Fermi Gamma-ray Space Telescope

The Point Spread Function (PSF)

The point spread function for the LAT is a function of an incident photon's energy and inclination angle, and the event class. Futhermore, the PSF is defined in terms of a scaled-angular deviation:

Equation for angular deviation

Equation for scaled angular deviation

Where is the reconstructed direction and is the true direction of the photon.

The scale factor describes the first order variation of the PSF with energy, namely that at lower energies the PSF is dominated by multiple scattering, improves with energy, and at high energies is dominated by the spatial resolution of the LAT Silicon Tracker. If E is expressed in MeV, the scale factor is:

Energy dependent scaling factor

Parameters c0 and c1 have distinct values for Front and Back converting events, while β is shared between front and back. All of these parameters are stored in the PSF fits file that is part of the IRFs, in the PSF_SCALING_PARAMS table.

c0 front c1 front c0 back c1 back beta version
P7TRANSIENT_V6 5.80e-2 3.77e-4 9.60e-2 1.30e-3 0.800 Monte Carlo
P7SOURCE_V6 2.45e-2 5.68e-4 4.18e-2 1.34e-3 0.778 In-Flight
P7CLEAN_V6 2.47e-2 5.99e-4 4.00e-2 1.32e-3 0.778 In-Flight
P7ULTRACLEAN_V6 2.32e-2 6.71e-4 3.96e-2 1.57e-3 0.777 In-Flight
P7SOURCE_V6MC 5.80e-2 3.77e-4 9.60e-2 1.30e-3 0.800 Monte Carlo

In-Flight PSF and Monte Carlo PSF

For some time, the LAT team has been reporting that the Monte Carlo simulations underestimated the PSF at energies of a few GeV. Accordingly they have developed a technique to characterize the PSF using flight data, a short summary of this technique is provided below. This characterization required a large integration time on the sources. As a result, the in-flight version of the PSF does not include the dependence on the theta angle that is included in the P7TRANSIENT_V6 and P7SOURCE_V6MC IRFs.

Once the overall energy scaling has been accounted for, additional variations in the PSF are defined for a default binning of 4 energy bins per decade, from 1.25 to 5.75 in logarithmic MonteCarlo energy, 8 angle bins, equally spaced from 0.2 to 1.0 in cos(θMC). For each bin, the scaled angular deviation is calculated as described above. Note that although the in-flight version of the PSF use the same binning as the Monte Carlo version, since the theta dependence is not modeled, all of the theta bins at a given energy use the same PSF parameters.

The functional form of the analytical description of the LAT's Point Spread Function is derived and adapted from XMM's (usually described as a "King" function). The function is:

Equation for the PSF functional form

Note that this is normalized to 1 when integrated from 0 to infinity, and that the integral includes a extra factor of 2πx, because it is performed over solid angle.

To allow for a tail in the distribution we allow for the possibilty to have two King functions with different parameters:

Equation for the PSF including a tail in the distribution

The sigmas and gammas are stored in the PSF FITS table as SCORE,STAIL,GCORE,GTAIL. Because of the arbitrary normalization used in fitting the PSF function, fcore must be extracted from NTAIL and the other parameters. NCORE is an arbitrary overall normalization factor.

Equations for the NTAIL parameter

Note also that the in-flight versions of the PSF use a single King function, so for those fcore is simply 1.

Note that the earlier P6_V3 IRFs used a slightly different scheme. In that schema the two sigma values were equal and stored as SIGMA in the FITS table. The fraction of events in the King function describing the tail was fixed by matching the components at xb = sqrt(20) * σ.

Equation for the PSF including a tail in the distribution

Derivation of the In-Flight PSF

The in-flight versions of the PSF are derived from the angular distribution of photons around bright point sources in LAT data at high energy. A dedicated paper describing the employed technique is being completed and will soon be available. A short summary follows.

At the lower end of the interval (above 1 GeV) the analysis is based on the Vela pulsar, at the higher end of the interval it is based on a stacking analysis of a suitable sample of bright AGNs.

Front and back converting photons are treated separately. At each energy the distribution of detected photons around the source position is parametrized, taking into account the background counts from other astrophysical sources and the true background hits. Containment levels are evaluated and the PSF is described in terms of a single King function (see above) to match the observed containment. Where the two methods overlap AGN data and pulsar data are in good agreement. Due to the statistical uncertainties a single function is used to describe the PSF, i.e. there is no "tail" component; the derived parameters are smoothed within uncertainties by re-fitting the scaling parameters described above and rescaling the σ accordingly. This results in a smooth variation with energy for each parameter.

Any dependence of the PSF on the theta angle, i.e. the inclination from the LAT boresight, is at the moment ignored, so the estimate included in flight versions of the IRFs is an average weighted over two years of observations.

Derivation of the Monte Carlo PSF

The Monte Carlo versions of the PSF are derived by first constructing a histogram of the scaled angular deviation for each bin in logarithmic Monte Carlo energy and cos(θMC). Here is an example of such a histogram for the bin centered at 7.5 GeV, and 30 degress for Front events.

Distribution of scaled angular deviation

The next step of the process is to remove the extra factor of x (or r) from the solid angle integral by converting this histogram to a density histogram by dividing the contents of each bin by the bin width. In fact, a short cut is to divide by the bin center, since that is proportional to the bin width for a logarithmic scale, and the normalization is arbitrary. The resulting density histogram is then fit to extract the PSF parameters for that bin

Distribution of scaled angular deviation

Examples from Monte Carlo version of P7SOURCE_V6 IRFs

To illustrate the behavior of the PSF and of the effects that we are explicitly averaging over, we show plots of the Monte Carlo version of the P7SOURCE_V6 IRFs, which are not being released. Note that released P7SOURCE_V6 IRFs do not include any theta dependence.

Quantity Plot. Front(left) and Back(right)
R68% in terms of scaled deviation Table of values for the scale 68% contaiment
R95%/R68% Table of values for the ratio of 95% to 68% containment

Effect of Ignoring the Theta Dependence of the PSF

Since the in-flight version of the PSF averages over the acceptance of the LAT, it will provide unbiased, if slightly less than optimal, results for long observation times (longer than a few weeks), where we observe the source with a distribution of observing angles. However, for shorter integration periods, non-uniformities in the observing profile will mean that the averaged PSF may either over or underestimate the true containment.

Data Samples Observing Profiles:
Fraction of the observation time at a given angle for each of the samples.
 Containment Curves
Fraction of the observation weighted containment that falls within the average 68% and 95% for each of the samples
2 year time intervals:
for 13 different directions seprated by 15 degrees in DEC
Crab:
for 60 different 12 hour intervals

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