For a King PSF, the asymptotic expression at high flux of the count rate (Eq. 5) of clean (not piled-up) monopixels, can be written explicitly in the following way:

(B1) | ||

(B2) |

The expectation value of the total count rate *M _{1}* (Eq. 6)
is somewhat more difficult to handle
analytically but can also be written asymptotically in the following way:

(B3) | ||

(B4) |

after splitting the integral in two, performing the same variable change
(with instead of for the first half),
integrating by parts the and
the 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 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 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:

(B5) | ||

(B6) | ||

(B7) |

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