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- The diffuse response quantities are proportional, up to an energy-dependent factor
(i.e., the spectrum), to the probability densities of a given event for the corresponding source models. IfWiki Markup Latex {latex}$s(E)${latex}
is the spatial distribution of the diffuse component, then the diffuse response isWiki Markup Latex {latex}$\tilde{S}(\hat{p})${latex}
Latex Wiki Markup {latex} \newcommand{\phat}{{\hat{p}}} \newcommand{\phatp}{{\hat{p}^\prime}} \newcommand{\E}{{\epsilon}} \newcommand{\Ep}{{\E^\prime}} \begin{eqnarray} d_0(\Ep, \phatp) &= &\int d\phat \tilde{S}(\phat) P(\phatp; \E, \phat, t) A(E, \phat, t) D(\Ep; \E, \phat, t)\\ &= &\int d\phat \tilde{S}(\phat) P(\phatp; \Ep, \phat, t) A(\Ep, \phat, t) \end{eqnarray} Here, $\phat$ and $\E$ are true photon direction and energy, $t$ is the arrival time, primes indicate measured quantities, $P$ is the PSF, $A$ is the effective area, and $D$ is the energy dispersion (taken to be a delta function in energy in the second line). {latex}
- In gtsrcmaps the spatial distribution is multiplied by the time-integrated exposure,
and convolved with the mean PSF,Wiki Markup Latex $E${latex}$E${latex}
:Wiki Markup Latex {latex}$P_{\rm avg}${latex}
The lhs of the above equation is plotted vs the rhs below for extragalactic (isotropic) and gll_iem_v02.fit diffuse models. The red curves are best fit power-laws with the index fixed to unity. Other than a constant factor of approximately 2 in both plots (2.04 and 1.95, respectively), the relation holds fairly well. Replacing the gtdiffrsp values in the FT1 file with the gtsrcmaps values does not result in a substantial change in the fit of the two components.Latex Wiki Markup {latex} \newcommand{\phat}{{\hat{p}}} \newcommand{\phatp}{{\hat{p}^\prime}} \newcommand{\E}{{\epsilon}} \newcommand{\Ep}{{\E^\prime}} \begin{eqnarray} E(\Ep, \phat) &=& \int dt A(\Ep, \phat, t)\\ P_{\rm avg}(\phatp; \Ep, \phat) &=& \frac{1}{E(\Ep, \phat)} \int dt A(\Ep, \phat, t) P(\phatp; \Ep, \phat, t)\\ d_1(\Ep, \phatp) &=& \int_{\Delta\phatp} d\phatp \int d\phat E(\Ep, \phat) P_{\rm avg}(\phatp; \Ep, \phat) \tilde{S}(\phat) \end{eqnarray} The integral over $\phatp$ is over the pixel size, $\Delta\phatp$. For an inertial pointing, $E$ is just the effective area times the livetime $\Delta t$ and we should have \begin{equation} d_0 = \frac{d_1}{\Delta t \Delta \phatp} \end{equation} {latex}
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