Fermi Gamma-ray Space Telescope

XSPEC Analysis of Fermi Data

The XSPEC-type spectral analysis with which most high energy astrophysicists are familiar is a maximum likelihood analysis undertaken under special conditions. As we will show, these conditions usually will not apply to the LAT data. For the discussion here, 'XSPEC analysis' will refer to spectral analysis using XSPEC or a similar tool.

When an XSPEC Analysis is Valid

In XSPEC analysis the data consist of counts binned by energy in one dimensional vectors. For an imaging mission the analyst accumulates all the counts from a region surrounding the source into a count spectrum; these counts are binned in energy, either on the ground, or by the detector hardware onboard the spacecraft. No distinction is made between counts that are very close to the source or those on the edge of the accumulation region; forming the spectrum has destroyed information—where the counts were detected. Some missions, such as RXTE, by design accumulate spectra only, with no imaging information.

Thus the fundamental data set for an XSPEC analysis is a one-dimensional vector of counts in energy bins. An estimate of the background is subtracted from this count spectrum, resulting in a vector of the counts assumed to result from the source. A model of the source's flux distribution is run through a model of the detector, producing a model vector of the counts that would have resulted if the source model were correct. The source model is then varied until the agreement between the observed and actual are sufficiently close (χ2 is minimized). This process assumes that the number of counts in each energy bin will have a normal (Gaussian) distribution with an expectation value equal to the model counts.

However, where the number of counts in an energy bin is small, as is often the case for LAT photon data, we need another statistic for finding the best fit, and we usually do not have a goodness of fit measure. In addition, for LAT analysis we want to consider where the counts fall relative to the point source—counts that are further from a source are less likely to originate from that source. Thus we want to use software that was constructed to consider multi-dimensional data.

The LAT Data and the Conditions for XSPEC Analysis

In most applications the analysis of LAT data must be multi-dimensionsal. The LAT PSF is large (~3.5 degrees) at low energy (~100 MeV), small (<0.15 degrees) at high energy (~10 GeV). With the LAT's large effective area (>0.8 m2), many sources will be detected near the analyzed source(s); their PSFs will merge at low energy. In addition, the LAT's FOV is large—usable counts can be accumulated from over 65 degrees off-axis—and with the large PSF, distant sources will influence the modeling of a given source. For example, to model a given source, the sources within a few PSF radii must be modeled since some of the counts from these sources will fall near the source of interest. But to model these sources, the sources a few PSF radii further out must be modeled. Obviously the influence of a source becomes attenuated with distance, but as a result of the large FOV, distant sources do influence the analysis. Finally, the spatially varying diffuse background must be included in the analysis. Therefore the analysis must be three dimensional—two spatial and one spectral.

The instrument response (PSF, effective area, energy resolution) is currently a function of energy, inclination angle (the angle between the source and the LAT normal) and photon category. Since the LAT is usually in sky-survey mode, a source will be observed at different inclination angles. Each count is therefore characterized by a different instrument response function (IRF). It is feasible to sum the counts from a given direction and form an IRF weighted by the fraction of the observation that occurred at a given inclination angle, but a great deal of information will be lost. For example, counts for which the PSF is small will be mixed with counts for which the PSF is large. Therefore, it is best to maintain the inclination angle as one of the observables characterizing a count.

Each count is therefore characterized by many observables, for example, the apparent energy, the apparent origin, the inclination angle, and the event category. If one forms bins with finite widths in each of these observable dimensions, then the number of bins will be very large. Even if many counts are accumulated, the number of counts in each bin will be small, in many cases only 0 or 1. Statistical treatments that assume that the distribution in each bin is Gaussian, or near Gaussian, are usually inadequate.

Thus the conditions under which an XSPEC analysis can be undertaken usually do not apply to the LAT data: the data usually must be treated as multi-dimensional and cannot be binned into one-dimensional bins or with sufficient numbers of counts per bin. However, XSPEC analysis of LAT data would be justified when the data can be binned into a one dimensional count spectrum with sufficient counts per energy bin. An XSPEC analysis should be valid for a few strong sources that dominate any surrounding sources (e.g. a high galactic latitude AGN in a flaring state) and for fluent gamma-ray bursts (few background counts are expected within the PSF during the short burst duration).

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