Pierrick Martin / ctools

Commit 8e0a833086eadb40ac712db069cef8f4b1322e1d

Authored by Jürgen Knödlseder
1 parent 63473779

Add information about computation of mean PSF to Fermi/LAT User Manual

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doc/source/users/user_manual/irf_lat.rst
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... ... @@ -61,3 +61,68 @@ The parameters :math:`n_\mathrm{c}`, :math:`n_\mathrm{t}`, :math:`s_\mathrm{c}`,
61 61 :math:`\gamma_\mathrm{t}` depend on energy :math:`E` and off-axis angle
62 62 :math:`\theta`.
63 63 Energy dispersion is so far not implemented.
  64 +
  65 +
  66 +Event types
  67 +~~~~~~~~~~~
  68 +
  69 +The LAT events are partitioned into exclusive event types that for Pass 6 and
  70 +Pass 7 data correspond to pair conversions located in either the front or the
  71 +back section of the tracker. For Pass 8 the event partitioning has been
  72 +generalised to other event types. For each event type, a specific response
  73 +function exists that will be designated in the following with the superscript
  74 +:math:`\alpha`.
  75 +
  76 +
  77 +Livetime cube
  78 +~~~~~~~~~~~~~
  79 +
  80 +The livetime cube is a means to speed up the exposure calculations in a
  81 +Fermi/LAT analysis and contains the integrated livetime as a function of sky
  82 +position and inclination angle with respect to the LAT z-axis.
  83 +This livetime, denoted by :math:`\tau(p,\theta)`, is the time that the LAT
  84 +observed a given position on the sky at a given inclination angle, and includes
  85 +the history of the LAT's orientation during the entire observation period.
  86 +A Fermi/LAT livetime cube includes also a version of the livetime information
  87 +that is weighted by the livetime fraction (i.e. the ratio between livetime and
  88 +ontime) and that allows correction of inefficiencies introduced by so-called
  89 +ghost events, and that we denote here by :math:`\tau_\mathrm{wgt}(p,\theta)`.
  90 +
  91 +
  92 +Mean point-source PSF
  93 +~~~~~~~~~~~~~~~~~~~~~
  94 +
  95 +GammaLib, the library that is underlying ctools, natively implements the
  96 +computation of the mean PSF for point sources.
  97 +The exposure for a given sky direction :math:`p`, photon energy :math:`E` and
  98 +event type :math:`\alpha` is computed using
  99 +
  100 +.. math::
  101 + X^\alpha(p, E) = f_1^\alpha(E) \int_{\theta} \tau(p,\theta) \,
  102 + A_\mathrm{eff}^\alpha(E,\theta) \, d\theta
  103 + + f_2^\alpha(E) \int_{\theta} \tau_\mathrm{wgt}(p,\theta)
  104 + \, A_\mathrm{eff}^\alpha(E,\theta) \, d\theta
  105 +
  106 +The exposure weighted point spread function is computed using
  107 +
  108 +.. math::
  109 + \mathrm{\it PSF}^\alpha(\delta|p,E) =
  110 + f_1^\alpha(E) \int_{\theta} \tau(p,\theta) \,
  111 + A_\mathrm{eff}^\alpha(E, \theta) \,
  112 + \mathrm{\it PSF}^\alpha(\delta|E,\theta) \, d\theta
  113 + + f_2^\alpha(E) \int_{\theta} \tau_\mathrm{wgt}(p,\theta) \,
  114 + A_\mathrm{eff}^\alpha(E,\theta) \, \mathrm{\it PSF}^\alpha(\delta|E,\theta) \,
  115 + d\theta,
  116 +
  117 +where :math:`f_1^\alpha(E)` and :math:`f_2^\alpha(E)` are energy and event
  118 +type dependent efficiency factors.
  119 +
  120 +The mean point spread function for a point source is computed using
  121 +
  122 +.. math::
  123 + \overline{\mathrm{\it PSF}}(\delta|p,E) =
  124 + \frac{\sum_\alpha \mathrm{\it PSF}^\alpha(\delta|p,E)}
  125 + {\sum_\alpha X^\alpha(p,E)}
  126 +
  127 +where the sum is taken over all event types :math:`\alpha`.
  128 +
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