Likelihood Diffuse response tests

Comparison of gtdiffrsp output with gtsrcmaps values

• The diffuse response quantities are proportional, up to an energy-dependent factor (
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  $s(E)$

  ), to the probability densities of a given event for the corresponding source models. If
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  $\tilde

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  (\hat

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  )$

  is the spatial distribution of the diffuse component, then the diffuse response is
  
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  \newcommand

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  {{\hat

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  }}
  \newcommand

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  {{\hat

  ^\prime}}
  \newcommand

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  \epsilon
  \newcommand

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  \E^\prime
  \begin

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  d_0(\Ep, \phatp) &= &\int d\phat \tilde

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  (\phat) P(\phatp; \E, \phat, t) A(E, \phat, t) D(\Ep; \E, \phat, t)
  &= &\int d\phat \tilde

  (\phat) P(\phatp; \Ep, \phat, t) A(\Ep, \phat, t)
  \end

  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).

• In gtsrcmaps the spatial distribution is multiplied by the exposure,
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  $E$

  and convolved with the mean PSF:
  
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  \newcommand

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  {{\hat

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  }}
  \newcommand

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  {{\hat

  ^\prime}}
  \newcommand

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  \epsilon
  \newcommand

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  \E^\prime
  \begin

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  E(\Ep, \phat) &=& \int dt A(\Ep, \phat, t)
  P_

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  (\phatp; \Ep, \phat) &=& \frac

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  \int dt A(\Ep, \phat, t) P(\phatp; \Ep, \phat, t)
  d_1(\Ep, \phatp) &=& \int_

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  \int d\phat E(\Ep, \phat) P_

  (\phatp; \Ep, \phat) \tilde

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  (\phat)
  \end

  $E$ is the time-integrated exposure and $P_

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  $ is the average PSF. 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
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  d_0 = \frac

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  \end