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Appendix B: Asymptotic expressions for measured count rates

  At high flux, the general method to evaluate integrals (5) and (6) is to change variables to the local photon flux $\lambda = \Lambda g(x,y)$.I give here a few example calculations for an axisymmetric PSF g(r).

For a King PSF, the asymptotic expression at high flux of the count rate $M_1^{\rm t}$ (Eq. 5) of clean (not piled-up) monopixels, can be written explicitly in the following way:
g(r) & = & \frac{\beta/2-1}{\pi r_0^2} \;
 ... \Gamma\left(1-\frac{2}{\beta}\right) \: \Lambda^{\frac{2}{\beta}}\end{eqnarray} (B1)
after changing variables to $v = \gamma_1 \lambda$ and neglecting terms in $\mathrm e^{- \gamma_1 \Lambda g(0)}$. $\Gamma$ denotes the Gamma function here.

The expectation value of the total count rate M1 (Eq. 6) is somewhat more difficult to handle analytically but can also be written asymptotically in the following way:
M_1 & {\simeq }& \pi r_0^2 
 ...left(1-(1-\frac{\alpha_1}{\gamma_1})^{\frac{2}{\beta}}\right)^{-1}\end{eqnarray} (B3)

after splitting the integral in two, performing the same variable change (with $\gamma_1-\alpha_1$ instead of $\gamma_1$ for the first half), integrating by parts the $v^{-\frac{2}{\beta}-1}$ and the $\mathrm e^{-v}$ terms and noting that the two infinities at v = 0 cancel out exactly.

This proves that for a PSF following a King profile the rates increase at high flux as $\Lambda^{2/\beta}$ while the pile-up fraction tends to a (small) constant. The transition to this regime is when the incoming flux at the center gets higher than $\lambda_0$of Eq. (4) and saturates.

For a Gaussian PSF (which has wings decreasing as fast as can be imagined for a reasonable PSF) the same kind of analytical exercise can be carried out:
g(r) & = & \frac{1}{2 \pi r_0^2} \: 
 \exp\left(-\frac{1}{2} \l...
 ...q & \frac{\alpha_1/\gamma_1}
after performing the same variable change as in (23) (but with a different g(r)). (26) is exact. In (27) terms in $\mathrm e^{- \gamma_1 \Lambda g(0)}$ were neglected. The main contribution to this integral comes from the difference between the two infinities at v = 0, which can appear as

\lim_{\epsilon \to 0} \; 
 ...} \; = \;
 \ln\left(\frac{\gamma_1}{\gamma_1-\alpha_1}\right). \end{displaymath}

This shows that even for a Gaussian PSF the count rate never decreases as the flux increases, but tends to a constant instead. Because of the rapidly decreasing wings a Gaussian PSF leads to a higher pile-up fraction, reaching 6% for the worst case of only monopixel patterns.

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