Measuring B-V on the DS in B and V images with different PSFs: The Case For Forcing them!

We have detected some image-pairs where the B and V halo appear to cancel. We have plenty of other image-pairs where they evidently do not cancel. This is seen by structure on the sky – DS sky in particular as well as ‘slopes’ across the DS in the direction towards the terminator near the BS. We here investigate whether it is possible to ‘force’ the image with a narrower PSF to acquire a halo that resembles the broader image’s halo so that cancellation can occur.

If the differences between the PSF width-parameters (alfa) is small we expect to convolve the sharper image with a quite narrow PSF that only broadens the halo a little. We will use PSFs that are based on our ‘canonical PSF’ and simply raise it to a large number – this will generate a more and more delta-function like peak.

We identify centered pairs of B and V images that are close in time; identify the image with the narrower PSF – by using previously fitted alfa values; we then broaden the sharper image, while conserving flux, and inspect the difference between the B and V images. The results of such a trial is shown in the Figure:

Here is a less fuzzy pdf file of the above plot:

We see 15 panels showing a ‘slice’ through the disc centre, of B-V (magnitudes). Each panel has an increasing power (alfa) used to raise the canonical PSF to – at first we raise it to 1.5, then to 1.6, and so on. A smaller power will broaden the PSF a lot and thus give us ‘fuzzier’ images. We expect that when the image is too fuzzy we will see slopes in the B-V slice – just as when two observed images in a pair are subtracted and one has a broader PSF than the other. As alfa increases in the figure we reach a point where it is so large that the sharper image no longer is made appreciable more fuzzy – in the limit of large alfa it becomes an almost identical copy of the original image – hence, towards the end of the sequence of plots we essentially see what the slice across the original B-V image looked like.

In each image we notice the DS to the left and the BS to the right – both situated between the two vertical dashed lines, denoting edge of disc. We see a ‘level difference’ between the DS and the BS. We notice the slope of the DS – upwards for large alfa and downwards for smaller alfa – e.g. at alfa=1.7. Somewhere near alfa=1.8 and before alfa=1.9 the slope is horizontal.

We expect the real DS to be ‘flat’ [we need to check this against realistic models] so when alfa is near 1.85 it seems we have induced enough additional fuzziness in the sharper of the two images that the halos cancel and the level difference between DS and BS is what it is in reality, [We need to test this on images with known properties: Student Project!].

We notice also that the level difference depends on alfa, even when alfa has gone past the ‘now DS is flat’ point. This is a warning that incorrect estimation of
the limiting alfa value may give us spurious results for delta(B-V)! We estimate the slope on the DS (between vertical dotted lines) and look for the values of alfa where the DS slope is not statistically different from zero at the 3 sigma limit. We also do this for the BS. We find midpoints of the fitted DS line and the fitted BS line (red lines in uppermost plot) and take the difference b├ętween these, forming a sort of ‘+/- 3 sigma interval of confidence’ for the B-V difference between BS and DS. This is plotted here:

Here is the pdf:

We see thaat the generous +/-3 sigma interval allows quite a range in B-V – and also that if we do not estimate the right alfa in the above procedure – but make it too large then we will underestimate B-V.

For the 3-sigma limits we squint at the above graph and read off upper and lower B-V limits for the BS/DS difference: 0.5 to 0.7.

All of the above has been done for an average of an 11-row slice across the middle of the disc. The method could in general be extended to treat broader strips, or the whole ds – planes could be fitted instead of lines.
First, though, it is necessary to understand if the above is even correct! We need to look at artificial images of the Moon. These should have their own “B-V colours” then we fold the B and V images with seperate PSFs and try to force one to be as fuzzy as the other, and so on. The actual slope of the DS, when fitted with lines or planes could then be estimated and in fact used as the goal slopes for the above procedure. Work work work!