At the moment we are displaying our results by providing plots of ratios of the DS/BS ratio in observations, to that in models. We do this chiefly to get rid of common factors – such as solar irradiance and distance-related geometry.

What remains in such a ‘ratio of ratios’ are the effects of:

1) The Earth’s actual albedo,
2) the model’s Earth albedo,
3) Earth’s reflectance (real vs modelled),
4) lunar surface albedo’s in the reference patches (real ratio vs model ratio)
5) effects due to the choice of lunar reflectance model.

We are really only interested in 1.

2 is an assumed value so that the results we get for terrestrial albedo are relative to that choice. We use a value of 0.31.

3 is a choice – we expect that any errors made in this choice will be seen as a phase-dependency in the results, and we can therefore control or at least understand it. Earth is more Lambertian than the Moon. The Earth has edge-darkening, the Moon has very little. We use a Lambertian model for an otherwise uniform Earth.

4 is observable, but only with difficulty – you need a good total lunar eclipse, then the albedos in the two patches on the Moon – or their ratio – can be measured. BBSO has done this. We have not (yet). In the model we make a choice, based on which lunar albedo map we use. As long as it is fixed the results will be relative to that choice. Perhaps we can use published images of total lunar eclipses to extract the ratio?

5 is a choice – we have models for Lambertian reflectance, as well as the Hapke 63 model and other, as yet untested, more advanced reflectance models. We expect that incompletenesses in these models will be seen as a phase-dependency in the results.

Of the above only some could possibly induce a phase dependency: 3 and 5.

We have reduced all data using both lunar models based on the Lamertian reflectance and the Hapke 63 model. We show them next – look here for a discussion of what we are actually showing: ‘ratio of ratios’ and all that:

(as before, plot needs to be downloaded as it does not show up well on this blog).

The plot is for 5 filters with the EFM method applied. The left column is the ratio of DS/BS in obs to the same in model, while the right column is DS/total in obs relative to models, where ‘total’ means the disk-integrated brightness of all source counts (plus a few stars that we can ignore!).

The first page is for the Lambertian lunar reflectance. The last page is for the Hapke 63 reflectance.

We notice that left and right columns are quite similar. We notice that the ‘jump up’ at angles corresponding to about 40 degrees from New Moon is much smaller in the Hapke 63 model results. Inspection of the files that correspond to the individual points in the ‘jump up’ and those next to the jump, reveals that the ‘jumped up’ points have a different processing history: they are the result of coadding and averaging single image sequences, while the rest are stacks of observations that were averaged. Why these should be different is unclear, as yet.

We conclude that there was an effect of lunar reflectance model on our results – and that Hapke 63 is better than Lambert. This is not surprising as the Moon is well-known not to be Lambertian in its reflectance.

So – we are beginning to see observations constrain theory!

There is still some scatter to account for, and we shall return to this mattrer, using the estimates of nightly seeing available from the measured alfa.

There is also some ‘slope’ in the result wrt phase – so the EFM method has a success rate that is phase dependent. WHile the second-best method (BBSO logarithmic; not shown here) has some ‘upturn’ towards Full Moon (center of plot), the EFM has an even slope down towards FM. Perhaps some empirical fine-tuning of the method will remove this problem too.