In the last plot in this entry http://iloapp.thejll.com/blog/earthshine?Home&post=375 we found the rise in fitted albedo unreasonable. Let us review what we have:
1) The albedos are derived as a scaling-parameter inside the Earth model we use. This model assumes the Earth is a Lambertian sphere with uniform albedo.
2) We use Hapke 63 to describe the reflectance of the Moon, and Clementine albedos for the lunar albedo.
3) We fit the DS edge – adjusting parameters in the synthetic lunar-image model so that a good fit is found between observed DS-edge profile and model DS-edge.
4) We see a 30-40% rise in albedo as Moon approaches Full (and Earth New).
5) We see a much weaker dependence of time-of-day in albedo. From considerations of the Opposition Surge (which may be inadequately represented in Hapke 63) we expected a drop in albedo towards end of day, and a rise at the start of day. We do see weak signs of the drop at the end of the day, but at the start it is ambiguous.
5) makes me think the problem is not the Opposition Surge in Hapke 63. 4) makes me think it is the total flux of the synthetic models that depend incorrectly on lunar phase. 1) makes me think that perhaps the Earth is not very Lambertian and that the discrepancy becomes much worse, particularly towards the crescent New Earth (Full Moon).
If scattered light is incorrectly included in the fit at the edge we would find a larger fitted albedo as lunar phase approached Full Moon, as we do – but there is NO WAY the fitting method is making a 30-40% bias-like error – it would be blatantly obvious when the fits were inspected.
What’s going on? I would like to compare the observed run of total fluxes against the model-based expectation.
First we fit the edges of realistic but synthetic lunar images (made with Chris ‘syntheticmoon.f’ code):
Here models with albedo=0.30 have been edge-fitted – just like observations. We see that the edge-fitting method is completely able to retrieve terrestrial albedo.
So, there is a problem of some sort with the total flux of synthetic models. Let us look directly at the total flux from the synthetic images:
Here are shown (black) lunar model image magnitudes (adjusted to just about the right V magnitude at Full Moon) against lunar phase [FM=0], and expressions fro the same from Allen (red: 1955 edition, orange: 1975 edition), and from the JPL ephemeris system (green). JPL is based on Allen 1975 but adjusts fro actual distance to the Moon. Only ou rmodels allow for libration – hence the East-West assymetry in the black symbols.
It seems our synthetic images have a smaller range in total flux against phase than do the Allen and JPL expressions. By 1.5-2 magnitudes for some phase ranges, relative to Full Moon flux – or Full Moon is less bright by 1.5-2 magnitudes than are the Allen expressions, relative to intermediary phases. Our Moon images, relatively speaking, lack flux as we approach Full Moon.
In normalized-image edge fitting this will cause the model expression that is fitted to have been scaled by too small a number, making the step-to-sky at the DS edge too large, so albedo should compensate during the fitting by adjusting down. We see the opposite …
Could it be that our near Full Moon albedos are all right? But they are so large – 0.4 and more at small phases: but what do we know? Perhaps we are making a relative error in estimating what the ‘answer should be’?