In using the fitting of model to observations at the edge of the lunar DS we have the following situation:
It is seen that the 'edge' of the observed image is not 'sharp' - it slopes over perhaps 3-4 columns on either side of the midpoint of the jump from sky to disc. The models often have problems matching that low slope and tend to be 'sharper' at the edge, at 'best fit'. We fit +/- 50 columns on either side of the edge so more pixels on the sky participate in driving the fit than does the 'edge-pixels'.
The shape of the edge is driven by convolution with the PSF - and the PSF core dominates in setting the sharp details. The slope of the DS sky halo is given by the wings of the PSF.
There is an opposition between fitting edge and halo slope - lower alfa would broaden the PSF and 'lift' the halo wings, increasing the slope at the same time as 'dulling' the edge of the model edge. With more pixels driving the fit at the sky halo the edge is bound to be ignored - thus too large alfa's probably are the result. The pedestal added to the model image to match sky brightness is competing with alfa's effect on the sky halo - part of the level of the sky halo is provided by the pedestal value while part is given by the effect of alfa on the PSF itself.
1) Exponentiation of the canonical PSF to power alfa may not preserve the shape of the PSF near the core in the required way.
2) More weight could be given to the 'edge pixels'.
3) Slope of the edge is not just due to rounding by the PSF - there is also some 'jitter' from co-addition of 100 frames so alfa is used to account for more than the effects of the PSF.
4) The fitted pedestal is sometimes negative - it should probably always be positive due to sky brightness after bias subtraction, but the bias varies by 1 count due to the temperature effect and is scaled, as described elsewhere. The pedestal is often near -0.1 counts - are we sure that the bias has been correctly scaled to the level of 0.1 counts?