diff --git a/doc/source/users/user_manual/irf_lat.rst b/doc/source/users/user_manual/irf_lat.rst index 97230ce..7b97fc4 100755 --- a/doc/source/users/user_manual/irf_lat.rst +++ b/doc/source/users/user_manual/irf_lat.rst @@ -61,3 +61,68 @@ The parameters :math:`n_\mathrm{c}`, :math:`n_\mathrm{t}`, :math:`s_\mathrm{c}`, :math:`\gamma_\mathrm{t}` depend on energy :math:`E` and off-axis angle :math:`\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 +:math:`\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 :math:`\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 :math:`\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 :math:`p`, photon energy :math:`E` and +event type :math:`\alpha` is computed using + +.. math:: + 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 + +.. math:: + \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 :math:`f_1^\alpha(E)` and :math:`f_2^\alpha(E)` are energy and event +type dependent efficiency factors. + +The mean point spread function for a point source is computed using + +.. math:: + \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 :math:`\alpha`. +