Fermi Gamma-ray Space Telescope

Binned GRB Spectral Analysis

Method

Gamma-ray burst spectral analysis takes advantage of the unique properties of this phenomenon: a relatively short transient from a point source. For the LAT there will usually not be competing emission from other sources in the field-of-view, while the GBM cannot distinguish spatially between burst photons and those originating from other sources. Therefore, the analysis is one dimensional: we determine the input burst flux from the apparent energies of the events that triggered the detector (for the purposes of this discussion we call these events 'counts'). In binned spectral analysis the apparent energies are binned. For the GBM there is little choice because the measured energies are binned by the hardware, while for the LAT we will assume there are sufficient counts per bin.

For the LAT we select all the counts from a region 1-2 PSF-radii around the burst position from the time range which includes the burst; these counts are then binned into energy channels. These counts should all originate from the burst because estimates of the non-burst event rate predict about one count within a PSF-radius per minute.

For the GBM the origin of the counts is unknown, and therefore all the counts from the burst's time range are selected. The counts consist of photons originating from the burst and background from other sources, both astrophysical and instrumental. The background is usually estimated from the count rates before and after the burst. The GBM will have already binned the counts into predetermined energy channels.

The selected counts may be further binned in time. The result is then a series of count spectra that will be analyzed. In general each count spectrum is fitted independently.

Consider a count spectrum ci, where the index i runs over energy channels. This count spectrum is the sum of the burst flux convolved with the detector response and the background bi. We sample the photon flux striking the detector in different energy channels fj, where the index j runs over energy channels (NOT necessarily the same channels as the count spectrum!). The response function can be simplified into a mapping between the photon's true energy and the count's apparent energy. With the counts and the fluxes expressed as vectors, the response function is a matrix Dij, the 'Detector Response Matrix' (DRM) in the burst community (frequently called the 'RSP'). The resulting matrix equation is

ci = Dijfj+bi

where summation over j is assumed. Since Dij is not a square matrix, and even if it is, it is usually nearly singular, this equation cannot be solved by inverting Dij but requires 'forward folding.' Note that for the LAT bi~0 but for the GBM bi is substantial, and in many channels will dominate the burst counts.

In forward folding a model flux vector f'j is folded through the response, resulting in a model count spectrum c'i. The underlying model flux is usually an analytic function (e.g., a power law) with a small number of spectral parameters (e.g., normalization and spectral index for a power law flux model). The model c'i is compared to the observed ci, and then a new model flux vector f'j is calculated, usually by varying the spectral parameters. This iterative process ends when the model c'i is sufficiently close to the observed ci, resulting in best-fit spectral parameters.

'Sufficiently close' is usually determined by minimizing χ2. A sufficiently small value of χ2, e.g., comparable to the number of degrees-of-freedom, indicates that the fit is satisfactory. If the number of counts per bin is not large enough to assume that they are drawn from a Gaussian distribution, then the Cash statistic should be used instead of χ2.

If two or more detectors observe the same burst at the same time, then the counts recorded by each detector resulted from the same input burst spectrum. Thus, we can require that the count spectra for each detector be fit by the same flux model. The result is a joint fit.

LAT Analysis

The LAT data consist of photons. To use the techniques described above, these events must first be binned. The steps in the analysis are as follows:

  • Extract the photons from a region around the burst at the time of the burst. The selection of the photons occurs both in the extraction of the data from the database and using the gtselect tool.
  • Bin the photons with gtbin. You choose the energy bins either interactively (bins that are uniform in photon energy or the logarithm of the photon energy) or through a binning file. The time bins can be also be chosen interactively or through a binning file. The time bins can be uniform in time, have a constant signal-to-noise ratio per bin, or be Bayesian Blocks. The binning files are created either by a previous run of gtbin or using the gtbindef utility. The output of the binning is a PHA or PHAII file, standard FITS filetypes.
  • Create the DRM with gtrspgen. The output is a RSP file, also a standard FITS filetype.
  • Fit the resulting spectra with an analysis tool such as XSPEC. The inputs are the PHA and RSP files created above. Note, no background file!

The analysis thread 'Analysis of LAT Burst Observations' leads you through this analysis step-by-step.

GBM Analysis

The GBM data also consists of individual events (here called 'counts'). Once again, they must be binned.

  • Bin the counts with gtbin. The GBM detectors will have already binned the counts in energy. Once again the time bins can be chosen interactively or through a binning file. Three interactive time binning methods are available: uniform in time; constant signal-to-noise ratio per bin; or Bayesian Blocks. The binning files are created either by a previous run of gtbin or using the gtbindef utility. The output of the binning is a PHA or PHAII file, standard FITS filetypes.
  • Fit the resulting spectra with an analysis tool such as XSPEC. The input are the PHA or PHAII file created above and the RSP and background files provided with the burst data.

The analysis thread Analysis of GBM Burst Observations leads you through this analysis step-by-step.

Joint Fits

A major hurdle for joint fitting has always been getting spectra from different detectors with the same time bins. But because the Fermi data are event lists, we can just bin the data with the same time bins. The binning tool gtbin can output a file with the time bins used to bin an event list, and can read a binning file to bin an event list. Therefore:

  • Bin the data from one detector, for example using constant signal-to-noise ratio bins. These time bins should be output in a binning file.
  • Use the resulting binning file to bin the event data from the other detectors.
  • Fit the spectra simultaneously with an analysis tool such as XSPEC; most such tools are capable of performing joint fits. In most cases the relative normalization between detectors should be added as a fitted parameter to compensate for errors in the effective areas of the different detectors.

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