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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} |
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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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