3).Substituting equation A.3 into equation A.2 givesS · ftrans ·{Kw + C2+[1 − Kw + C2]· fPb}. (A.5)It is common practice to define the “effective area” of a detector as the ratio of the detectedcounts/s to the number of incident photons/cm2/s. This ratio has units of area (commonly cm2),and another way to think of it is that it is the area of the detector multiplied by the “quantumefficiency” of the detector (or the probability that a photon will be detected by the detector) forthat particular incident photon direction. Now, photons with more energy have a greater chanceof passing through the detector undetected Likewise, a flux of photons coming at the detector“straight on” will “see” a larger cross-sectional area of the detector than a flux of photons comingin at an angle. For these reasons, the effective area is a function of energy and also of angle. Inthe case we are considering, then, the counts/s that the detector would record would be equal tothe flux of photons incident on it multiplied by its effective area:S · ftrans ·{Kw + C2+[1 − Kw + C2]· fPb}·Aeff (A.6)239(where Aeff is the effective area of the detector). Now, in addition to those photons that happento be coming from the source we are interested in, the detector is also recording “background”counts. These are counts that result from photons from other sources, from photons scatteredinto the detector from the shield or other nearby material or that simply manage to pass throughthe shield, and from stray cosmic protons, electrons, or other particles that deposit energy in thedetectors. We will call the count rate from these sources B. Adding this to the total count ratemeasured by the detector, we obtainRd = S · ftrans ·{Kw + C2+[1 − Kw + C2]· fPb}·Aeff +B. (A.7)Now, to generate a “mask-weighted” count rate for the entire array, we add the count ratesof all the individual detectors together, weighted by their mask-weighting factors:R =∑detectorsRd · wd=∑detectors[S · ftrans ·{Kwd + C2+[1 − Kwd + C2]· fPb}·Aeff,d +Bd]· wd, (A.8)where the subscript d indicates that the variable applies to a particular detector.If we assume that both Aeff and B are the same for every detector, then this equation240simplifies quite nicely:R ≈ S · ftrans · Aeff ·12∑detectors(Kw2d + Cwd + 2fPbwd − fPbKw2d − CfPbwd)+B ·∑detectorswd= S · ftrans · Aeff ·12K(∑detectorsw2d)+ S · ftrans · Aeff ·12C(∑detectorswd)+ S · ftrans · Aeff ·122fPb(∑detectorswd)− S · ftrans · Aeff ·12fPbK(∑detectorsw2d)− S · ftrans · Aeff ·12CfPb(∑detectorswd)+B ·(∑detectorswd)= S · ftrans · (1 − fPb) · Aeff ·12K(∑detectorsw2d)(A.9)since by definition∑wd = 0 (because of how the constant C was chosen). Note that all of theterms that include the background counts B have cancelled out.Parenthetically,∑w2d is approximately equal to a constant multiplied by the number ofdetectors in the sum. This constant, which depends only upon the distance between the sourceand the array, is approximately 0.54 for a source at infinity.By default, K is chosen so that12K∑detectorsw2d =1Cangle, (A.10)where Cangle is a geometric correction factor. It is intended to account for effect of photons beingincident on the edge of the detectors and on the different cross sectional area of the detector thatis illuminated when the photons are incident at different angles, so thatAeff ≈ Aeff,on−axis · Cangle. (A.11)241In practical terms,Cangle = cos θz+ min[0.15, 0.05 ·∣∣∣zsxs∣∣∣]· cos θx+ min[0.15, 0.05 ·∣∣∣zsys∣∣∣]· cos θy, (A.12)where θx, θy, and θz are the angles between the line joining the source and the detector, and thex, y, and z axes, respectively, and xs, ys, and zs are the coordinates (relative to the center ofthe array) of the source. This factor has been defined in such a way that it may be applied tocalibration sources near the detector and to astrophysical sources at infinity, in which case theratios in the equation are still finite.With K chosen according to Equation A.10, the mask-weighted count rate becomesS · ftrans · (1 − fPb) · Aeff,on−axis (A.13)Note that at low photon energies, fPb ≈ 0, so the mask-weighted count rate is just the expectedcount rate for a single detector that is illuminated from directly above and is not shadowed by themask. At higher photon energies, fPb approaches 1, so that as the mask becomes more and moretransparent, the mask-weighted count rate decreases to 0.By defining K in this way (in terms of Cangle), the mask-weighted count rate is approxi-mately the same regardless of the incident angle of the photons, so that as the spacecraft slews,the count rate will not be affected by it (much).B does not actually have to be the same for every detector for it to cancel out of theexpression. It just needs to satisfy∑B · w = 0 this is true if there is an equal probability thatsome detector will have a value of B ·w as there is that one will have a value of −B ·w (that is, Band w are “uncorrelated”), and if there are a large number of detectors. Both of these conditionshold for our situation.242Appendix BGlossary of TermsAfterglow The light that comes from the position of a gamma-ray burst long after the gamma-ray burst itself ends. The gamma-ray burst only lasts for a matter of seconds, whereas theafterglow can last for days or even weeks. The afterglow can be seen in x-rays, optical light,and radio wavelengths, all of which have longer wavelengths than gamma rays.Amati relation A relationship between Esrcpeak (the energy of peak power in the source frame)and Eiso (the total amount of gamma-ray energy emitted by the burst if it emitted the sameamount in all directions) that was discovered by Amati and others. [1] They found thatgamma-ray bursts exhibiting a large Esrcpeak also tend to exhibit a large Eiso. In particular,they found that Esrcpeak ∼ E0.52±0.06iso .BAT (“Burst Alert Telescope”) One of the three telescopes aboard the Swift spacecraft. TheBAT, which is sensitive to gamma rays, detects the gamma-ray bursts and x-ray flashes andrecords their spectra and light curves. It also determines their position in the sky and sendsthat position to the spacecraft so that Swift can slew the other two telescopes to point atthe afterglow.BATSE (“Burst and Transient Source Experiment”) An instrument aboard the ComptonGamma-Ray Observatory spacecraft launched in 1991. BATSE’s primary purpose was todetermine the distribution of gamma-ray bursts in the sky by determining the locations of alarge number of bursts to within a few degrees. During the course of its nine-year mission,BATSE made measurements and found positions of over 2700 gamma-ray bursts.BeppoSAX An Italian-Dutch satellite launched in 1996. It contained instruments that coulddetect and localize gamma-ray bursts to 2-3 arcmin as well as instruments that could measurethe afterglow emission. BeppoSAX was the first satellite to detect an afterglow, which was243a major breakthrough in GRB astronomy. It continued to observe and make contributionsuntil it re-entered the earth’s atmosphere in April 2003.Break A time (or energy) at which the light curve (or spectrum) abruptly changes from one powerlaw index to a different power law index. In a plot where the axes are logarithmic, this lookslike two straight lines with different slopes that meet. The place where they meet is the“break”.Chandra An x-ray telescope launched into orbit in 1999. It is the most sophisticated and sensitivex-ray telescope built to date. As such, it is well-suited to observing x-ray afterglows ofgamma-ray bursts. As of this writing, Chandra is still in operation.Eγ Pronounced “Ee-gamma”. The total energy emitted by a gamma-ray burst.Eiso Pronounced “Ee-eye-so”. The “isotropic equivalent” energy emitted by a gamma-ray burst—that is, the total energy emitted if the amount radiated toward the Earth were radiated inall directions. If the radiation all comes from jets in which the radiation is uniform, thenEγ = Eiso(1 − cos(∆θ)), where ∆θ is the half-angle of the jet.Eobspeak Pronounced “Ee-peak-observed” or just “Ee-peak”. The photon energy at which the great-est amount of energy is radiated, as measured in the reference frame of the telescope (seeFigure 2.6).Esrcpeak Pronounced “Ee-peak-source” or “Ee-peak in the source frame”. The photon energy atwhich the greatest amount of energy is radiated, as measured in the reference frame of theburst. Esrcpeak = Eobspeak(1 + z).Gaussian Distribution A function of the form1σ√2πe−(x−µ)22σ2 . (B.1)This function describes a bell-shaped curve that peaks at x = µ and has a width of ≈ σ.244Ginga A Japanese x-ray astronomy mission that orbited the Earth from 1987 through 1991.Gamma-ray bursts detected by Ginga resembled those detected by BATSE, except that thepeak energies of the bursts extended to lower photon energies.Gamma Ray Photons that have more than about 10 keV of energy each. These are the highestenergy photons in the electromagnetic spectrum.GCN (“Gamma-ray burst Coordinates Network” A system used to transmit informationabout a newly-detected gamma-ray burst to scientists and instruments all over the world inseconds. This enables others to quickly begin observing the burst and the afterglow. GCNcirculars provide information about specific observations. Throughout this work, sourcesthat are cited as “GCN” followed by a number refer to these circulars, and can be found athttp://gcn.gsfc.nasa.gov/gcn3 archive.html.GRB (“Gamma-Ray Burst”) A burst of gamma-rays originating from a distant galaxy thatlasts anywhere from a few milliseconds to a few minutes and then disappears, never toreappear in the same place again.HETE-2 (“High Energy and Transient Explorer”) The first satellite dedicated to observ-ing GRBs. It was launched in October 2000 and, as of this writing, is still in operation.keV (“kiloelectron-volt”) An electron-volt (eV) is the amount of energy an electron gains bytraveling through a potential difference of 1 volt. It is often used to describe very smallquantities of energy, like the energy levels within atoms. X-ray and gamma-ray photonenergies are often given in units of keV (thousands of electron-volts). Sometimes, gamma-ray photon energies are even expressed in MeV (millions of electron-volts) or larger units.Lorentz Factor Γ (“Gamma”) A quantity related to the speed of an object:Γ =1√1 − v2c2, (B.2)where v is the velocity of the object, and c is the speed of light. For an object at rest (v = 0),Γ = 1. As v approaches c, Γ gets larger and larger, approaching infinity when v = c. This245quantity comes up often in special relativity.Order of Magnitude This is just a fancy way of saying “a factor of 10”. If A is “an order ofmagnitude” smaller than B, that means that A is about a factor of 10 smaller than B. If it is“many orders of magnitude” smaller, that means it is many factors of 10 (like, for instance,1000 or 10000) times smaller.Photon A “bundle” (or “quantum”) of light. A photon can be thought of as a tiny, masslessparticle that carries a certain amount of energy.Power law A function of the form y = axk, where k is called the “power law index”. Spectraand light curves often follow a power law.Redshift Light originating from a star that is moving away from us has a longer wavelengthwhen measured on Earth than it had in the star’s own reference frame. This is known as a“redshift”. The redshift (z) of an object is defined asz =∆λλ(B.3)where λ is the wavelength of light measured at the source, and ∆λ is the difference betweenthe wavelength measured on earth and the wavelength measured at the source. By thisdefinition, z = 0 would correspond to a source that is neither moving away from nor towardthe earth. Galaxies farther from us tend to move away from us at a faster speed than galaxiescloser to us. For that reason, redshift is often used as a surrogate measure for distance—objects farther from us have a higher redshift.Spectrum The number of photons emitted (or detected) as a function of energy. “Photon”spectra (the number of photons emitted as a function of energy) are often given in units ofphotons/cm2/s. “Count” spectra (the number of detected photons as a function of energy)are often given in units of counts/s.Trigger The moment when a gamma-ray burst is first detected. It gets its name from the factthat when a telescope first detects a burst, it often initiates a specific set of procedures or246measurements (for instance, when BAT detects a burst, it calculates its position and initiatesa slew). The detection of the burst “triggers” this set of procedures. The time of the afterglowis usually measured relative to the trigger time.UVOT (“Ultra-Violet and Optical Telescope”) One of the three telescopes aboard the Swiftspacecraft. The UVOT makes measurements of the ultra-violet and optical afterglows thatsometimes accompany gamma-ray bursts and x-ray flashes.Vela A series of satellites launched in 1963 to monitor the Earth and the nearby solar system forsigns of violations of the 1963 nuclear test ban treaty between the United States and theSoviet Union. In 1967, these satellites were the first to detect gamma-ray bursts.X-Ray Photons that have between about 0.1 and about 10 keV of energy each (though this energyrange is sometimes defined a little bit differently).XRF (“X-Ray Flash”) An event similar to a gamma-ray burst, but in which most of the emittedenergy is in x-rays (that is, in photons with energies below about 10 keV).XRR (“X-Ray Rich Gamma-Ray Burst”) A gamma-ray burst for which a substantial frac-tion of the emitted energy is in x-rays (that is, in photons with energies below about 10keV).XRT (“X-Ray Telescope”) One of the three telescopes aboard the Swift spacecraft. The XRTmakes measurements of the x-ray afterglows that accompany gamma-ray bursts and x-rayflashes.247BIBLIOGRAPHY[1] L. Amati, F. Frontera, M. Tavani, J. J. M. in ’t Zand, A. Antonelli, E. Costa, M. Feroci,C. Guidorzi, J. Heise, N. Masetti, E. Montanari, L. Nicastro, E. Palazzi, E. Pian, L. Piro,P. Soffitta, “Intrinsic spectra and energetics of BeppoSAX Gamma-Ray Bursts with knownredshifts”, A&A 390, 81 (2002).[2] L. Amati, F. Frontera, J. J. M. in ’t Zand, M. Capalbi, R. Landi, P. Soffitta, L. Vetere, L. A.Antonelli, E. Costa, S. Del Sordo, M. Feroci, C. Guidorzi, J. Heise, N. Masetti, E. Montanari,L. Nicastro, E. Palazzi, L. Piro, “Prompt and afterglow X-ray emission from the X-Ray Flashof 2002 April 27”, A&A 426, 415 (2004).[3] R. L. Aptekar, D. D. Frederiks, S. V. Golenetskii, V. N. Ilynskii, E. P. Mazets, V. N. Panov, Z.J. Sokolova, M. M. Terekhov, L. O. Sheshin, T. L. Cline, D. E. 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