Reading Between the Lines:Modelling the Hydrogen Emission ofT Tauri StarsThomas Josiah Garratt WilsonSubmitted by Thomas J. G. Wilson to the University of Exeter as a thesis for the degree of Doctorof Philosophy in Physics, September, 2022.This thesis is available for Library use on the understanding that it is copyright material and thatno quotation from the thesis may be published without proper acknowledgement.I certify that all material in this thesis which is not my own work has been identified and that nomaterial has previously been submitted and approved for the award of a degree by this or anyother University.Signed: . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Thomas J G. WilsonDate: . . . . . . . . . . . . . . . . .13/09/2022Abstract1st Supervisor: Prof. Sean Matt2nd Supervisor: Prof. Tim HarriesT Tauri stars are young low-mass stellar objects. Studying how the progenitorsof Sun-like stars form is vital to our understanding of stellar evolution and planetaryformation. However, themechanisms bywhich these stars accretematter, powerwinds, andshed angular momentum remains elusive. Too distant and dim to probewith direct imagingor interferometry, the inner disks of T Tauri stars must be studied using spectroscopy.However, the atomic spectra of these stars are complex and highly variable. Radiativetransfer modelling offers a powerful tool for understanding the physical processes thatcreate T Tauri stars’ enigmatic hydrogen emission lines.In this thesis, I use the state-of-the-art radiative transfer code TORUS to exploresynthetic T Tauri spectra. I present a detailed description of the radiative transfer codeand the T Tauri model. Moreover, I develop and test a new polar stellar wind frameworkapplicable to both axisymmetric and non-axisymmetric T Tauri simulations. In addition,I explore the effects of Stark and turbulent broadening. The optical and near-IR spec-troscopy of 29 T Tauri stars is compared with a grid of synthetic line profiles created usingTORUS . The archival T Tauri spectra, obtained with VLT/X-Shooter, provide simulta-neous coverage of many optical and infrared hydrogen lines. The observations exhibitmorphologies similar to those seen in other studies. The synthetic line profile grid wascomputed for the hydrogen transitions of HU, PaV, PaW, and BrW for a fiducial T Taurimodel that included axisymmetric magnetospheric accretion and a polar stellar wind. Fur-thermore, I use the Reipurth classification scheme to study observed and synthetic profilemorphology distributions.I show that the modelled infrared lines are narrower than the observations by≈ 80 kms−1. In other words, the mean width of the synthetic lines is only ≈ 50 percent of the observed mean width. Additionally, the models predict a significantly higherproportion (≈ 90 per cent) of inverse P-Cygni profiles. The radiative transfer modelssuggest that the frequency of P-Cygni profiles depends on the ratio of mass loss to massaccretion rates, and blue-shifted sub-continuum absorption is predicted for mass-lossrates as low as 10−12 " yr−1. Several modifications to the T Tauri models are explored,including the effects of rotation and turbulence on the synthetic hydrogen line profiles.Finally, I present six 3D T Tauri models that include non-axisymmetric accretionflows and wind. For each model, line profiles are calculated at different azimuthal values,allowing me to study the rotational variability of the synthetic spectra. The modelsexhibit patterns of variability, correlation, and morphology, as seen in observations andprevious theoretical studies. It was seen that non-axisymmetric accretion flows couldreduce the frequency of inverse P-Cygni profiles and increase the HW10% of the infraredlines by ≈ 60 kms−1. However, the width of both the infrared lines and HU increasedsimultaneously. Additionally, the 3D models predict trends in the variation of equivalentwidth and profile half-width that could be tested against T Tauri time-series spectroscopy.The radiative transfer code developed in this work and the models presented inthis thesis provide a modern foundation for future synthetic T Tauri line modelling. Themodels were show to successfully reproduced the hydrogen line profile characteristicssimilar to those of T Tauri stars. Moreover, the radiative transfer modelling predictedtrends in variability and profile morphology that observations could verify. However, thisthesis also highlights several problems with the current models and discusses how futurework may solve these obstacles so that radiative transfer modelling can be used effectivelyas an effective diagnostic tool for T Tauri systems.2ContentsList of Figures 6List of Tables 8Declaration 9Acknowledgements 101 Introduction 111.1 Accretion and mass-loss process of T Tauri stars . . . . . . . . . . . . . . 131.2 Spectroscopic analysis of T Tauri stars . . . . . . . . . . . . . . . . . . . 182 Radiative Transfer 222.1 Introduction to TORUS . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Radiative transfer calculations . . . . . . . . . . . . . . . . . . . . . . . 232.2.1 Statistical equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 252.2.2 Synthetic line profiles . . . . . . . . . . . . . . . . . . . . . . . . 272.2.3 Stark broadening . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3 T Tauri model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.1 Accretion funnel . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.1.1 Accretion shock hot-spot . . . . . . . . . . . . . . . . . 352.3.1.2 Accretion temperature . . . . . . . . . . . . . . . . . . 372.3.1.3 Photospheric continuum . . . . . . . . . . . . . . . . . 372.3.1.4 Stellar wind . . . . . . . . . . . . . . . . . . . . . . . . 382.3.1.4.1 Wind density . . . . . . . . . . . . . . . . . . 392.3.1.4.2 Wind velocity . . . . . . . . . . . . . . . . . 402.3.2 3D geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.4 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.4.1 Magnetospheric accretion only . . . . . . . . . . . . . . . . . . . 452.4.2 Magnetosphere and stellar wind . . . . . . . . . . . . . . . . . . 47CONTENTS3 Observations 503.1 X-Shooter instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 X-Shooter data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2.1 Telluric corrections . . . . . . . . . . . . . . . . . . . . . . . . . 533.2.2 Continuum and Gaussian fitting . . . . . . . . . . . . . . . . . . 553.3 Presentation of hydrogen spectra . . . . . . . . . . . . . . . . . . . . . . 563.3.1 Reipurth classification . . . . . . . . . . . . . . . . . . . . . . . 613.3.2 Line profile characteristics . . . . . . . . . . . . . . . . . . . . . 634 A Comparison of Synthetic and Observed Spectra 664.1 Synthetic line profile grid . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2 Comparison of profile features . . . . . . . . . . . . . . . . . . . . . . . 724.2.1 Reipurth classification . . . . . . . . . . . . . . . . . . . . . . . 794.2.2 P-Cygni profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.3 Modified T Tauri model . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3.1 Turbulent line broadening . . . . . . . . . . . . . . . . . . . . . . 844.3.2 Rotational line broadening . . . . . . . . . . . . . . . . . . . . . 864.3.3 Blue-shifted emission . . . . . . . . . . . . . . . . . . . . . . . . 904.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925 Non-Axisymmetric 3D Radiative Transfer 985.1 3D model geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.2 Variability and morphology of 3D non-axisymmetric spectra . . . . . . . 1025.2.1 Quantitative analysis of the variability of time-series spectra . . . 1045.2.1.1 Accretion and temperature comparison . . . . . . . . . 1115.2.1.2 Geometry comparison . . . . . . . . . . . . . . . . . . 1125.2.1.3 Mass-loss rate comparison . . . . . . . . . . . . . . . . 1125.2.2 Correlation analysis of time-series spectra . . . . . . . . . . . . . 1135.3 Inverse P-Cygni profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.4 Variability of the line profile widths . . . . . . . . . . . . . . . . . . . . 1175.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206 Conclusions and Future Outlook 1246.1 Future Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Bibliography 131A Atlas of Line Profiles from the Axisymmetric Accretion Models 1364CONTENTSB Spectral Variability Figures 146C Auto-Correlation Maps 1535List of Figures1.1 Cartoon of a T Tauri star . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Magnetohydrodynamic T Tauri simulation . . . . . . . . . . . . . . . . . 172.1 An overview of the capabilities of TORUS . . . . . . . . . . . . . . . . . 242.2 Ray distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Datacube and spectrum from TORUS . . . . . . . . . . . . . . . . . . . 312.4 A 3D density plot of the T Tauri model . . . . . . . . . . . . . . . . . . . 332.5 T Tauri model diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.6 Temperature, density, and velocity of the magnetosphere . . . . . . . . . 362.7 Kurucz model atmospheres . . . . . . . . . . . . . . . . . . . . . . . . . 392.8 Stellar wind V-law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.9 Stellar wind velocity components . . . . . . . . . . . . . . . . . . . . . . 432.10 Offset magnetic dipole model . . . . . . . . . . . . . . . . . . . . . . . . 442.11 Comparison of radiative transfer models . . . . . . . . . . . . . . . . . . 462.12 Comparison of different wind models . . . . . . . . . . . . . . . . . . . . 483.1 X-Shooter echelle spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 523.2 Full X-Shooter spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3 Continuum and Gaussian Fitting . . . . . . . . . . . . . . . . . . . . . . 573.4 Hydrogen line correlation matrix . . . . . . . . . . . . . . . . . . . . . . 583.5 Spectra of T Tauri stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.6 Reipurth Classification of Observed Stars . . . . . . . . . . . . . . . . . 633.7 Observed profile characteristics . . . . . . . . . . . . . . . . . . . . . . . 644.1 Grid of synthetic models . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Distribution of synthetic line centres . . . . . . . . . . . . . . . . . . . . 724.3 Convolution of synthetic spectra to observed resolution . . . . . . . . . . 734.4 Line profile best fit examples . . . . . . . . . . . . . . . . . . . . . . . . 744.5 Distribution of HW10% for different hydrogen transitions . . . . . . . . . 764.6 FWHM versus HW10% . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.7 Ratio of PaV to BrW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80LIST OF FIGURES4.8 Distribution of synthetic Reipurth types . . . . . . . . . . . . . . . . . . 814.9 Signal-to-noise ratios of the spectra . . . . . . . . . . . . . . . . . . . . . 824.10 The fraction of P-Cygni profiles . . . . . . . . . . . . . . . . . . . . . . . 834.11 The effect of turbulent broadening on the FWHM and HW10%. . . . . . . 854.12 Grid of turbulently broadened line profiles . . . . . . . . . . . . . . . . . 874.13 The effect of rotation on the line profiles . . . . . . . . . . . . . . . . . . 894.14 Blue-side HW10% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.1 A cartoon of the 3D accretion and wind model . . . . . . . . . . . . . . . 1005.2 HU images of the T Tauri model . . . . . . . . . . . . . . . . . . . . . . 1025.3 PaV images of the T Tauri model . . . . . . . . . . . . . . . . . . . . . . 1035.4 3D non-axisymmetric line profiles . . . . . . . . . . . . . . . . . . . . . 1055.5 Variance of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.6 Auto-correlation maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.7 Inverse P-Cygni profiles versus phase . . . . . . . . . . . . . . . . . . . . 1175.8 The HW10% at different phases . . . . . . . . . . . . . . . . . . . . . . 1185.9 The change in HW10% and equivalent width versus phases . . . . . . . . 121A.1 Atlas of the synthetic axisymmetric line profiles . . . . . . . . . . . . . . 137B.1 Variance of the 3D models . . . . . . . . . . . . . . . . . . . . . . . . . 147C.1 Auto-correlation maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1537List of Tables3.1 The T Tauri stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2 Stars classified as weak line T Tauri stars . . . . . . . . . . . . . . . . . . 613.3 Observed spectral line characteristics . . . . . . . . . . . . . . . . . . . . 654.1 Parameters of the grid of synthetic models . . . . . . . . . . . . . . . . . 684.2 Best fit model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3 Stellar rotation rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.4 Blue side HW10% differences . . . . . . . . . . . . . . . . . . . . . . . 925.1 3D model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101DeclarationThe work contained in Chapters 2, 3, and 4 incorporates research published in Wilsonet al. (2022). The work was undertaken by myself with contributions from S. Matt, T.Harries, and G. Herczeg. The material in Chapter 5 is yet to be published.AcknowledgementsFirst, I would like to acknowledge my supervisors, Sean Matt and Tim Harries, for theirhelp and guidance in my work. Thank you for the inspiration and knowledge you gaveme. I would also like to thank Gregory Herczeg for his council and valuable input intomy research. Furthermore, I extend my gratitude to the whole AWESoMeStars team andthe members of the Exeter Astrophysics Department for being such a friendly, supportive,and exciting group to work with.The research contained in this thesis was made possible due to funding from theEuropean Research Council (ERC) under the European Union’s Horizon 2020 researchand innovation program (grant agreement No. 682393; AWESoMeStars: Accretion,Winds, and Evolution of Spins and Magnetism of Stars; http://empslocal.ex.ac.uk/AWESoMeStars). My research has made use of the services of the ESO ScienceArchive Facility, including observations collected at the European Southern Observatoryunder ESO programme 084.C-1095(A). The simulations presented in this thesis made useof the University of Exeter High-Performance Computing (HPC) facility. I also madeuse of the SIMBAD database, operated at CDS, Strasbourg, France (Wenger et al. 2000).Furthermore, I would like to thank the developers of the Python packages Pandas (Rebacket al. 2020) and Matplotlib (Hunter 2007) that I used extensively in this thesis for dataanalysis and presentation.I owe tremendous gratitude to my parents. I could not have undertaken this journeywithout the incredible education, perspective, and thirst for knowledge you have given mesince my earliest years. Thank you for the opportunities and wisdom you have gifted me.Lastly, I’m immensely grateful to my partner, Diva Drissell. Not only for helpingme proofread each chapter and teaching me how to use commas but also for always beingthere, supporting me and being patient in times of great stress.Chapter 1Introduction“Dwell on the beauty of life. Watch the stars,and see yourself running with them.”Marcus Aurelius, ‘Meditations’.Since our first ancestors, just under half of our existence has been spent living in thedark. The blackness of the sky and the sparks of light that appear each night to bejewel ithave been a constant companion to all life on Earth. The stars of the night sky have playedan intrinsic role in shaping our understanding of the world in which we exist. Beforewe knew what stars were, they provoked our imagination, inspiring our songs, stories,and beliefs. Early cultures identified stars with their gods and used their movements asprophecies of the future. Moreover, since ancient times, humans have been using thepatterns and motion of stars to track the passage of time and to navigate across land andthe ocean. The study of stars is one of the oldest sciences, which has its beginnings atthe root of many cultures. Astronomy, in its modern form, is far removed from its ancientorigins but still plays an integral role in our society. Not only does astronomy inspireus to look outwards, but it pushes the boundaries of science and technology. It givesus an insight into the immensity of the universe, entwining our view of Earth with thevastness and beauty of space. As astronomy has progressed, we have discovered manyextraordinary phenomena; however, millennia on, the study of stars still provides us withintriguing discoveries that reveal secrets to our genesis.Although stars are the most prominent and numerous objects observable in our sky,investigating them is a non-trivial problem. The difficulties arise from a disparity of scale.With terrestrial objects, we can generally poke things with a stick, look inside them, orwatch them change over time. However, stars can live for billions of years, and the closeststar to Earth, the Sun, is more than 150 million kilometres away. Therefore, we cannotobserve a star over its entire lifetime to see how it changes and until very recently, whenCHAPTER 1. INTRODUCTIONthe Parker Solar Probe dived through Sun’s corona (Kasper et al. 2021), we had neverpoked one with a stick.To overcome these challenges, astronomers analyse the star’s movement, spectra,and neighbours to learn about their internal workings, and they use large samples of starsto piece together the different phases of their lives. For instance, the Sun is already over4.5 billion years old. It is known to be in its main-sequence, a period characterised bythe fusion of hydrogen into helium in the core, a phase that will last for approximately 10billion years. In contrast, the pre-main-sequence evolution of Sun-like stars is brief, onthe order of 10 million years, yet encompasses a broad range of physics that determinestheir future evolution. The time when these stars are free from an enshrouding cloud ofdebris and before they reach the main-sequence is only a few million years, making themrare to observe and, therefore, difficult to study. However, determining the mechanismsthat govern the formation of young stars is pivotal to our understanding of the evolution ofstars, their disks, and, ultimately, the formation of life-bearing planets such as the Earth.The progenitors of Sun analogues are known as T Tauri stars and were originallydefined by Joy (1945). T Tauri stars are low-mass . 2 M, young (∼ 1 to 10 Myrs)pre-main-sequence stars that have strong magnetic fields (Johns-Krull 2007) on the orderof a few kG. T Tauri stars are generally divided into two subgroups: classical T Tauri starsand weak line T Tauri stars. Classical T Tauri stars have an observed excess of infraredand ultraviolet continuum, and they show strong emission lines that can have complexkinematic profiles (Reipurth et al. 1996). Their continuum and line luminosity can be inexcess of the photospheric luminosity and, furthermore, they display an enigmatic patternof variability across the spectral regime (Hartmann et al. 2016). Weak line T Tauri starshave a similar age to classical T Tauri stars but show neither strong emission lines norexcess infrared emission. The strong hydrogen HU line, observed in classical T Tauristars, correlates with the presence of an accretion disk; weak line T Tauri stars are thoughtto be not accreting.For decades, the observed infrared excesses of T Tauri stars were taken as evidencefor the presence of circumstellar disks (Mendoza V. 1968; Strom et al. 1989) and coron-agraphic observations of T Tauri stars confirmed the presence of disk and outflows (e.g.Schneider et al. 2003). Furthermore, interferometric observations show the existence ofa circumstellar disk around T Tauri stars, for example, Simon et al. (2000), and morerecently, the survey by Anthonioz et al. (2015). However, the inner gaseous disk regionsof T Tauri stars, where the dust is sublimated, are small ≈ 0.1 AU. Additionally, T Tauristars are relatively faint, for example, in comparison to their larger counterparts, HerbigAe/Be stars. Combined, the size and the brightness of T Tauri stars make interferometricobservations of the inner disks challenging. Whilst modern interferometric observations121.1. ACCRETION AND MASS-LOSS PROCESS OF T TAURI STARScan probe the outer disk regions of T Tauri stars, for example, GW Orionis (Kraus et al.2020), the inner disk regions and the processes of accretion are poorly constrained byobservations.Figure 1.1: A cartoon of a T Tauri Star. The image shows the circumstellar disk, the magneto-spheric accretion, and accretion shock zone. The possible mass outflows from the system are alsomarked. A magnetospheric ejection is indicated to be forming in an inflated magnetic field line.1.1 Accretion and mass-loss process of T Tauri starsThe manner by which accreting matter moves from the disk to the stellar surface providesan interesting problem in T Tauri stars. The original disk-boundary layer model theorisedthat the accretion disk extended down to the star, where the material was rapidly decel-erated, transferring angular momentum to the star (e.g. Hartmann et al. 1994). However,observations of T Tauri stars show that they are generally slow rotators (e.g. Bouvieret al. 2014). Therefore, there must be some mechanism to remove the angular momentumfrom the star, a process that is difficult to reconcile with the disk-boundary layer model.Furthermore, some of the observed spectra exhibit inverse P-Cygni features (e.g. Reipurthet al. 1996; Folha & Emerson 2001; Edwards et al. 1994); red-shifted sub-continuum ab-sorption created by gas moving away from the observer, absorbing emission from beyondit. The boundary layer model cannot explain the broad high-velocity wings of the atomicspectra or the inverse P-Cygni profiles seen in some of the observed spectra.13CHAPTER 1. INTRODUCTIONCamenzind (1990) and Koenigl (1991) proposed that the accretion mechanism issimilar to that of neutron stars, as theorised by Ghosh et al. (1977). In this formalism,known as magnetospheric accretion, the magnetic field truncates the disk at a radius 'mi,where the magnetic and material stresses are of the same order. The accreting materialfree-falls from the disk to the stellar surface along the magnetic field, creating an accretioncolumn or funnel. The in-falling matter thermalises the kinetic energy as it impacts thestellar surface, forming a shock zone or “hot-spot”. The enormous energy release heatsthe gas to ≈ 106 K and dissipates the energy as X-rays and extreme ultraviolet. Theshort-wavelength radiation is quickly reabsorbed in the optically thick accretion gas andre-emitted as longer wavelength radiation. This emission could be responsible for thehigh blue and UV excesses observed from T Tauri stars (e.g. see Muzerolle et al. 1998b;Hartmann et al. 2016, and references therein). The presence of an extendedmagnetosphereallows for mechanisms that transport angular momentum away from the star, which canhelp explain the observed slow T Tauri rotation (Bouvier et al. 2014). Furthermore,magnetospheric accretion can explain the inverse P-Cygni profiles. The accreting gasfalling away from the observer at high velocity can absorb photospheric, continuum, andline emission from the star and magnetosphere.Bouvier et al. (2007) firmly established the funnel-flow magnetospheric accretionparadigm as the generally accepted model for accretion in classical T Tauri stars. Blue-shifted sub-continuum absorption (the classical P-Cygni profile) is also observed andinterpreted as evidence of the presence of mass outflows (e.g. Edwards et al. 2006; Kwanet al. 2007). Fig. 1.1 shows a cartoon depicting the standard model of T Tauri accretionand outflow. The figure shows the accreting disk, which sublimates from dust to gas atsome radius. The magnetic fields of the star truncate the inner disk, and the gas free fallsonto the star along the closed field lines. The figure points to three forms of mass outflow:stellar wind from the poles, disk wind, and magnetospheric ejections.The truncation radius of the circumstellar disk is expected to be at the point wherethemagnetic and gas ram pressures roughly equate. For example, Koenigl (1991) theorisedthat the truncation radius 'mi of the disk would be where the magnetic stress is strongenough to remove the angular momentum of the Keplerian flow in a narrow region. Thiscan be quantified as'mi =V`47∗(2"∗)17 ¤"27acc(1.1)where `∗ is the magnetic dipole moment of the star, "∗ and ¤"acc are the mass of thestar and the mass accretion rate of the disk, respectively. The parameter V . 1 is aproportionality coefficient that encapsulates effects such as the magnetic field geometry,141.1. ACCRETION AND MASS-LOSS PROCESS OF T TAURI STARSenergy, and coupling to the disk. The material must be coupled to the magnetic field totruncate the disk. For instance, if the inner disk . 0.1 AU has a temperature of & 103 K,then collisional ionisation should be sufficient to create a field-gas coupling (Koenigl1991). For fiducial T Tauri values of "∗ = 0.5 M, a stellar radius '∗ = 2 R, and amagnetic field of ≈ 1 – 2 kG, the truncation radius can be estimated to be ≈ 4 – 6 '∗at an accretion rate of ¤"acc = 10−8 M year−1. Typical deduced T Tauri parameters andaccretion rates lead to a truncation radii of ∼ 2–6 '∗ (cf. Muzerolle et al. 2001; Hartmannet al. 2016).The accreting matter will free fall from the truncation radius to the stellar surface,releasing the kinetic energy, which is assumed to radiate away. Calvet & Gullbring (1998)showed that the shock region forms near the photosphere, and the energy could be radiatedaway. Therefore, the excess luminosity from the accretion !acc can be used to approximatethe accretion rate of the star. The accretion rate can be determined from the accretionluminosity using the relationship (Calvet & Gullbring 1998)!acc ="∗ ¤"acc'∗(1 − '∗'mi). (1.2)During their evolution, the T Tauri star accretion rates are observed to be in therange of 10−9 to 10−7 Myr−1 (Muzerolle et al. 2001). Material accreting from the diskwill transfer angular momentum to the star, spinning it up. The spin-up torque on the starcan be quantised as (Matt & Pudritz 2005b)gacc = ¤"acc√"∗'mi . (1.3)Combined with the fact that the T Tauri stars are still contracting, the stars are expectedto spin up to break-up velocities in ∼ 106 yrs (Zanni & Ferreira 2009). However, T Tauristars are generally slow rotators; a significant fraction of the stars have rotation rates ofone to ten days, well below their rotation limit (Bouvier et al. 1993). Therefore, since theaccretion and contraction processes are expected to spin up the T Tauri stars, and theyare not usually observed to rotate near their break-up velocities, there must be an efficientmechanism by which the angular momentum is removed.The process that brakes the rotation of T Tauri stars is still debated. Camenzind(1990) and Koenigl (1991) proposed that star-disk interactions could provide a spin-down torque. They suggested that the closed magnetic field lines threading the disk atradii greater than the corotation radius (where it is rotating slower than the star) exerta torque on the stellar surface, effectively braking its rotation. However, analytical andmagnetohydrodynamic simulations show that the differential velocity between the star15CHAPTER 1. INTRODUCTIONand disk twists the magnetic fields, inflating them and opening the magnetosphere (e.g.Uzdensky et al. 2002; Matt & Pudritz 2005a). This effect greatly reduces the spin-downtorque efficiency of the extended magnetosphere (see Ireland et al. 2021, and referencestherein). Zanni & Ferreira (2009) concluded that for their numerical example, the spin-down torque from the star-disk magnetic connection could only remove 10 per cent of thespin-up torque from the accretion.Another mechanism for transporting angular momentum away from the stars arisesfrom the opening and reconnecting of the magnetic field lines. This effect is caused bythe inflation of magnetic fields induced by the difference in rotation velocities between thestar and disk (e.g. Zanni & Ferreira 2009; Ireland et al. 2021). The mechanism is knownas magnetospheric ejections and can carry both angular momentum and mass away fromthe disk and the star. Magnetospheric ejections can provide a spin-up or spin-down torquedepending on the relative location of the truncation and corotation radii. Zanni & Ferreira(2013) were able to show that for a mass-loss rate of 1–2 per cent of the accretion rate,magnetospheric ejections could balance 20–30 per cent of the accretion torque.A further proposed mechanism to remove angular momentum is through the in-teraction of the magnetic field and a stellar wind (e.g. Matt & Pudritz 2005b, 2008a).Blue-shifted sub-continuum absorption (P-Cygni) features provide observational evidenceof disk and stellar winds from T Tauri stars (e.g. Edwards et al. 2003, 2006). Typical massoutflow rates from T Tauri stars are on the order of ∼ 0.01 ¤"acc (Hartigan et al. 1995), butcan be as high as ∼ 0.2 ¤"acc (Cranmer 2008). A stellar wind coupled to open magneticfield lines can produce a spin-down torque on the stellar surface.As the star rotates, the wind is accelerated by the magnetic field, pushing it torotate with the star, transporting angular momentum to the wind. Matt & Pudritz (2005b)characterised this stellar wind torque asgsw = −^ ¤"swΩ∗'2 (1.4)for a stellar wind mass-loss rate ¤"sw and a stellar rotation rate Ω∗. ^ is a dimensionlessquantity that accounts for the geometry of the wind. The Alfvén radius ' is the distanceat which the wind velocity becomes super-Alfvénic. Beyond ', the magnetic interactionsbetween the wind and the star are reduced. The spin-down torque from a stellar wind canbe orders of magnitude greater than that provided by star-disk interactions (e.g. Matt &Pudritz 2008b).However, Matt & Pudritz (2008b) showed that to balance accretion and wind torque,a mass-loss rate of ∼ 10 per cent of the mass accretion rate is required. A mass-loss ratethis high is difficult to achieve unless the wind is powered by a significant proportion161.1. ACCRETION AND MASS-LOSS PROCESS OF T TAURI STARS(∼ 36 per cent) of the accretion power (Zanni & Ferreira 2009; Matt & Pudritz 2008b).Mass-loss rates of this size have been identified, but it is unclear what proportion of thesewinds originate from the star or disk (Matt & Pudritz 2008b). Furthermore, while thereis a strong observed correlation between the accretion rate and the mass-loss rate, the1.0vK,0 10 20 30 40 50Radius [R ]402002040Radius [R]65432101log 10()RmiFigure 1.2: A logarithmic density colour map of a magnetohydrodynamic T Tauri model createdfrom data produced by Ireland et al. (2021). The magnetic field is represented by the white lines.The red lines surround the area associated with magnetospheric ejections. The green arrowsindicate the material velocity normalised by the Keplerian velocity. The magnified insert showsthe magnetospheric accretion and the truncation radius 'mi = 2.41 '∗.17CHAPTER 1. INTRODUCTIONcause of this connection is debated. A limited amount of research has explored how theaccretion energy could power a stellar wind. Cranmer (2008) and Cranmer (2009) showedthat turbulent magnetic waves could work as a power transfer mechanism but could onlyachieve efficiency on the order of ¤"sw/ ¤"acc ≈ 1 per cent.Fig. 1.2 shows a snapshot of a magnetohydrodynamic T Tauri model, created fromthe simulations of Ireland et al. (2021) and is representative of modern T Tauri magne-tohydrodynamic simulations. The model shows a mass density map of an axisymmetricsimulation of a T Tauri star. The white lines indicate the magnetic field lines, which nearthe star are closed and truncate the circumstellar disk, forming a magnetosphere. Despitethe simulations being initialised with north-south symmetry, the accretion funnel predomi-nantly prefers one hemisphere over the other, creating a non-axisymmetric magnetosphere.During the simulation run-time, the accretion funnel may remain in one hemisphere orswitch as magnetic perturbations disturb it (Ireland et al. 2021, 2022). In Fig. 1.2, redboundary lines indicate the area corresponding to magnetospheric ejections. The figureshows magnetic field lines that are being stretched by the interaction of star and disk untilthey are disconnected and reconnected. The green arrows indicate the relative speed ofthe material in multiples of the star’s Keplerian velocity. Beyond a radius of ∼ 6 '∗, thefield lines threading the disk become open; in other words, they are stretched, effectivelyextending to infinity. A slow disk wind is seen flowing from the disk. Magnetocentrifugalforces may drive such a wind, accelerating matter along the open field lines where theangle between the disk and the field is less than 60◦ (Blandford & Payne 1982). Fig. 1.2also shows a fast thermally driven stellar wind flowing from the poles of the star.1.2 Spectroscopic analysis of T Tauri starsDue to the physical size, distance, and brightness of T Tauri stars, direct and even interfer-ometric observations of the inner disks remain challenging. Therefore, our understandingof the mechanisms of accretion and outflow in these stars must be inferred indirectly fromalternative observations. One such observational metric is the analysis of atomic spectrallines associated with the release of accretion energy. The shock regions produce nearX-ray, ultraviolet, optical, and infrared continuum emission. However, broad optical andinfrared emission lines are most commonly associated with the extended magnetosphericaccretion (Hartmann et al. 2016). The spectral lines contain the kinematic informationrequired to understand these stars’ accretion and outflow processes. However, the ob-served spectra are complex, and disentangling the contributions from different T Tauricomponents is challenging. While extensive magnetohydrodynamic modelling of T Tauristars has given valuable insight into the possible accretion and wind processes, better181.2. SPECTROSCOPIC ANALYSIS OF T TAURI STARSobservational diagnostics are needed to constrain the theory of T Tauri evolution. To dothis, the radiative emission of these stars needs to be modelled to decipher the observedline profiles.Complex radiative transfer models were first employed to study magnetosphericaccretion in T Tauri stars by Hartmann et al. (1994). They used a two-level hydrogenmodel to show that magnetospheric accretion could explain the strong emission lines andblue excess of several T Tauri stars. Muzerolle et al. (1998b) further built on the T Taurimodel of Hartmann et al. (1994) by using a 2D radiative transfer code that performed amultilevel statistical equilibrium calculation for eight hydrogen levels. They concludedthat the BrW line profiles could be reproduced for embedded objects and that the Balmerline flux was consistent with observations. Muzerolle et al. (1998b, 2001) used radiativetransfer simulations to constrain the temperature of the accretion column to the range6000–10000 K and determined that Stark broadening may contribute to the observedbroad HU wings (±500 kms−1). Folha & Emerson (2001) noted that the radiative transfermagnetospheric accretionmodels produced infrared lines (Paschen and Brackett) that werenarrower than their observations by ≈ 100 kms−1.Kurosawa et al. (2006) introduced a disk wind and a polar stellar wind into theirT Tauri radiative transfer simulations. They showed that the observed blue-shifted sub-continuum absorption profiles, interpreted as evidence of the presence of mass outflow(e.g. Edwards et al. 2006; Kwan et al. 2007), could be reproduced by their wind models.Additionally, using wind and magnetospheric accretion models, Kurosawa et al. (2006)were able to produce the seven classical line profile classes defined by Reipurth et al.(1996). Furthermore, Kurosawa et al. (2011) showed that a bipolar stellar wind wouldproduce broad P-Cygni absorption due to its high velocity and narrow P-Cygni absorptionwas the product of a slow disk wind. Both of these forms of P-Cygni profiles are observed(e.g. Edwards et al. 2006); although, it is not clear in which physical paradigms theseoutflow mechanisms are dominant or whether they coexist.Another consideration for magnetospheric accretion is that the magnetic fields ofT Tauri stars are never perfectly aligned with their poles (e.g. McGinnis et al. 2020). Amagnetic obliquity will likely cause the accretion funnel to favour specific longitudes ofthe ascending nodes (e.g. Espaillat et al. 2021), restricting the azimuthal angle of the mag-netosphere. A non-axisymmetric accretion funnel is likely to reduce the frequency of theobserved inverse P-Cygni profiles and cause significant line and continuum variability asthe hot-spots and accretion columns rotate with the star. Symington et al. (2005a) explorednon-axisymmetric radiative transfer models and the rotational phase line variability. Theyshowed that introducing a magnetic dipole offset induced greater line variability than wasobserved. Kurosawa et al. (2005) used similar models to compare synthetic spectra with19CHAPTER 1. INTRODUCTIONtime series PaV spectroscopy from SU Aurigae and were able to reproduce some of theobserved velocity and equivalent width variability. Other studies have used magnetohy-drodynamic models and post-processed them using radiative transfer simulations. Forexample, Kurosawa et al. (2008) and Kurosawa & Romanova (2012) successfully repro-duced many of the commonly observed hydrogen line profiles and predicted similar linevariability. Some studies have used magnetohydrodynamic models of stars with well-constrained stellar parameters to create radiative transfer models to study the observedemission. For instance, Alencar et al. (2012) successfully used a non-axisymmetric accre-tion model based on a magnetohydrodynamic simulation to perform a radiative transfersimulation of V2129 Oph* in HU and HV. Despite some similarities between the modelledemission and the observed spectroscopy, the authors could not reproduce the red-shiftedabsorption accurately.SummaryT Tauri stars present an interesting problem in that their accretion and mass-outflowmechanisms remain elusive even with decades of observations and modelling. Modernradiative transfer simulations offer an opportunity to unlock the origins of the complexspectra. However, despite the relative success of radiative transfer studies, the T Taurimodels still have multiple free parameters and chronic degeneracy. It is unclear how todisentangle the effects of geometry, accretion, and mass loss on the observed spectra.Consequently, although the evidence supports the magnetospheric paradigm, radiativetransfer models need development. Therefore, while radiative transfer simulations canprovide valuable insight into the spectra of T Tauri stars, there is still room for significantimprovement of the models and their parameterisation.The primary problems facing radiative transfer simulations are as follows. Theconstraints on the magnetospheric temperature and accretion rates are still vague becauseof a lack of a comprehensive theory for the heating of the accreting gas. Both stellarand disk winds have been modelled, but the dominant geometry, mass-loss rate, andtemperature are still argued. Folha & Emerson (2001) noted that the synthetic infraredlines produced were too narrow. Furthermore, 3D radiative transfer simulations based onmagnetohydrodynamic T Tauri simulations struggle to reproduce the observed red-shiftedatomic line components. In this thesis, I use and develop a new generation of T Taurimodels for the radiative transfer code TORUS , incorporating both polar stellar winds andan accelerated Monte Carlo processes. I explore a grid of synthetic hydrogen transitionsusing these models and compare them to the same atomic spectra from 29 T Tauri stars.The comparison aims to determine whether a range of accretion rates, mass-loss rates,201.2. SPECTROSCOPIC ANALYSIS OF T TAURI STARSand temperatures based on observed parameters can reproduce the ensemble of observedT Tauri spectral characteristics.This thesis aims to achieve two primary research goals. Firstly, I will scrutinisethe radiative transfer T Tauri models presented in this thesis and consider possible mod-ifications to improve the diagnostic capabilities via a comparison to observed spectra.Secondly, I will extend the 2D dimensional models to 3D and attempt to evaluate hownon-axisymmetric accretion influences the spectral characteristics of the optical and in-frared hydrogen lines. This research was undertaken to further the current theories on theevolution of Sun-like stars by improving T Tauri radiative transfer models.In Chapter 2, the radiative transfer code TORUS (Harries et al. 2019) is presentedand the new developments of the T Tauri model are described. Next, in Chapter 3, I reduceand show a set of hydrogen spectra from 29 T Tauri stars. Then, in Chapter 4, I examinethe observed spectra in comparison to a grid of synthetic hydrogen line profiles producedusing TORUS, focusing on the profile morphologies. Finally, in Chapter 5, I explore3D non-axisymmetric T Tauri models and consider the rotational phase variability of thesynthetic spectra.21Chapter 2Radiative TransferRadiative transfer simulations attempt to reproduce the flow of electromagnetic energythrough and around different media, the characterisation of which can reveal the com-plex physical process at play in the interaction of radiation and matter. Furthermore, itcan be used to predict the observational characteristics of different theoretical physicalphenomena. At the heart of these simulations is the radiative transfer equation,dadga= (a − a (2.1)where ga is the optical depth of the medium at frequency a. The specific intensity a isthe radiative energy flux per solid angle per frequency bandwidth, and (a is the sourcefunction, the ratio of the local absorption and emission coefficients. For all but the mostsimple systems, modelling the passage of light through a medium is a complex problemwhich has to be solved numerically (see Steinacker et al. 2013). The complexity of thesystem arises because the radiation field a is coupled to the state of the medium (a bymicrophysical processes such as temperature, ionisation, and excitation. This coupling canbe time-dependent and non-local due to the communication of physically distant regionsvia their radiation fields (Harries et al. 2019).This chapter introduces the radiative transfer code TORUS in § 2.1 and discussesthe theory behind the simulation of radiation around stars in § 2.2. The geometry andparameters of the T Tauri models used to create the synthetic spectra are discussed in § 2.3.Additionally, in § 2.4, we present two small grids of synthetic line profiles to validate theresults of TORUS and our T Tauri model.2.1. INTRODUCTION TO TORUS2.1 Introduction to TORUSTORUS is a radiation transfer and radiation hydrodynamics code designed around thecore tenet of flexibility. Originally written by Tim Harries, the code has been notablydeveloped by Neil Symington, Ryuichi Kurosawa, Dave Acreman, and TomHaworth. Thecore capabilities and programme flow are described in Fig. 2.1. The physical variablesare stored in a grid structure. To balance efficiency with the accuracy of the solution,TORUS uses an adaptive mesh refinement (AMR) to vary spatial resolution in differentcomputation domains. For example, areas of high optical depth, such as the T Tauriaccretion funnel, require smaller grid cells to resolve accurately than the significantlyless dense stellar wind that is far from the star. There are several physical processesthat TORUS can simulate, including radiative dust equilibrium, radiative hydrodynamics,photoionisation, and statistical molecular and atomic equilibrium. The states of theradiation field a and medium (a are then solved iteratively using either Monte Carlophoton packet tracing or ray tracing. TORUS can output several products, includingStokes intensity maps, spectral energy distributions (SED), and line spectra.2.2 Radiative transfer calculationsTORUS has previously been used tomodel the emission fromTTauri stars (e.g. Symingtonet al. 2005a; Kurosawa et al. 2006, 2011). However, the treatment of the radiation fieldthat is used in this work has diverged significantly from prior versions. The latest exactintegration method used to calculate the atomic (and molecular) statistical equilibriumis based on the accelerated Monte Carlo scheme developed by Hogerheĳde & van derTak (2000). This implementation was presented for molecular statistical equilibrium inTORUS by Rundle et al. (2010).The original TORUS code calculated the atomic statistical equilibrium using theSobolev approximation and geometric dilution based on thework ofKlein&Castor (1978).The line profileswere then determined using traditionalMonteCarlo radiative transfer. TheMonteCarlo formalism uses photon packets originating from randompositions (simulatingthe spontaneous emission) in the field that travel on random paths until they either escapethe field or are absorbed. Other photon sources, such as stars and background radiation,are treated using separate photon packets. The passage of photon packets through a cellcontributes to the local radiation field.The accelerated Monte Carlo scheme of Hogerheĳde & van der Tak (2000) is basedon a cell-centric model, in which rays are propagated to random positions in a cell fromrandom directions. Solving the radiative transfer equation along these rays to find the23CHAPTER 2. RADIATIVE TRANSFERGrid basedcodeSPHparticlelistA photonsourceAnanalyticalmodelTORUScheckpointGridAdaptive mesh refinement using a tree.TORUS can use 1D spherical, 2D/3D cylindrical, or 1D/2D/3D CartesiancoordinatesAtomic statisticalequilibriumDust radiativeequilibriumPhotoionisationequilibriumMolecularstatisticalequilibriumHydrodynamicsRadiationhydrodynamicsPhotodissociationFinal Radiative transferSobolev/LVGray tracingCo-moving frameray tracingPolarised MonteCarlo radiationSpectralenergydistributiondata cubeLinespectrumStokesintensityimageInput from(Micro)physicsOutputsFigure 2.1: An overview of the capabilities of TORUS as described by Harries et al. (2019).242.2. RADIATIVE TRANSFER CALCULATIONSlocal radiation field accelerates the convergence of the simulation in areas of significantoptical depth. In a full co-moving frame simulation, the rays are propagated from infinityor other intersection points (stars, disk, etc.) to the cell, the integrated radiation fieldalong the rays is then used to calculate the statistical equilibrium. However, this isconsiderably computationally expensive and often the problem is simplified by using theSobolev approximation (Sobolev 1957).The Sobolev approximation works for a medium with large velocity gradients. Itassumes that radiative interactions at any point are driven by the local vicinity due to theshift in the resonant frequencies of the emitting and absorbing atoms due to the velocitygradient (Grinin 2001). The Sobolev length or characteristic length B0 of the local vicinityis given by the distance from a point such that the change in resonant frequency due to thevelocity gradient is equal to the half-width of the spectral line profile given by the thermalor turbulent velocity ECB0 =ECdE/dB (2.2)where dE/dB is the velocity gradient. If the velocity gradient is approximated as E/',where ' is the characteristic size of the system and E is the characteristic large-scalevelocity, the Sobolev length can be estimated by B0 ≈ '/(E/EC).The radiative transfer method used by TORUS in this thesis computes the lineprofiles in two stages. First, TORUS calculates the level populations using the Sobolevapproximation and assuming statistical equilibrium. Second, TORUS computes the lineprofiles using full-frame co-moving ray tracing. This method is known as Sobolev withexact integration (SEI) (Harries et al. 2019).2.2.1 Statistical equilibriumTORUS calculates the level populations assuming the Sobolev approximation based on themethod of Klein & Castor (1978). The electron energy level populations of the hydrogenatoms are solved for 15 levels, with 3 more held in local thermal equilibrium (LTE). Thestatistical equilibrium rate equation is a balance of '!;the net transition from the ;th levelto lower levels, the transition from the ;th level to higher levels '*;, and the recombinationand ionisation rates '';and ';, respectively. These components are balanced for eachlevel, so that'!; + '*; + ''; − '; = 0. (2.3)25CHAPTER 2. RADIATIVE TRANSFERUsing the equations that define each term, the statistical equilibrium rate equation is asfollows.∑;D[#; (;D + ;DJ;D + #4;D) − #D (D;J;D + #4D;)]+ #∗D∫ ∞aD4cℎa0D (a)(2ℎa322+ a)exp(− ℎa:)6) da + #4D:− #D[4c∫ ∞aD0D (a)ℎaa da + #4D:]= 0 (2.4)where ;D, ;D are the Einstein coefficients and ;D is the collision rate. #D is thepopulation of level D in statistical equilibrium and #∗D the level population given by theSaha-Boltzmann equation for an electron density #4 and temperature )6. a is the angleaverage continuum mean intensity and term 0D (a) refers to the photoionisation crosssection of level D at frequency a. The term : refers to the continuum state. The angle-averaged profile-weighted intensity of the radiation field in the line transitions between ;and D is J;D and is determined using the Sobolev escape probability theory. J;D gives theprobability that a photon is absorbed in the transition of ; → D (Hubeny 2013) and for; < D it is given by (cf. Harries et al. 2019; Kurosawa et al. 2011),J;D =(1 − V;D) 2ℎa3;D22[6D6;#;#D− 1]−1+ V2,;D 2,;D . (2.5)Here, 6 represents the level degeneracy, 2,;D the line frequency continuum intensity, andthe variables V;D and V2,;D are the Sobolev escape probabilities given by the expressionsV;D =14c∮4c1 − e−g;Dg;DdΩ , (2.6)andV2,;D =∫Ωdisk1 − e−g;Dg;DdΩ . (2.7)The continuum escape probability V2,;D is calculated by integrating the Sobolev opticaldepth g;D across the solid angle subtended by the stellar photosphere Ωdisk, see § 2.3.1.3.The Sobolev optical depth for a velocity field v projected along the unit vector n is givenbyg;D (n) =c42<42(6; 5;D) 1E1n · ∇v(#;6;− #D6D). (2.8)262.2. RADIATIVE TRANSFER CALCULATIONSWhere<4 and 4 are the electronmass and charge, respectively. 5;D is the oscillator strengthof the line transition. Only hydrogen is assumed to be present, so the electron density is#4 = # ( H)+. Thus, the conservation equation is=max∑==1#= +=max+3∑===max#∗= + # ( H)+ =d 0.1 and dga < 20. Furthermore, if the velocity gradient acrossthe cell would cause the line resonance to be traversed or a is close to the line resonance,the number of segments is set to 20.The emission and absorption coefficients are affected by the local velocity, as achange in relative velocity could Doppler shift the line into resonance. This effect iscontained within the line profile function qa. If the intrinsic line broadening is negligible,then the profile is defined by thermal Doppler broadening and qa is a Gaussian profile. Foroptically thick lines such as HU, pressure broadening becomes important and a normalisedVoigt profile is adopted, which is a convolution of a Gaussian and a Lorentzian (cf.Vernazza et al. 1973a). The Voigt profile is given as qa = c−1/2(0, H)where(0, H)≡ 0c∫ ∞−∞e−H′2(H − H′)2 + 02 dH′ . (2.16)Here, H = (a − a0) /Δa , H′ = (a′ − a0) /Δa and 0 = Γ4cΔa where a0 is the line centrefrequency. Δa is the Doppler linewidth of hydrogen given byΔa =a02√2) <+ E2turb (2.17)here < is the hydrogen mass and E2turb is the turbulent broadening factor (de la Cruz Ro-dríguez & van Noort 2017), set to zero for part of the present study. The damping constantΓ is discussed in § 2.2.3.15 0 15Photosphere2.5 0.0 2.5Magnetosphere150 0 150Wind6 0 6Extra0.0 0.2 0.4 0.6 0.8 1.0Radius [R ]0.00.20.40.60.81.0Figure 2.2: The distribution rays used in to calculate the line profiles. Note the aspect of the y-axisand x-axis are equal.TORUS uses a set of rays that sample the photosphere, magnetosphere, and stellarwind. The rays are uniformly distributed radially and a random offset is applied to29CHAPTER 2. RADIATIVE TRANSFERthe azimuthal distribution. Fig. 2.2 shows a typical arrangement of rays sampling thephotosphere, magnetosphere, and stellar wind. For instance, the models presented inChapter 4.2 used a total of 270000 rays spread over the 250 × 250 spatial bins; 10000 forthe photosphere arranged with 100 radial points and 50 azimuthal points within a radiusof ≤ '∗ and another 100 × 50 rays between '∗ and 20'∗. The magnetosphere and windhave 100 × 50 rays each, and a further four rays per bin of the datacube frame are addedto fill in any gaps in the coverage.When TORUS uses the SEI method, it spends most of the computation time cal-culating the outputs. For example, the mean computer uptimes for the models presentedin Chapter 4 were ≈ 12 min to determine level populations and ≈ 40 min to calculate asingle data cube with 300 velocity bins and 250 × 250 spatial bins. For these datacubes,TORUS used 100 cores spread over five to six nodes with MPI parallelisation.The datacubes output by TORUS contain position-position-wavelength flux mapscentred around the central wavelength. Using the Doppler relationship, the wavelength _axis can be converted to velocity E relative to the central wavelength _0,E =2_0(_ − _0) . (2.18)By summing the total intensities of each position-position frame of the datacube andplotting this against the frame velocity, a spectral density plot can be created. This isdemonstrated in Fig. 2.3, which shows a series of eight compound datacube frames andthe spectrum they create. The original datacube had 400 wavelength bins, and the figureshows eight frames, which are compound images of 80 wavelength bins. The figure alsoshows the spectrum (or line profile) created from the datacube, and the contribution of theframes shown to the line profile. The line profile is plotted as the continuum-normalisedflux versus relative velocity. The flux is normalised by the first wavelength frame of thedatacube, which is assumed to be at the continuum level. Care was therefore taken toensure the full extent of the line profiles were covered by the wavelength range of thedatacubes, and the initial frames did not have any line emission in them.Because the T Tauri models used in this work contain large velocity fields, theradiation field contains complex emission and absorption components from different areasof the mesh. For example, in Fig. 2.3, in frames two and three, the hot spot and star areobscured by the stellar wind moving towards the observer. The wind absorbs the emissionand creates the blue-shifted sub-continuum absorption feature (P-Cygni profile) seen inthe spectrum.302.2. RADIATIVE TRANSFER CALCULATIONS1000 750 500 250 0 250 500 750 1000Velocity [kms 1]36Relative Flux0 1 2 3 4 5 6 710 5 10 4 10 3Surface Brightness[erg s 1cm 2Hz 1sr 1]0 1 2 34 5 6 7Figure 2.3: The top 8 frames are compound images of 80 wavelength bins from TORUS datacubecentred around HU. The colour scale indicates the HU flux, the brightest emission originates fromthe accretion hot-spot. The bottom subplot shows the line profile created from the datacube. Thenumbers correspond to the datacube frames and indicate their contribution to the spectrum.2.2.3 Stark broadeningStark broadening is caused by the local electric microfield of the plasma that perturbs theemission and absorption frequencies of the atomic energy levels. It is believed that Starkbroadening is important for modelling hydrogen Balmer lines (Muzerolle et al. 1998b,2001). Stark broadening is incorporated into the line transfer equations using the dampingconstant Γ, such thatΓ = rad + vdW(#H I1016) ()5000)0.3+ stark(#e1012)b(2.19)31CHAPTER 2. RADIATIVE TRANSFERwhere rad, vdW, and stark are the radiative, van der Waals, and Stark broadening half-widths. #H I is the neutral hydrogen density and ) is the gas temperature in Kelvin.For HU, the following broadening parameters were adopted from Luttermoser & Johnson(1992) rad = 8.2 × 10−3 Å, vdW = 5.5 × 10−3 Å, and stark = 1.47 × 10−2 Å. b is thestark broadening type, Kurosawa et al. (2006) used a non-linear stark broadening, whereb = 2/3. However, according to Griem (1964) and Vernazza et al. (1973b) hydrogenexhibits a linear Stark broadening b = 1. Therefore, in this thesis, we adopt the approachof Luttermoser & Johnson (1992); Kurosawa et al. (2011) and Vernazza et al. (1981) anduse linear Stark broadening.Stark broadening has a significant impact on the HU line widths at high accretionrates and funnel flow temperatures; see § 2.4. For lines other than HU, I exploredtwo methods to determine the Stark Broadening coefficients. Firstly, I determined theapproximate line half-widths using the analytical solution proposed by Sutton (1978).Secondly, I used the extended Stark broadening tables of Lemke (1997) to calculate thehalf-widths of the hydrogen lines. For the infrared lines (PaW, PaV, and BrW), I concludedthat the Stark broadening had a negligible effect on the synthetic line profiles. Thedominant broadening effects come from the large velocity fields of the T Tauri model.2.3 T Tauri modelTORUS uses an adaptive mesh refinement (AMR) system to create a numerical grid thatrepresents the physical space of the simulation (see Fig. 2.4). The majority of the modelspresented in this work use a 2.5D coordinate system that is axisymmetric around the pole.However, TORUS can also use a 3D geometry for non-axisymmetric models, see § 2.3.2and Chapter 5. The 2.5D geometry is defined using a 2D density field and a 3D velocityfield. The third component is calculated symmetrically for a given azimuthal angle.The T Tauri model populates the mesh with a star, an accretion flow, and a stellarwind; see Fig. 2.5. The disk used in our model is assumed to be geometrically thin andoptically thick, any rays that intersect it are terminated at the disk, and 0 = 0. The density,temperature, and velocity fields are defined as outlined in § 2.3.1 and § 2.3.1.4. The star,located at its origin, is defined by its radius '∗, mass "∗, and effective temperature )eff .The coarseness of the grid with dimensions # is refined so that it starts with #2 cells,which are split so that each cell contains a mass of ≤ 1010 g. The cells are subdivided nomore than 10 times. The total size of the simulation space was set to be a 3D cylindricalmesh with a height and radius of 400 × 1010 cm (0.27 AU).Fig. 2.4 shows a 3D density plot of the T Tauri model. A disk is illustrated; however,this is purely artistic and is included to indicate the position. In the simulation, the disk322.3. T TAURI MODEL0.02 AU10 15 10 14 10 13 10 12 10 11Density [gcm 3]Figure 2.4: A 3D density plot of the T Tauri model, showing the density of the magnetosphereand the stellar wind. Note, the density colour scale applies purely to the magnetosphere and stellarwind. The accretion hot-spot highlighted in red on the surface of the star. A slice of the mesh isshown to illustrate the AMR system.does not have any physical height or density. The colour of the star indicates the surfacetemperature, and the accretion hot-spot can be clearly seen. A slice of the AMR mesh isalso displayed, which illustrates how the AMR system creates a finer grid for the higher-density regions of the magnetosphere and wind. Fig. 2.5 is a diagram of the model andthe physical parameters used to define the geometry of the T Tauri model.2.3.1 Accretion funnelMagnetospheric accretion is the theory that the material accreting from the circumstellardisk to the star follows the magnetic field lines of the T Tauri star. The idea was first33CHAPTER 2. RADIATIVE TRANSFERFigure 2.5: Diagram of the T Tauri Magnetospheric accretion model. The shaded regions indicatethe areas populated with hydrogen in the TORUS model. Not to scale.proposed by Camenzind (1990) and Koenigl (1991), based on the mechanism developedbyGhosh et al. (1977) for accretion around neutron stars. However, observational evidenceof magnetospheric accretion was reported prior to the development of these concepts (e.g.Bertout & Bouvier 1988). In this formalism, the magnetic field truncates the accretiondisk at some radius '<8 where the material and magnetic stresses balance each other. Inthe case of an axially aligned dipole field, the accreting material is lifted from the diskand free flows onto the star, sandwiched between two dipole field lines (see Fig. 2.5). Inthis thesis, we used the geometry of the accretion flow as described by Hartmann et al.(1994). MHD simulations predict a twisting of the magnetic field (e.g. Uzdensky et al.2002) due to differential velocities between the star and disk; however, for simplicity, themagnetic field in our model is purely poloidal. The accreting matter flows freely along thefield lines under gravitational acceleration. Where the accretion funnel impacts the star’ssurface, it creates a shock region known as the hot-spot.The accretion funnel follows the magnetic dipole streamline given byA = '0 sin2(\) (2.20)342.3. T TAURI MODELwhere '0 is the radius at the equator of the streamline that intersects the stellar surface atcolatitude \. The component of the poloidal magnetic field at radius A is? (A, \) = `A−3√4 − 3 sin2(\) (2.21)where ` is the magnetic dipole moment. Finding the unit vector of ? and converting toCartesian coordinates gives the components of the accretion flow unit vectorv̂p =1√4 − 3 sin2 \(3 sin(\)√1 − sin2 \x̂ + (2 − 3 sin2 \)ẑ). (2.22)The flow speed at A is calculated from the change in potential energy,E? (A, \) =√√2"∗(1'∗− sin2 \A), (2.23)where '∗ is the stellar radius. The velocity of the flow is vp = E?v̂p. For an accretion rate¤"acc the density at radius A is given to be (cf. Hartmann et al. 1994)d(A) =¤"acc4c(1/'<8 − 1/'<>)A−5/2√2"∗√4 − 3 sin2 \√1 − sin2 \(2.24)The velocity and density profiles can be seen in Fig. 2.6. In the figure, the density,velocity, and temperature (see § 2.3.1.2) are plotted for four field lines following the formof equation 2.20. The lines are a little uneven at lower radii because the values are lifteddirectly from the mesh without interpolation. The density is initially high, but as thematerial accelerates, the density rapidly decreases and then it begins to increase again asthe field lines constrict the flow.2.3.1.1 Accretion shock hot-spotThe hot-spot temperature )hs is calculated from the potential energy liberated by theaccreting material. The surface of the star is divided into a grid, each with its own SED,for which a Kurucz (1979) model atmosphere is used (see § 2.3.1.3). If a source gridcell coincides with the accretion funnel, the mass flux is determined from the AMR cellsdirectly adjacent. The mass flux ¤"cell, accretion velocity E, and the cell area cell can beused to determine the cell temperature,)hs =[E2 ¤"cell2fcell] 14, (2.25)35CHAPTER 2. RADIATIVE TRANSFER60007000Temp. [K]0.10.2Density [gcm3×1010]1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75Radius [R ]0150Velocity [kms1 ]R0 = 2.28 R R0 = 2.44 R R0 = 2.60 R R0 = 2.76 RFigure 2.6: The three panels show the magnetospheric temperature (top), density (middle), andvelocity (bottom) along four dipole field lines. The data is extracted from a TORUS mesh with amaximum accretion temperature of 7500 K and an accretion rate of log ¤"acc = −7. The wobblesin the lines are artefacts of the numerical grid.here f is the Stefan–Boltzmann constant. The temperature is used to determine a black-body SED, which is added to the photospheric SED. For example, in the instance of astar with a mass of 0.5 M, a radius of 2 R, and with accreting material falling from adistance of 3 '∗ to the stellar surface intersecting at angle of 55◦ with accretion rates of¤"acc = 10−7, 10−8, and 10−9 Myr−1, the hotspot temperatures will be )hs ≈ 6780, 3810,and 2140 K, respectively.By using a blackbody, it is assumed that the line emission and absorption from thehot spot-are negligible. The model assumes that the energy radiated away from the shockzone is quickly reabsorbed in the optically thick accretion gas and re-emitted at longerwavelengths, as characterised by Calvet & Gullbring (1998). The accretion regions are362.3. T TAURI MODELalso expected to have line emission (e.g. Dodin 2018) that will contribute to the observedspectra; however, we neglect this due to the lack of available models. A complete radiativeequilibrium treatment of the shock region andmagnetosphere is necessary to fully constrainthis effect.2.3.1.2 Accretion temperatureThe synthetic emission lines are very sensitive to the temperature distribution in the ac-cretion funnel. However, the magnetospheric heating mechanisms are poorly understood,although it is assumed that they are magnetically driven; for instance, by the dissipationof Alfvén waves. The self-consistent thermal model produced by Martin (1996) usedadiabatic compression to heat the accretion columns. However, Muzerolle et al. (1998b)showed that the model could not produce line profiles consistent with observations. Thus,the temperature is provided as an input parameter and the code does not solve for radiativeequilibrium.The temperature profiles adopted for this work are based on those used by Hartmannet al. (1994), who created a temperature structure using a volumetric heating rate ∝ A−3.The dissipation of Alfvén waves from inhomogeneous accretion could, for example, causethis heating rate. A schematic radiative cooling rate is used to balance the heating. Thetemperature structure used in our models is interpolated and scaled from the values givenin fig. 6 of Hartmann et al. (1994). More details on heating and cooling rates can be foundin Hartmann et al. (1982). The profiles are scaled by an input parameter of the maximumtemperature in the accretion flow )acc. A temperature profile from a TORUS model with)acc = 7500 K is shown in Fig. 2.6. As gas falls towards the star from the disk, thetemperature rapidly increases as the density decreases and then begins to get cooler as thefield lines constrict and the density increases. The temperature drops near the disk and thestar because the increase in density enhances the radiative cooling.Without a self-consistent model of the accretion heating mechanism, the modelcan only be indicative of the temperature distribution. Muzerolle et al. (2001) furtherconstrained these models by determining the range of)acc that creates realistic line profilesfor different accretion rates. Furthermore, in the words of L. Hartmann,“our approach is merited if the adoption of a smooth, plausible temperature distri-bution results in line profiles which agree with observations” (Hartmann et al. 1994)2.3.1.3 Photospheric continuumThe photospheric continuum radiation from the star is calculated by interpolating themodel atmospheres of Kurucz (1979). The Kurucz (1979) model atmospheres are com-37CHAPTER 2. RADIATIVE TRANSFERputed for a broad range of effective temperatures and stellar gravities, from a statisticaldistribution of nearly a million atomic lines. TORUS interpolates between the nearestmodels to determine an atmosphere that fits the input stellar parameters. Fig. 2.7 showsseven different models for a temperature range of 3500 – 6000 K and surface gravity oflog (6∗/cms−2) = 3.5 which is close to the fiducial stellar gravity used in our models oflog (6/cms−2) = 3.53. The locations of the primary hydrogen lines (HU, PaV, PaW, andBrW) studied in this work are highlighted by the grey bars. The effective temperature ofthe fiducial star is 4000 K (see Chapter 4.1), at this temperature there is no significant con-tinuum absorption or emission near the HU, PaV, PaW, and BrW hydrogen lines. However,at temperatures of 5000 K and above, there begins to be significant absorption present forthe HU, PaV, PaW, and BrW transitions.It is assumed that radiation from the high-temperature )hs shock region created bythe magnetosphere is absorbed and re-emitted by the cooler surrounding gas. As such, thehot-spot emission is treated simply as an addition to the continuum emission. See § 2.3.1for a further discussion of the shock region. The continuum emission at a frequency a fromthe hot-spot is calculated as the sum of the Kurucz model atmosphere and the blackbodyspectral radiance a,hs given bya,hs (a, )hs) =2ℎa3221eℎa:B)hs − 1, (2.26)here, the terms have their usual meanings.2.3.1.4 Stellar windWe implemented a new stellar wind in TORUS that builds on the work by Kurosawa et al.(2011). Their model used a radial stellar wind launched from a spherical cap above thestellar surface so that a large opening angle could be achieved without interacting with theaccretion funnel. However, the densest and radiatively significant part of the stellar windshould be closest to the star. The new wind geometry presented in this thesis, launchesthe outflow from the stellar surface, specifically, from the polar regions at latitudes higherthan the accretion hot-spot. The wind and the accretion funnel do not intersect becausethe wind follows the dipole magnetic field lines until it reaches a specified opening angleΘBF, at which point the wind becomes radial (see Fig. 2.5). The parameterisation of ΘBFautomatically determines the radius 'F> where the wind transitions from following thedipole lines to becoming radial. This geometry approximates the wind behaviour observedin MHD simulations (e.g. Zanni & Ferreira 2009; Kurosawa & Romanova 2012; Irelandet al. 2021). Similarly to previous studies (e.g. Kurosawa et al. 2006, 2011), the wind is382.3. T TAURI MODEL2500 5000 7500 10000 12500 15000 17500 20000 22500 25000Wavelength [Å]Relative Flux3500K4000K4250K4500K5000K5500K6000KFigure 2.7: Kurucz (1979) model atmospheres for a surface gravity of log (6∗/cms−2) = 3.5 andseven different temperatures close to fiducial effective temperature used in Chapter 4. The verticalgrey bars indicate the position of the four hydrogen lines (HU, PaW, PaV, and BrW) studied inChapter 3 and Chapter 4.assumed to be isothermal and uses a V-law velocity profile.2.3.1.4.1 Wind densityThe density of the stellarwind is determined from thewindmass-loss rate ¤"BF, a parameterset at run time. The density of the wind dBF is proportional to ¤"BF/(EBF,A(A)) where(A) is the surface at radius A through which the wind is moving with a radial velocityof EBF,A . The area through which the wind is moving at a radius A depends on if thewind is moving with the magnetic field or flowing radially from the star. In the range'∗ < A ≤ 'F>, the wind is assumed to be contained within a diverging flux tube. The39CHAPTER 2. RADIATIVE TRANSFERfield strength of a magnetic dipole diverges at a rate of 1/A3. Accordingly, the density isgiven by,dBF (A) =12¤"swEBF,A∗('∗A)3(2.27)where ∗ is the surface area of the star from which the wind is launched. For radii greaterthan 'F>, the surface area (A) increases as a spherical cap (radial flow) with increasingradius. Hence,(A) = 2cA2(1 − cosΘopen) (2.28)and so the density decrease is given by,dBF (A) =12¤"BFEBF,AF>('F>A)2. (2.29)Here, F> is the area of the spherical cap with radius 'F>. The value 'F> is the radius atwhich the wind velocity becomes radial such that'F> = U('<> + X) sin2Θopen. (2.30)The terms X and U are input parameters that allow the wind coupling region to be adjusted.By default, these values are set to zero and unity, respectively. It should be noted, thatwhile value 'F> is analogous to the Alfvén radius, 'F> is set solely by the geometry ofthe wind and magnetosphere and it is not based on the physical parameters of the Alfvénradius.2.3.1.4.2 Wind velocityThe wind velocity EBF is composed of two components: the poloidal velocity Epol andthe toroidal velocity Etor. The most commonly used parameterization of poloidal windvelocity is the so-called V-law (cf. Castor & Lamers 1979). The V-law gives the magnitudeof the poloidal component of the stellar wind velocity at a radius A to beEpol(A) = Emin + (Emax − Emin)(1 − '∗A) V(2.31)where Emin and Emax are the starting and end velocities of the wind. The free parameter Vdetermines the steepness or acceleration of the wind. To obtain a physical grounding forthe wind velocity, we used an MHD radial wind velocity profile (L. G. Ireland, personalcommunication, 5 March 2020), from Ireland et al. (2021) to fit equation 2.31. TheMHD model had parameters similar to those of the fiducial radiative transfer model402.3. T TAURI MODELwith a truncation radius '<8 = 2.41 R. A least-squares regression fit was used to fitequation 2.31 to the MHD velocity profile. The lower bound of Emin = 1 kms−1 wasforced to match the launch velocity of the MHD wind. The fit was made using thePython module scipy.optimize.curve_fit. This determined that a value of V = 2.98best suited the MHD velocity profile. In comparison, Kurosawa et al. (2011) used a valueof V = 0.5 for their wind models.Fig. 2.8 shows the MHD velocity profile and the best fit. The fitted line closelyresembles theMHDmodel velocity, which suggests that, although, the V-law is a relativelysimple approach, it is an appropriate approximation for our model. The figure shows thedifference in velocity and density profiles for V = 0.5 and V = 2.89. The higher V valueproduces a wind that accelerates slower, which creates a denser wind closer to the stellarsurface. This dense wind close to the stellar surface will allow for greater absorption ofthe star’s continuum and hot-spot emission.The toroidal velocity component is derived from the star’s rotation rate at thecolatitude of wind launch. It assumes that the wind and magnetic field are perfectlycoupled, and that the wind corotates as a rigid body with the star out to the wind corotationradius 'F>. Beyond this, the wind conserves the angular momentum and the toroidalcomponent of the velocity decreases proportionally to 1/A.Fig. 2.9 shows the wind’s poloidal, toroidal, and total velocity components alongwith the total velocity magnitude. The components shown here are for a star with a massof 0.5 M and a radius of 2 R with an equatorial rotation rate of 40.2 kms−1, whichis approximately 15% of the stars breakup velocity. Additionally, Fig. 2.9 shows threeimages that display the wind velocity for x-y positions around the star. The star is locatedin the top left of each frame and the individual frames show the poloidal, toroidal, andmagnitude of the velocity. The colour displays the wind velocity around the star assumingthere is no magnetosphere. It can be seen that the poloidal component dominates the windvelocity magnitude apart from at radii close to the star. Even then, the largest toroidaleffect is seen at high colatitude, an area that is mostly filled by the magnetosphere. Thetoroidal velocity has a more significant effect once the rotation rate is upped to ≈ 50%of the break up velocity. Rotation and the effect it has on the synthetic line profiles arediscussed further in Chapter 4.2.3.2 3D geometryThe T Tauri model presented in § 2.3 can also be extended to full 3D simulations inTORUS . This enables us to create synthetic spectra from non-axisymmetric models. Forexample, this is useful for modelling a magnetosphere that does not fully extend around41CHAPTER 2. RADIATIVE TRANSFER1 10 100020406080Velocity [kms1 ]Best Fit: = 2.89MHD Radial Velocity1 100100200300400velocity [kms1 ]1 100e+002e-125e-127e-121e-11Density [gcm3 ]0.0 0.2 0.4 0.6 0.8 1.0Radius [R ]0.00.20.40.60.81.0Beta = 0.5 Beta = 2.89Figure 2.8: The top panel shows the fit of the velocity V-law, equation 2.31 and a wind velocityprofile from an MHD simulation. A value of V = 2.89 was determined to be good match usinga least-squares regression fit. The lower two panels shows the difference in velocity and densitybetween beta values of V = 2.89 and V = 0.5, the latter being the value adopted by Kurosawa et al.(2011).the star. This is a potentially likely scenario because the magnetic fields of T Tauri starsare rarely aligned with their poles (e.g. McGinnis et al. 2020). In the case of the magneticdipole being offset from the rotation axis, the accreting matter will preferentially flow inhalf of each alternate hemisphere (e.g. Long et al. 2011). Moreover, the axisymmetricMHD simulations of Ireland et al. (2021) show that the magnetospheric flow tends toprefer one hemisphere and may change as permutations of the magnetic field develop. Thedominance of accretion in one hemisphere in an axisymmetric MHD model can be seenin Fig. 1.2.The TTauri model is extended to 3D inTORUS by building the density, temperature,422.3. T TAURI MODEL2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Radius [R ]050100150200250300350Velocity [kms1 ]Vtor Vpol |V|0 5 10 15 20Poloidal05101520Radius [R]0 5 10 15 20Toroidal0 5 10 15 20Magnitude0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 50 100 150 200 250 300 350Velocity[kms 1]Figure 2.9: The lower frame shows the poloidal velocity, the toroidal velocity, and the velocitymagnitude of the stellar wind. The rotation rate is 15% of the stars breakup velocity. The upperthree panels display the wind velocity components as a 2D heat map. The star is located in theupper left corner.and velocity fields in a 3D AMR mesh using the prescription laid out in § 2.3.1 and§ 2.3.1.4. The complexity arises if a magnetic dipole offset is required. When the axisof the magnetic dipole is offset from the rotation axis (z-axis), the magnetosphere and thewind must similarly be rotated. Fig. 2.10 shows a slice of a 3D Cartesian T Tauri modelfrom TORUS with a 10◦ dipole offset as indicated by the inclined black line. By default,when the magnetic dipole offset is greater than zero, the magnetosphere is limited to twoopposing half-hemispheric accretion funnels; one in either hemisphere. The positionsof the accretion funnels are chosen to minimise the distance from the disk to the star;see Fig. 2.10. The width of the two accretion funnels can be set as an input parameter.Previous iterations of TORUS have been used to model similar setups (e.g. Esau et al.43CHAPTER 2. RADIATIVE TRANSFERDisk0.01 AU10 14 10 13 10 12 10 11Density [gcm 3]Figure 2.10: A slice of a 3D TORUS mesh where the magnetic dipole axis (inclined black line)has be rotated around the y-axis by 10◦. The colour map denotes the density of the wind andmagnetosphere. The arrows show the model velocity vector field.2014; Symington et al. 2005a). However, these studies did not use the new implementationof radiative transfer described in § 2.2 or include a stellar wind.The new stellar wind model in TORUS (see § 2.3.1.4) can also be used in a non-axisymmetric 3D model. The initial stage of the wind follows the magnetic field, which isrotated to match the magnetic dipole offset. After which, the wind becomes radial relativeto the rotation axis. The wind flows along the field lines until it reaches the openingangle Θopen; which is relative to the rotation rather than to the dipole axis. If an offset isapplied, the magnetic dipole rotates around the y-axis by an angle b. The disk remainsfixed, therefore the length of the field lines is changed and the path of the wind must be442.4. MODEL VALIDATIONaltered. The wind opening radius equation 2.30 is re-defined as,'F> = U'max sin2Θopen. (2.32)The term 'max is the radius at which the outermost magnetic streamline intersects the disk,given by,'max ='<> + Xsin2 \′0. (2.33)Where \′0 is the angle at which the field line intersects the disk in the rotated frame ofreference. If \0 = c/2 then,sin2 \′0 =1(1 + tan2 b cos2 q′) (2.34)here q′ is the azimuthal angle in the rotated frame of reference. The input parametersU and X can be set to increase 'F> and the gap between the wind and magnetosphere,respectively.The poloidal velocity vector field is shown in Fig. 2.10 overlaying a density map ofthe magnetosphere and wind. The mass-loss rate used in this simulation is 10−8Myr−1and an accretion rate of 10−7Myr−1. The change from dipolar to radial velocity at 'F>can be seen. Assuming the wind and magnetic field are perfectly coupled, it is reasonableto assume that there would be minimal movement of the wind perpendicular to the fieldlines. The adaptive mesh refinement works as described in § 2.3 and a slice of the meshcan be seen in the figure.2.4 Model validationTo validate the new radiative transfer routines implemented, I ran two separate modelgrids. The first contained only a magnetosphere and no outflow. The second includedour new stellar wind model. The results were compared with previous radiative transfersimulations from literature. I was able to show that the new routines implemented inTORUS performed as expected and produced line profiles with morphologies, widths, andintensities similar to those seen in the literature. The results of the code validation arediscussed below.2.4.1 Magnetospheric accretion onlyWeusedTORUS to reproduce the simpleHUmagnetospheric onlymodel presented in fig. 3of Kurosawa et al. (2006). The model used in fig. 3 of Kurosawa et al. (2006) is identical45CHAPTER 2. RADIATIVE TRANSFER2.04.06.010 70.91.11.210 81.01.01.06500 K10 92.55.07.50.81.01.21.50.91.11.27500 K2.55.07.52.04.06.00.81.01.28500 K600 0 6002.55.07.5600 0 600Velocity [kms 1]5.010.015.0Kurosawa et al.(2006)TORUSNon-LinearTORUSLinear600 0 6001.62.43.29500 K0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Relative FluxFigure 2.11: A comparison of synthetic HU line profiles produced by Kurosawa et al. (2006)(black lines) and TORUS . The models have magnetospheric accretion only and no form ofmass outflows. Profiles arrange by accretion rate (columns) in values of Myr−1 and accretionmaximum temperature (rows) for a viewing colatitude of 55◦. Two different Stark broadeningparameterisations are used by TORUS ; linear (blue dot-dashed) and nonlinear (pink dashed) Starkbroadening. Line strength increases with temperature and accretion rate, top-right to bottom-left.to the setup described in § 2.3.1, and no outflows are included. The models were runwith three different accretion rates of 10−7, 10−8, 10−9 Myr−1, four accretion maximumtemperatures of 6500, 7500, 8500, 9500 K, and computed at an inclination of 55◦. Themodel used a stellar mass of 0.5 M, a stellar radius of 2 R, and a magnetospherestretching from '<8 = 2.2 '∗ to '<> = 3 '∗. All other parameters are identical tothose presented in Table 4.1. The only difference is that Kurosawa et al. (2006) used anonlinear Stark broadening coefficient. Our TORUS model was run with both linear andnonlinear Stark broadening. In other words, the same fundamental physics is used by both462.4. MODEL VALIDATIONKurosawa et al. (2006) and TORUS . This comparison therefore highlights the influenceof the upgrades to TORUS discussed in § 2.2.A comparison of the results produced by Kurosawa et al. (2006) and TORUS can beseen in Fig. 2.11. In the figure, it can be seen that the majority of the spectra share similarwidths, shapes, and intensities. In general, the HU lines produced by TORUS are slightlybroader, with peak intensities higher than those presented by Kurosawa et al. (2006). Athigher temperatures and accretion rates, the non-linear Stark broadening produces spectrawith marginally higher intensities at the line centre.The general line profile trends shown by Kurosawa et al. (2006) are reflected by ourTORUS models. The line strength and width increase with the accretion temperature andrates. The strongest lines for a given temperature are seen for the highest accretion rateslog ¤"acc = −7. An exception to this trend is seen in Fig. 2.11 for the line profile producedwith an accretion rate of log ¤"acc = −8 and )acc = 9500 K. The line profile has a greaterrelative peak flux than the profile produced with an accretion rate of log ¤"acc = −7. Thisis because the continuum level from the hotspot is increased by the greater accretion rate,reducing the peak flux relative to the continuum. We see, in Fig. 2.11, that the width ofthe profiles increases with the rate and temperature of the accretion; it is unaffected by thecontinuum level.2.4.2 Magnetosphere and stellar windThe above comparisons did not include any outflows. Therefore, to assess our stellarwind geometry (§ 2.3.1.4), we ran an additional test to compare our model with the stellarwind geometry used by Kurosawa et al. (2011). The results are shown in Fig. 2.12.TORUS was used to run two models with identical stellar and magnetospheric parametersas those defined for fig. 10 of Kurosawa et al. (2011). Namely "∗ = 0.5 M, '∗2 R,'<8 = 2.2 '∗ and '<> = 3 '∗, all other stellar and magnetospheric parameters areas described in Table 4.1. The first models had an identical wind as Kurosawa et al.(2011); the wind launched from 3R∗ above the star’s surface. The second model usedour wind geometry with the outflow being launched from the star’s surface. We adoptedthe same outflow par wind processes, better181.2. SPECTROSCOPIC ANALYSIS OF T TAURI STARSobservational diagnostics are needed to constrain the theory of T Tauri evolution. To dothis, the radiative emission of these stars needs to be modelled to decipher the observedline profiles.Complex radiative transfer models were first employed to study magnetosphericaccretion in T Tauri stars by Hartmann et al. (1994). They used a two-level hydrogenmodel to show that magnetospheric accretion could explain the strong emission lines andblue excess of several T Tauri stars. Muzerolle et al. (1998b) further built on the T Taurimodel of Hartmann et al. (1994) by using a 2D radiative transfer code that performed amultilevel statistical equilibrium calculation for eight hydrogen levels. They concludedthat the BrW line profiles could be reproduced for embedded objects and that the Balmerline flux was consistent with observations. Muzerolle et al. (1998b, 2001) used radiativetransfer simulations to constrain the temperature of the accretion column to the range6000–10000 K and determined that Stark broadening may contribute to the observedbroad HU wings (±500 kms−1). Folha & Emerson (2001) noted that the radiative transfermagnetospheric accretionmodels produced infrared lines (Paschen and Brackett) that werenarrower than their observations by ≈ 100 kms−1.Kurosawa et al. (2006) introduced a disk wind and a polar stellar wind into theirT Tauri radiative transfer simulations. They showed that the observed blue-shifted sub-continuum absorption profiles, interpreted as evidence of the presence of mass outflow(e.g. Edwards et al. 2006; Kwan et al. 2007), could be reproduced by their wind models.Additionally, using wind and magnetospheric accretion models, Kurosawa et al. (2006)were able to produce the seven classical line profile classes defined by Reipurth et al.(1996). Furthermore, Kurosawa et al. (2011) showed that a bipolar stellar wind wouldproduce broad P-Cygni absorption due to its high velocity and narrow P-Cygni absorptionwas the product of a slow disk wind. Both of these forms of P-Cygni profiles are observed(e.g. Edwards et al. 2006); although, it is not clear in which physical paradigms theseoutflow mechanisms are dominant or whether they coexist.Another consideration for magnetospheric accretion is that the magnetic fields ofT Tauri stars are never perfectly aligned with their poles (e.g. McGinnis et al. 2020). Amagnetic obliquity will likely cause the accretion funnel to favour specific longitudes ofthe ascending nodes (e.g. Espaillat et al. 2021), restricting the azimuthal angle of the mag-netosphere. A non-axisymmetric accretion funnel is likely to reduce the frequency of theobserved inverse P-Cygni profiles and cause significant line and continuum variability asthe hot-spots and accretion columns rotate with the star. Symington et al. (2005a) explorednon-axisymmetric radiative transfer models and the rotational phase line variability. Theyshowed that introducing a magnetic dipole offset induced greater line variability than wasobserved. Kurosawa et al. (2005) used similar models to compare synthetic spectra with19CHAPTER 1. INTRODUCTIONtime series PaV spectroscopy from SU Aurigae and were able to reproduce some of theobserved velocity and equivalent width variability. Other studies have used magnetohy-drodynamic models and post-processed them using radiative transfer simulations. Forexample, Kurosawa et al. (2008) and Kurosawa & Romanova (2012) successfully repro-duced many of the commonly observed hydrogen line profiles and predicted similar linevariability. Some studies have used magnetohydrodynamic models of stars with well-constrained stellar parameters to create radiative transfer models to study the observedemission. For instance, Alencar et al. (2012) successfully used a non-axisymmetric accre-tion model based on a magnetohydrodynamic simulation to perform a radiative transfersimulation of V2129 Oph* in HU and HV. Despite some similarities between the modelledemission and the observed spectroscopy, the authors could not reproduce the red-shiftedabsorption accurately.SummaryT Tauri stars present an interesting problem in that their accretion and mass-outflowmechanisms remain elusive even with decades of observations and modelling. Modernradiative transfer simulations offer an opportunity to unlock the origins of the complexspectra. However, despite the relative success of radiative transfer studies, the T Taurimodels still have multiple free parameters and chronic degeneracy. It is unclear how todisentangle the effects of geometry, accretion, and mass loss on the observed spectra.Consequently, although the evidence supports the magnetospheric paradigm, radiativetransfer models need development. Therefore, while radiative transfer simulations canprovide valuable insight into the spectra of T Tauri stars, there is still room for significantimprovement of the models and their parameterisation.The primary problems facing radiative transfer simulations are as follows. Theconstraints on the magnetospheric temperature and accretion rates are still vague becauseof a lack of a comprehensive theory for the heating of the accreting gas. Both stellarand disk winds have been modelled, but the dominant geometry, mass-loss rate, andtemperature are still argued. Folha & Emerson (2001) noted that the synthetic infraredlines produced were too narrow. Furthermore, 3D radiative transfer simulations based onmagnetohydrodynamic T Tauri simulations struggle to reproduce the observed red-shiftedatomic line components. In this thesis, I use and develop a new generation of T Taurimodels for the radiative transfer code TORUS , incorporating both polar stellar winds andan accelerated Monte Carlo processes. I explore a grid of synthetic hydrogen transitionsusing these models and compare them to the same atomic spectra from 29 T Tauri stars.The comparison aims to determine whether a range of accretion rates, mass-loss rates,201.2. SPECTROSCOPIC ANALYSIS OF T TAURI STARSand temperatures based on observed parameters can reproduce the ensemble of observedT Tauri spectral characteristics.This thesis aims to achieve two primary research goals. Firstly, I will scrutinisethe radiative transfer T Tauri models presented in this thesis and consider possible mod-ifications to improve the diagnostic capabilities via a comparison to observed spectra.Secondly, I will extend the 2D dimensional models to 3D and attempt to evaluate hownon-axisymmetric accretion influences the spectral characteristics of the optical and in-frared hydrogen lines. This research was undertaken to further the current theories on theevolution of Sun-like stars by improving T Tauri radiative transfer models.In Chapter 2, the radiative transfer code TORUS (Harries et al. 2019) is presentedand the new developments of the T Tauri model are described. Next, in Chapter 3, I reduceand show a set of hydrogen spectra from 29 T Tauri stars. Then, in Chapter 4, I examinethe observed spectra in comparison to a grid of synthetic hydrogen line profiles producedusing TORUS, focusing on the profile morphologies. Finally, in Chapter 5, I explore3D non-axisymmetric T Tauri models and consider the rotational phase variability of thesynthetic spectra.21Chapter 2Radiative TransferRadiative transfer simulations attempt to reproduce the flow of electromagnetic energythrough and around different media, the characterisation of which can reveal the com-plex physical process at play in the interaction of radiation and matter. Furthermore, itcan be used to predict the observational characteristics of different theoretical physicalphenomena. At the heart of these simulations is the radiative transfer equation,dadga= (a − a (2.1)where ga is the optical depth of the medium at frequency a. The specific intensity a isthe radiative energy flux per solid angle per frequency bandwidth, and (a is the sourcefunction, the ratio of the local absorption and emission coefficients. For all but the mostsimple systems, modelling the passage of light through a medium is a complex problemwhich has to be solved numerically (see Steinacker et al. 2013). The complexity of thesystem arises because the radiation field a is coupled to the state of the medium (a bymicrophysical processes such as temperature, ionisation, and excitation. This coupling canbe time-dependent and non-local due to the communication of physically distant regionsvia their radiation fields (Harries et al. 2019).This chapter introduces the radiative transfer code TORUS in § 2.1 and discussesthe theory behind the simulation of radiation around stars in § 2.2. The geometry andparameters of the T Tauri models used to create the synthetic spectra are discussed in § 2.3.Additionally, in § 2.4, we present two small grids of synthetic line profiles to validate theresults of TORUS and our T Tauri model.2.1. INTRODUCTION TO TORUS2.1 Introduction to TORUSTORUS is a radiation transfer and radiation hydrodynamics code designed around thecore tenet of flexibility. Originally written by Tim Harries, the code has been notablydeveloped by Neil Symington, Ryuichi Kurosawa, Dave Acreman, and TomHaworth. Thecore capabilities and programme flow are described in Fig. 2.1. The physical variablesare stored in a grid structure. To balance efficiency with the accuracy of the solution,TORUS uses an adaptive mesh refinement (AMR) to vary spatial resolution in differentcomputation domains. For example, areas of high optical depth, such as the T Tauriaccretion funnel, require smaller grid cells to resolve accurately than the significantlyless dense stellar wind that is far from the star. There are several physical processesthat TORUS can simulate, including radiative dust equilibrium, radiative hydrodynamics,photoionisation, and statistical molecular and atomic equilibrium. The states of theradiation field a and medium (a are then solved iteratively using either Monte Carlophoton packet tracing or ray tracing. TORUS can output several products, includingStokes intensity maps, spectral energy distributions (SED), and line spectra.2.2 Radiative transfer calculationsTORUtebalancing filters) for each spectrograph.• Wavelength calibration, using Ar, Hg, Ne, and Xe Penray lamps.• Background removal and spectra extraction.• Flux calibration, using the standard spectrophotometric star GD-71 (Manara et al. 2016).I further reduced the X-Shooter spectra by correcting for telluric contamination § 3.2.1and extracting and fitting hydrogen lines § 3.2.2.3.2.1 Telluric correctionsGround based telescopes, such as VLT, observe through the Earth’s atmosphere, theconstituentmolecules of which absorb or emit radiation that contaminates the astronomicalspectra. These atmospheric features are called telluric lines and they are primarily aproblem in the near-infrared regions of the spectrum (Fischer et al. 2016). Water vapourin the troposphere and stratosphere produces the greatest effect, however, molecules ofcarbon dioxide (CO2), methane (CH4), nitrous oxide (N2O), and ozone (O3) also producesignificant telluric features. Due to the nature of the variable weather conditions andmolecular abundances, the telluric features are unpredictable. Traditionally, to removetelluric features, science spectra are divided by the spectra of a photometric standard starobserved as close as possible in time and airmass to the target. However, the observationof this calibration data is expensive and requires a standard star to be visible at a similartime and airmass as the target. A more recent method for removing telluric features isto use synthetic transmission spectra created using a radiative transfer code, based onatmospheric conditions at the time of observation. This is the method that was used inthis chapter. To remove telluric contamination, I employed the Molecfit code from ESO(Smette et al. 2015; Kausch et al. 2015).Fig. 3.2 is an example of the raw and telluric corrected spectra from X-Shooter. Thespectra of all threeX-Shooter arms are plotted, showing the full observedwavelength range.53CHAPTER 3. OBSERVATIONSTable 3.1: The 29 T Tauri stars used in this work. Here, the notes “B” or “T” denote that the staris listed in the literature as a binary or triple system respectively; a value of the angular separationis given where available. The complete SIMBAD name for Stars named T-# are Ass-Cha-T-2-#.The values and notes are adopted from: 1 Manara et al. (2016), 2 Schneider et al. (2018), 3 Manaraet al. (2014), 4 McGinnis et al. (2015), and 5 Rugel et al. (2018).Name Type )eff + Mass log¤" Notes Ref.[K] [mag] [ M] [ Myr−1]ESO-Ha-562 M0.5+M0.5 3705 3.4 0.66 -9.3 B 0.28" 1IQ-Tau M0.5 3850 1.7 0.99 -8.2 1V354-Mon K4 4900 ∼ 1.4 -8 2T-33 K0 5110 2.7 0.95 -8.7 T 2.4" 1T-11 K2 4900 0.8 1.32 8.3 B 3VW-Cha K7+M0 4060 1.9 1.24 -7.9 B 0.66" 1V409-Tau K5 4350 0.51 1.22 -7.8 3CR-Cha K0 5110 1.3 1.78 -8.7 1T-52 K0 5110 1.0 1.4 -7.4 B 11.6" 1T-38 M0.5 3780 1.9 0.71 -9.4 1KV-Mon K4 ∼ ∼ 1.31 ∼ 4Sz-22 K5 4350 3.2 1.09 -8.4 T 17.6" 1T-4 K7 4060 0.5 1.03 -9.5 1DG-Tau K5 4350 2.2 1.4 -7.7 1T-23 M4.5 3200 1.7 0.33 -8.3 1RECX-12 M3 3410 ∼ 0.29 -9.8 5Cha-Ha-6 M6.5 2935 0.1 0.1 -10.3 1T-12 M4.5 3200 0.8 0.23 -8.8 1CT-Cha-A K5. 4350 2.4 1.4 -6.9 1ESO-HA 442 M2 ∼ ∼ 0.36 ∼ 4CHX18N K2 4900 0.8 1.17 -8.1 1TW-Cha K7. 4060 0.8 1 -9 B 1RECX-6 M3 3415 ∼ 0.15 -11 5T-49 M3.5 3340 1.0 0.36 -7.6 B 24.4" 1T-35 K7 4060 2.9 0.96 -8.9 3T-45a K7 4060 1.1 0.97 -9.9 B 28.3" 1RECX-9 M4.5 3085 ∼ 0.15 -10.4 5T-24 M0 3850 1.5 0.91 -8.7 1Hn-5 M5 3125 0.0 0.16 -9.3 1Note. ESO-HA 442 is also known as CID-0223980048.The pink-shaded areas indicate the position of the four hydrogen lines HU, PaV, PaW, andBrW. The areas of the spectra most affected by telluric features are 9000–10000, 11000–12300, 13000–15000, 17000–21000, and > 23000 Å. In particular, PaV (12821.6 Å) andPaW (10941.1 Å) are at the edges of these regions. Telluric contamination introducessignificant noise and distorts the continuum. Molecfit applied minimal corrections in543.2. X-SHOOTER DATAthe regions of HU and BrW because they lie in portions of the spectra that are relativelyfree of telluric features, as can be seen in Fig. 3.2.5000 7500 10000 12500 15000 17500 20000 22500 25000Wavelength [Å]-1e-140e+001e-142e-143e-14Flux [ergcm2 s1 Å1 ]Molecfit RawFigure 3.2: The raw and telluric feature corrected X-Shooter spectra. The pink shaded bars showthe region of the four hydrogen lines extracted from the spectra. From left to right, these are HU,PaW, PaV, and BrW.3.2.2 Continuum and Gaussian fittingOnce the telluric corrections had been applied, the regions of the hydrogen transitions(_0 ± 60 Å) of interested were extracted from the full spectra. The line profiles werethen further reduced by fitting the nearby continuum and creating a Gaussian fit to theprofiles. The continuum fit is used to normalise the continuum to unity and remove anycontinuum features. Gaussian rather than spline fits were used for the emission profile toavoid problems with noise and to conform to standard data reduction methods.55CHAPTER 3. OBSERVATIONSThe profiles were normalised by fitting a third-order polynomial to the nearbycontinuum. A demonstration of the continuum fit can be seen in the top panel of Fig. 3.3.The grey areas of the figure indicate the regions that were excluded from the fitting routine.An interactive Python routine was programmed so that up to three separate areas couldbe selected (by clicking on the figure) to fit the continuum; this allowed line features to beexcluded from the fit.For the line profile fit, up to fourGaussianswere used. Amaximumof twoGaussianswere used to fit the main emission feature, while a further two were used to fit the sub-continuum absorption. The fitting was done using the astropy.modeling module. Themiddle panel of Fig. 3.3 shows an example Gaussian fit and the central region (white) towhich the Gaussian curves were fitted. Up to two such regions could be specified duringthe process to avoid fitting the Gaussians to spectral contaminates. The routine prioritisedreproducing the line width and relative peak intensity.The Gaussian fit was then used to determine several characteristics of the lineprofile. These were: full width at half maximum (FWHM), half width at 10% maximum(HW10%), equivalent width (,_), relative peak intensity, and the position of the line peak.Fig. 3.3 gives an example of the FWHM, HW10% and ,_. The equivalent width is thewidth of a rectangle with a height from the continuum to zero that has an area equal to theintegrated spectral line. Absorption lines are taken to have positive equivalent widths andemission as negative. In Fig. 3.3, the equivalent width is denoted by a vertical blue hashedbar (the full height is not shown), the area of which is equal to the blue hashed area underthe line profile. The plotted bar has a width of 37.8 kms−1, however, traditionally theequivalent width is given in the units of wavelength, and hence it is shown as,_ = −2.7Å.3.3 Presentation of hydrogen spectraThe extracted and reduced hydrogen spectra from X-Shooter can be seen in Fig. 3.5. Thefigure shows the normalised line profiles for the target stars arranged in order of HU peakintensity. Each column displays the spectra for the individual objects listed in Table 3.1 atfour different wavelengths: HU, PaV, PaW, and BrW.Veiling is the ratio of excess flux to photospheric flux. The excess flux from theaccretion and shock zones changes the measured equivalent widths of the optical andinfrared lines (Folha & Emerson 1999). The excess emission increases the continuumlevel and because the continuum level is used to normalise the line profiles, a highercontinuum level will result in weaker relative emission and absorption. The scattering oflight by the interstellar medium (ISM) is known as ISM extinction. The ISM extinctionhas a greater effect on shorter wavelengths, leading to a reddening of the observed spectra;563.3. PRESENTATION OF HYDROGEN SPECTRA0.0 0.2 0.4 0.6 0.8 1.0Velocity [kms 1]0.00.20.40.60.81.0Figure 3.3: A diagram showing the continuum fit (top) to an observed profile, the Gaussian fit(middle) and extracted profile characteristics (bottom). Greyed areas in the top and middle panelsindicate sections not used for fitting. The vertical blue hashed bar in the bottom panel indicatesthe equivalent width, for the purpose of clarity the full height is not shown and the bar is shiftedup vertically.the ratio of long to short-wavelength emission is increased by the extinction. The ISMextinction will influence the observed spectra fluxes. The star’s Cardelli et al. (1989)extinction law reddening corrections are given in Table 3.1. However, the effects ofveiling and extinction are not addressed in this thesis. Instead, the focus is on the linemorphology and width; characteristics independent of veiling and extinction.The strengths of the HU and infrared lines do not correlate well, as can be seenin Fig. 3.4. The figure shows the Spearmans’s rank correlation matrix of the equivalentwidth’s between the different transitions; values close to unity suggest a good correlation.The different infrared lines correlate strongly with each other, but poorly with HU. How-57CHAPTER 3. OBSERVATIONSH Pa Pa BrHPaPaBr10.41 10.42 0.97 10.33 0.91 0.91 10.4 0.5 0.6 0.7 0.8 0.9 1.0Spearman's rank correlation coefficientFigure 3.4: The correlation matrix between the equivalent widths of the different hydrogentransitions. The values are the Spearman’s rank correlation coefficient, values close to unitysuggest a strong correlation.ever, the disparity seen in Fig. 3.5 cannot be accurately quantified because the effects ofveiling and extinction have not been accounted for. Some stars with strong HU emission,e.g. Cha Ha 6, CID 0223980048 (ESO-HA 442), and Ass Cha T 2-33 have negligibleemission in the Paschen and Brackett lines. The lack of BrW lines in some spectra, whilethe PaV and PaW are in emission, indicates that for these stars the population of level sevenis negligible compared to the populations of five and six. Using the Boltzmann and Sahaequations, we can calculate the hydrogen temperature at which the fraction in state seven iscommensurate with states six and five to be ≈ 3700 K and ≈ 4500 K, respectively. We canthen surmise that the PaV and PaW emission from these stars are emitted from hydrogen ata temperature of < 4500 K. However, temperatures this low cannot explain the observedHU emission lines (Muzerolle et al. 1998b).Four of the observed stars are classified as weak line T Tauri stars, according to theclassification scheme of White & Basri (2003), because their half widths at 10 percentmaximum (HW10%) are less than 135 kms−1. The stars are RECX-6, Ass Cha T 2-45a,Ass Cha T 2-4, and RECX-12. In this work, I adopt the HW10% criterion because White& Basri (2003) showed it to be a reliable method of distinguishing between classical andweak line T Tauri stars and the HW10% criterion is unaffected by veiling.In § 3.3.1, I discuss the classification and distribution of the different line profilestructures. In § 3.3.2, I analyse the widths and general characteristics of the lines.583.3. PRESENTATION OF HYDROGEN SPECTRA125Ass Cha T 2 23Ass Cha T 2 49ESO Ha 562Ass Cha T 2 12Cha Ha 6VW ChaAss Cha T 2 35Hn 5CID 0223980048DG TauSz 22Ass Cha T 2 52TW ChaHIQ Tau1.01.8Pa12Pa-50005001.01.3Br0.00.20.40.60.81.0Velocity [kms1 ]0.00.20.40.60.81.0Relative FluxFigure3.5:ThenormalisedX-Shooterspectraoftheobjects(columns)listedinTable3.1,atfourdifferentwavelengths:HU;6565Å,PaW;10938Å,PaV;12818Å,andBrW;21655Å.Therelativefluxisplottedasafunctionofvelocity,andeachcolumnhasawidthof±500kms−1 .ThestarsarearrangedfromlefttorightbyHUpeakintensity.59CHAPTER 3. OBSERVATIONS1 8KV MonAss Cha T 2 11Ass Cha T 2 38CT Cha ACR ChaRECX 9RECX 12RECX 6CHX18NAss Cha T 2 4Ass Cha T 2 24Ass Cha T 2 33v409 tauAss Cha T 2 45aHV354 Mon1.01.4Pa1.01.5Pa-50005001.01.2Br0.00.20.40.60.81.0Velocity [kms1]0.00.20.40.60.81.0Relative FluxFigure3.5:Continued.ThestarsarearrangedfromlefttorightbyHUpeakintensity.Notethedifferenty-axisrange.603.3. PRESENTATION OF HYDROGEN SPECTRATable 3.2: The half width at 10% maximum (HW10%) of the star’s HU line profiles. Those starswith a width less than 135 kms−1 are classified as weak line T Tauri stars (WTTS) according to theapproach of White & Basri (2003).Name HW10% WTTS[kms−1]ESO-Ha-562 197.1IQ-Tau 187.3V354-Mon 178.2T-33 214.7T-11 191.0VW-Cha 209.3V409-Tau 283.3CR-Cha 220.8T-52 280.7T-38 219.2KV-Mon 228.9Sz-22 301.3T-4 125.0 YDG-Tau 271.3T-23 160.2RECX-12 85.7 YCha-Ha-6 136.8T-12 159.8CT-Cha-A 272.8ESO-HA 442 166.3CHX18N 192.2TW-Cha 184.3RECX-6 63.1 YT-49 172.3T-35 205.5T-45a. 92.8 YRECX-9 149.1T-24 218.2Hn-5 161.23.3.1 Reipurth classificationThe hydrogen line profiles presented show a broad spectrum of different complex struc-tures. The profile morphologies are important because they provide insight into thephysical processes present in the T Tauri stars. Reipurth et al. (1996) devised a schemeto classify the HU T Tauri line profiles that not only simplified the data but also reflectedthe underlying physics of the spectra. The system established by Reipurth et al. (1996)characterises the spectra by the position and strength of a secondary peak relative to61CHAPTER 3. OBSERVATIONSthe primary peak. The interplay between the peaks is assumed to be the product of asymmetric emission line and one or more absorbers.Following the scheme described by Reipurth et al. (1996), the emission line profileswere classified as follows.• Type I: symmetric profiles with few or no features.• Type II: two peaks, the second greater than half the strength of the primary.• Type III: two peaks, the second being less than half the strength of the primary.• Type IV: sub-continuum absorption with no further significant emission beyond theabsorption.The types are subdivided into B and R because the secondary peak or absorption featurecan be either blue or red-shifted relative to the primary peak. The classifications areillustrated in the top panels of Fig. 3.6. The types IVB and IVR are equivalent to the P-Cygni and inverse P-Cygni profiles, respectively. The classification scheme is extended tothe infrared lines because similar structures are seen for the higher lines and the underlyingphysics is assumed to be the same. Furthermore, the system has been used to categoriseT Tauri spectra in previous studies, for example, Folha & Emerson (2001) and Kurosawaet al. (2006).Fig. 3.6 shows the Reipurth classification for the observations. Line profiles wereclassified by eye. The seven types of profile morphology defined by Reipurth et al. (1996)are seen in the data. For spectra in which the Reipurth type was not easily determineddue to noise levels, the profiles were not classified and excluded from the results ofthis paper. Therefore, these unclassified spectra introduce uncertainty into the Reipurthdistribution. The number of unclassified spectra are: 0 HU, 10 PaV, 14 PaW, and 18BrW. The sample of T Tauri stars is comparatively small, and a significant proportion ofthe infrared lines could not be classified. Consequently, the statistical significance of thedistribution is ambiguous; however, it exhibits a similar distribution of morphologies asthose seen in previous T Tauri studies (e.g. Folha & Emerson 2001; Reipurth et al. 1996).The distribution concurs with previous results showing that the majority (≈ 43 per cent)of profiles are type I and the second highest population of Reipurth types for the infraredlines are type IVR.There are several cases of inverse P-Cygni profiles in our data, the majority of whichare seen for PaV. There are two notable stars that contain prominent inverse P-Cygnisignatures across all infrared lines. Ass Cha T 2-35 has strong inverse P-Cygni absorptionin all of the infrared lines, whereas the HU line only displays a red-shifted secondary623.3. PRESENTATION OF HYDROGEN SPECTRApeak. Ass Cha T 2-49 has a narrow inverse P-Cygni absorption in the infrared lines and nosignificant HU red-shifted absorption. The single example of blue-shifted sub-continuumabsorption is seen in the HU spectra of Ass Cha T 2-52, which has a narrow P-Cygniabsorption profile consistent with a disk wind (Kurosawa et al. 2011). The infrared linesof Ass Cha T 2-52 exhibit no discernible blue-shifted absorption. None of the infraredlines exhibit discernible blue-shifted absorption.IV B III B II B I II R III R IV R0246810H Pa Pa BrIV B III B II B I II R III R IV R0.0 0.2 0.4 0.6 0.8 1.0Reipurth Class0.00.20.40.60.81.0# of ProfilesFigure 3.6: The distribution of Reipurth classification from the X-Shooter spectra. The results aresubdivided by hydrogen emission line and example profiles are shown along the top.3.3.2 Line profile characteristicsFig. 3.7 shows the distribution of the central velocity of the primary peak, FWHM,HW10%, and the equivalent width as histograms subdivided by different categories of63CHAPTER 3. OBSERVATIONSFigure 3.7: Histograms of the velocity of the line peak, the FWHM, the HW10%, and theequivalent width. The histograms are subdivided into the following categories: HU, infrared lines,type IV, and types I–III.observed spectra. The mean values for these subdivisions are shown in Table 3.3, alongwith the associated standard deviation. Generally, the HU profiles are wider than theinfrared lines and have a greater equivalent width. The distribution of widths for HUis wider than for the infrared lines and the subsets have a large overlap. The centralvelocity of the primary peak is on average slightly (−8 kms−1) blue-shifted from the restvelocity for the infrared lines. The PaV and BrW profiles presented by Folha & Emerson(2001) exhibited similar FWHM distributions and generally had slightly blue-shifted peakvelocities. The mean HU peak velocity is 35 kms−1, however, the distribution is bifurcatedinto two groups, one blue-shifted from the rest velocity and the other red-shifted.The width of types I–III covers a broad range, but type IV profiles are on average643.3. PRESENTATION OF HYDROGEN SPECTRATable 3.3: The mean values of the primary peak’s central velocity, FWHM, HW10%, and equiv-alent width for four different subdivisions of the observed spectra. The standard deviation of eachvalue is shown in brackets.Category Peak Vel. FWHM HW10% ,_kms−1 kms−1 kms−1 ÅType IV -6 (60) 164 (74) 139 (59) -8 (17)Type I–III 9 (51) 184 (75) 170 (57) -17 (26)HU 35 (48) 208 (87) 191 (58) -38 (31)PaV, PaW, BrW -8 (45) 169 (62) 157 (52) -3 (3)narrower by ≈ 26 kms−1. The mean peak velocity is blue-shifted for type IV profilesand red-shifted for types I–IV, although they have similar standard deviations of 59 and57 kms−1, respectively. It is expected that the type IV profiles are narrower and blue-shifted because the inverse P-Cygni features reduce the line width, centring the emissionslightly blueward.SummaryThis section presented the hydrogen line profiles of 29 different T Tauri stars observedusing the X-Shooter instrument at the VLT. I explored the general characteristics of theprofiles and saw that the HU line profiles were poorly correlated with the near-infraredlines of PaV, PaW, and BrW. At least four of the stars can be classified as weak line T Taurifrom the HU half-width at 10%.Using the line profile classification scheme set out by Reipurth et al. (1996), I foundthat our data produced a distribution of Reipurth types similar to previous studies ofhydrogen lines. The most common type in our observations is type I for both HU and theinfrared lines. The infrared lines have the highest proportion of inverse P-Cygni profiles(type IV R), and conversely, do not exhibit P-Cygni profiles (type IV B). This is becausethe generally low optical depth of the infrared lines creates absorption features only inhigh-density regions, such as the accretion funnel.In the next chapter, I compare the observations to a grid of synthetic line profilesproduced using the radiative transfer model (Chapter 2). The ensemble of the syntheticline profiles is compared with the general profile characteristics observed in this chapter,and it is seen that several discrepancies exist.65Chapter 4A Comparison of Synthetic andObserved SpectraIn this chapter, I present a grid of synthetic line profiles (§ 4.1) produced usingTORUS andthe T Tauri model discussed in Chapter 2. Line profiles were produced using a set ofparameters (see Table 4.1) intended to broadly span the range of predicted accretion ratesand temperatures in T Tauri stars, based on observational constraints (e.g. Muzerolle et al.2001).The radiative transfer models have many unconstrained parameters, and the lineprofiles are highly degenerate. Therefore, in this chapter, I analyse the width and mor-phology of the ensemble of synthetic line profiles. I then compare our grid to the spectraof the 29 T Tauri stars presented in Chapter 3.4.1 Synthetic line profile gridThe parameters used to create the synthetic line profile grid are described below. Themodels contain a single source star with an effective temperature of )eff = 4000 K, a massof "∗ = 0.5 M, and a radius of '∗ = 2.0 R. These parameters were chosen to beconsistent with earlier work, for example, Muzerolle et al. (1998b, 2001); Kurosawa et al.(2006). Our grid covers a wide range of accretion rates of 10−7, 10−8, and 10−9 Myr−1with mass-loss rates of 0.1, 0.01, and 0.001 of the accretion rate. These mass-loss rateswere selected to be in the range required to provide a significant spin-down torque on thestar (Matt & Pudritz 2005b) and are consistent with the range of values estimated fromjets (Nisini et al. 2018). The source star parameters and the wind and magnetosphericgeometries were not varied. For each model setup, line profiles were computed for threeinclinations: 20◦, looking down the pole through the wind; 60◦, just outside of the wind’s4.1. SYNTHETIC LINE PROFILE GRIDopening angle; and 80◦, looking along the disk through the magnetosphere. The lineprofiles were computed for the four hydrogen transitions presented in Chapter 3 of HU,PaV, PaW, and BrW. The full list of parameters and their respective values can be found inTable 4.1.Fig. 4.1 shows a subset of line profiles in the grid; only models with a wind temper-ature of 8000 K are included. The full atlas of the line profile can be seen in Appendix A.The figure arranges the line profiles by accretion rate and maximum magnetospheric tem-perature, subdivided by the hydrogen transition. The synthetic lines exhibit a broad rangeof different morphologies. HU has the highest peak intensity, and the subsequent strongestlines are PaV, PaW, and BrW, respectively. However, ≈ 7 per cent of the BrW lines at aninclination of 20◦ have a greater peak intensity than PaV and PaW. These lines occur foran accretion temperature of 9500 K and an accretion rate of 10−7 Myr−1. In general, theHU lines are wider than the infrared lines and some have broad wings > 1000 kms−1. Themajority (≈ 83 per cent) of the infrared lines display red-shifted sub-continuum absorp-tion. Blue-shifted sub-continuum absorption is not seen at high inclination, where theviewing angle is along the disk outside of the wind opening angle.The general trends seen in the grid are outlined as follows. Higher accretion ratesincrease the line intensity. A high accretion rate of 10−6 Myr−1 was tested and the mod-els produced structured HU absorption superposed on a broad (±1000 kms−1) emissionprofile. Only a limited number of models were run with an accretion rate of10−6 Myr−1and the profiles are not included in this work. The line profiles intensity also increaseswith higher accretion flow temperature. The HU lines are dominated by Stark broadeningfor accretion flow temperatures of at least 7500 K at accretion rates of 10−8 Myr−1. Theeffect of Stark broadening is that the line width increases with temperature and accretionrate. For a temperature of 9500 K and an accretion rate of 10−7 Myr−1, the HU profilesare Stark broadened to havewings greater than±1000 kms−1. In contrast, the infrared linesare almost completely unaffected by Stark broadening. The intensity and equivalent widthof the infrared lines increases significantly with greater accretion rates and temperature.On the other hand, the full width at half maximum (FWHM) is less strongly affected bythe accretion rate and temperature. The mean equivalent widths of the synthetic profilesare HU, −34.5; PaV, −2.3; PaW, −0.6; and BrW, −2.3.HU exhibits inverse P-Cygni profiles for the accretion rates of 10−8 Myr−1 when)acc ≤ 7500 K and for 10−9 Myr−1 when )acc ≤ 8500 K. For accretion rates andtemperatures above this, the Stark broadening smooths over the absorption features. Forthe infrared lines, the inverse P-Cygni profiles are exhibited irrespective of the inclination,accretion rate, or temperature.Fig. 4.2 shows the distribution of line peak velocities for the four different hydrogen67CHAPTER 4. A COMPARISON OF SYNTHETIC AND OBSERVED SPECTRATable4.1:ParametersofthegridofsyntheticmodelsParameterValueUnitDescription"∗0.5MStarmassStar'∗2.0RStarradius)eff4000KEffectivestellartemperatureAccretionFunnel'mi2.2'∗Truncationradiusandinnermagnetosphereconnectionpoint'mo3.0'∗Outerconnectionpointofmagnetosphere)acc6500,7500,8500,9500KMaximumtemperatureofmagnetosphere¤"acc10 −7,10 −8,10 −9M yr −1Magnetosphericaccretionrate+rot0.0kms −1RotationvelocityΘsw50degStellarwindopeningangleEmin10kms −1VelocityofwindatlaunchStellarEmax400kms −1MaximumwindvelocityWindV2.89Windvelocityscaling¤"sw0.1,0.01,0.001¤"accMass-lossrateasafractionofaccretionrate)sw6000,8000,10000KIsothermalwindtemperatureSyntheticinc20,60,80degCo-latitudeofsyntheticprofileviewingangleProfile_6562,10938,12818,21655ÅObservedwavelengths:HU,PaW,PaV,andBrWrespectively684.1. SYNTHETIC LINE PROFILE GRID10705H500050012Pa12Pa500050012Br1080.51H50005000.51Pa0.51Pa50005000.51Br6500 K1090.51H50005000.9751Pa0.9751Pa50005000.9751Br05H500050012Pa12Pa500050012Br0.511.5H50005000.51Pa0.51Pa50005000.51Br7500 K0.751H50005000.9751Pa0.9751Pa50005000.9751Br010H500050012Pa12Pa500050012Br510H500050012Pa12Pa500050012Br8500 K0.7511.25H50005000.9751Pa0.9751Pa50005000.9751Br010H500050024Pa24Pa500050024Br1020H5000500123Pa123Pa5000500123Br9500 K2.55H500050011.25Pa11.25Pa500050011.25Br0.00.20.40.60.81.0Velocity [kms1 ]0.00.20.40.60.81.0Relative Flux206080Figure4.1:AsampleofthelineprofilesproducedbyTORUS.Thespectraareshownfor¤ "sw=0.1¤ "accand)sw=8000K.Thelinesarearrangedby¤ "acc(columns)and)acc(rows).Eachsectionissubdividedbythehydrogentransitionandshowsthethreedifferentinclinations(colours).They-axesofPaW,PaV,andBrWforeachsetshareacommonrange.IPCprofilesarepresentinmostinfraredlines.Forhydrogentransitionsataninclinationof80◦(outsideofthestellarwindopeningangle),thereislessblue-shiftedabsorptionandtherearenoP-Cygnifeatures.HUprofilesarebroad,withsomewingsexceeding500kms−1 .69CHAPTER 4. A COMPARISON OF SYNTHETIC AND OBSERVED SPECTRA1070 5H50005001 2Pa1 2Pa50005001 2Br1080.75 1H50005000.975 1Pa0.975 1Pa50005000.975 1Br6500 K1090.8 1H50005000.99975 1Pa0.99975 1Pa50005000.99975 1Br2.5 57.5H50005001 2Pa1 2Pa50005001 2Br11.5H50005000.9 1Pa0.9 1Pa50005000.9 1Br7500 K0.75 11.25H500050011.002Pa11.002Pa500050011.002Br5 10H50005001 2 3Pa1 2 3Pa50005001 2 3Br5 10H50005001 2Pa1 2Pa50005001 2Br8500 K0.5 11.5H50005000.99 11.01Pa0.99 11.01Pa50005000.99 11.01Br5 10H50005002.5 5Pa2.5 5Pa50005002.5 5Br0 20H50005001 2 3Pa1 2 3Pa50005001 2 3Br9500 K2.5 5H500050011.25Pa11.25Pa500050011.25Br0.00.20.40.60.81.0Velocity [kms1]0.00.20.40.60.81.0Relative Flux206080Figure4.1:Continued.Modelspresentedherefor¤"sw=0.01¤"acc and)sw=8000K.704.1. SYNTHETIC LINE PROFILE GRID1072.557.5H500050012Pa12Pa500050012Br1080.7511.25H500050011.002Pa11.002Pa500050011.002Br6500 K1090.97511.025H500050011.00002Pa11.00002Pa500050011.00002Br510H500050012Pa12Pa500050012Br12H50005000.91Pa0.91Pa50005000.91Br7500 K0.7511.25H500050011.002Pa11.002Pa500050011.002Br510H5000500123Pa123Pa5000500123Br510H500050012Pa12Pa500050012Br8500 K0.511.5H50005000.9911.01Pa0.9911.01Pa50005000.9911.01Br2.557.5H50005002.55Pa2.55Pa50005002.55Br020H5000500123Pa123Pa5000500123Br9500 K2.55H500050011.25Pa11.25Pa500050011.25Br0.00.20.40.60.81.0Velocity [kms1 ]0.00.20.40.60.81.0Relative Flux206080Figure4.1:Continued.Modelspresentedherefor¤ "sw=0.001¤ "accand)sw=8000K.71CHAPTER 4. A COMPARISON OF SYNTHETIC AND OBSERVED SPECTRA100 80 60 40 20 0 20 40 60Peak Velocity [M yr 1]020406080100CountBr Pa Pa HFigure 4.2: The distribution of synthetic line centres for the different hydrogen transitions. Theinfrared line centres are blue shifted, whereas the HU line centres are bi-modal with both red andblue-shifted peaks.transitions. The infrared lines havemean peak velocities of−20.6,−16.5, and−14.5 kms−1for PaW, PaV, and BrW, respectively. Whereas, the mean peak velocity for HU is −3 kms−1.However, the distribution is divided into two sections of red-shifted peaks with a meanof ≈ 20 kms−1 and blue-shifted peaks with a mean of ≈ −35 kms−1. The red shiftedpeaks are the product of the high accretion rate models where log ¤"acc = −7 (the units ofMyr−1 are emitted for brevity) and the blue shifted peaks of the low accretion rate models(log ¤"acc = −9). Models with an accretion rate of log ¤"acc = −8 have peak velocities fairlyevenly distributed from −60 to 40 kms−1. The effect of Stark broadening is minimal atlow accretion rates, which gives rise to greater red-shifted absorption, effectively shiftingthe peak velocity blueward.In the following sections, I present a comparison of the characteristics (§ 4.2) andthe structure (§ 4.2.1) of the ensemble of synthetic line profiles and the observed T Tauristars.4.2 Comparison of profile featuresTo analyse the synthetic and observed emission as an ensemble, I calculated the fullwidth at half maximum (FWHM), the half width at 10% peak maximum (HW10%),equivalent width ,_, the peak intensity, and the velocity of the peak intensity of thesynthetic profiles. The FWHM and HW10% were calculated by fitting the synthetic lineprofiles with a linear spline using the Python module scipy.interpolate.interp1d.724.2. COMPARISON OF PROFILE FEATURES500 0 500246H200 0 2000.81.21.6Pa200 0 2000.81.21.6Pa200 0 2000.81.21.6BrR=24900 R=18400 R=116000.0 0.2 0.4 0.6 0.8 1.0Velocity [kms 1]0.00.20.40.60.81.0Relative FluxFigure 4.3: Convolution of synthetic spectra created with a velocity resolution of ΔE =12.04 kms−1 (' = 24900) to the observed resolutions of ' = 18400 for HU and ' = 11600for PaW, PaV, and BrW.The trapezium integrator provided by scipy.integrate.trapz was used to determinethe area under the line profile to calculate ,_. The synthetic profiles are smooth withnegligible small scale fluctuations; therefore, a spline can accurately represent the lineprofile and a Gaussian fit is not required.The synthetic line profileswere calculated for velocity increments ofΔE = 12.04 kms−1,which corresponds to a spectral resolution of ' = 24900. The observations were observedat resolutions of ' = 18400 (ΔE = 16.3 kms−1) for HU and ' = 11600 (ΔE = 25.9 kms−1)for the infrared lines. I explored the effect of applying a Gaussian convolution to thesynthetic spectra to reduce their resolution to that of the observed spectra. However,the change in the morphology of the line profiles, particularly the FWHM and HW10%was negligible when the synthetic spectra were convolved. Fig. 4.3 shows the effect ofconvolving the synthetic spectra to the resolution of the observations using a Gaussian ker-nel. Therefore, the TORUS native resolution was kept and the resolution of the syntheticspectra were not convolved hereafter.Although our model grid was not set up to fit any particular star, it is informative todirectly compare the synthetic line profiles with the observed line profiles. As an example,I show the spectra of three stars (Sz 22, DG Tau, and Ass Cha T 2-52) in Fig. 4.4 anda corresponding model selected to match the HU line profiles. The models were chosenusing a j2 fit test to find the closest HU profiles from the grid. The parameters of thechosenmodels are shown in Table 4.2. Themodels use our fiducial values of)eff = 4000 Kand "∗ = 0.5M, whilst the effective photospheric temperature for Sz 22 is 4350 K, for73CHAPTER 4. A COMPARISON OF SYNTHETIC AND OBSERVED SPECTRA612Sz 22 DG TauHT-521.21.8Pa12Pa500 0 5000.91.2500 0 500 500 0 500BrTORUS X-Shooter0.0 0.2 0.4 0.6 0.8 1.0Velocity [kms 1]0.00.20.40.60.81.0Relative FluxFigure 4.4: Three of the best cases of the grid matching the observed line profiles. The modelshave been chosen because they best reproduce the observed HU height and width. The observedspectra have been adjusted to compensate for each system’s radial velocity.744.2. COMPARISON OF PROFILE FEATURESDG Tau it is 4350 K, and for Ass Cha T 2-52 it is 5110 K (Manara et al. 2016; Nguyenet al. 2012). The stellar masses for Sz 22, DG Tau, and Ass Cha T 2-52 are 1.09, 1.4,and 1.4 M, respectively (Manara et al. 2016). The observed inclinations of DG Tauand Ass Cha T 2-52 are < 30◦ (Isella et al. 2010) and 35 ± 10◦ (Hussain et al. 2009),respectively. For Sz 22 the circumstellar disk is thought to be viewed edge on, suggestinga high inclination (Schmidt et al. 2013), which agrees with the fitted model. For DGTau and Ass Cha T 2-52 the fitted models suggest a medium inclination compared to theobserved low inclinations. However, the synthetic grid is course and the fitted modelsonly crudely match the observed line profiles, especially for DG Tau and Ass Cha T 2-52.A close fit of the inclinations is not expected.A caveat to direct comparison is that the veiling of the observed lines has not beenconsidered. Whereas the effect of reddening is negated by the continuum normalisation,the influence of veiling is not. Nevertheless, Fig. 4.4 highlights a trend seen in the syntheticspectra grid. Namely, that while our method selects models with similar strengths andwidths in HU to that observed, the models fail to reproduce the observed infrared lines.The synthetic infrared lines are too structured and narrow. The tendency of the ensembleof widths being too narrow for synthetic infrared lines is discussed below.Table 4.2: The model parameters used in Fig.4.4Object Sz 22 DG Tau T-52log ¤"acc −8 −8 −8)acc 9500 K 9500 K 9500 Klog ¤"BF −9 −9 −9)BF 10000 K 10000‘K 8000 KInclination 80◦ 60◦ 60◦Fig. 4.5 compares the distributions of synthetic and observed HW10% for the fourhydrogen transitions. The upper quartile, the median, and the lower quartile are shownas horizontal dashed lines. The distribution of synthetic HU widths is much broader thanthat observed, ranging from close to zero to more than 1100 kms−1. This broad range isa result of our parameter grid sampling a relatively uniform coverage of line profiles inthis range. The right-hand figure shows a subset of our grid constrained such that onlymodels with an HU HW10% between 60–350 kms−1 are included. By constraining thesynthetic models so that the HU HW10% lies in a range similar to that observed, it is easierto compare the ensemble of line profiles across the different hydrogen transitions. Evenwith the HU distribution constrained to match the observed widths, the median width ofthe synthetic infrared line is narrower than that in the observed sample. The differencesin the median value of HW10% for the synthetic and observed PaW, PaV, and BrW lines75CHAPTER 4. A COMPARISON OF SYNTHETIC AND OBSERVED SPECTRA105011001150Full GridHPaPaBr0 50100150200250300HPaPaBr0 50100150200250300350Filtered Grid0.00.20.40.60.81.0Hydrogen Line0.00.20.40.60.81.0HW10% [kms 1]TORUSX-ShooterFigure4.5:TheHW10%distributionofthesyntheticgridandobservationsareshownforeachspectralline.Theleftpanelshowsthefullgrid(notethesplitoftheyaxis),andtherightpanelshowsthefilteredgridwhere60≤HW10%HU<350.Horizontallinesindicatethe25th,median,and75thpercentiles764.2. COMPARISON OF PROFILE FEATURESare 96, 64, and 80 kms−1, respectively. In other words, for a range of models that predictline widths of HU in the same range as observations, the predicted widths for the infraredlines are systematically too narrow by ≈ 80 kms−1. In § 4.3, I explore what modificationscan be made to the models to bring these synthetic lines into better agreement with theobserved sample.Fig. 4.6 shows a plot of FWHM versus HW10% for the synthetic and observedspectra. Each point indicates a separate line profile; the colour and size indicate theaccretion rate. The White & Basri (2003) HU HW10% distinction between the weak lineand the classical T Tauri stars is shown by the vertical dotted line. For HU, the FWHMand HW10% are strongly correlated for both observations and models. Both the syntheticand the observed data show that the widths depend on the accretion rate. However, thetemperature of the accretion funnel and the viewing inclination have a significant influenceon the widths of the models, scattering the FWHM and the HW10% for a given accretionrate. The relationship between FWHM and HW10% for a single Gaussian isFWHMHW10%=2√2 ln 2√2ln 10≈ 1.1 , (4.1)displayed in Fig. 4.6 by inclined solid black lines. The observed sample is scatteredaround this relationship. The synthetic HU spectra tend to have a smaller FWHM fora given HW10% than expected from the Gaussian relationship. The deviation from theGaussian FWHM to HW10% relationship is due to Stark broadening. The synthetic HUlines take on a Voigt profile when Stark broadening is dominant (see Chapter 2.2.3). TheVoigt form has wider wings and a narrower peak than a Gaussian curve. This effect isnot observed in the higher lines, where Stark broadening is negligible. The widths of theobserved line profiles were measured using a Gaussian fit; therefore, the deviation awayfrom the expected FWHM to HW10% relationship for the observed data is due to theconvolution of the multiple Gaussians components used.For PaW and BrW, the observations exhibit a tighter correlation between the FWHMand HW10% than for PaV. The PaV observations exhibit a larger scatter, suggesting adivergence away from the Gaussian form and more structure in the line profiles. This iscorroborated by the fact that PaV has the highest frequency (63 per cent) of non-type Iprofiles (see § 4.2.1). In contrast to HU, the FWHM and HW10% of the infrared lines inthe observations and models exhibit little correlation with the accretion rate. The modelsshown in Fig. 4.6 have been selected so that the range of HW10% widths for HU roughlycorresponds to the range of observational data. However, these models predict widths forthe three higher lines that are, on average, much narrower than the observed widths. Ingeneral, for an accretion rate of log ¤"acc = −7 the synthetic lines have a lower ratio of77CHAPTER 4. A COMPARISON OF SYNTHETIC AND OBSERVED SPECTRA0150300H080160240Pa0 50 100 150 200 250 300 350 4000100200300Pa0 50 100 150 200 250 300 350 4000100200300BrTORUS X-Shooter-11-10-9-8-7log Macc0.0 0.2 0.4 0.6 0.8 1.0HW10% [kms 1]0.00.20.40.60.81.0FWHM [kms1 ]Figure 4.6: The FWHM vs. HW10% for observed (black outlined stars) and TORUS data(coloured hexagons). The colour denotes the accretion rate. The vertical dashed line shows the HUsplit between WTTS and CTTS (White & Basri 2003). The diagonal black line is the analyticalGaussian relationship between the FWHM and the HW10%.784.2. COMPARISON OF PROFILE FEATURESFWHM/HW10% than other accretion rates.Fig. 4.7 compares the ratio of the peak PaV to BrW versus the accretion rate for thesynthetic and observed line profiles. Note that all synthetic PaV and BrW profiles in ourcomplete grid are included in this figure. Because our T Tauri observations (see Chapter 3)are not corrected for veiling and ISM extinction, there is an uncertainty to the line ratios.Therefore, I have also included the line ratios of a sample of classical T Tauri stars fromMuzerolle et al. (2001) and Muzerolle et al. (1998a). The observed line ratios of bothdata sets are in accord and typically have values between three and seven. Furthermore,the ratios are constant across a wide range of accretion rates (log ¤"acc = −6.3 to −9.4).The synthetic profiles show that as the accretion rate decreases, the ratio of PaV to BrWincreases because the emission of BrW decreases more rapidly. To duplicate the line ratiosobserved, higher temperatures are needed at lower accretion rates, which was also seen byMuzerolle et al. (2001). The constant line ratio observed across the range of accretion ratesindicates that there may be a mechanism for increasing the temperature of the accretionflow as the accretion rate decreases.4.2.1 Reipurth classificationFollowing the framework given by Reipurth et al. (1996) and outlined in Chapter 3.3.1,the synthetic line profiles were classified using a Python routine. The code worked byfinding peaks and troughs using a gradient search and using them to classify the lineswith the following order of precedence: IV, I, II, and III. The programme defines typeIV as being a sub-continuum absorption with no further emission greater than 0.01 ofthe maximum line flux. If the profile contains both red and blue-shifted absorption, thedeepest feature is used to classify the line. Many profiles have type II or III features alongwith type IV. To avoid any classification bias, the synthetic spectra were classified usingthe same order of precedence as the X-Shooter observations.Fig. 4.8 displays the results of the classification of synthetic spectra as hashed barsthat overlay the distribution of the observed Reipurth types. The distribution of types forthe synthetic HU lines shows a distribution similar to the observations, with the majorityconcentrated in type I. The HU distribution differs significantly from the observations fortype IVR. Inverse P-Cygni profiles are seen in 35 per cent of the syntheticHU lines; whereas,only 3.4 per cent of the observations (1 spectra) exhibit this morphology. However, the HUinverse P-Cygni profiles are only present in models that would be classified as Weak LineT Tauri stars, based on the HU 10 per cent width criteria of White & Basri (2003). IPCprofiles are seen in the spectra of one of the 29 TTauri stars (Reipurth et al. (1996) observedtwo out of a sample of 43), approximately 3.4 per cent. From Fig. 4.5, it can be seen that79CHAPTER 4. A COMPARISON OF SYNTHETIC AND OBSERVED SPECTRA9.5 9.0 8.5 8.0 7.5 7.0 6.5log Macc [M yr 1]100101102Pa/BrTacc [K]6500750085009500Figure 4.7: Ratio of PaV to BrW versus accretion rate the X-Shooter data (pink stars) the T Taurisample of Muzerolle et al. (2001) (pink circles) and the synthetic line profiles (hexagons). Thehorizontal dotted lines indicate the PaV/BrW ratios of 1, 2, and 10. The colour of the hexagonsindicate the magnetospheric temperature.despite the uniform grid of parameters, the HW10% distribution is not uniform. A greaternumber of profiles are produced with a lower HW10%, these profiles predominantly haveinverse P-Cygni features and account for the excess of type IVR profiles. Blue-shiftedsub-continuum absorption (type IV B) is relatively rare for HU and only seen in 7 per centof synthetic profiles; see § 4.2.2 for further discussion.Unlike the relative agreement seen for HU, synthetic infrared lines predict a signifi-cantly higher frequency of P-Cygni profiles; ≈ 98 per cent are of type IVR or IVB. However,across all three infrared lines, the mean frequency of IPC profiles in the X-Shooter sampleis ≈ 23 per cent and none of the infrared observations exhibit blue-shifted sub-continuumabsorption. Comparatively, Folha & Emerson (2001) saw a frequency of type IVR profiles804.2. COMPARISON OF PROFILE FEATURESIV B III B II B I II R III R IV R0.00.20.40.60.8H Pa Pa BrIV B III B II B I II R III R IV R0.0 0.2 0.4 0.6 0.8 1.0Reipurth Class0.0.20.40.60.81.0Fraction of ProfilesFigure 4.8: Relative fraction of Reipurth types. The coloured bars indicate the results for the X-Shooter observations and the hashed bars for the synthetic TORUS grid. The results are subdividedby hydrogen transition and an example profile of each type is shown at the top. The hashed barsover plotted are the classification of the corresponding hydrogen transition.of 34 and 20 per cent for PaV and PaW, respectively. Furthermore, they did not observeany type IVB profiles in their sample of 50 T Tauri stars. For the chosen parameters of ourmodel grid, the synthetic line profiles show a much higher occurrence of deep-absorptionfeatures than that present in our observed sample. I explore possible modifications to themodel in § 4.3 and further discuss them in § 4.4.The mean signal-to-noise ratio (SNR) of the X-Shooter spectra is ≈ 70 for the VISarm and≈ 108 for the NIR. These values are taken as themean over the whole VIS andNIRarms of X-Shooter. The mean ratio of the peak emission to the depth of the sub-continuumabsorption (EAR) for the synthetic type IV profiles is ≈ 3 and ≈ 4 for the optical andinfrared lines, respectively, see Fig. 4.9. Hence, the sub-continuum absorption features in81CHAPTER 4. A COMPARISON OF SYNTHETIC AND OBSERVED SPECTRA0 60 120 180 240 300 360SNR061218NIRVIS0 3 6 9 12 15 18EAR0100200300HPaPaBr0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0CountFigure 4.9: Left-hand panel shows the distribution of SNR for the VIS and NIR arms of X-Shooter.The right-hand panel shows the distribution of the ratios of the peak emission to the depth of thesub-continuum absorption (EAR) for the different hydrogen wavelengths in the synthetic data.the synthetic grid are 18 – 25 times greater than the observed noise level. Consequently,the effect of the noise is not expected to significantly influence the Reipurth classificationsof the synthetic profiles.4.2.2 P-Cygni profilesBlue-shifted sub-continuum absorption (a P-Cygni profile) is created in our model by theoutflowing gas and is a good tracer of the stellar wind. In the observed T Tauri stars,there is a single P-Cygni profile for the HU spectra of Ass Cha T 2-52. As can be seen inFig. 4.8, the frequency of P-Cygni profiles in our grid is ≈ 10 per cent. The blue-shiftedsub-continuum absorption features in our grid are broad (≈ 100 kms−1), consistent withline profiles produced by the stellar wind models of Kurosawa et al. (2011).P-Cygni profiles are seen in our grid at all three inclination of 20◦, 60◦, and 80◦,but the majority is seen at 20◦. HU P-Cygni profiles are primarily (92 per cent) seenfor an inclination 20◦; the other eight per cent at an inclination of 60◦. The depth ofthe blue-shifted absorption increases with the wind temperature and the largest fractionof P-Cygni profiles are seen with a wind temperature of 104 K. The line emission peakis inversely correlated with the wind temperatures. This is particularly apparent for theinfrared lines because the higher energy levels five, six, and seven become depopulatedbefore the upper HU level (three). This inverse relationship does not hold at a high wind824.2. COMPARISON OF PROFILE FEATUREStemperature (2×104 K), where the wind emission dominates the emission of the accretionfunnel. Fig. 4.10 presents the fraction of synthetic lines that exhibit P-Cygni profiles as-7 -8 -9-8-9-10-11-12logMsw [Myr1 ]0.00 0.05 0.10 0.15 0.20 0.25Frac of PC-7 -8 -90.000.150.30 H-7 -8 -90.000.150.30 Pa-7 -8 -90.000.150.30 Pa-7 -8 -90.000.150.30 Br0.0 0.2 0.4 0.6 0.8 1.0log Macc [M yr 1]0.00.20.40.60.81.0Frac of PC10% Macc 1% Macc 0.1% MaccFigure 4.10: The right hand panels is a heatmap showing the total fraction of P-Cygni profilesfor accretion rate versus mass-loss rate in the synthetic grid. The diagonal lines indicate constantmass-loss rates relative to the accretion rate. The left hand panel shows the fraction of P-Cygniprofiles for different mass-loss rate subdivided by hydrogen transition.a function of the rates of accretion and mass loss. It can be seen that the frequency ofP-Cygni profiles is correlated with the ratio of ¤"BF/ ¤"acc and not ¤"BF alone. For example,at a high accretion rate of log ¤"acc = −7, P-Cygni profiles are only observed at mass-lossrates of log ¤"BF = −8 and −9. Whereas, for an log ¤"acc = −8 P-Cygni profiles are seenfor mass-loss rates of log ¤"BF = −9, −10 and −11 with decreasing frequency. Similarresults are seen for an accretion rate of log ¤"acc = −9. P-Cygni profiles were produced formass-loss rates as low as 10−12 Myr−1.Additionally, Fig. 4.10 shows the fraction of P-Cygni profiles for different accretion83CHAPTER 4. A COMPARISON OF SYNTHETIC AND OBSERVED SPECTRArates at the three fractions of the mass-loss rate, subdivided by hydrogen transition.Although a mass loss of 10% ¤"acc generally produces the highest fraction of P-Cygniprofiles, at an accretion rate of log ¤"acc = −7 a mass-loss rate of 1% ¤"acc creates a largerfrequency of HU P-Cygni profiles. Amass-loss rate of 0.1% ¤"acc has the lowest proportionof P-Cygni profiles, and they are seen primarily for an accretion rate of log ¤"acc = −9. Thehigher optical depth of HU should make it a good tracer of the low-density stellar wind.However, in contrast to the expected and observed P-Cygni frequency, the HU profilesproduce P-Cygni profiles with a frequency lower than that of the infrared lines.4.3 Modified T Tauri modelThe model grid does not adequately reproduce the observed frequency of the infrared lineprofile structures or the observed range of profile widths (§ 4.2). In the following section, Iexplore the effects of a few different modifications to the models in an attempt to discoverthe possible reasons for the discrepancies. In § 4.3.1, the effects of introducing turbulentbroadening are examined. The influence of rotation on synthetic lines is introduced in§ 4.3.2. Finally in § 4.3.3, the widths are re-evaluated, considering only the blue-shiftedside, to determine the consequence of the high proportion of sub-continuum absorptionon the widths of the line profiles.4.3.1 Turbulent line broadeningIt is expected that there will be unresolved small-scale motion in the T Tauri systems on topof the bulk gas flows. For instance, from unsteady accretion or wind flow, instabilities inthe accretion, or post-accretion shock gas. The effect of Stark broadening on the infraredlines is negligible and so it was considered that the small-scale motion in conjunctionwith Stark broadening could broaden the observed lines and reduce the frequency ofsub-continuum absorption features.To explore the effect of turbulent broadening on the infrared line widths, the ef-fect of this small-scale motion was incorporated into the models. I assume that theunresolved velocity is everywhere in the grid and acts like a form of turbulence, i.e. aMaxwell–Boltzmann velocity distribution. I used a value of Eturb = 100 kms−1, which isapproximately the largest difference in the median HW10% of the observed and syntheticinfrared lines; see § 4.2. There is little physical justification for the turbulent velocity tobe as high as 100 kms−1, other than it is the required velocity to sufficiently broaden theensemble of line profiles.Fig. 4.11 shows the FWHM versus the HW10% for the Stark and turbulently broad-844.3. MODIFIED T TAURI MODEL0 50 100 150 200 250 300 350 4000250500H0 50 100 150 200 250 300 350 4000150300Pa0 50 100 150 200 250 300 350 4000150300Pa0 50 100 150 200 250 300 350 4000150300BrStark Turbulent X-Shooter-9-8-7log Macc0.0 0.2 0.4 0.6 0.8 1.0HW10% [kms 1]0.00.20.40.60.81.0FWHM [kms1 ]Figure 4.11: The effect of combining 100 kms−1 of turbulent broadening with Stark broadening.Grey squares are the original Stark only broadened profiles from TORUS , and the colouredhexagons are the Stark and turbulent broadened profiles. The grey lines connect the two differentmodels.85CHAPTER 4. A COMPARISON OF SYNTHETIC AND OBSERVED SPECTRAened lines and the Stark-only broadened lines. Similarly to Fig. 4.6, the profiles are filteredso that only models with a HU HW10% between 60 − 400 kms−1 are included. Turbulentbroadening increases the HU line too much, pushing the median value of HW10% to≈ 320 kms−1. The effect on the infrared lines is to bring the range of predicted values tobe largely overlapping with the observed data. Turbulent broadening increases the numberof synthetic lines that have HW10% greater than 400 kms−1 by 4%.The full set of turbulently broadened line profiles can be seen in Fig. 4.12. The figureshows the line profiles organised by spectral line (columns) and inclination (rows). Theeffect of the turbulence is to create a boxier profile; steep sides with flat tops. The addedturbulent velocity increases the resonant range of the transition wavelength, broadeningboth the emission and absorption features. This “top-hat” morphology is not seen in theobserved data. The distribution of the Reipurth classifications is not significantly changedby the turbulence and a high proportion of inverse P-Cygni profiles are still predicted. Theboxier profiles increase the ratio between the FWHM and the HW10%, creating a steeperrelationship than seen for a Gaussian or a Voigt profile.In conclusion, the turbulent broadening produces an infrared range of line widthsin better agreement with the observations. This modification causes the width of the HUlines to become boxy, a morphology not reflected in the observations. A smaller value ofEturb will reduce the top-hat like morphology but will not increase the width of infraredline profiles sufficiently. Furthermore, the frequency of sub-continuum absorption is notsignificantly changed, suggesting that incorporating unresolved small-scale motions intothe simulations cannot solely account for the lack of width in the infrared lines. This isespecially true since there is no strong physical motivation for the turbulent motion to beas high as 100 kms−1.4.3.2 Rotational line broadeningThe line profiles presented in § 4.1 are computed for a non-rotating T Tauri model. Inprinciple, the Doppler shift from rotational motion should broaden the line profiles. Thus,to explorewhether this improved the fit of themodel to the data, I ran a small grid of rotatingmodels with a wind velocity as prescribed in Chapter 2.3.1.4.2. The magnetosphere istreated as a solid body that rotates with the star. The models were computed at two rotationvelocities, 0.05 Ebr (13.4 kms−1) and 0.5 Ebr (134 kms−1), where Ebr is the stellar break-upvelocity, taken to be the balance of gravitational and centrifugal forces without consideringthe effects of an equatorial bulge such thatEbr =√"'−1∗ . (4.2)864.3. MODIFIED T TAURI MODEL0369H0.40.81.21.6Pa0.61.21.82.4Pa0.60.91.21.520Br03690.81.01.21.40.81.21.60.901.051.201.35601000500050010000481240020002004000.751.001.251.5040020002004000.81.21.62.040020002004001.001.251.50800.00.20.40.60.81.0Velocity [kms1 ]0.00.20.40.60.81.0Relative FluxlogMacc=-7logMacc=-8logMacc=-9Figure4.12:TurbulentlybroadenedTORUSlineprofilesarrangedbywavelength(column)andinclination(row).Theprofilesareboxywitha’top-hat’likeappearanceandthesub-continuumabsorptionfeaturesarebroadened.Notethattherangesontheaxesvary.Modelspresentedhereforaccretiontemperaturesof)acc=6500and7500K.87CHAPTER 4. A COMPARISON OF SYNTHETIC AND OBSERVED SPECTRA0 8 16 24H0.81.62.43.2Pa1 2 3 4Pa1.53.04.520Br0 5 10 151.01.52.01.01.52.02.50.81.62.4601000500050010000 4 8 1240020002004000.91.21.51.840020002004001.21.82.440020002004001.21.62.0800.00.20.40.60.81.0Velocity [kms1]0.00.20.40.60.81.0Relative FluxlogMacc =-7logMacc =-8logMacc =-9Figure4.12:Continued.Modelspresentedhereforaccretiontemperaturesof)acc=8500and9500K.884.3. MODIFIED T TAURI MODEL062036H800.81.20.81.2Pa121.21.8Pa400 200 0 200 4000.91.21.5400 200 0 200 4000.91.2Br0.0 0.2 0.4 0.6 0.8 1.0Velocity [kms 1]0.00.20.40.60.81.0Relative FluxNo rotation 5% break-up 50% break-upFigure 4.13: The effect of rotation on line profiles. The profiles are shown for three differentrotation rates: no rotation (solid line), 0.05 Ebr (pink dashed line), and 0.5 Ebr (blue dash-dottedline). The profiles are arranged by viewing inclination (columns) and wavelength (rows). At highrotation rates, the profiles are flattened and broadened.T Tauri stars are typically slow rotators (e.g. Bouvier et al. 2014). However, I includea model with a rotation rate of 0.5 Ebr so as to be able to compare the effect of anextreme case with a more realistic rotation rate. The models have an accretion rateof log ¤"acc = −7, a mass-loss rate of ¤"sw = 0.1 ¤"acc, a maximum accretion funneltemperature of )acc = 7500 K, and a stellar wind temperature of )sw = 8000 K. Fig. 4.13shows the line profiles for the rotating and the non-rotating case.The profiles are smoothed and broadened by rotation. However, this effect isnegligible unless the rotation is at a significant proportion of the stellar break-up velocity.For example, a rotation velocity of 0.05 Ebr had no effect on HU, while for PaW and PaV, thepeak intensity was reduced by ≈ 10 per cent for an inclination of 80◦. On the other hand,89CHAPTER 4. A COMPARISON OF SYNTHETIC AND OBSERVED SPECTRAa rotation rate of 0.5 Ebr significantly broadens the synthetic profiles when viewed from ahigh colatitude. The peaks are shifted away from the rest velocity as the hot, high-densitygas in the magnetosphere dominates the emission. Estimates of E sin 8 and the rotation as afraction of the break-up velocity for the X-Shooter stars are shown in Table 4.3. V354-Monhas the highest rotation rate in the data set with a velocity of E sin 8 = 40.9 kms−1, only11 per cent of the star’s break-up velocity. The rotation rates in our observed data set arenot large enough to cause significant line broadening. Therefore, rotational broadeningcannot explain the mismatch between the observed and synthetic line widths.4.3.3 Blue-shifted emissionMore than 80 per cent of the synthetic infrared line profiles have inverse P-Cygni features,significantlymore frequent than the∼ 23 per cent seen in the observations; see § 4.2.1. Thered-shifted sub-continuum absorption reduces the width of the emission profiles. Thus, ifthe model were modified to reduce the frequency of red-shifted sub-continuum absorption,we can expect that the average line widths would also increase. A possible mechanismfor removing the red-shifted absorption is by introducing a non-axisymmetric accretionflow where the accreting column only periodically intersects with the line of sight. Forexample, Esau et al. (2014) used a non-axisymmetric accretion flow to reproduce theBalmer variations seen for AA Tau successfully; this effect is discussed further in § 4.4.To estimate howmuch the inverse P-Cygni absorption might affect the line widths inour models, I calculated the HW10% values using only the blue side of each synthetic lineprofile (HW10%Blue). I employed a linear interpolation of the line that was blue-shiftedfrom the peak to determine values of the profile width. Fig. 4.14 shows the HW10%Blueversus the true HW10%. The diagonal line (black dashed) shows the one-to-one ratio;points below the diagonal indicate that the width decreased when considering only the blueside. The points are coloured, denoting the inclination of the model. For a viewing angleof 20◦, the majority of line widths decreased when only the blue-shifted emission wasconsidered. The blue sides of these modelled lines are narrower than the red because ofthe strong blue-shifted absorption from the stellar wind, which lies along the line of sightat this inclination. Fig. 4.14 also shows the distribution of the width change (Δ HW10%)as a series of histograms. The mean change in HW10% for all models with inclinationof 60◦ or 80◦ is 48 kms−1. When only models with a low mass-loss rate (0.001 ¤"acc)are considered, the average increase in width is ≈ 50 kms−1 for inclinations of 60◦ and80◦. The change in measured HW10% is consistent across all four hydrogen lines, HUis broadened similarly to the infrared lines. The mean changes in width for differentinclinations and hydrogen transitions are tabulated in Table 4.4.904.3. MODIFIED T TAURI MODELTable 4.3: Stellar rotation ratesTable showing the rotation rates for the observed X-Shooter stars. References are: 1Nguyen et al. (2012), 2 Baxter et al. (2009), 3 Frasca et al. (2015), 4 Kounkel et al.(2019), 5 Malo et al. (2014), 6 Manara et al. (2016), 7 Schneider et al. (2018), 8 Manaraet al. (2014), 9 McGinnis et al. (2015), and 10 Rugel et al. (2018).Name !∗ '∗ + sin 8 + sin 8+1A Reference[ L] [ R] [ kms−1]ESO-Ha-562 0.12 0.8 40 0.1 1,6IQ-Tau 0.75 1.9 14.4 0.05 1,6V354-Mon 1.67 1.8 40.9 0.11 2,7T-33 0.69 1.1 16.5 0.04 3,6T-11 1.45 1.7 14.1 0.04 3,8VW-Cha 1.64 2.6 13 0.04 1,6V409-Tau 1.17 1.5 34 0.09 1,8CR-Cha 3.26 2.3 35 0.09 1,6T-52 2.55 2.0 28 0.08 1,6T-38 0.13 0.8 18.7 0.05 1,6KV-Mon ∼ ∼ 24.25 ∼ 4,9Sz-22 0.51 1.3 ∼ ∼ 6T-4 0.43 1.3 12.4 0.03 1,6DG-Tau 1.55 2.2 24.7 0.07 1,6T-23 0.32 1.8 9.6 0.05 3,6RECX-12 0.23 1.4 6.4 0.03 5,10Cha-Ha-6 0.07 1.0 ∼ ∼ 6T-12 0.15 1.3 10.7 0.06 1,6CT-Cha-A 1.5 2.2 7.8 0.02 1,6ESO-HA-442 ∼ ∼ ∼ ∼CHX18N 1.03 1.4 26.5 0.07 1,6TW-Cha 0.38 1.2 11.3 0.03 1,6RECX-6 0.1 0.9 ∼ ∼ 10T-49 0.29 1.6 8.2 0.04 1,6T-35 0.33 1.2 21 0.05 1,8T-45a 0.34 1.2 12.4 0.03 1,6RECX-9 0.095 1.1 ∼ ∼ 10T-24 0.4 1.4 10.5 0.03 1,6Hn-5 0.05 0.8 7.8 0.04 1,6The above analysis suggests that if the models were modified in some way to reducethe prevalence of inverse P-Cygni absorption, the synthetic line profiles would becomesignificantly wider by an average of ≈ 48 kms−1. A mean increase of ≈ 80 kms−1 (see§ 4.2) is needed to shift the HW10% of the synthetic infrared lines into agreement with theobservations. Therefore, while this effect cannot account for the whole disparity of widthson its own, when combined with other mechanisms, such as turbulence and rotation, it91CHAPTER 4. A COMPARISON OF SYNTHETIC AND OBSERVED SPECTRATable 4.4: The mean differences between HW10% and HW10%Blue.Inclination HU PaW PaV BrW[kms−1]20◦ −0.7 −5.9 −4.3 −5.760◦ 40.1 40.7 39.8 41.780◦ 52.8 57.4 54.1 58.4may be sufficient.4.4 DiscussionA significant caveat to our comparison of the observed and synthetic line profiles is thepossibility that our observed data are incongruous and not representative of the consensusof T Tauri stars. However, our sample of 29 stars appears to be consistent with otherobservational data. For example, the sample has a fraction of Reipurth classes similar tothe distributions reported by Reipurth et al. (1996) for HU and Folha & Emerson (2001) forPaV and BrW. Furthermore, the peaks of the PaV and BrW lines are slightly blue shifted, inagreement with Folha & Emerson (2001), although the mean FWHM of our line profilesis smaller than the mean FWHM reported by Folha & Emerson (2001). The lines profilesobserved showed little correlation between the strength of HU and the strength of theinfrared lines. No correction for veiling has been applied to the spectra in Fig. 3.5 makingthis result difficult to interpret. However, some stars, e.g. Cha Ha 6, have HU emissionbut no infrared emission. This demonstrates that the fourth electron level is relativelypopulated in some stars, while the gas does not have a significant proportion of hydrogenwith electrons in levels five, six and seven.A second caveat is that our grid of synthetic line profiles was not adjusted toagree with any particular observation; a singular stellar mass, radius, and temperaturewere used. Broadly speaking, the synthetic line profile morphology is determined by themagnetosphere’s geometry, accretion rate, and temperature. Consequently, our grid wasdesigned to encompass the broad range of parameters predicted for T Tauri accretion andoutflow (e.g. Hartmann et al. 1994; Muzerolle et al. 1998b, 2001; Kurosawa et al. 2006;Lima et al. 2010; Kurosawa & Romanova 2012). In turn, our T Tauri sample has anextensive range of accretion rates and stellar parameters, making it an appropriate initialdata set for a comparison with the ensemble of synthetic spectra.Our models predict a greater number of HU inverse P-Cygni profiles than observed.However, our grid of models can reproduce the observed fraction of HU Reipurth typesafter the lines classified as weak line T Tauri stars (≈ 40 per cent) are removed. Themodels924.4. DISCUSSION0250500750100003006009008060200501001502000100200300050100150200010020030004080120HW10% [kms1 ]0801602404004080H4004080Pa4004080Pa4004080 HW10%Br0.00.20.40.60.81.00.00.20.40.60.81.0HW10%Blue [kms1]Figure4.14:ComparisonofthetrueHW10%andhalfwidthmeasuredonlyforblue-shiftedemission.Thedasheddiagonallinehasagradientofunity;thepointsaboveitarebroaderwhenonlytheblue-shiftedemissionisconsidered.Thecolourandstyleofthepointsdenotetheinclinationofthemodel.Ontherightarehistogramsshowingthedifferenceofthetwomeasuredwidths.Theverticallinesindicatetherespectivemeans:20◦dotted,60◦dashed,and80◦solid.93CHAPTER 4. A COMPARISON OF SYNTHETIC AND OBSERVED SPECTRAcan be classified as Weak Line T Tauri stars based on the equivalent width criterion of10 Å (Herbig & Bell 1988) or the HW10% criteria of White & Basri (2003) to the sameeffect. By removing the weak line T Tauri models, I discard the HU models that containred-shifted sub-continuum absorption. Hence, for our model grid, inverse P-Cygni profilesare only produced by models that would be classified as Weak Line T Tauri stars. Thisresult is substantiated byKurosawa et al. (2006) who demonstrated that the inverse P-Cygniprofiles in HU were only produced in ≈ 5 per cent of their sample. Moreover, in thesecases, the accretion rate was low enough that the lines would be consistent with a weakline T Tauri classification.The same models that correctly reproduce the observed HU Reipurth types predicta very high frequency of inverse P-Cygni profiles for PaV, PaW, and BrW. For HU, Starkbroadening smooths out the synthetic spectra, masking the absorption features (Muzerolleet al. 2001). However, for the infrared lines, the effect of Stark broadening is negligibleat the temperatures and densities expected in T Tauri accretion. A possible reason forthe dearth of observed inverse P-Cygni profiles is non-axisymmetric accretion flow. Amagnetic obliquity from the stellar pole could cause the accretion funnel to form “curtains”that rotate with the star. Inverse P-Cygni profiles would then be seen intermittently whenthe accretion curtain intersects the line of sight. Observations support this effect; forexample, McGinnis et al. (2020) measured a magnetic obliquity of 18◦+8−7 for DK Tauand observed an inverse P-Cygni profile for two of the eight epochs. Accretion funnel“curtains” have also been explored in radiative transfer studies. For instance, Symingtonet al. (2005a) used a 3D non-axisymmetric model to reproduce some of the observedvariability seen in T Tauri stars, but in general, the models produced results that were toovariable. Esau et al. (2014) successfully modelled variable Balmer emission from AA Tauwith a non-axisymmetric accretion model, and Alencar et al. (2012) reproduced Balmerlines from V2129 Oph over several stellar rotational cycles. Further analysis of T Tauritime-series spectroscopic surveys would be helpful in better quantifying the variability ofinverse P-Cygni profiles across multiple wavelengths.In addition to the red-shifted features of the magnetosphere, some interesting blue-shifted features were created by the stellar wind. The stellar wind produced the character-istic broad sub-continuum absorption feature noted by Kurosawa et al. (2011). P-Cygnifeatures occur in ≈ 10 per cent of the synthetic profiles, whereas P-Cygni profiles arerare in our observations, with none exhibited for the infrared lines. The infrared lines areoptically thinner than HU, and therefore less sensitive to the lower-density wind. P-Cygniprofiles were also observed to be rare by Folha & Emerson (2001) and Edwards et al.(2006). The reason why these mass-outflow signatures are scarce in the observationscould be due to the geometry, temperature, or mass-loss rate. Geometry should signif-944.4. DISCUSSIONicantly reduce the frequency of fast wind absorption since these winds are likely to becollimated (Edwards et al. 2006) and therefore may not intersect the line of sight at highinclinations. HU is optically thicker than the infrared lines, making it a better tracer oflow-density gas, such as the wind. However, the infrared lines of our grid produced ahigher frequency of P-Cygni profiles than HU, ∼ 11 and ∼ 8 per cent, respectively. Thismay be due to Stark broadening effects smoothing out the absorption features for HU andthe relatively high density (≈ 5× 10−12 g cm−3) gas at the foot of the wind being opticallythick for PaW, PaV, and BrW. Lima et al. (2010) showed that for HU a mass-loss rate fromthe inner disk of at least 10−9M yr−1 was needed for the blue-shifted absorption featuresto be shown. However, for our polar stellar wind, the models produced HU blue-shiftedsub-continuum absorption for mass-loss rates as low as 10−12M yr−1. Additionally, ourresults suggest that the frequency of synthetic P-Cygni profiles depends on the ratio ofmass loss to mass accretion, not the mass-loss rate alone. This is because higher accretionrates increase magnetospheric and hot-spot emission, which masks the absorption featuresfrom the wind.Muzerolle et al. (2001) noted that the ratio of PaV to BrW was consistent for a sampleof T Tauri stars at different accretion rates. The ratio of the peak fluxes of PaV to BrWfrom our sample of T Tauri stars was compared with the sample from fig. 15 of Muzerolleet al. (2001) and found to be in agreement despite the lack of veiling and ISM extinctioncorrections to our spectra. Similarly to the results of Muzerolle et al. (2001), our modelsshow that for the synthetic line ratio of PaV to BrW to correspond to the observed lineratio, the magnetospheric temperature must increase as the accretion rate decreases. Thisis because the emission of BrW abates rapidly with the decrease in density produced bylower accretion rates. There must be a mechanism in the T Tauri systems that links theaccretion temperature to the accretion rate.Another factor seen in § 4.2 is that the infrared widths (FWHM and HW10%) ofthe synthetic and observed data do not agree. The models that produce HU profiles thatlie within the observed range create Paschen and Brackett spectra that are too narrow by≈ 80 kms−1. Alencar & Basri (2000) suggested that rotation and turbulence may playan important role in the formation of the width of the line profiles. In agreement withMuzerolle et al. (2001), our results demonstrated that stellar rotation has a negligible effecton the line widths at the rotation rates seen in our observed sample. Furthermore, thelevel of turbulent motion needed to broaden the synthetic lines sufficiently to account forthe difference in infrared line widths is very high (≈ 100 kms−1). Not only is there noapparent physical justification for such a considerable value of turbulence, but the spectraproduced are broad and flat-topped, a morphology not seen in observations.In addition to turbulence and rotation, I attempted to quantify the reduction in the95CHAPTER 4. A COMPARISON OF SYNTHETIC AND OBSERVED SPECTRAwidth of the spectra due to red-shifted absorption by calculating the width using only theblue-shifted emission; an approach similar to that adopted by Alencar et al. (2012) tocompensate for poorly fitting red-shifted absorption. As expected (Bouvier et al. 2007;Alencar et al. 2012), our analysis suggested that a significant change in themodel geometrythat could remove the inverse P-Cygni absorption, such as non-axisymmetric accretion,would account for a substantial increase in the infrared line widths. Alone, such a methodmay not be sufficient to account for the disparity between the synthetic and observedhydrogen profiles. However, if the models were to include some combination of stellarrotation, turbulence, and non-axisymmetric accretion, the synthetic profiles may be shiftedinto accord with the observations. Conversely, Folha & Emerson (2001) showed that, fortheir sample of PaV and BrW T Tauri spectra, the FWHM of Reipurth type I and typeIVR profiles were not significantly different. This would indicate that spectra with inverseP-Cygni profiles are generally broader than those without them or that the sub-continuumabsorption does not significantly narrow the spectra. There is not a sufficient populationof type IVR’s in our sample to confirm this result.Another factor not considered is the line emission from the accretion shock region.This work treats the accretion shock zone as a black body heated by the release ofgravitational potential energy that adds to the continuum. However, the accretion regionswill also have line emission (e.g. Dodin 2018) that will contribute to the observed spectra.A complete radiative equilibrium treatment of the shock region and magnetosphere isnecessary to constrain this effect fully, which requires the development of a self-consistenttheory of the heating and cooling mechanisms in the magnetosphere.SummaryI presented a grid of synthetic T Tauri line profiles, computed using the radiative transfercode TORUS . The models are based on the magnetospheric accretion configurationoriginally developed by Hartmann et al. (1994) and developed further by Muzerolle et al.(1998b, 2001); Symington et al. (2005a); Kurosawa et al. (2006); Lima et al. (2010);Kurosawa et al. (2011). Our models also included a polar stellar wind launched fromthe star’s surface, a more realistic development to the stellar wind used by Kurosawaet al. (2011). I compared our ensemble of models to the spectra of 29 T Tauri stars.The medium-resolution spectra covered a broad range of wavelengths that allowed me tosimultaneously compare the HU, PaV, PaW, and BrW lines. The following conclusions aredrawn.• The observations exhibited similar Reipurth classifications, widths, and mean line964.4. DISCUSSIONcentres to those seen in other studies (e.g. Folha & Emerson 2001; Reipurth et al. 1996;Edwards et al. 2006).• For HU, our synthetic line profile grid was able to reproduce the observed distributionof Reipurth classifications and line widths. However, for the same models, the infraredlines could not reproduce the observed emission. The modelled lines were too narrowby ≈ 80 kms−1, and the majority (≈ 90 per cent) had inverse P-Cygni profiles.• The ratio of synthetic PaV to BrW peak intensities showed that higher magnetospherictemperatures are needed at lower accretion rates to match the observed infrared lineratios.• I explored the effect of rotation and turbulence and determined that they could notsufficiently broaden the synthetic infrared line profiles or remove the inverse P-Cygnifeatures. The red-shifted sub-continuum absorption was determined to narrow theHW10% of the lines by ≈ 48 kms−1.• The polar stellar wind produced the characteristic broad P-Cygni profiles as suggested byKurosawa et al. (2011). Our models produced blue-shifted sub-continuum absorptionwith mass-loss rates as low as 10−12 " yr−1. Furthermore, our results suggested thatthe frequency of P-Cygni profiles depends on the ratio ofmass loss tomass accretion rate;the increased emission from higher accretion rates masks the blue-shifted absorption.Despite the higher optical depth of HU, synthetic infrared lines produced a higherfrequency of P-Cygni profiles than HU.In the next chapter, radiative transfer simulations of non-axisymmetric T Taurimodels are presented and consideration is made to the phase-dependent variability seenin the synthetic line profiles.97Chapter 5Non-Axisymmetric 3D RadiativeTransferIn Chapter 4, I presented axisymmetric T Tauri models; however, such a system is a specialcase. It is more probable that the magnetic and rotation axes are not aligned but offset by atleast a few degrees. For example,McGinnis et al. (2020) saw amean offset of 11.4◦ for theirsample of 10 classical T Tauri stars with an RMSdispersion of 5.4◦. McGinnis et al. (2020)measured the offset by monitoring the radial velocity variations, produced by the accretionshock, of the He I _5876 Å line. Magnetohydrodynamic models show that even whenthe magnetic field and rotation axes are aligned, the accretion funnel preferentially flowsthrough one hemisphere over the other. Magnetic perturbations can shift the flow betweenhemispheres (Ireland et al. 2021, 2022). Therefore, amagnetic obliquity necessarily causesa non-axisymmetric magnetosphere, where the accretion flow adopts the most energy-efficient path to accrete from the disk to the star. Magnetohydrodynamic simulations ofT Tauri stars with an offset magnetic dipole commonly predict a paradigm in which thealternate accretion columns (azimuthally constrained) form on either side of the star inopposite hemispheres (e.g. Romanova et al. 2003; Long et al. 2011).Non-axisymmetric accretion flow may account for some of the observed variabilityof T Tauri stars. As the T Tauri stars rotate, the accretion columns and hot spots movethrough the line of sight, altering the observed continuum and line emission. For instance,inverse P-Cygni profiles are only expected when the accretion column intersects theline of sight. In Chapter 4, I speculated that a reduction in the frequency of syntheticinverse P-Cygni profiles could explain the narrowness of the infrared line ensemble. Non-axisymmetric accretion flow may, by reducing red-shifted absorption, bring the widths ofthe infrared line profiles into better agreement with the observations.Symington et al. (2005a) presented the first 3D radiative transfer models of T Tauristars with non-axisymmetric accretion flows. The authors showed that by confining theazimuthal extend of the accretion columns, themodels produced spectral variations similarto those observed in T Tauri stars. However, Symington et al. (2005a) concluded that therotational variability produced by the radiative transfer simulations was greater than thatseen in the observations. In particular, they stated that the accretion model suggested bymagnetohydrodynamic simulations (two accretion columns, one in each hemisphere oneither side of the star) produced excessive variability at odds with observations. Kurosawaet al. (2005) compared time-series PaV spectroscopy of SUAurigaewith non-axisymmetricaccretion radiative transfer simulations. Their models used either a magnetosphere formedby modifying an axisymmetric accretion curtain by tilting it 10◦ relative to the rotationaxis or by removing three equally spaced 10◦ gaps. They showed that their models couldreproduce the variability of the equivalent width and the red-shifted absorption. Kurosawaet al. (2008) used the results of a magnetohydrodynamic model mapped onto a radiativetransfer grid to study the induced variability of HU, PaV, and BrW. Kurosawa et al. (2008)showed that, in contrast to the conclusions of Symington et al. (2005a), the syntheticspectra predicted line variability in good agreement with observations. Additionally, theirresults demonstrated that the spectra of Folha & Emerson (2001) were consistent withmodels where the magnetic offset was small (< 15◦).Clearly, 3D non-axisymmetric radiative transfermodels are necessary for adequatelysimulating T Tauri accretion. For instance, bothAlencar et al. (2012) and Esau et al. (2014)successfully reproduced some of the time-series spectroscopy variability of V2129 Ophand AA Tau, respectively, by using radiative transfer simulations based on magnetohy-drodynamic models. Dmitriev et al. (2019) presented 3D radiative transfer models of TTauri stars with an inclined magnetic dipole and introduced magnetospheric heating fromaccretion shock, based on the work of Dodin (2018). Using their new models, Dmitrievet al. (2019) demonstrated rotational modulation of synthetic spectra and line profiles thatwere similar to the theoretical results of Muzerolle et al. (1998b) and Muzerolle et al.(2001). However, extending the models to three dimensions only increases the degeneracyof the synthetic spectra and adds further free parameters to the already complex models.Therefore, a better understanding of how the geometry and accretion parameters shapespectral variability is paramount.This chapter examines several non-axisymmetric 3D T Tauri models and the vari-ability of the hydrogen spectra produced at different viewing inclinations, azimuthal angles,and wavelengths. In § 5.1, I present the models and the parameters used. § 5.2 analyzesthe spectra and considers their variability. In § 5.3 and § 5.4, I attempt to quantify thechanges in the Reipurth type and the width of the line profile in different rotational phases.Finally, in § 5.5, we discuss the results and attempt to quantify whether the 3D geometry99CHAPTER 5. NON-AXISYMMETRIC 3D RADIATIVE TRANSFERFigure 5.1: A cartoon showing the 3D accretion and wind model. The left-hand diagram showsa side view of the magnetosphere and wind, indicating the offset magnetic axis. The right-handdiagram gives a top-down view of the magnetosphere and indicates the azimuthal constraint iacc.The labelled regionsA andB correspond to the same accretion funnels in the left and right diagram.Note that the cartoons are not to scale, and the wind is not included in the right-hand cartoon.may address the problems seen in Chapter 4.5.1 3D model geometryThis chapter investigates the variability of hydrogen spectra induced by azimuthally struc-tured accretion flows. To this effect, a magnetic axis offset b was introduced into thefiducial T Tauri model used in Chapter 4. Fig. 5.1 shows a cartoon representation ofthe magnetosphere and stellar wind model used in this chapter. The accretion flow takesthe shortest distance along the dipole field lines from the accretion disk to the stellarsurface. By default, TORUS creates two accretion curtains, one in either hemisphere,with an azimuthal angular width iacc = 180◦. The stellar wind is mostly unchanged bythe dipole offset, moving along the dipole field lines until the specified opening angle Θswis achieved, at which point the wind’s velocity becomes radial.This setup of the model is similar to the “A30” model used by Symington et al.(2005a), who concluded that such a model produced spectra “sufficiently at odds withspectroscopic observations that this form of MHD model may be rejected”. However,despite the findings of Symington et al. (2005a), these models are extensively predicted bymagnetohydrodynamic simulations (e.g. Romanova et al. 2003; Long et al. 2011; Irelandet al. 2021) and have been subsequently used to relatively successfully reproduce observed1005.2. VARIABILITY AND MORPHOLOGY OF 3D NON-AXISYMMETRIC SPECTRAT Tauri spectral variability (e.g. Kurosawa et al. 2005; Alencar et al. 2012; Esau et al.2014). Therefore, this chapter further explores this form of accretion and the line profilesproduced.Table 5.1: The parameters of the 3D models.Model log¤"acc ¤"sw )acc iacc b[Myr−1] [ ¤"acc] [K] [deg] [deg]0 −7 0.1 7500 180 101 −7 0.1 7500 90 102 −7 0.1 7500 30 103 −7 0.01 7500 30 104 −8 0.1 8500 30 105 −8 0.01 8500 30 10In this chapter, I present six differentmodels all with amagnetic obliquity of b = 10◦,the model parameters are shown in Table 5.1. Three models (0, 1, and 2) have an accretionrate log ¤"acc = −7, a magnetospheric maximum temperature of 7500 K and a mass-lossrate of 0.1 ¤"acc. These models explore the effect of the azimuthal width on the syntheticspectra and have values of iacc = 180◦, 90◦, and 30◦, respectively. Model 3 has a lowermass-loss rate of ¤"sw = 0.01 ¤"acc and was run to explore the effects of the stellar windwhere log ¤"acc = −7 and iacc = 30◦. The last two models (4 and 5) have a lower accretionrate of log ¤"acc = −8, but a higher maximum accretion temperature of 8500 K. Thehotter temperature is so that the models can reproduce the typically observed HU strength(Muzerolle et al. 2001). Models 4 and 5 have a narrow accretion curtain of iacc = 30◦and different mass-loss rates of 0.1 ¤"acc and 0.01 ¤"acc, respectively. All of the modelshave a slow rotation speed of 13.4 kms−1 (five per cent of their break of velocity) andan isothermal wind temperature of 8000 K. All other input parameters are as defined inTable 4.1.For eachmodel, 30 different viewing angles around the model were chosen, using allcombinations of three colatitudes of 20◦, 60◦, and 80◦ at ten different azimuthal angles laround the z-axis. At each position, synthetic spectra were computed for the four hydrogentransitions HU, PaV, PaW, and BrW. The z-axis is the rotational axis or pole of the models.Therefore, different values ofl are analogous to the rotational phase of the T Tauri model.101CHAPTER 5. NON-AXISYMMETRIC 3D RADIATIVE TRANSFERPhase: 0.0600 0 6000.0080.0120.0160.020Phase: 0.1600 0 600Phase: 0.2600 0 600Phase: 0.3600 0 600Phase: 0.4600 0 600Phase: 0.5600 0 6000.0080.0120.0160.020Phase: 0.6600 0 600Phase: 0.7600 0 600Phase: 0.8600 0 600Phase: 0.9600 0 60010 6 10 5 10 4 10 3 10 2H Surface Brightness[erg s 1cm 2Hz 1sr 1]0.0 0.2 0.4 0.6 0.8 1.0Velocity [kms 1]0.00.20.40.60.81.0Flux [ergcm2 s1 ]Figure 5.2: HU surface brightness colour maps, which show model 4 at an inclination of 60◦ andten different phases. Below each image is the integrated line profile. The spectra are not continuumnormalised, and as the hot-spot rotates into view, the continuum level increases. The peak flux isseen when the accretion columns are perpendicular to the line of sight.5.2 Variability and morphology of 3D non-axisymmetricspectraThe line profiles produced by the 3D non-axisymmetric models have a high degree ofphase variability and exhibit large changes in morphology when viewed from differentangles. The following sections outline the trends in variability and morphology seen inthe synthetic spectra. An example of rotational variability can be seen in Figs. 5.2 and5.3. The figures show the synthetic images and line profiles of model 4with an inclinationof 60◦ for HU and PaV. The PaW and BrW lines are not shown as they are similar toPaV. The line profiles are not normalised and share a common flux axis; the continuum1025.2. VARIABILITY AND MORPHOLOGY OF 3D NON-AXISYMMETRIC SPECTRAPhase: 0.0600 0 6000.0180.0190.020Phase: 0.1600 0 600Phase: 0.2600 0 600Phase: 0.3600 0 600Phase: 0.4600 0 600Phase: 0.5600 0 6000.0180.0190.020Phase: 0.6600 0 600Phase: 0.7600 0 600Phase: 0.8600 0 600Phase: 0.9600 0 60010 6 10 5 10 4 10 3 10 2Pa Surface Brightness[erg s 1cm 2Hz 1sr 1]0.0 0.2 0.4 0.6 0.8 1.0Velocity [kms 1]0.00.20.4.60.81.0Flux [ergcm2 s1 ]Figure 5.3: The same model and inclination as as Fig. 5.2 but for the hydrogen transition PaV.level increases as the hot spot rotates into view. The strongest emission is seen whenthe accretion columns are aligned to be perpendicular to the line of sight, lessening themagnetospheric absorption. The narrower the magnetosphere, the higher the gas density.This increases the surface brightness of the accretion gas. For example, the HU surfacebrightness of the magnetosphere with an accretion rate of log ¤"acc = −7 and iacc = 30◦ is≈ 10 times greater than the photospheric surface brightness; a similar magnitude of surfacebrightness as the hot-spot. As iacc increases, the strength of the magnetospheric surfacebrightness decreases. For models 4 and 5 which have an accretion rate of log ¤"acc = −8,the stellar continuum has a higher surface brightness than the magnetosphere. This isalways the case for the infrared lines, where the photospheric surface brightness is severalorders of magnitude greater than the line flux in all the models presented here.Fig. 5.4 shows the complete atlas of the line profiles produced by the six models.For each model, there are 120 spectra arranged by wavelength and inclination. The ten103CHAPTER 5. NON-AXISYMMETRIC 3D RADIATIVE TRANSFERspectra produced at different phases are plotted. Phases 0.0 and 0.5 are shown in bold,highlighting the phases when the magnetosphere is aligned with the observer. The HUline shows more extended emission, originating further away from the star in regions ofmore rarefied gas. This is especially apparent for the stellar wind, where PaV, PaW, andBrW emission from the outflow is limited to within 1 – 2 stellar radii. However, blue-shifted absorption and emission features are seen at an inclination of 20◦ in all the modelspresented here.5.2.1 Quantitative analysis of the variability of time-series spectraA common metric used by observers to quantify the variability of time series spectra isthe temporal variance spectrum (TVS) (Fullerton et al. 1996). The method statisticallydetermines the variance at each velocity bin of the series of spectra using a weightedfunction inversely proportional to the signal-to-noise level of the continuum. However,given that our synthetic spectra have no continuum noise, an alternative method is used.In this work, we quantify variance versus velocity using a range-normalised root meansquare deviation (RMSD) distribution. This is defined for the 8th velocity bin asRMSD8 =(5̄8,max − 5̄8,min)−1 √√∑#9 ( 5̄8 − 58 9 )2#, (5.1)where 5̄ is the mean line intensity across the # rotational phases. 58 9 is the flux in the 8thvelocity bin of the 9 th spectra. 5̄8,max and 5̄8,min are the maximum and minimum intensitiesof the mean profile, respectively. Fig. 5.5 shows, for each inclination and wavelength,the mean line profile, the RMSD distribution, and an image showing the quotient of thespectra at each phase. The continuum level versus rotational phase is also plotted. Theminimum and maximum of the grey scale is given for each image. The flux of the meanline profile is normalised by the continuum. The spectra of model 3 are shown in Fig. 5.5,similar figures for models 0, 1, 2, 4, and 5 can be seen in Appendix B.BrW exhibits the strongest spectral variability relative to the mean profile range. Thenext strongest in decreasing order are PaW, PaV, and HU. The HU flux is significantlygreater than the infrared, and the percentage change in the profiles is smaller than therelative change in the infrared lines. Generally, the largest variability is at the line centrefor all wavelengths and inclinations. For an inclination of 20◦, the RMSD distributionsof the lines exhibit three peaks: one at the rest velocity, one blue-shifted, and the otherred-shifted. These three points of increased variability are created by absorption from thestellar wind and the magnetosphere. Because of the magnetic offset, the magnetosphere1045.2. VARIABILITY AND MORPHOLOGY OF 3D NON-AXISYMMETRIC SPECTRA0.004.008.0012.00H0.501.001.502.00Pa0.801.602.40Pa0.601.201.802.4020Br2.505.007.500.901.201.501.801.201.602.001.001.251.501.756040004003.006.009.0030003000.801.201.6030003001.001.502.0030003001.001.251.50800.00.20.40.60.81.0Velocity [kms1 ]0.00.20.40.60.81.0Normalized FluxFigure5.4:Thelineprofilesfrommodel0arearrangedbyinclination(rows)andwavelength(columns).Thephases0.0(solid)and0.5(dashed),whichare180◦apart,arehighlightedbold.105CHAPTER 5. NON-AXISYMMETRIC 3D RADIATIVE TRANSFER0.003.006.009.00H0.801.602.40Pa0.801.602.40Pa0.801.602.403.2020Br3.006.009.001.001.502.001.201.802.401.201.602.002.406040004004.008.0012.0030003001.001.502.0030003001.201.802.4030003001.201.602.00800.00.20.40.60.81.0Velocity [kms1]0.00.20.40.60.81.0Normalized FluxFigure5.4:Continued.Lineprofilesformodel1.1065.2. VARIABILITY AND MORPHOLOGY OF 3D NON-AXISYMMETRIC SPECTRA0.003.006.009.00H0.801.602.40Pa0.801.602.40Pa0.601.201.802.4020Br3.006.009.001.201.802.403.001.602.403.201.001.502.002.506040004004.008.0012.0030003001.201.802.403.0030003001.602.403.2030003001.001.502.002.50800.00.20.40.60.81.0Velocity [kms1 ]0.00.20.40.60.81.0Normalized FluxFigure5.4:Continued.Lineprofilesformodel2.107CHAPTER 5. NON-AXISYMMETRIC 3D RADIATIVE TRANSFER1.503.004.506.00H1.001.502.002.50Pa1.201.602.002.40Pa1.201.602.0020Br1.503.004.501.001.502.001.201.602.000.901.201.501.806040004001.503.004.5030003001.201.602.0030003001.201.602.0030003001.001.251.501.75800.00.20.40.60.81.0Velocity [kms1]0.00.20.40.60.81.0Normalized FluxFigure5.4:Continued.Lineprofilesformodel3.1085.2. VARIABILITY AND MORPHOLOGY OF 3D NON-AXISYMMETRIC SPECTRA0.801.602.40H0.920.961.001.04Pa0.880.961.04Pa0.981.001.0220Br1.001.502.002.500.900.951.001.050.880.961.041.120.930.960.991.026040004001.201.602.002.4030003000.880.961.0430003000.901.001.1030003000.920.961.001.04800.00.20.40.60.81.0Velocity [kms1 ]0.00.20.40.60.81.0Normalized FluxFigure5.4:Continued.Lineprofilesformodel4.109CHAPTER 5. NON-AXISYMMETRIC 3D RADIATIVE TRANSFER1.001.502.002.50H1.001.041.08Pa0.961.041.121.20Pa0.991.001.021.0320Br1.001.502.002.500.900.961.020.880.961.041.120.920.961.006040004001.201.602.0030003000.800.880.961.0430003000.800.901.001.1030003000.920.961.001.04800.00.20.40.60.81.0Velocity [kms1]0.00.20.40.60.81.0Normalized FluxFigure5.4:Continued.Lineprofilesformodel5.1105.2. VARIABILITY AND MORPHOLOGY OF 3D NON-AXISYMMETRIC SPECTRAintersects with the line of sight to the hot-spot in phases 0.3 – 0.7, creating the red-shiftedRMSD peak. At 80◦, the magnetosphere rotates through the line of sight, increasingthe variability of red-shifted absorption and emission. Thus, the RMSD distribution isred-shifted at this high inclination.The synthetic continuum emission in the spectra is simulated only for the star,hotspot and magnetosphere; see Chapter 2. Infrared continuum emission from the diskis not included and the rotational phase changes in the continuum arise solely from thehotspot and accretion column. The continuum level of the spectra increases when theaccretion hot-spot is visible, peaking at a phase of 0.5. For an inclination of 60◦, the hot-spot on the star’s far side creates a secondary continuum peak at phase 0.0. The change inthe magnitude of the continuum is calculated according to the following equationΔ< = −2.5 log105max5min. (5.2)where 5max and 5min are the maximum and minimum continuum values in the series. Thechange in the continuum near HU is dependent on the inclination; it increases at highercolatitudes. For an inclination 20◦ and an accretion rate of log ¤"acc = −7, the continuumnear HU changes in magnitude by ≈ 0.28. However, at 80◦, the change in magnitude is≈ 0.7. For the lower accretion rate of log ¤"acc = −7, the maximum magnitude change forHU is ≈ 0.2 at 80◦ and half that at 20◦. The change in the magnitude of the continuumnear the infrared lines is smaller but follows a trend similar to HU. The mean changes inthe magnitude of the continuum near the infrared lines are: ≈ 0.16, ≈ 0.12, and ≈ 0.06for PaW, PaV, and BrW, respectively.5.2.1.1 Accretion and temperature comparisonThe models 2 and 4 have the same geometry but different accretion rates and temperatures.For the infrared lines, reducing the accretion rate increases the variability of the spectrarelative to the intensity of the line profile. At an inclination of 20◦, the peak value ofthe RMSD for model 2 is ≈ 50 per cent of that for model 4. However, this decreases to≈ 10 per cent at higher inclinations. There is a similar change in the variability of the HUprofiles but to a lesser extent. At 80◦, a difference of 40 per cent is observed between theRMSD’s of the two models. Models 3 and 5 exhibit comparable trends in variability.The increased variability seen in models 4 and 5, relative to 2 and 3, is red-shifted.Whereas, the rotational variability of the blue-shifted side of the line profiles is unchangedby the different accretion rates. The infrared line profiles of models 4 and 5 at a phaseof 0.5 and inclinations of 60◦ and 80◦ show no emission, only sub-continuum absorption;see Fig. 5.4. This is because the line emission for PaV, PaW, and BrW is dominated by the111CHAPTER 5. NON-AXISYMMETRIC 3D RADIATIVE TRANSFERmagnetospheric absorption of the photospheric and hot-spot continua. On the other hand,in the models 2 and 3, the higher accretion rate of log ¤"acc = −7 creates accretion funnelswith significant line emission that dominates the continuum absorption. For HU, thereis an increase in the rotational variability as the accretion rate decreases; however, at thephases calculated, the line profiles all had emission components.Models with accretion rates of log ¤"acc = −8 show significantly less change intheir continua than models with an accretion rate of log ¤"acc = −7. For instance, theHU continuum magnitude change of models 2 and 4 at an inclination of 80◦ are ≈ 0.6and ≈ 0.2, respectively. The differences in the continua are due to the stronger hot-spotcontinuum emission, created by the higher accretion rate.5.2.1.2 Geometry comparisonFor the infrared lines, the smaller the azimuthal extent of the accretion curtain iacc, thelesser the phase variability of the red-shifted emission. This is because the wider theaccretion column, the greater the proportion of red-shifted absorption at each rotationalphase. For example, stronger and more frequent red-shifted absorption features are seenin model 0 than in model 2. In general, the larger the value of iacc, the broader theRMSD spectrum. At an inclination of 60◦, the narrower accretion columns produce morevariability. Whereas, at 80◦, the wider accretion columns create a higher peak RMSD. ForHU, the effect of iacc on the RMSD spectra is negligible. The HU RMSD distributionshave strong red-shifted peaks at 80◦ and blue-shifted peaks at an inclination of 60◦. Thisis due to the increased sensitivity of HU to the stellar wind.The continuum levels of the HU spectra have a consistent variation and magnitudechange across the phases, independent of iacc. However, the wider the accretion columns,the stronger the continuum is at all phases for the infrared lines. Additionally, the changein magnitude of the continuum increases with iacc. At an inclination of 60◦, model 0 doesnot show a secondary peak in the continuum at phase 0.0. This is because a hot-spot ineither the upper or lower hemisphere is visible at all phases.5.2.1.3 Mass-loss rate comparisonModels 2 and 3 have different mass-loss rates of 0.1 ¤"acc and 0.01 ¤"acc, respectively.Similarly, models 4 and 5 have the same ratios of mass loss to mass accretion rates.The variability of the models increases as the mass-loss rate reduces. This effect is morepronounced betweenmodels 2 and 3, which have the larger accretion rate of log ¤"acc = −7.The wind absorbs emission from the magnetosphere and hot-spot, creating blue-shiftedabsorption features in the line profiles. The wind’s emission and absorption decrease with1125.2. VARIABILITY AND MORPHOLOGY OF 3D NON-AXISYMMETRIC SPECTRAlower mass-loss rates as the wind density reduces.5.2.2 Correlation analysis of time-series spectraA common method for analysing the often complex changes in the morphology of time-series spectra are auto-correlation images. Using the technique pioneered by Johns (1994)and Johns & Basri (1995), auto-correlation images or maps are created to examine thecorrelation of the variability of each velocity bin with the rest of the spectra. Thiscorrelation is quantified by the coefficient 8 9 for the 8th row and 9 th column as8 9 =1##∑<=1((<8 − (̄8) ((< 9 − (̄ 9)f8f9. (5.3)Where # is the number of spectra in the series and (<8 is the signal of the 8th velocitybin of the 0 kms−1) is anti-correlated withthe blue side of the line (< 0 kms−1). As the single accretion column rotates, it shifts frombeing blue-shifted to red-shifted, translating the peak emission from negative to positivevelocity. Another trend is that the width of the blocks of correlation and anti-correlationincreases with the inclination. This is because for the higher inclinations there is greaterline-sight velocity from the accretion flow.At an inclination of 20◦, models 0 to 3 exhibit a red-shifted band (≈ 100 kms−1 to150 kms−1, the width and position vary betweenmodels) of anti-correlation with the entireprofile. The anti-correlation band arises because the accretion column is orientated suchthat at certain phases, it absorbs emission from the photosphere and hot-spot. This causes anarrow absorption band to appear in the line profiles as the overall line emission increasessimultaneously. This effect is not seen in models 4 and 5, where the lower accretionrate reduces the column density, decreasing the absorption and emission properties ofthe magnetosphere. Furthermore, at 20◦, many of the images show an anti-correlation113CHAPTER 5. NON-AXISYMMETRIC 3D RADIATIVE TRANSFER0.000.050.10RMSDH0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.18Min 0.812 4Flux0.0100.0160.0220.00.10.2Pa0.00.10.20.30.40.50.60.70.80.9Max 1.33Min 0.731.01.52.00.0150.0210.0270.000.080.16Pa0.00.10.20.30.40.50.60.70.80.9Max 1.25Min 0.741.01.52.00.0170.0230.0290.000.150.30Br0.00.10.20.30.40.50.60.70.80.920Max 1.27Min 0.751.21.60.0160.0220.0280.000.150.30RMSD0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.43Min 0.411.53.0Flux0.0080.0140.0200.00.20.40.00.10.20.30.40.50.60.70.80.9Max 1.37Min 0.641.21.51.80.0150.0210.0270.00.20.40.00.10.20.30.40.50.60.70.80.9Max 1.31Min 0.681.21.51.80.0170.0230.0290.00.20.40.00.10.20.30.40.50.60.70.80.960Max 1.33Min 0.701.001.251.500.0160.0220.0280.00.10.2RMSD0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.27Min 0.43Velocity [kms1]1.53.0Flux0.0040.0100.0160.00.20.40.00.10.20.30.40.50.60.70.80.9Max 1.50Min 0.69Velocity [kms1]1.21.50.0090.0150.0210.000.150.300.00.10.20.30.40.50.60.70.80.9Max 1.40Min 0.72Velocity [kms1]1.21.50.0110.0170.0230.000.250.500.00.10.20.30.40.50.60.70.80.980Max 1.42Min 0.76Velocity [kms1]1.01.21.40.0100.0160.022MinMeanMaxFigure5.5:Spectrafrommodel3thatarearrangedbyinclinationandwavelength.Theplotsshowthemeanspectrum(bottom)andthequotientofthespectrumateachrotationalphase(greyscaleimage).ThetopplotdisplaysthenormalisedRMSDdistribution,andtheplotontheright-handsideshowsthecontinuumlevelateachphase.1145.3. INVERSE P-CYGNI PROFILESAnti-correlation Correlation2000200H2000200200 0 2002000200Pa200 0 200Pa200 0 20020Br60200 0 200800.0 0.2 0.4 0.6 0.8 1.0Velocity [kms 1]0.00.20.40.60.81.0Velocity [kms1 ]Figure 5.6: Auto-correlation maps for model 2. White areas correspond to a strong correlation andblack areas strong anti-correlation. The images are symmetrical along the x-y diagonal. The mapsare arranged by wavelength and inclination. The PaV, PaW, and BrW lines share similar patterns foreach inclination. For instance, at 80◦, the red side of the infrared lines is anti-correlated with theblue side.between the line centre (±25−50 kms−1) and the wings. This effect is not seen as stronglyin the HU profiles, which more commonly show a blue-shifted anti-correlation.The above evaluation is intended to provide only a rudimentary analysis of thesecorrelation maps, and a complete analysis is beyond the scope of this thesis. I discuss theauto-correlation maps and briefly compare them with those from literature in § 5.5.5.3 Inverse P-Cygni profilesOne of the problems with the grid of synthetic line profiles presented in Chapter 4 was that≈ 90 per cent of the infrared line profiles were classified as type IVR (inverse P-Cygni).This is a significantly greater value than the observed frequency of inverse P-Cygni profiles(≈ 35 per cent) seen in both in our sample of T Tauri stars and that of Folha & Emerson(2001). In Chapter 4, it was proposed that a possible method to reduce the frequency115CHAPTER 5. NON-AXISYMMETRIC 3D RADIATIVE TRANSFERof the inverse P-Cygni profiles was to use non-axisymmetric accretion. It was postulatedthat the induced rotational variation from such a model might reduce the frequency ofthe red-shifted sub-continuum absorption. To quantify this effect, I used a Python scriptto classify the spectra at each phase into the types defined by Reipurth et al. (1996). InChapter 4, the strongest absorption feature was used to distinguish between types IVRand IVB for profiles that contained both red and blue-shifted absorption. Whereas, inthis section, only type IVR profiles are considered, and as such, any spectra that containred-shifted sub-continuum absorption are classified as IVR. Fig. 5.7 shows two histograms(optical and infrared) representing the fraction of inverse P-Cygni profiles seen in oursynthetic spectra at different phases. The bars are sub-divided by the azimuthal extentof the accretion column iacc. It is immediately apparent that the occurrence of inverseP-Cygni profiles is centred around phase 0.5, where the magnetosphere is directly in theobserver’s line of sight. The maximum fraction of spectra from all six models that showred-shifted sun-continuum absorption at any given phase is 0.5 for HU and 0.23 for thecombined infrared lines. For HU (optical), the red-shifted sub-continuum absorption isonly seen in our spectra at phases of 0.4 – 0.6. The type IVR HU profiles primarily comefrom models 4 and 5 where the accretion rate is log ¤"acc = −8. The lower accretion ratereduces the influence of Stark broadening on the line profiles.For the infrared lines PaV, PaW, and BrW, the distribution of inverse P-Cygni profilesis centred around a phase of 0.5. Models 0, 1, 2, and 3 have a total proportion of inverseP-Cygni profiles of 0.47, 0.29, 0.13, and 0.17, respectively. As the value of iacc decreases,the phases at which the magnetosphere intersects the line of sight are reduced. This canbe seen in Fig. 5.7, where the inverse P-Cygni profiles are predicted for the phases 0.2– 0.8 for iacc = 180◦. Whereas, the inverse P-Cygni profiles occur only between phases0.3 to 0.7 for iacc = 90◦ and 30◦. The lower accretion rate models 4 and 5 have a totalproportion of infrared P-Cygni profiles of 0.49 and 0.5, respectively. In these models,the fraction of inverse P-Cygni profiles is produced equally between the phases 0.3 to0.7. The lower accretion rate produces less line emission, increasing the absorption ofthe continuum. There is also evidence to suggest that reducing the mass-loss rate from0.1 to 0.01 ¤"acc increases the proportion of inverse P-Cygni profiles; however, additionalmodels are needed to verify this result.The mean fractions of infrared inverse P-Cygni profiles for all phases and modelsare 0.44, 0.41, and 0.16 for the inclinations 80◦, 60◦, and 20◦, respectively. For HU,inverse P-Cygni profiles are only predicted at inclinations of 80◦ and 60◦ and both with amean fractional occurrence of 0.08.1165.4. VARIABILITY OF THE LINE PROFILE WIDTHS0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.000.150.300.45Optical0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.000.050.100.150.20Infrared30 90 1800.0 0.2 0.4 0.6 0.8 1.0Phase0.0.20.40.60.81.0FractionFigure 5.7: The fraction of synthetic line profiles with inverse P-Cygni features for all modelsand inclinations at the ten different phases. The left-hand figure shows the fraction for HU andthe right-hand figure for the infrared lines. The fraction is subdivided by the azimuthal extend ofthe accretion column. The wider the accretion funnel, the more phases contain inverse P-Cygnifeatures. The most significant fraction occurs at a phase of 0.5 when the near-side accretion columnis between the observer and the star.5.4 Variability of the line profile widthsIn Chapter 4, it was seen that the synthetic spectra grid could reproduce the observedHU line widths. However, the same models produced infrared lines that, on average,had HW10% approximately 80 kms−1 narrower than the observed T Tauri spectra. Thissection considers how the widths of the line profiles change when the 3D models arerotated. We quantify the width using the HW10% and the equivalent width.Fig. 5.8 shows two polar plots displaying the HW10% (radius) at different anglesaround the z-axis for HU and PaV. The line profiles from models 2–5 were determined foran inclination of 60◦. The greyed sections indicate the location of the accretion columns;the darker section is the column on the near side of the disk. In the figure, it can be seenthat the narrowest spectra occur when the magnetosphere is directly in the line of sight, anangle of l = 180◦ corresponding to a phase of 0.5. The narrowing of the line is causedby the increased continuum absorption of the magnetosphere. However, for an accretionrate of log ¤"acc = −7 and mass-loss rate of 0.1 ¤"acc, the narrowest PaV spectra occursat l = 72◦ and 252◦. This is where the accretion columns are side on, minimising thevelocity component of the accreting gas relative to the viewing angle.Fig. 5.9 shows the residual change of the HW10% and the equivalent width of the117CHAPTER 5. NON-AXISYMMETRIC 3D RADIATIVE TRANSFER0°45°90°135°180°225°270°315°Hlog Macc = 7, Msw = 0.1 Macclog Macc = 7, Msw = 0.01 Macclog Macc = 8, Msw = 0.1 Macclog Macc = 8, Msw = 0.01 Macc0°45°90°135°180°225°270°315°Pa0100200300400500600700HW10% [kms1 ]050100150200250HW10% [kms1 ]Figure 5.8: Polar plots showing the HW10% (radius) at different phases for HU (top) and PaV(bottom). The plots show spectra observed at an inclination of 60◦ from models with iacc = 30◦.The dark grey section indicates the near side magnetosphere, and the light grey is the far side.1185.4. VARIABILITY OF THE LINE PROFILE WIDTHSHU and infrared spectra. The residual is calculated by subtracting the smallest width fromthe phase series. The figure shows the mean change and the minimum and maximumranges. As the azimuthal extent of the accretion column decreases, so does the changein the infrared HW10%. Reducing the accretion rate of the models, increases the changein the infrared HW10% by ≈ 2 times and reduces the change seen in HU by 100 kms−1.The change in the HU HW10% in model 3 is significantly greater than in model 2, theonly difference being a lower rate of mass loss. However, there is a negligible differencebetween the HW10% residuals of models 4 and 5. Therefore, the change in HW10% iscoupled with the accretion and mass-loss rates. Spectra produced at a low inclination(20◦) exhibit the least change in HW10%. Changes in equivalent width follow trendssimilar to those of HW10%. Decreasing iacc, increases the change in the equivalent widthat different phases, as does increasing the accretion rate.The HW10% is narrowest for HU and the infrared lines when the magnetosphere isbetween the observer and the star; a phase of 0.5. Whilst this orientation should give thelargest width due to the magnetospheric velocity, the absorption from the accretion funneldominates and significantly reduces the line width. The broadest infrared HW10% occursat a phase of 0.0, when the near-side accretion column is on the star’s far side. The sameis true for the HU spectra produced by models with an accretion rate of log ¤"acc = −8.However, when the accretion rate is log ¤"acc = −7, a secondary HW10%minimum occursat a phase of 0.0 caused by increased absorption of emission from the accretion curtain bythe stellar wind. The effect is more noticeable at higher mass-loss rates because the windis denser, and lower values of iacc which increases the surface brightness of the accretionfunnel.Similar to the HW10%, the equivalent width is narrowest when the near-side mag-netosphere is directly aligned with the observer at a phase of 0.5. The absorption featuresand continuum level are maximised in this orientation, reducing the equivalent width.When iacc = 180 and the inclination is 20◦ or 60◦, the broadest equivalent width isseen at a phase of 0.0 when the accretion funnel is behind the star. In this orientation,the maximum velocity broadening is obtained from the accretion flow, absorption by themagnetosphere and wind are minimised, and there is no increase in the continuum level.However, a secondary equivalent width minimum forms at a phase of 0.0 as iacc decreases.This secondary equivalent width minimum does not appear in the spectra produced foran inclination of 20◦ when log ¤"acc = −7. The secondary equivalent width minima arecaused by a combination of strong absorption features and an increase in the continuumlevels due to the visibility of the far-side hot-spot. In models 4 and 5, where log ¤"acc = −8,the secondary equivalent width minimum is seen for all inclinations. This is because ofthe deep absorption features seen in the spectra produced by these models at a phase of119CHAPTER 5. NON-AXISYMMETRIC 3D RADIATIVE TRANSFER0.0. The equivalent width at a phase of 0.0 is correlated with the inclination, accretionrate, mass-loss rate, and the azimuthal extent of the magnetosphere.The width variability of the infrared lines is at least dependent on the accretion rate,mass-loss rate, and inclination. From our models, it can be seen that changes in the lineHW10% of up to ≈ 240 kms−1 are possible by introducing non-axisymmetric accretionflow. However, both the infrared lines and HU get broader when the accretion flow is notin the line of sight. For an accretion rate of log ¤"acc = −7, the majority of the HU spectraare broadened more than the infrared lines. On the other hand, when log ¤"acc = −8,the infrared lines are broadened more than HU; this may be due to the effect of Starkbroadening. Therefore, while the non-axisymmetric accretion flow broadens the infraredlines sufficiently to account for the HW10% disparity observed in Chapter 4, the effectalso broadens HU shifting the distributions out of agreement with the observations.5.5 DiscussionThe six 3D models presented in this chapter provide a preparatory study into the spectralmodulation caused by the rotation of non-axisymmetric T Tauri models. The parametersare not adjusted to fit any specific observations but are taken as illustrative cases fromthe synthetic grid described in Chapter 4 and magnetohydrodynamic simulations (e.g.Romanova et al. 2003; Long et al. 2011; Alencar et al. 2012; Ireland et al. 2021). Thischapter builds on the work of Symington et al. (2005a) and Kurosawa et al. (2008), whopresentedmodels with accretion columns in alternate hemispheres. This chapter augmentsprevious radiative transfer studies by including a mass outflow in the T Tauri models inthe form of a polar stellar wind.The 3D models produced spectra with similar profile characteristics and morpho-logical variability to that seen in T Tauri observations, (e.g. McGinnis et al. 2020; Alencaret al. 2018). From the limited number of models presented, it is apparent that the syntheticspectra’s rotational phase modulation depends on the magnetospheric temperature and ac-cretion rate, inclination, wavelength, and mass-loss rate. Our results show that decreasingthe accretion rate increases the line variability; the line profiles of model 4 and 5 have agreater relative change than models 0 – 3. In contrast, Johns & Basri (1995) showed thatfor their sample of seven T Tauri stars, the profile variance increased with the accretionrate. Further radiative transfer models are needed to unravel the effects of the accretionrate and magnetospheric temperature.The primary variability of the synthetic spectra lies near the line centre. However,the red-side line variability is generally more significant than that on the blue-side due tostrong emission and absorption by the magnetosphere. Alencar et al. (2001) noted that1205.5. DISCUSSION0306090 Model 0061218Model 00306090Model 10102030 Model 104080120Model 201530Model 2080160240 Model 3081624Model 3050100150 Model 40.00.81.62.4 Model 40.5 0.0 0.5 1.0 1.5050100150 Model 5Infrared H0.5 0.0 0.5 1.0 1.50123Model 50.0 0.2 0.4 0.6 0.8 1.0Phase0.00.20.40.60.81.00.2 0.4 0.6 0.8 1.0Residual HW10% [kms1 ]0.0 0.2 0.4 0.6 0.80.00.20.40.60.81.0Residual Equivalent Width [Å]Figure 5.9: Plots showing the residual change in HW10% (left) and the equivalent width (right)of HU (pink) and the infrared lines PaV, PaW, and BrW (black). The mean change for all theinclinations and wavelengths is illustrated by the solid line and the maximum and minimum rangedenoted by the shaded area. The x-axis shows two full phases from −0.5 to 1.5.121CHAPTER 5. NON-AXISYMMETRIC 3D RADIATIVE TRANSFERfor the pole-on spectra of DR Tau, the hydrogen line variability was mostly blue-shifted.In our models, spectra produced at a low inclination (pole-on) have increased blue-shiftedvariance due to the stellar wind. The spectral variability seen in the models depends on themass-loss rate, and a lowermass loss relative to the accretion rate significantly increases thespectral variability. Additionally, at high inclinations, the azimuthally constrained mag-netosphere changes from being blue-shifted to red-shifted, creating substantial variabilityon both sides of the synthetic spectra.An analysis of the correlation between different components of the line profiles wasundertaken using auto-correlation images. The images reproduced share similar patternsto those created by radiative transfer models from literature (Symington et al. 2005a andKurosawa et al. 2005). Models 1, 2, and 3 showed an anti-correlation between the redand blue sides of the spectra. This is caused by the rotation of the magnetosphere throughthe line of sight. Similar features can be seen in some cross-correlation maps of observedspectral series (e.g. Pouilly et al. 2020; Alencar et al. 2012; Oliveira et al. 2000). InAlencar et al. (2018), the authors noted that the HU line centre was anti-correlated to thehigh-velocity blue wing; a feature they could not explain. A similar anti-correlation isseen in the HU maps of models 4 and 5 at inclinations of 60◦ and 80◦. In our models,this relationship is caused by the rotation of the narrow magnetosphere. The maximumline intensity is achieved when the accretion flow is perpendicular to the line of sight,minimising absorption by the magnetosphere. At this point, the line intensity is centredbecause the velocity components of the accretion flow relative to the observer are at aminimum. When the accretion column is on the far side of the star, the dominant lineemission source is from the magnetosphere and is strongly blue-shifted as it falls towardsthe star. The emission from the near side accretion column is predominately blockedby the disk. As the star rotates, the accretion column emission becomes red-shifted;simultaneously, the red-shifted absorption of the line and continuum increase. Theseeffects work in tandem to maintain a peak line centre close to zero. Additionally, as thehot-spot becomes visible, the blue-shifted sub-continuum absorption from the stellar windincreases, further decreasing the blue-side emission strength. Combined, these effectsproduce correlation maps that show the line centre to be anti-correlated with the blue-side wing. Although this may not be the cause of the correlation seen in Alencar et al.(2018), it is useful to see how the correlation images may be used to diagnose the complexphenomenon seen in the time-series variability of T Tauri spectra.In Chapter 4, it was shown that the primary problem with the synthetic line profilegrid was that the infrared line widths were too narrow when compared to the observedspectra. One of the proposed solutions was that non-axisymmetric accretion might lowerthe frequency of synthetic inverse P-Cygni profiles. In § 5.3, we saw that the mean1225.5. DISCUSSIONoccurrence rate of infrared inverse P-Cygni profiles in all models and inclinations was≈ 34 per cent. This is considerably lower than predicted for axisymmetric models (seeChapter 4) and similar to the observed frequency. These sub-continuum absorptionfeatures depend on the phase, inclination, accretion rate, and columnwidth. The associatedpredicted phase-modulated change in the width of the spectra was similarly dependent onthe model, wavelength, and inclination. The models produced changes in the infraredHW10% of up to 170 kms−1 with a mean change of ≈ 60 kms−1. These values couldaccount for the disparity of width seen between the synthetic and observed infrared lines.However, both the width of the infrared and HU lines change concurrently. Therefore,while the infrared widths can be brought into agreement with the observed range ofHW10%, the HU spectra are shifted out of accord with the observed range. For modelswith log ¤"acc = −7, the HU lines change more than the infrared lines, probably due to theeffect of Stark broadening. At a lower accretion rate of log ¤"acc = −8, the infrared linesshow a greater change in width over the phase. Further study should be undertaken toexplore this effect for a wide range of accretion rates and temperatures to determine howthe ensemble of models behave.Lastly, themodels suggested that theminima of the equivalentwidth and theHW10%occur when the velocity broadening should be at its highest. In addition, a secondaryminimum of width occurs at a half phase from the narrowest width. This secondaryreduction in width is created by combinations of an increased continuum level and strongerabsorption from the accretion flow and wind. The feature is strongly dependent on theinclination, accretion rate, mass-loss rate, and the azimuthal extent of the accretion column.Time-series spectroscopic surveys of T Tauri stars may provide observational evidence ofthis predicted feature.SummaryIn conclusion, this chapter presents a preparatory radiative transfer study of six 3D T Taurimodels that include non-axisymmetric accretion and polar stellar winds. The modelsexhibit patterns of variability, correlation, and morphology, as seen in observations andprevious theoretical studies. The models produce a phase modulation of the HW10%and equivalent width within a range similar to that observed (e.g. Alencar et al. 2012).Furthermore, the 3D models predicted two points of minimum equivalent width andHW10% created by the interplay of absorption and continuum level. Further work needsto be undertaken to create a more extensive set of synthetic spectra from 3D models andto compare them to existing or new times-series of hydrogen spectra.123Chapter 6Conclusions and Future OutlookRadiative transfer simulations present an invaluable tool for understanding the complexand variable spectroscopic observations of T Tauri stars. While modern radiative transfercodes are well tested and understood, the exact form the T Tauri models should takeremains an enigma. Magnetospheric accretion is the accepted paradigm for circumstellaraccretion, but the heating mechanisms and geometry of the funnel flow are still poorlyconstrained. Observations show that there are mass outflows from T Tauri stars, but it isstill unknown whether the dominant flows come from the star, disk, or magnetosphere.These unknowns lead to a large number of free parameters in T Tauri radiative transfersimulations and degeneracy of the synthetic spectra produced as similar spectra can becreated with multiple different inputs. Our theoretical understanding of T Tauri stars isrefined by constraining the radiative transfer input parameters. This is vital for providingbetter observational diagnostic tools and for developing our theory of T Tauri stellarevolution. Understanding how T Tauri stars accrete matter, drive outflows, and managethe angular momentum problem is integral to establishing how low-mass stars evolve andproduce planetary systems.Radiative transfer studies in the literature have produced synthetic atomic spectrasimilar to the observed line profiles from T Tauri stars. A significant limitation to thesestudies is that they are plagued by a plethora of free parameters and unknowns. Forinstance, radiative transfer simulations show that the line profiles are susceptible to thetemperature distribution of the accretion funnel. However, the heating mechanism of theaccretion funnel is unknown. Current simulations use a model proposed by Hartmannet al. (1994) that is based on a volumetric heating rate balanced by radiative cooling.Furthermore, there are few modern radiative transfer studies of T Tauri stars, despite anincrease in the availability of high-resolution spectroscopy. To address this, in this thesis, Ihave presented new developments to a state-of-the-art radiative transfer code TORUS andthe associated T Tauri model, which includes a polar stellar wind.Nominally, this thesis uses the radiative transfer code TORUS and verifies theatomic hydrogen line transfer calculations for a 2.5D and 3D non-axisymmetric T Taurimodel. However, the work also builds on T Tauri radiative transfer studies from theliterature, implementing a more physically realistic stellar wind geometry and comparingthe synthetic line profiles to multiple simultaneously observed hydrogen transition lines.Through this comparison, it was shown that current radiative transfer models could notconcurrently reproduce the observed HU and infrared spectra. This led to the explorationof several modifications to the T Tauri model and the initial investigation into a set of3D non-axisymmetric models, including a polar stellar wind. The 3D radiative transferstudy led to several T Tauri rotational variability predictions, preparing the field for furtherstudy. Moreover, the thesis presents an atlas of HU, PaV, PaW, and BrW line profiles for abroad range of accretion rates, mass-loss rates, temperatures, and viewing inclinations.During thework of this thesis, the non-LTE atomic statistical equilibrium calculationroutines of TORUSwere verified by reproducing results similar to those of Kurosawa et al.(2006). Furthermore, the polar stellar wind model was tested against the wind modelsof Kurosawa et al. (2011). The more physically realistic wind model presented in thisthesis produced spectra noticeably different from the results seen in the literature. Notably,compared to Kurosawa et al. (2011), our model produced a greater HU emission close tothe line centre and deeper blue-shifted sub-continuum absorption features in the infraredlines. This justifies using the more complex stellar wind, and I suggest that future radiativetransfer studies implement a similar wind model.Using TORUS and the T Tauri model, I compared a large grid of synthetic hydrogenline profiles to a set of spectra from29TTauri stars observedwith theX-Shooter instrumenton VLT. Due to the broad wavelength range of X-Shooter, I was able to compare thesimultaneously observed HU, PaV, PaW, and BrW line profiles with synthetic spectra ofthe same hydrogen transitions. T Tauri stars are spectroscopically variable; therefore, ifmultiple lines are to be compared accurately, they must be observed concurrently. Whilstthe ensemble of HU line profiles produced spectra with the same range of widths andmorphologies as seen in the observations, the infrared lines (PaV, PaW, and BrW) createdprofiles that had a mean HW10% narrower than the observed spectra by ≈ 80 kms−1.Moreover, the synthetic infrared spectra showed an extremely high frequency (≈ 90 percent) of inverse P-Cygni profiles, whereas the observed prevalence is ≈ 30 per cent (e.gFolha & Emerson 2001). A caveat to this comparison is that the observed sample ofT Tauri stars is statistically ambiguous. This is because the sample is comparativelysmall and the simulation grid was not designed to directly probe the parameter spacecovered by the observations. However, the sample does show a very similar distribution of125CHAPTER 6. CONCLUSIONS AND FUTURE OUTLOOKReipurth types, as seen in other studies (e.g Reipurth et al. 1996; Folha & Emerson 2001).Furthermore, the observations highlight that the majority of lines should be symmetrical,and only approximately 25 per cent of infrared lines should show inverse P-Cygni profiles.To increase the width of the infrared line profiles and reduce the frequency of inverseP-Cygni profiles, we explored three modifications to our radiative transfer model. Firstly,it was shown that the broadening of the line profiles due to the stellar rotation has anegligible effect on the synthetic spectra unless the rotation rate is a significant proportionof the stellar break-up velocity. Secondly, I investigated whether small-scale motions in theaccretion and wind flows could increase the width of the infrared line profiles. To do this, Iincorporated the small-scale motions as turbulence using a Maxwell–Boltzmann velocitydistribution. While increasing the turbulence could broaden the lines to the observed range,there was no physical justification for the large value of turbulence required (∼ 100 kms−1).Moreover, the resulting line profiles were boxy (broad peaks and bases), a morphology notcommonly observed in T Tauri spectra. Lastly, I considered whether the high frequency ofinverse P-Cygni profiles decreased the mean ensemble width of the infrared line profiles.For the inclinations of 60◦ and 80◦, it was shown that considering only the blue side ofthe line profiles, the mean HW10% increased by 48 kms−1. However, at an inclinationof 20◦ (within the wind opening angle), the widths decreased when only the blue-sideemission was considered due to greater absorption by the stellar wind. Therefore, itwas concluded that, independently, none of the above modifications could fully bringthe synthetic infrared lines into an agreement with the observed range of profile widths.Nevertheless, the combination of rotation, turbulence, and a reduced inverse P-Cygnifrequency may be sufficient to explain the discrepancy in the widths of the spectra.The high frequency of inverse P-Cygni profiles can be reduced by introducingnon-axisymmetric accretion flow. For instance, if there is an obliquity to the magneticfield so that it is misaligned with the stellar pole, the accretion funnel will favour onehemisphere. In fact, the accretion column may prefer to flow in one hemisphere evenwhen the magnetic and rotation axes align (Ireland et al. 2021, 2022). If the accretioncolumn is non-axisymmetric, the line and continuum emissions are expected to vary as thestar rotates. This has previously been shown in several radiative transfer simulations, forexample, Symington et al. (2005a), Kurosawa et al. (2005), and Kurosawa et al. (2008).In this thesis, I expanded upon the literature by including a stellar wind in the 3D models,analysing the effect of mass loss and accretion rate, and quantifying the change in profilemorphology. It was seen that on average, inverse P-Cygni profiles occur only in ≈ 34 percent of the infrared profiles and, as expected, are phase-dependent. This is close to theobserved frequency of inverse P-Cygni profiles (Folha & Emerson 2001).I showed that, by using fiducial 3D radiative transfer T Tauri models, I could produce126synthetic hydrogen spectra that had similar morphologies and time-series variation asseen in observations. The rotational variability of the line profiles is dependent on thewavelength, inclination, accretion rate, and mass-loss rate of the models. As the accretionrate decreased, the change in the line profiles in different phases increased. The phasechange in HW10% and the equivalent width of the synthetic line profiles are similar inrange to the observed time-series data. The auto-correlation maps produced for our lineprofiles agree with the general areas of correlation and anti-correlation seen in previousradiative transfer studies and some observations. The auto-correlation maps also allowedme to make a prediction as to the origin of the anti-correlation between the high-velocityblue wing of HU and its line centre observed in LkCa 15 (Alencar et al. 2018). The meanchange in HW10% of the infrared lines is ≈ 60 kms−1, less than the necessary widthincrease seen in Chapter 4. Only six models are presented, and the maximum changein the infrared HW10% over the rotational phase is ≈ 170 kms−1. However, both theinfrared and HU spectra are broadened concurrently. This shifts the infrared lines intobetter agreement with the observed range of widths while simultaneously making the HUprofiles too wide. The effect is exacerbated by Stark broadening. A more extensive setof models is needed with a wide range of accretion rates temperatures to determine if anon-axisymmetric geometry can solve the narrow infrared lines.In conclusion, the 3Dmodels presented in this thesis build on the work of Symingtonet al. (2005a) and Esau et al. (2014) by focusing on a magnetospheric geometry with accre-tion columns in alternate hemispheres and exploring this paradigm for a range of accretionrates and azimuthal extents. Furthermore, the models also include a non-axisymmetricstellar wind, the mass-loss rate of which has a significant effect on the spectral variabilityand line correlations. Lastly, the models predict line width and morphology variations,which could be verified with current or future time-series data. Specifically, it wouldrequire a time-series monitoring of T Tauri stars (e.g. Sousa et al. 2021; Pouilly et al.2020; Alencar et al. 2018) over a few rotation periods using a high spectral resolution andbroad wavelength coverage, ideally producing flux-calibrated spectra.The observed PaV to BrW flux ratio for T Tauri stars, both in our data and in thatof Muzerolle et al. (2001), is constant across a wide range of accretion rates. I showedthat for the models to produce synthetic line profiles with a PaV to BrW flux ratio similarto the observed stars, the magnetospheric temperature must increase as the accretion ratedecreases; this confirmed the results of Muzerolle et al. (2001). Finally, it was seen thatthe frequency of the P-Cygni profiles was correlated not just with the mass-loss rate butwith the ratio of mass loss to mass accretion. This is because the increased emissionfrom higher accretion rates masks the blue-shifted absorption. A further effect seen in oursynthetic hydrogen profiles is that, despite the larger optical depth of HU, the frequency127CHAPTER 6. CONCLUSIONS AND FUTURE OUTLOOKof the P-Cygni profiles was higher in the infrared hydrogen lines.In summary, in this thesis, I demonstrate that the T Tauri axisymmetric radiativetransfer models cannot simultaneously reproduce the observed distributions of width andmorphology for HU, PaV, PaW, and BrW. The addition of rotation and turbulence tothe fiducial models cannot sufficiently broaden the infrared lines, while the rotation andturbulent velocities are constrained to physically realistic values. By introducing a non-axisymmetric magnetosphere, the half-widths of the synthetic spectra can be increasedby a mean of 50 per cent. However, both the infrared and HU lines are broadenedsimultaneously. Finally, from the radiative transfer simulations, several predictions havebeen made. For instance, the frequency of P-Cygni profiles, the sensitivity of the hydrogenspectra to a stellarwindmass-loss rate, and the variability of theHW10%and the equivalentwidth. Although there are still improvements to be made with the models developed inthis thesis, the research presented shows the current state of the modern radiative transfersimulations and pushes the field closer to a better understanding of T Tauri spectroscopy.6.1 Future OutlookAs this thesis has demonstrated, while radiative transfer simulations can provide valuableinsight into the radiative emission of T Tauri stars, some phenomena are challengingto reproduce. The first and most tempting recourse is to throw more physics into thecomputational melting pot. However, the radiative transfer models of T Tauri stars alreadysuffer from a surfeit of free parameters. Adding further elaborations has the potential toincrease the degeneracy of the simulated emission without adding any real insight intothe origins of the observed spectra. Moreover, given a sufficient number of parameters, amodel can be constructed to fit any observed spectra. Therefore, the addition of furthercomplexity to T Tauri radiative transfer models must be done carefully and, if possible,with observational and theoretical constraints.Nevertheless, several areas of improvement and research should be investigated infuture radiative transfer studies. These can be broadly split into two categories: observa-tions of T Tauri stars to add constraints to the models and developments to the radiativetransfer simulations. Future radiative transfer models would benefit from constraintsgained from the following comparison, either from pre-existing data or new observations.• Further comparisons should be made between synthetic line profiles and large datasets of high-resolution, optical, infrared, and UV spectra. For example, from theGIARPS infrared spectrometer. Simultaneous observations of multiple atomic tran-sition lines are desirable to remove uncertainty from the spectral variability. Further-1286.1. FUTURE OUTLOOKmore, as an improvement over the spectra presented in this thesis, the observationsshould be corrected for veiling and extinction. This would allow for comparisonsbetween the flux and the equivalent width of the observed and synthetic data.• High-cadence time-series spectroscopy of T Tauri stars is necessary for constrainingthe 3D non-axisymmetric magnetospheric radiative transfer models. The observa-tions should be used to test predictions such as the variability of the equivalent widthand the HW10%.• Spectro-interferometry surveys and spectro-astrometry of T Tauri spectra couldprobe the geometry of the magnetosphere and winds by providing sub-seeing spatialinformation. This may add constraints to the size and location of line-formingregions around the T Tauri star.• Observations characterising the surfacemagnetic fields of T Tauri stars, for example,ZDI maps, may further constrain the magnetospheric geometry.Future radiative transfer models should attempt to reduce the degeneracy of the syntheticspectra and further curtail their free parameters by constraints based on T Tauri observa-tions. However, subsequent studies should consider the following areas of research.• In this thesis, I considered the four hydrogen transition lines of HU, PaV, PaW,and BrW. However, future work would benefit from modelling more lines in bothhydrogen and helium. For example, the He I _10830 Å triplet, is optically thick andvery sensitive to absorption from winds and the magnetosphere (e.g. Kwan et al.2007). To this end, TORUS already contains much of the necessary code to extendthe models to the helium lines, making this an ideal project for future work.• An outstanding issue with the T Tauri model is the unknown heating mechanismsof the accretion column and its temperature distribution. Martin (1996) developeda self-consistent theory of heating using adiabatic compression, but it was shownby Muzerolle et al. (1998b) that it did not produce line profiles in agreement withthe observations. Therefore, it is imperative that work is undertaken to create a self-consistent theory of heating in the accretion column. Codes such asTORUS can thenbe used to test heating mechanisms by performing radiative equilibrium calculationsto determine magnetospheric temperature distributions and emission.• Further radiative transfer simulations may benefit from including continuum emis-sion from dust in the circumstellar disk, which is an important factor for infraredlines, in particular, BrW.129CHAPTER 6. CONCLUSIONS AND FUTURE OUTLOOK• The model presented in this thesis considers the hot-spot as producing only contin-uum emission. However, line emission is expected from these regions (e.g. Dodin2018) and its inclusion may be significant in the simulated spectra.• The models could be used to create synthetic spectro-astrometric and spectro-interferometric observations to give insight into the spatial separation and orientationof the line emission regions.• Spectropolarimetry has been used to show that the T Tauri emission lines originatefrom within a structured magnetic field (e.g. Symington et al. 2005b). A possibleextension to such observations is to use the radiative transfer simulations to computethe Stokes vectors. Adding a magnetic field configuration to the grid (e.g. frommagnetohydrodynamic models) would allow for the simulation of synthetic circularspectropolarimetry and more accurate diagnostics of observed magnetic fields.Finally, several studies of T Tauri line emission have used the output of a magne-tohydrodynamic simulation to map the density and velocity distributions onto a radiativetransfer grid to produce synthetic line profiles (e.g. Kurosawa et al. 2008; Kurosawa &Romanova 2012; Alencar et al. 2012). These forms of simulation offer the benefits ofboth dynamic and radiative modelling. However, the temperature distributions of themagnetohydrodynamic models are not realistic, as they lack a comprehensive treatment ofradiative heating and cooling. 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Rev., 210, 109135Appendix AAtlas of Line Profiles from theAxisymmetric Accretion ModelsThe full set of line profiles that are generated in Chapter 4 using an 2.5D axisymmetric TTauri model.107010H50005000.511.5Pa12Pa500050012Br10802.5H50005000.51Pa0.51Pa50005000.51Br6500 K1090.511.5H50005000.751Pa0.51Pa50005000.81Br05H50005000.511.5Pa12Pa50005000.511.5Br02.5H50005000.51Pa0.51Pa50005000.51Br7500 K0.511.5H50005000.751Pa0.51Pa50005000.81Br010H50005000.511.5Pa12Pa500050012Br010H50005000.751Pa0.511.5Pa50005000.751Br8500 K0.511.5H50005000.751Pa0.51Pa50005000.91Br010H500050012Pa123Pa5000500123Br020H500050011.5Pa12Pa500050011.5Br9500 K24H50005000.751Pa0.751Pa50005000.91Br0.00.20.40.60.81.0Velocity [kms1 ]0.00.20.40.60.81.0Relative Flux206080FigureA.1:AsampleofthelineprofilesproducedbyTORUS.Thespectraareshownfor¤ "sw=0.1¤ "accand)sw=10000K.Thelinesarearrangedby¤ "acc(columns)and)acc(rows).Eachsectionissubdividedbyhydrogentransitionandshowsthethreedifferentinclinations(colours).137APPENDIX A. ATLAS OF LINE PROFILES FROM THE AXISYMMETRIC ACCRETION MODELS1070 5H50005000.75 11.25Pa1 2Pa500050011.25Br1080.5 11.5H50005000.75 1Pa0.5 1Pa50005000.9 1Br6500 K10900.5 1H500050000.5 1Pa11.005Pa500050000.5 1Br0 5H50005000.75 11.25Pa1 2Pa500050011.5Br11.5H50005000.75 1Pa0.5 1Pa50005000.9 1Br7500 K0.75 1H50005000.99 1Pa0.975 1Pa50005000.998 1Br0 10H500050011.5Pa1 2Pa50005001 2Br5 10H500050011.25Pa1 2Pa500050011.2Br8500 K0.5 11.5H50005000.995 1Pa0.98 1Pa50005000.998 1Br5 10H50005001 2Pa1 2 3Pa50005002 4Br10 20H50005001 2Pa1 2 3Pa50005001 2Br9500 K2.5 5H50005000.9 11.1Pa11.25Pa50005000.975 11.025Br0.00.20.40.60.81.0Velocity [kms1]0.00.20.40.60.81.0Relative Flux206080FigureA.1:Continued.Modelspresentedherefor¤"sw=0.01¤"acc and)sw=10000K.1381072.557.5H500050011.5Pa12Pa500050011.5Br1080.7511.25H50005000.991Pa0.9751Pa50005000.99751Br6500 K1090.951H50005000.9999751Pa0.9999511.00005Pa50005000.999991Br2.557.5H500050011.5Pa12Pa500050011.5Br12H50005000.9751Pa0.91Pa500050011.005Br7500 K0.7511.25H500050011.0005Pa11.002Pa500050011.0001Br510H500050012Pa123Pa500050012Br510H500050011.5Pa12Pa500050011.25Br8500 K0.511.5H50005000.99511.005Pa0.9911.01Pa500050011.001Br2.557.5H5000500123Pa24Pa50005002.55Br020H500050012Pa123Pa500050012Br9500 K2.55H50005000.911.1Pa11.25Pa50005000.97511.025Br0.00.20.40.60.81.0Velocity [kms1 ]0.00.20.40.60.81.0Relative Flux206080FigureA.1:Continued.Modelspresentedherefor¤ "sw=0.001¤ "accand)sw=10000K.139APPENDIX A. ATLAS OF LINE PROFILES FROM THE AXISYMMETRIC ACCRETION MODELS1070 5H50005000.75 11.25Pa1 2Pa500050011.25Br1080.5 1H50005000.75 1Pa0.5 1Pa50005000.9 1Br6500 K1090.5 1H50005000.99 1Pa0.975 1Pa50005000.9975 1Br0 5H500050011.5Pa1 2Pa500050011.5Br0.5 11.5H50005000.75 1Pa0.5 1Pa50005000.9 1Br7500 K0.75 1H50005000.99 1Pa0.975 1Pa50005000.9975 1Br0 10H500050011.5Pa1 2Pa50005001 2Br5 10H500050011.25Pa1 2Pa500050011.2Br8500 K0.75 11.25H50005000.99 1Pa0.975 1Pa50005000.9975 1Br0 10H50005001 2Pa1 2 3Pa50005002 4Br10 20H50005001 2Pa1 2 3Pa50005001 2Br9500 K2.5 5H50005000.9 11.1Pa11.25Pa50005000.975 11.025Br0.00.20.40.60.81.0Velocity [kms1]0.00.20.40.60.81.0Relative Flux206080FigureA.1:Continued.Modelspresentedherefor¤"sw=0.1¤"acc and)sw=8000K.14010705H500050011.5Pa12Pa500050011.25Br1080.751H50005000.991Pa0.9751Pa50005000.99751Br6500 K1090.81H50005000.999951Pa0.999751Pa50005000.9999751Br2.557.5H500050011.5Pa12Pa500050011.5Br11.5H50005000.9751Pa0.91Pa500050011.005Br7500 K0.7511.25H500050011.0005Pa11.002Pa500050011.0001Br510H500050012Pa123Pa500050012Br510H500050011.5Pa12Pa500050011.25Br8500 K0.511.5H50005000.99511.005Pa0.9911.01Pa500050011.001Br510H5000500123Pa123Pa50005002.55Br020H500050012Pa123Pa500050012Br9500 K2.55H50005000.911.1Pa11.25Pa50005000.97511.025Br0.00.20.40.60.81.0Velocity [kms1 ]0.00.20.40.60.81.0Relative Flux206080FigureA.1:Continued.Modelspresentedherefor¤ "sw=0.01¤ "accand)sw=8000K.141APPENDIX A. ATLAS OF LINE PROFILES FROM THE AXISYMMETRIC ACCRETION MODELS1072.5 57.5H500050011.5Pa1 2Pa500050011.5Br1080.75 11.25H500050011.0005Pa11.002Pa500050011.0001Br6500 K1090.975 11.025H500050011.00001Pa11.00002Pa50005000.999999 1 1Br5 10H50005001 2Pa1 2Pa500050011.5Br1 2H50005000.975 1Pa0.9 1Pa500050011.005Br7500 K0.75 11.25H500050011.0005Pa11.002Pa500050011.0001Br5 10H50005001 2Pa1 2 3Pa50005001 2Br5 10H500050011.5Pa1 2Pa500050011.25Br8500 K0.5 11.5H50005000.995 11.005Pa0.99 11.01Pa500050011.001Br2.5 57.5H50005001 2 3Pa2 4Pa50005002.5 5Br0 20H50005001 2Pa1 2 3Pa50005001 2Br9500 K2.5 5H50005000.9 11.1Pa11.25Pa50005000.975 11.025Br0.00.20.40.60.81.0Velocity [kms1]0.00.20.40.60.81.0Relative Flux206080FigureA.1:Continued.Modelspresentedherefor¤"sw=0.001¤"acc and)sw=8000K.14210705H500050011.5Pa12Pa500050011.25Br1080.7511.25H500050011.0005Pa11.002Pa500050011.0001Br6500 K1090.97511.025H500050011Pa11.00002Pa50005000.9999991Br05H500050011.5Pa12Pa500050011.5Br12H50005000.9751Pa0.91Pa500050011.005Br7500 K0.7511.25H500050011.0005Pa11.002Pa500050011.0001Br010H500050011.5Pa12Pa500050012Br510H500050011.5Pa12Pa500050011.25Br8500 K0.511.5H50005000.99511.005Pa0.9911.01Pa500050011.001Br010H5000500123Pa123Pa500050024Br020H500050012Pa123Pa500050012Br9500 K2.55H50005000.911.1Pa11.25Pa50005000.97511.025Br0.00.20.40.60.81.0Velocity [kms1 ]0.00.20.40.60.81.0Relative Flux206080FigureA.1:Continued.Modelspresentedherefor¤ "sw=0.1¤ "accand)sw=6000K.143APPENDIX A. ATLAS OF LINE PROFILES FROM THE AXISYMMETRIC ACCRETION MODELS1072.5 57.5H500050011.5Pa1 2Pa500050011.5Br1080.75 11.25H500050011.0005Pa11.002Pa500050011.0001Br6500 K1090.975 11.025H500050011.00001Pa11.00002Pa50005000.999999 1 1Br2.5 57.5H500050011.5Pa1 2Pa500050011.5Br1 2H50005000.975 1Pa0.9 1Pa500050011.005Br7500 K0.75 11.25H500050011.0005Pa11.002Pa500050011.0001Br5 10H50005001 2Pa1 2 3Pa50005001 2Br5 10H500050011.5Pa1 2Pa500050011.25Br8500 K0.5 11.5H50005000.995 11.005Pa0.99 11.01Pa500050011.001Br5 10H50005001 2 3Pa1 2 3Pa50005002.5 5Br0 20H50005001 2Pa1 2 3Pa50005001 2Br9500 K2.5 5H50005000.9 11.1Pa11.25Pa50005000.975 11.025Br0.00.20.40.60.81.0Velocity [kms1]0.00.20.40.60.81.0Relative Flux206080FigureA.1:Continued.Modelspresentedherefor¤"sw=0.01¤"acc and)sw=6000K.1441072.557.5H500050011.5Pa12Pa500050011.5Br1080.7511.25H500050011.0005Pa11.002Pa500050011.0001Br6500 K1090.97511.025H500050011.00001Pa11.00002Pa50005000.99999911Br510H500050012Pa12Pa500050012Br12H50005000.9751Pa0.91Pa500050011.005Br7500 K0.7511.25H500050011.0005Pa11.002Pa500050011.0001Br510H500050012Pa123Pa500050012Br510H500050011.5Pa12Pa500050011.25Br8500 K0.511.5H50005000.99511.005Pa0.9911.01Pa500050011.001Br2.557.5H5000500123Pa24Pa50005002.55Br020H500050012Pa123Pa500050012Br9500 K2.55H50005000.911.1Pa11.25Pa50005000.97511.025Br0.00.20.40.60.81.0Velocity [kms1 ]0.00.20.40.60.81.0Relative Flux206080FigureA.1:Continued.Modelspresentedherefor¤ "sw=0.001¤ "accand)sw=6000K.145Appendix BSpectral Variability FiguresFigures showing the mean line profile, the RMSD, the continuum level, and quotientspectra for the 3D non-axisymmetric models presented in Chapter 5.0.000.020.04RMSDH0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.73Min 0.400510Flux0.0110.0170.0230.000.060.12Pa0.00.10.20.30.40.50.60.70.80.9Max 1.34Min 0.620.61.21.80.0180.0240.0300.000.040.08Pa0.00.10.20.30.40.50.60.70.80.9Max 1.26Min 0.69120.0200.0260.0320.00.10.2Br0.00.10.20.30.40.50.60.70.80.920Max 1.35Min 0.611.01.52.00.0180.0240.0300.000.050.10RMSD0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.16Min 0.6836Flux0.0080.0140.0200.00.10.20.00.10.20.30.40.50.60.70.80.9Max 1.16Min 0.810.91.21.50.0170.0230.0290.000.080.16 0.00.10.20.30.40.50.60.70.80.9Max 1.17Min 0.821.01.52.00.0190.0250.0310.00.10.20.00.10.20.30.40.50.60.70.80.960Max 1.18Min 0.831.001.251.500.0170.0230.0290.000.060.12RMSD0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.37Min 0.77Velocity [kms1 ]369Flux0.0040.0100.0160.000.150.30 0.00.10.20.30.40.50.60.70.80.9Max 1.28Min 0.73Velocity [kms1 ]0.91.21.50.0100.0160.0220.00.10.20.00.10.20.30.40.50.60.70.80.9Max 1.30Min 0.75Velocity [kms1 ]1.01.52.00.0120.0180.0240.000.150.30 0.00.10.20.30.40.50.60.70.80.980Max 1.20Min 0.80Velocity [kms1 ]1.001.251.500.0110.0170.023MinMeanMaxFigureB.1:Spectrafrommodel0,arrangedbyinclinationandwavelength.Thebottomplotshowsthemeanspectrum,thegrey-scaleimageisthequotientofthespectrumateachrotationalphase.ThetopplotdisplaysRMSDdistribution.Theplottotheerightshowsthechangeincontinuumlevelacrossthephase.Theminimumandmaximumrangeofeachthegrey-scaleimagesisindicated.147APPENDIX B. SPECTRAL VARIABILITY FIGURES0.000.030.06RMSDH0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.95Min 0.500 4 8Flux0.0100.0160.0220.000.050.10Pa0.00.10.20.30.40.50.60.70.80.9Max 1.44Min 0.590.81.62.40.0170.0230.0290.000.050.10Pa0.00.10.20.30.40.50.60.70.80.9Max 1.36Min 0.701 20.0190.0250.0310.00.10.2Br0.00.10.20.30.40.50.60.70.80.920Max 1.44Min 0.560.81.62.40.0170.0230.0290.000.080.16RMSD0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.36Min 0.563 6 9Flux0.0090.0150.0210.00.10.20.00.10.20.30.40.50.60.70.80.9Max 1.22Min 0.761.21.62.00.0160.0220.0280.000.080.160.00.10.20.30.40.50.60.70.80.9Max 1.21Min 0.761.21.82.40.0180.0240.0300.00.10.20.00.10.20.30.40.50.60.70.80.960Max 1.24Min 0.791.21.62.00.0170.0230.0290.000.080.16RMSD0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.33Min 0.68Velocity [kms1]4 8Flux0.0040.0100.0160.00.10.20.00.10.20.30.40.50.60.70.80.9Max 1.20Min 0.70Velocity [kms1]1.21.62.00.0100.0160.0220.00.10.20.00.10.20.30.40.50.60.70.80.9Max 1.23Min 0.68Velocity [kms1]1.21.82.40.0110.0170.0230.00.10.20.00.10.20.30.40.50.60.70.80.980Max 1.20Min 0.76Velocity [kms1]1.21.60.0100.0160.022MinMeanMaxFigureB.1:Continued.Model1.1480.000.060.12RMSDH0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.81Min 0.55048Flux0.0100.0160.0220.000.050.10Pa0.00.10.20.30.40.50.60.70.80.9Max 1.36Min 0.680.81.62.40.0150.0210.0270.000.040.08Pa0.00.10.20.30.40.50.60.70.80.9Max 1.22Min 0.810.81.62.40.0170.0230.0290.000.080.16Br0.00.10.20.30.40.50.60.70.80.920Max 1.32Min 0.630.61.21.80.0160.0220.0280.000.040.08RMSD0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.38Min 0.5448Flux0.0090.0150.0210.000.15 0.00.10.20.30.40.50.60.70.80.9Max 1.30Min 0.781.21.82.40.0150.0210.0270.000.080.16 0.00.10.20.30.40.50.60.70.80.9Max 1.23Min 0.791.62.40.0170.0230.0290.000.15 0.00.10.20.30.40.50.60.70.80.960Max 1.24Min 0.821.21.62.00.0160.0220.0280.000.080.16RMSD0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.25Min 0.72Velocity [kms1 ]510Flux0.0040.0100.0160.000.080.16 0.00.10.20.30.40.50.60.70.80.9Max 1.31Min 0.73Velocity [kms1 ]1.21.82.40.0090.0150.0210.000.060.12 0.00.10.20.30.40.50.60.70.80.9Max 1.18Min 0.74Velocity [kms1 ]1.62.40.0110.0170.0230.00.10.20.00.10.20.30.40.50.60.70.80.980Max 1.28Min 0.75Velocity [kms1 ]1.01.52.00.0100.0160.022MinMeanMaxFigureB.1:Continued.Model2.149APPENDIX B. SPECTRAL VARIABILITY FIGURES0.000.050.10RMSDH0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.18Min 0.812 4Flux0.0100.0160.0220.00.10.2Pa0.00.10.20.30.40.50.60.70.80.9Max 1.33Min 0.731.01.52.00.0150.0210.0270.000.080.16Pa0.00.10.20.30.40.50.60.70.80.9Max 1.25Min 0.741.01.52.00.0170.0230.0290.000.150.30Br0.00.10.20.30.40.50.60.70.80.920Max 1.27Min 0.751.21.60.0160.0220.0280.000.150.30RMSD0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.43Min 0.411.53.0Flux0.0080.0140.0200.00.20.40.00.10.20.30.40.50.60.70.80.9Max 1.37Min 0.641.21.51.80.0150.0210.0270.00.20.40.00.10.20.30.40.50.60.70.80.9Max 1.31Min 0.681.21.51.80.0170.0230.0290.00.20.40.00.10.20.30.40.50.60.70.80.960Max 1.33Min 0.701.001.251.500.0160.0220.0280.00.10.2RMSD0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.27Min 0.43Velocity [kms1]1.53.0Flux0.0040.0100.0160.00.20.40.00.10.20.30.40.50.60.70.80.9Max 1.50Min 0.69Velocity [kms1]1.21.50.0090.0150.0210.000.150.300.00.10.20.30.40.50.60.70.80.9Max 1.40Min 0.72Velocity [kms1]1.21.50.0110.0170.0230.000.250.500.00.10.20.30.40.50.60.70.80.980Max 1.42Min 0.76Velocity [kms1]1.01.21.40.0100.0160.022MinMeanMaxFigureB.1:Continued.Model3.1500.000.060.12RMSDH0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.67Min 0.620.81.6Flux0.0030.0090.0150.00.10.2Pa0.00.10.20.30.40.50.60.70.80.9Max 1.03Min 0.970.961.000.0130.0190.0250.00.10.2Pa0.00.10.20.30.40.50.60.70.80.9Max 1.07Min 0.940.961.040.0160.0220.0280.000.150.30Br0.00.10.20.30.40.50.60.70.80.920Max 1.02Min 0.990.9901.0050.0160.0220.0280.00.20.4RMSD 0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.38Min 0.701.21.6Flux0.0030.0090.0150.00.51.00.00.10.20.30.40.50.60.70.80.9Max 1.04Min 0.880.9901.0051.0200.0130.0190.0250.000.250.50 0.00.10.20.30.40.50.60.70.80.9Max 1.08Min 0.861.001.041.080.0160.0220.0280.00.61.20.00.10.20.30.40.50.60.70.80.960Max 1.02Min 0.930.9961.0021.0080.0160.0220.0280.000.150.30RMSD0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.38Min 0.71Velocity [kms1 ]1.21.6Flux0.0000.0060.0120.00.81.60.00.10.20.30.40.50.60.70.80.9Max 1.06Min 0.83Velocity [kms1 ]0.9901.0050.0080.0140.0200.00.40.80.00.10.20.30.40.50.60.70.80.9Max 1.10Min 0.80Velocity [kms1 ]0.991.021.050.0100.0160.0220.00.81.60.00.10.20.30.40.50.60.70.80.980Max 1.04Min 0.91Velocity [kms1 ]0.9921.0001.0080.0100.0160.022MinMeanMaxFigureB.1:Continued.Model4.151APPENDIX B. SPECTRAL VARIABILITY FIGURES0.000.15RMSDH0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.35Min 0.711.01.52.0Flux0.0030.0090.0150.00.20.4Pa0.00.10.20.30.40.50.60.70.80.9Max 1.04Min 0.961.0001.0251.0500.0130.0190.0250.00.20.4Pa0.00.10.20.30.40.50.60.70.80.9Max 1.08Min 0.911.021.081.140.0160.0220.0280.00.20.4Br0.00.10.20.30.40.50.60.70.80.920Max 1.02Min 0.981.0051.0200.0160.0220.0280.00.20.4RMSD0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.41Min 0.611.21.6Flux0.0030.0090.0150.00.51.00.00.10.20.30.40.50.60.70.80.9Max 1.05Min 0.860.9901.0051.0200.0130.0190.0250.00.40.80.00.10.20.30.40.50.60.70.80.9Max 1.09Min 0.850.991.021.050.0160.0220.0280.00.61.20.00.10.20.30.40.50.60.70.80.960Max 1.03Min 0.921.0001.0080.0160.0220.0280.00.20.4RMSD0.00.10.20.30.40.50.60.70.80.9PhaseMax 1.43Min 0.63Velocity [kms1]1.21.5Flux0.0000.0060.0120.00.81.60.00.10.20.30.40.50.60.70.80.9Max 1.07Min 0.79Velocity [kms1]0.9901.0051.0200.0080.0140.0200.00.61.20.00.10.20.30.40.50.60.70.80.9Max 1.12Min 0.76Velocity [kms1]1.0001.0250.0100.0160.0220.00.81.60.00.10.20.30.40.50.60.70.80.980Max 1.04Min 0.90Velocity [kms1]0.9921.0001.0080.0100.0160.022MinMeanMaxFigureB.1:Continued.Model5.152Appendix CAuto-Correlation MapsAuto-correlation maps of the 3D T Tauri models with non-axisymmetric accretion andwind that are presented in Chapter 5.Anti-correlation Correlation2000200H2000200200 0 2002000200Pa200 0 200Pa200 0 20020Br60200 0 200800.0 0.2 0.4 0.6 0.8 1.0Velocity [kms 1]0.00.20.40.60.81.0Velocity [kms1 ]Figure C.1: Auto-correlation map for model 0. White areas correspond to a strong correlationand black areas strong anti-correlation. The images are symmetrical along the x-y diagonal.APPENDIX C. AUTO-CORRELATION MAPSAnti-correlation Correlation2000200H2000200200 0 2002000200Pa200 0 200Pa200 0 20020Br60200 0 200800.0 0.2 0.4 0.6 0.8 1.0Velocity [kms 1]0.00.20.40.60.81.0Velocity [kms1 ]Figure C.2: Model 1154Anti-correlation Correlation2000200H2000200200 0 2002000200Pa200 0 200Pa200 0 20020Br60200 0 200800.0 0.2 0.4 0.6 0.8 1.0Velocity [kms 1]0.00.20.40.60.81.0Velocity [kms1 ]Figure C.3: Model 2155APPENDIX C. AUTO-CORRELATION MAPSAnti-correlation Correlation2000200H2000200200 0 2002000200Pa200 0 200Pa200 0 20020Br60200 0 200800.0 0.2 0.4 0.6 0.8 1.0Velocity [kms 1]0.00.20.40.60.81.0Velocity [kms1 ]Figure C.4: Model 3156Anti-correlation Correlation2000200H2000200200 0 2002000200Pa200 0 200Pa200 0 20020Br60200 0 200800.0 0.2 0.4 0.6 0.8 1.0Velocity [kms 1]0.00.20.40.60.81.0Velocity [kms1 ]Figure C.5: Model 4157APPENDIX C. AUTO-CORRELATION MAPSAnti-correlation Correlation2000200H2000200200 0 2002000200Pa200 0 200Pa200 0 20020Br60200 0 200800.0 0.2 0.4 0.6 0.8 1.0Velocity [kms 1]0.00.20.40.60.81.0Velocity [kms1 ]Figure C.6: Model 5158

The Enigma of Hydrogen Emission from T Tauri Stars

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The Enigma of Hydrogen Emission from T Tauri Stars

Abstract

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