Add information about Fermi/LAT and COMPTEL response to User manual

Jürgen Knödlseder
1 parent a587b6cb
Showing 4 changed files with 138 additions and 11 deletions
doc/source/users/user_manual/irf.rst
doc/source/users/user_manual/irf_comptel.rst
doc/source/users/user_manual/irf_cta.rst
doc/source/users/user_manual/irf_lat.rst
@@ -8,3 +8,5 @@ Instrument Response Functions
  
    irf_intro
    irf_cta
+   irf_lat
+   irf_comptel
+.. _um_irf_comptel:
+
+COMPTEL response functions
+--------------------------
+
+Formulation
+~~~~~~~~~~~
+
+A COMPTEL event is characterised by an instrument direction spanned by the
+angles :math:`(\chi, \psi, \bar{\varphi})`. :math:`(\chi, \psi)` is the
+direction of the photon after scattering in the upper detector plane, which is
+determined from the photon interaction locations in both detector planes, and
+
+.. math::
+   \bar{\varphi} = \arccos \left( 1 - \frac{m_\mathrm{e}c^2}{E_2} + \frac{m_\mathrm{e}c^2}{E_1+E_2} \right)
+
+is the Compton scattering angle as inferred from the energy deposits :math:`E_1`
+and :math:`E_2` in the upper and lower detector planes, respectively.
+The measured energy of the photon is estimated from the sum
+
+.. math::
+   E' = E_1 + E_2
+
+of the energy deposits in both detector planes. The probability that a photon
+which interacted in the upper detector plane will encounter a detector of the
+lower plane is described by :math:`DRG(\chi, \psi, \bar{\varphi})`, which also
+includes any cuts related to the removal of events coming from the Earth limb.
+
+The COMPTEL response is factorised using
+
+.. math::
+   R(p',E',t'|p,E,t) = \frac{DRX(p)}{T} \times DRG(\chi, \psi, \bar{\varphi})
+                       \times IAQ(\chi, \psi, \bar{\varphi} | p, E)
+
+where
+:math:`DRX(p)` is the exposure in units of :math:`cm^2 \, s`,
+:math:`T` is the ontime in units of :math:`s`, and
+:math:`IAQ(\chi, \psi, \bar{\varphi} | p, E)` quantifies the interaction
+probability for a Compton scattering in the upper detector plane followed by
+an interaction in the lower detector plane.
+
+
+We note that :math:`IAQ(\chi, \psi, \bar{\varphi} | p, E)` is azimuthally
+symmetric about the source direction, and the IAQ file is stored as a 2D FITS
+image providing the interaction probabilities as function of
+:math:`\bar{\varphi}` and :math:`\varphi_\mathrm{geo}` for a given energy
+range, where :math:`\varphi_\mathrm{geo}` is the angular separation between
+:math:`(\chi, \psi)` and :math:`p`.
@@ -3,31 +3,45 @@
 CTA response functions
 ----------------------
  
-Factorisation
-~~~~~~~~~~~~~
+Formulation
+~~~~~~~~~~~
  
 The instrument response functions for CTA (as well as the other IACTs such as
 H.E.S.S., VERITAS and MAGIC) are factorised into
-the effective area :math:`A_{\rm eff}(p, E, t)` (units :math:`cm^2`),
-the point spread function :math:`PSF(p' | p, E, t)`,
-and the energy dispersion :math:`E_{\rm disp}(E' | p, E, t)`
+the effective area :math:`A_{\rm eff}(p,E,t)` (units :math:`cm^2`),
+the point spread function :math:`PSF(p'|p,E,t)`,
+and the energy dispersion :math:`E_{\rm disp}(E'|p,E,t)`
 following
  
 .. math::
-    R(p', E', t' | p, E, t) =
-    A_{\rm eff}(p, E, t) \times
-    PSF(p' | p, E, t) \times
-    E_{\rm disp}(E' | p, E, t)
+    R(p',E',t'|p,E,t) = A_{\rm eff}(p,E,t) \times PSF(p'|p,E,t) \times
+                        E_{\rm disp}(E'|p,E,t)
  
 where
  
 .. math::
-   \int PSF(p' | p, E, t) dp' = 1
+   \int PSF(p'|p,E,t) \, dp' = 1
  
 and
  
 .. math::
-   \int E_{\rm disp}(E' | p, E, t) dE' = 1
+   \int E_{\rm disp}(E'|p,E,t) \, dE' = 1
+
+For the point spread function, two variants exist that both depend on energy
+and off-axis angle, but differ in the functional form that is used to describe
+the PSF. The first variant implements a superposition of three 2D Gaussian
+functions that are each characterised by a width and a relative amplitude.
+Alternatively, a King profile defined by
+
+.. math::
+   \mathrm{\it PSF}(p'|p,E,t) = \frac{1}{2 \pi \sigma^2}
+   \left( 1 - \frac{1}{\gamma} \right)
+   \left( 1 + \frac{1}{2 \gamma} \frac{\delta^2}{\sigma^2} \right)^{-\gamma}
+
+can be used, where :math:`\delta` is the angular separation between the true
+and measured photon directions :math:`p` and :math:`p'`, respectively,
+:math:`\sigma` describes the width and :math:`\gamma` the tail of the
+distribution.
  
  
 Format
+.. _um_irf_lat:
+
+Fermi/LAT response functions
+----------------------------
+
+Formulation
+~~~~~~~~~~~
+
+The instrument response functions for Fermi/LAT are factorised into
+the effective area :math:`A_{\rm eff}(p,\theta)` (units :math:`cm^2`),
+the point spread function :math:`PSF(\delta|E,\theta)`,
+and the energy dispersion :math:`E_{\rm disp}(E'|E,\theta)`
+following
+
+.. math::
+    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 :math:`\theta` is the inclination angle with respect to the LAT z-axis,
+and :math:`\delta` is the angular separation between the true and measured
+photon directions :math:`p` and :math:`p'`, respectively,
+
+.. math::
+   \int PSF(p'|p,E,t) \, dp' = 1
+
+and
+
+.. math::
+   \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:
+
+.. math::
+   \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
+
+.. math::
+   \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 :math:`n_\mathrm{c}`, :math:`n_\mathrm{t}`, :math:`s_\mathrm{c}`,
+:math:`s_\mathrm{t}`, :math:`\sigma`, :math:`\gamma_\mathrm{c}`,
+:math:`\gamma_\mathrm{t}` depend on energy :math:`E` and off-axis angle
+:math:`\theta`.
+Energy dispersion is so far not implemented.