Spectral Broadening Functions

Many astrophysical phenomena involving velocity elds produce broadening of spectral lines. Frequently, the cross-correlation function (CCF) is used to extract information about this broadening from segments of high resolution spectra. CCF has become the standard tool for handing extraction of radial-velocity and broaden-ing information from high resolution spectra as it permits integration of information which is common to many spectral lines into one function which is easy to calculate, visualize and interpret. However, it can be argued that this is not the best tool for applications such as determinations of metallicities ([Fe/H]), for nding locations of star spots on active stars or to study projected shapes of such distorted stars as contact binaries. For such applications, the proper broadening functions (BF) should be used. Properties of the BF’s are discussed in this paper with a stress on the fact that the CCF’s are not broadening functions. This note concentrates on the advantages of determining the BF’s through the process of linear inversion, preferably accomplished using the Singular Value Decomposition (SVD). 1. Convolution and cross-correlation This review attempts to have some practical value. Therefore, several examples of prac-tical applications will be given. Because the IDL programming language contains many relevant routines and oers one of the most concise notations, the examples will be given in this language. They will be marked by a symbolic prompt: IDL>. We should note that stellar spectra will be treated as simple, one-dimensional vectors. Convolution is an operation that the nature does for us. We seldom see ‘naked" functions; they are usually accessible as convolutions. These could be a convolution with the spectrograph’s instrumental prole or with the radial component of the micro-turbulence velocity eld in the stellar atmosphere or a rotational broadening function making spectrum of one star dierent relative to that of another star. Thus, instead o