Overview: GRB Spectral Analysis

The duration of the prompt burst emission – the ~100 keV component – is (relatively) short; at most 10's of seconds. Therefore, the LAT's pointing will not change significantly during the burst, and all the counts can be treated as having one response function. Within a PSF radius of the burst position, less than one non-burst count per minute is expected. The count rate over the FOV is ~2 Hz; the LAT's FOV is approximately 2 steradians; and the PSF has a radius of ~3.5 degrees at 100 MeV. Consequently, we expect ~0.01 Hz non-burst photons, or 0.7 cts/minute within a PSF radius. Therefore, we can treat all counts within 1-2 PSF radii as burst photons.

Multi-source spatial analysis is unnecessary for spectral analysis of LAT data, since:

1. all counts within a PSF radius of the burst originated in the burst, and

2. all the counts have the same response function.

However, spatial analysis might be necessary for localizing the burst.

All of the counts within a PSF radius, and within a given time range, can be binned into a count spectrum (apparent energy is the single dimension); and traditional spectral analysis can be applied to the resulting series of LAT count spectra.

Since both the LAT and GBM data consist of lists of counts, the same temporal binning for burst data from both detector types can be selected; joint fits on the binned one dimensional spectra can then be performed.

Binned GRB Spectral Analysis

Method

GRB spectral analysis takes advantage of the unique properties of the phenomenon of a relatively short point source transient. Normally, the LAT there will not experience competing emission from other sources in the field-of-view. Therefore, the analysis is one dimensional: the input burst flux is determined from the apparent energies of the events that triggered the detector which, for the purposes of this discussion, these events will be called 'counts'. It is also assumed that there are sufficient counts per bin.

For the LAT all the counts from a region 1-2 PSF-radii around the burst position are selected from the time range which includes the burst; these counts are then binned into energy channels.

Notes:

• For the LAT, 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 c_i, 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 b_i. We sample the photon flux striking the detector in different energy channels f_j, 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 D_ij, the 'Detector Response Matrix' (DRM) in the burst community, or the 'RSP' or 'RMF' in the X-ray astrophysics community. The resulting matrix equation is

c_i = D_ijf_j+b_i

where summation over j is assumed. Since D_ij is not a square matrix, and even if it is, it is usually nearly singular; this equation cannot be solved by inverting D_ij but requires 'forward folding.' Note that for the LAT b_i~0, but for the GBM b_i 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 c_i, 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 c_i, resulting in best-fit spectral parameters.

'Sufficiently close' is usually determined by minimizing chi2. A sufficiently small value of chi², 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 chi².

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:

1. 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 in using the gtselect tool.

2. Bin the photons with gtbin.

Energy bins are either chosen interactively (bins that are uniform in photon energy or the logarithm of the photon energy), or through a binning file.

Time bins can be also be chosen interactively, or through a binning file, and can be:

• uniform in time.
• have a constant signal-to-noise ratio per bin, or be
• Bayesian Blocks.

Note: The binning files are created either by a previous run of gtbin, or by using the gtbindef utility. The output of the binning is a standard PHA or PHAII .fits file.

3. Create the DRM with gtrspgen.

The output is a standard RSP .fits file.

4. Fit the resulting spectra with an analysis tool such as XSPEC. The inputs are the PHA and RSP files created above. (No background file!)

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; 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.

Joint Fits

A major hurdle for joint fitting has always been getting spectra from different detectors with the same time bins; however, because the Fermi data are event lists, data can be binned 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.

Note: The relative normalization between detectors should normally be added as a fitted parameter to compensate for errors in the effective areas of the different detectors.