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:

$$\begin{eqnarray} g(r) & = & \frac{\beta/2-1}{\pi r_0^2} \; \left(1+(r/r_0)^2\ri... ... \Gamma\left(1-\frac{2}{\beta}\right) \: \Lambda^{\frac{2}{\beta}}\end{eqnarray}$$ (B1) (B2)

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:

$$\begin{eqnarray} M_1 & {\simeq }& \pi r_0^2 \left(\frac{\frac{\beta}{2}{-}1}{\... ...left(1-(1-\frac{\alpha_1}{\gamma_1})^{\frac{2}{\beta}}\right)^{-1}\end{eqnarray}$$ (B3) (B4)

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:

$$\begin{eqnarray} g(r) & = & \frac{1}{2 \pi r_0^2} \: \exp\left(-\frac{1}{2} \l... ...q & \frac{\alpha_1/\gamma_1} {\vert\ln(1-\alpha_1/\gamma_1)\vert}\end{eqnarray}$$ (B5) (B6) (B7)

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

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

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.