Fermi/LAT response functions

Formulation

The instrument response functions for Fermi/LAT are factorised into the effective area \(A_{\rm eff}(p,\theta)\) (units \(cm^2\)), the point spread function \(PSF(\delta|E,\theta)\), and the energy dispersion \(E_{\rm disp}(E'|E,\theta)\) following

\[R(p',E',t'|p,E,t) = A_{\rm eff}(p,\theta) \times PSF(\delta|E,\theta) \times E_{\rm disp}(E'|E,\theta)\]

where \(\theta\) is the inclination angle with respect to the LAT z-axis, and \(\delta\) is the angular separation between the true and measured photon directions \(p\) and \(p'\), respectively,

\[\int PSF(p'|p,E,t) \, dp' = 1\]

and

\[\int E_{\rm disp}(E'|p,E,t) \, dE' = 1\]

The instrument response function is independent of time.

Two functional forms are available for the point spread function which are both composed of a superposition of two King functions:

\[\mathrm{\it PSF}_1(\delta | E, \theta) = n_\mathrm{c} \left( 1-\frac{1}{\gamma_\mathrm{c}} \right) \left( 1 + \frac{1}{2\gamma_\mathrm{c}} \frac{\delta^2}{\sigma^2} \right)^{-\gamma_\mathrm{c}} + n_\mathrm{t} \left( 1-\frac{1}{\gamma_\mathrm{t}} \right) \left( 1 + \frac{1}{2\gamma_\mathrm{t}} \frac{\delta^2}{\sigma^2} \right)^{-\gamma_\mathrm{t}}\]

and

\[\mathrm{\it PSF}_3(\delta | E, \theta) = n_\mathrm{c} \left( \left( 1-\frac{1}{\gamma_\mathrm{c}} \right) \left( 1 + \frac{1}{2\gamma_\mathrm{c}} \frac{\delta^2}{s_\mathrm{c}^2} \right)^{-\gamma_\mathrm{c}} \right. \left. + n_\mathrm{t} \left( 1-\frac{1}{\gamma_\mathrm{t}} \right) \left( 1 + \frac{1}{2\gamma_\mathrm{t}} \frac{\delta^2}{s_\mathrm{t}^2} \right)^{-\gamma_\mathrm{t}} \right).\]

The parameters \(n_\mathrm{c}\), \(n_\mathrm{t}\), \(s_\mathrm{c}\), \(s_\mathrm{t}\), \(\sigma\), \(\gamma_\mathrm{c}\), \(\gamma_\mathrm{t}\) depend on energy \(E\) and off-axis angle \(\theta\). Energy dispersion is so far not implemented.

Event types

The LAT events are partitioned into exclusive event types that for Pass 6 and Pass 7 data correspond to pair conversions located in either the front or the back section of the tracker. For Pass 8 the event partitioning has been generalised to other event types. For each event type, a specific response function exists that will be designated in the following with the superscript \(\alpha\).

Livetime cube

The livetime cube is a means to speed up the exposure calculations in a Fermi/LAT analysis and contains the integrated livetime as a function of sky position and inclination angle with respect to the LAT z-axis. This livetime, denoted by \(\tau(p,\theta)\), is the time that the LAT observed a given position on the sky at a given inclination angle, and includes the history of the LAT’s orientation during the entire observation period. A Fermi/LAT livetime cube includes also a version of the livetime information that is weighted by the livetime fraction (i.e. the ratio between livetime and ontime) and that allows correction of inefficiencies introduced by so-called ghost events, and that we denote here by \(\tau_\mathrm{wgt}(p,\theta)\).

Mean point-source PSF

GammaLib, the library that is underlying ctools, natively implements the computation of the mean PSF for point sources. The exposure for a given sky direction \(p\), photon energy \(E\) and event type \(\alpha\) is computed using

\[X^\alpha(p, E) = f_1^\alpha(E) \int_{\theta} \tau(p,\theta) \, A_\mathrm{eff}^\alpha(E,\theta) \, d\theta + f_2^\alpha(E) \int_{\theta} \tau_\mathrm{wgt}(p,\theta) \, A_\mathrm{eff}^\alpha(E,\theta) \, d\theta\]

The exposure weighted point spread function is computed using

\[\mathrm{\it PSF}^\alpha(\delta|p,E) = f_1^\alpha(E) \int_{\theta} \tau(p,\theta) \, A_\mathrm{eff}^\alpha(E, \theta) \, \mathrm{\it PSF}^\alpha(\delta|E,\theta) \, d\theta + f_2^\alpha(E) \int_{\theta} \tau_\mathrm{wgt}(p,\theta) \, A_\mathrm{eff}^\alpha(E,\theta) \, \mathrm{\it PSF}^\alpha(\delta|E,\theta) \, d\theta,\]

where \(f_1^\alpha(E)\) and \(f_2^\alpha(E)\) are energy and event type dependent efficiency factors.

The mean point spread function for a point source is computed using

\[\overline{\mathrm{\it PSF}}(\delta|p,E) = \frac{\sum_\alpha \mathrm{\it PSF}^\alpha(\delta|p,E)} {\sum_\alpha X^\alpha(p,E)}\]

where the sum is taken over all event types \(\alpha\).