From 63473779cd1d4366caef70d91324751492ba7c59 Mon Sep 17 00:00:00 2001 From: Jürgen Knödlseder Date: Mon, 1 Apr 2019 16:55:13 +0200 Subject: [PATCH] Add information about Fermi/LAT and COMPTEL response to User manual --- doc/source/users/user_manual/irf.rst | 2 ++ doc/source/users/user_manual/irf_comptel.rst | 48 ++++++++++++++++++++++++++++++++++++++++++++++++ doc/source/users/user_manual/irf_cta.rst | 36 +++++++++++++++++++++++++----------- doc/source/users/user_manual/irf_lat.rst | 63 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 4 files changed, 138 insertions(+), 11 deletions(-) create mode 100755 doc/source/users/user_manual/irf_comptel.rst create mode 100755 doc/source/users/user_manual/irf_lat.rst diff --git a/doc/source/users/user_manual/irf.rst b/doc/source/users/user_manual/irf.rst index 787994b..a269b49 100755 --- a/doc/source/users/user_manual/irf.rst +++ b/doc/source/users/user_manual/irf.rst @@ -8,3 +8,5 @@ Instrument Response Functions irf_intro irf_cta + irf_lat + irf_comptel diff --git a/doc/source/users/user_manual/irf_comptel.rst b/doc/source/users/user_manual/irf_comptel.rst new file mode 100755 index 0000000..494850d --- /dev/null +++ b/doc/source/users/user_manual/irf_comptel.rst @@ -0,0 +1,48 @@ +.. _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`. diff --git a/doc/source/users/user_manual/irf_cta.rst b/doc/source/users/user_manual/irf_cta.rst index 1725b85..4e073f1 100755 --- a/doc/source/users/user_manual/irf_cta.rst +++ b/doc/source/users/user_manual/irf_cta.rst @@ -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 diff --git a/doc/source/users/user_manual/irf_lat.rst b/doc/source/users/user_manual/irf_lat.rst new file mode 100755 index 0000000..97230ce --- /dev/null +++ b/doc/source/users/user_manual/irf_lat.rst @@ -0,0 +1,63 @@ +.. _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. -- libgit2 0.21.4