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Prior to Likelihood v14r3, there had been a long-standing problem with computing diffuse responses (with **gtdiffrsp**) for diffuse sources that have FITS image templates for discrete sources, such as SNRs or molecular clouds. In summary, **gtdiffrsp** performs the following integral for each diffuse source component

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i{latex}i{latex} |

in the xml model definition:

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{latex}
\newcommand{\e}{{\varepsilon}}
\newcommand{\ep}{{\varepsilon^\prime}}
\newcommand{\phat}{{\hat{p}}}
\newcommand{\phatp}{{\hat{p}^\prime}}
\begin{equation}
\int d\phat S_i(\phat, \ep_j) A(\ep_j, \phat) P(\phatp_j; \ep_j, \phat)
\end{equation}
{latex} |

where

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{latex}$\hat{p}_j${latex} |

and

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{latex}$\varepsilon^\prime_j${latex} |

are the measured direction and measured energy of detected photon

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$j${latex}$j${latex} |

, and the integral is performed over (true) directions on the sky

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{latex}$\hat{p}${latex} |

.

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{latex}$A(...)${latex} |

and

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{latex}$P(...)${latex} |

are the effective area and point spread function, and I have made the approximation equating the true photon energy with the measured value,

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{latex}$\varepsilon = \varepsilon^\prime${latex} |

.

The problem arises for a discrete diffuse source when its spatial distribution

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{latex}$S_i(\hat{p})${latex} |

is only significantly different from zero far from the location of event

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$j${latex}$j${latex} |

. Operationally, the integral is evaluated using an adaptive Romberg integrator that samples the integrand at theta and phi values that are referenced to

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{latex}$\hat{p}_j${latex} |

. For very compact sources, the integrator will often miss the source entirely and evaluate the integral to zero; and unless the measured photon direction lies directly on a bright part of the extended source, the integral will usually not be very accurate, even if non-zero.

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