X-ray spectral Analysis xamples Problems Algorithm Model types Statistic Fitting procedure Practical problems with Chandra CCD data Disk N(N) Disk N(S) Nucleus Plume Based partly on the lecture given by Keith Arnaud (NASA/GSFC & UMCP) Pietsch et al. (2001) XMM-Newton PC PN Spectra XMM/Newton RGS spectrum of the nucleus region + plume (ULX) Disk: thermal temperature: 0.13-0.5 kev, assuming solar abundance A2125 Z=0.25 Red: 0.5-2 kev Green: 2-4 kev Blue: 4-8 kev
ACS Spectrum of A2125 Unfold spectrum kt ~ 4 kev N H ~ 2x 10 20 cm -2 Practical X-ray spectroscopy Can we start with this Most X-ray spectra are of moderate or low resolution (eg Chandra ACS or XMM-Newton PC). However, the spectra generally cover a bandpass of more than 1.5 decades in energy. Moreover, the continuum shape often provides important physical information. Therefore, most uses of X-ray spectra have to rely on a simultaneous analysis of the entire spectrum. and deduce this The Basic Problem Suppose we observe D() counts in channel (of N) from some source. Then : D() = T R(,) A() S() d T is the observation length (in seconds) R(,) is the spectral matrices --- the probability of an incoming photon of energy being registered in channel (dimensionless) A() is the energy-dependent effective area of the telescope and detector system (in cm 2 ) S() is the source flux at the front of the telescope (in photons/cm 2 /s/kev)
An example R(,) The Basic Problem escape escape photopeak photopeak fluorescence fluourescenc e D() = T R(,) A() S() d T, A() and R(,) are known. We need to solve this integral equation for S(). We can divide the energy range of interest into M bins and turn this into a matrix equation: D i = T R ij A j S j where S j is now the flux in photons/cm 2 /s in energy bin J. We want to find S j. The Basic Problem This has a solution: D i = T R ij A j S j S j =1/(T A j ) (R ij ) - But this does not work! The S j derived in this way are very sensitive to slight changes in the data D i. This is a great method for amplifying noise. 1 D i A (brief) Mathematical Digression This is known as the remote sensing problem and arises in many areas of astronomy as well as eg geophysics and medical imaging. n mathematics the integral is known as a Fredholm equation of the first kind. Tikhonov showed that such equations can be solved using regularization - applying a priori knowledge to damp the noise. A familiar example is maximum entropy but there are a host of others. Some of these have been tried on X-ray spectra - none have had any impact on the field. Forward-fitting arf rmf χ 2 This standard method of analyzing X-ray spectra comprises the following step: Calculate a model spectrum. Multiply this spectrum by the effective area. Further convolve it with an instrumental response matrix. Compare the result with the actual observed data by calculating some statistic. Modify the model spectrum and repeat till the best value of a chosen statistic is obtained. Forward-fitting (cont.) Works if the number of fitting parameters is reasonably small so that the model can be varied in some sensible fashion. The aim of the forward-fitting is then to obtain the best-fit and confidence ranges of these parameters.
Spectral fitting programs XSPC - part of HAsoft. General spectral fitting program with many models available. Sherpa-part of CAO. Multi-dimensional fitting program which includes the XSPC model library and can be used for spectral fitting. SPX - from SRON in the Netherlands. Spectral fitting program specializing in collisional plasmas and high resolution spectroscopy. SS - from the MT Chandra HTG group. Mainly intended for the analysis of grating data. Models All models are wrong, but some are useful! X-ray spectroscopic models are usually built up from individual components. These can be thought of as two basic types: Additive (an emission component e.g. blackbody, line, ) Multiplicative (something which modifies the spectrum e.g. absorption). Model = M 1 * M 2 * (A 1 + A 2 + M 3 *A 3 ) + A 4 Additive Models Basic additive (emission) models include : Blackbody and its variations (e.g., neutron stars) Power-law (AGNs, X-ray binaries) Collisional plasma (e.g, raymond, mekal; SNRs, the hot SM/GM) Gaussian lines Many more models: Accretion disks Comptonized plasmas Non-equilibrium ionization plasmas Multi-temperature collisional plasmas Multiplicative (and Other) Models Photoelectric absorption due to our Galaxy. Photoelectric absorption due to ionized material. High energy exponential roll-off. Other types of XSPC model components (e.g., convolution) which are used like a multiplicative model but perform more complicated operations on the current model. Roll Your Own Models Statistic. A simple XSPC model interface is available. You can write new models and fit them to your data. Specifically, ither write a subroutine that takes in the energies on which to calculate the model and writes out the fluxes (in photons/cm2/s). Or a table model - files of a standard format that contain model spectra, which can then be fit to data. χ 2 =Σ(d i -m i ) 2 /σ 2 i Convenient and goodness of fit To use the χ 2 statistic, the rule-of thumb is that each data point contains > 6 counts Thus rebinning of the data is often necessary (e.g., using the FTOOLS routine grppha). Cash Statistic C=-2log(L), where L=Π i p(d i m i ) Poisson likelihood is an unbiased estimator, whereas χ 2 can be biased, depending on the choice of No goodness of fit criterion C-statistic cannot be used to analyze backgroundsubtracted data, since the difference of two Poissondistributed variables is not Poisson distributed. The background and the source data can be fit simultaneously.
Finding the best-fit Finding the best-fit means minimizing the statistic value (e.g., χ 2 or C-statistic). Many algorithms available to do this in a computationally efficient fashion (see Numerical Recipes). Most methods used to find the best-fit are local i.e. they use some information around the current parameters to guess a new set of parameters. All these methods are liable to get stuck in a local minimum. Watch out for this! The more complicated your model and the more highly correlated the parameters then the more likely that the algorithm will not find the absolute best-fit. Global Minimization There are global minimization methods available - simulated annealing, genetic algorithms, - but they require many function evaluations (so are slow) and are still not guaranteed to find the true minimum. A new technique called Markov Chain Monte Carlo, which provides an intelligent sampling of parameter space, looks promising but it is not yet widely available (not yet available to XSPC). Confidence ntervals 64%, 90%, 99% contours corresponding to χ 2 = χ 2 min + (1, 2.7, and?) Final Advice and Admonitions The purpose of spectral fitting is to attain understanding, not fill up tables of numbers. Try to test whether you really have found the best-fit. X-ray data analysis is half science and half art. A fit only means that the model is consistent with the data: There are always alternative models that can fit the data equally well or better. Astrophysical intuition is absolutely important! Don t misuse the F-test. nergy resolution of the ACS CT correction for ACS- data The energy resolution of the S3 and 3 as a function of row number. These data were taken at -120C. Radiation damage of F chips the need for the CT correction before spectral analysis