With a 1-degree occulter I took an image of the Sun. A slice across that image, through the darkest spot on the occulter gives this profile, plotting from darkest spot and outwards:
A log-log plot of the radial profile of the occulted Sun, on August 1 2014, using f22 and 1/500 s exposure, and a 1-degree occulter. Pixel scale is 0.04 deg/pixel, so 1 degree is at 25 pixels and the 200 pixels are at 8 degrees. The image is from the R-plane, G and B are not shown.
The occuluter is the dark part at left stopping near pixel 20. The drop-off after pixel 200 is some roof or something like that, while the curve between pixels 20 and 200 are what the halo around teh Sun looks like. The CCD is 12-bit so saturation starts near 4000, and non-linear response, if any, before that. However, we see clearly that the dropoff due to scatter in the optics is faster than dropoff of halo intensity. This implies that optical scatter plays a smaller role in defining the halo than does atmospheric scattering.
Note that this is with a multi-element SIGMNA telelens - albeit on a rather hazy day.
Of interest for our own little earthshine telescope is to repeat the above on some crystal-clear day (hopefully approximating conditions on Mauna Loa) and see if the optics drop-off is also faster than atmospherics then.
On Mauna Loa, of course, the effects of optics are what they are independent of altitude while the atmosphere is MUCH clearer there - and there is less of it than in Frederiksberg.