Model factorisation

Sky models describe the spatial, spectral and temporal properties of a gamma-ray source using the following factorisation:

\[M(p,E,t) = M_{\rm spatial}(p | E,t) \times M_{\rm spectral}(E | t) \times M_{\rm temporal}(t)\]

(\(M(p,E,t)\) is given in units of \(photons \,\, cm^{-2} s^{-1} MeV^{-1} sr^{-1}\)). The spatial model component \(M_{\rm spatial}(p|E,t)\) is defined by the abstract GModelSpatial class, the spectral model component \(M_{\rm spectral}(E|t)\) is defined by the abstract GModelSpectral class and the temporal component \(M_{\rm temporal}(t)\) is defined by the abstract GModelTemporal class.

The spatial model component describes the energy and time dependent morphology of the source. It satisfies

\[\int_{\Omega} M_{\rm spatial}(p|E,t) \, d\Omega = 1\]

for all \(E\) and \(t\), hence the spatial component does not impact the spatially integrated spectral and temporal properties of the source (the integration is done here over the spatial parameters \(p\) in a spherical coordinate system). The units of the spatial model component are \([M_{\rm spatial}] = {\rm sr}^{-1}\).

The spectral model component describes the spatially integrated time dependent spectral distribution of the source. It satisfies

\[\int_{E} M_{\rm spectral}(E | t) \, dE = \Phi\]

for all \(t\), where \(\Phi\) is the spatially and spectrally integrated total source flux. The spectral component does not impact the temporal properties of the integrated flux \(\Phi\). The units of the spectral model component are \([M_{\rm spectral}] = {\rm cm}^{-2} {\rm s}^{-1} {\rm MeV}^{-1}\).

The temporal model component describes the relative variation of the source flux with respect to the mean value given by the spectral model component. The temporal model component is unit less.