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    Re: 1901 May, 22 Lunar example by French Navy Captain Arago
    From: Frank Reed
    Date: 2010 Feb 6, 20:24 -0800

    I wrote previously:
    "to get this order of magnitude improvement in lunar distances would require an accurate model of the lunar limb. The mountains and maria along the limb lead to undulations in the nearly circular outline of the Moon of roughly one arcsecond. The details vary over time in accordance with the "lunar libration" cycles. You cannot accurately analyze the timing of lunar occultations unless corrections are applied for the profile of the lunar limb. And in fact people who observe lunar occulations, like the members of IOTA, use lunar limb models on a regular basis to get accurate predictions. The limb models have been improved by the occultation observations, but fundamentally these current limb models are based on one which was developed c.1960-1965 in preparation for the Apollo program. An interesting footnote here is that models of the lunar surface are rapidly improving right now thanks to some recent spacecraft missions. So it should be possible to permanently remove this last uncertainty in lunar observations in the very near future."

    This has progressed significantly just in the past three months. The Japanese Space Agency released a complete DEM (digital elevation model) from their Kaguya mission's laser altimeter experiment just this past November so we now have available lunar limb models that are about ten times better than anything available before. This has big benefits for calculating the exact timings of occultations and solar eclipses as well as offering that final bit of accuracy that we might want for lunar distance calculations (once someone writes the code!). The amateur astronomers of IOTA have already begun switching over to these new limb models. There are details on this here:
    http://weblore.com/richard/IOTAMeeting2009.htm
    (search the article on Kaguya and be sure to scroll down to the limb profile graphic towards the end). Note this quote regarding the Kaguya limb profiles: "This high accuracy data now supercedes the older outdated Watts data".

    You can download an "easy" version of the Kaguya DEM data here:
    http://ltvt.wikispaces.com/Additional+Textures
    (click on the link in the second paragraph that says "click to retrieve at full size").

    Here's a nice long description of the old Watts data and how it is used in eclipse calculations:
    http://umbra.nascom.nasa.gov/eclipse/951024/text/lunar-limb-profile.html

    And finally, here's a standard eclipse diagram with the limb profile superimposed on an exaggerated scale:
    http://umbra.nascom.nasa.gov/eclipse/2003/figures/figure_1.7.gif
    Naturally since this was for an eclipse in 2003, it is based on the older Watts profiles.

    Again, note that the profile depends on the lunar libration angle. So basically you would have a function that gives the limb height in arcseconds with inputs of position angle around the limb, the libration in lunar latitude, and the libration in lunar longitude. For lunar distance observations you would probably want to do some sort of smoothing over a small range of position angles since the other object is generally "swung" along the limb to find the right contact.

    I think it was Antoine who said that he couldn't see putting something like this on a small calculating device, but consider this: you could include the value of the limb height in tenths of an arcsecond, which is more than necessary, for every degree around the limb, at every tenth of a degree of the two libration angles (13 degrees x 16.5 degrees) in about 8 megabytes of storage without even bothering with data compression. So not at all hard to do on any device today when you consider that you can buy a 2 gigabyte memory card for five dollars. At that rate, those 8 megabytes of storage would cost you exactly two cents!

    -FER


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