IRF Theory

Modeling the expected number of detected events

To model the expected number of events a gamma-ray source should produce on a detector one has to model its effect using an instrument responce function (IRF). In general, such a function gives the probability to detect a photon emitted from true position \(p_{\rm true}\) on the sky and true energy \(E_{\rm true}\) at reconstructed position \(p\) and energy \(E\) and the effective collection area of the detector at position \(p_{\rm true}\) on the sky and true energy \(E_{\rm true}\).

We can write the expected number of detected events \(N(p, E)\):

\[N(p, E) {\rm d}p {\rm d}E = t_{\rm obs} \int_{E_{\rm true}} {\rm d}E_{\rm true} \, \int_{p_{\rm true}} {\rm d}p_{\rm true} \, R(p, E|p_{\rm true}, E_{\rm true}) \times \Phi(p_{\rm true}, E_{\rm true})\]

where:

  • \(R(p, E| p_{\rm true}, E_{\rm true})\) is the instrument response (unit: \({\rm m}^2\,{\rm TeV}^{-1}\))

  • \(\Phi(p_{\rm true}, E_{\rm true})\) is the sky flux model (unit: \({\rm m}^{-2}\,{\rm s}^{-1}\,{\rm TeV}^{-1}\,{\rm sr}^{-1}\))

  • \(t_{\rm obs}\) is the observation time: (unit: \({\rm s}\))

The Instrument Response Functions

Most of the time, in high-level gamma-ray data (DL3), we assume that the instrument response can be simplified as the product of three independent functions:

\[R(p, E|p_{\rm true}, E_{\rm true}) = A_{\rm eff}(p_{\rm true}, E_{\rm true}) \times PSF(p|p_{\rm true}, E_{\rm true}) \times E_{\rm disp}(E|p_{\rm true}, E_{\rm true}),\]

where:

  • \(A_{\rm eff}(p_{\rm true}, E_{\rm true})\) is the effective collection area of the detector (unit: \({\rm m}^2\)). It is the product of the detector collection area times its detection efficiency at true energy \(E_{\rm true}\) and position \(p_{\rm true}\).

  • \(PSF(p|p_{\rm true}, E_{\rm true})\) is the point spread function (unit: \({\rm sr}^{-1}\)). It gives the probability of measuring a direction \(p\) when the true direction is \(p_{\rm true}\) and the true energy is \(E_{\rm true}\). Gamma-ray instruments consider the probability density of the angular separation between true and reconstructed directions \(\delta p = p_{\rm true} - p\), i.e. \(PSF(\delta p|p_{\rm true}, E_{\rm true})\).

  • \(E_{\rm disp}(E|p_{\rm true}, E_{\rm true})\) is the energy dispersion (unit: \({\rm TeV}^{-1}\)). It gives the probability to reconstruct the photon at energy \(E\) when the true energy is \(E_{\rm true}\) and the true position \(p_{\rm true}\). Gamma-ray instruments consider the probability density of the migration \(\mu=\frac{E}{E_{\rm true}}\), i.e. \(E_{\rm disp}(\mu|p_{\rm true}, E_{\rm true})\).

The implicit assumption here is that energy dispersion and PSF are completely independent. This is not totally valid in some situations.

These functions are obtained through Monte-Carlo simulations of gamma-ray showers for different observing conditions, e.g. detector configuration, zenith angle of the pointing position, detector state and different event reconstruction and selection schemes. In the DL3 format, the IRF are distributed for each observing run.

Further details on individuals responses and how they are implemented in gammapy are given in Effective area, Energy Dispersion and Point Spread Function.