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    Re: 1901 May, 22 Lunar example by French Navy Captain Arago
    From: Frank Reed
    Date: 2010 Jan 31, 16:40 -0800

    Antoine, in your original post on this topic, you wrote:
    "It is an interesting approach for which he claims an excellent suitability for even the smallest Limb distances, in order to remedy to a then known weakness of some of the more "classical" methods."

    And also:
    "it is a quite clever method. I still doubt whether it was actually used on board"

    It's an interesting development, yes. It reflects, not practice at sea, but a direction in which things might have gone. A number of French nautical astronomers had recognized that short distance Moon-star lunars are easier to shoot and benefit more from high magnification. This suggested a possibility of increasing the accuracy of lunar observations by an order of magnitude or so. Arago, and others, set themselves the task of setting up the mathematical tools to make this possible. It didn't happen in the end. There was "no market" for these advances.

    And you wrote:
    "One question though : are short limb distances still a limitation nowadays with the powerful computation software available on line ? For example, in your Lunar on-line computer Frank or for other on-line Lunar computers you might know of?"

    It depends slightly on how things are done. Part of the problem with short distance lunars is the geometry of clearing the distance. In the standard direct solution of the problem, we're basically extracting the cosine of the lunar distance at the end. Naturally for small distances, there is very little change in that cosine for a given change in lunar distance. The cosine of 1 degree differs from the cosine of 1d 1' by only 0.0000005 so we need to work with lots of significant digits to get an accurate result. Of course, for nearly any computing device available in the past twenty years, that's no problem at all. I am just old enough, though, to remember a time when a mainframe computer gave considerably worse values for trig functions than a handheld calculator, and there are still specialized applications where you might have trouble with this issue.

    Another issue with short distance lunars is the problem of comparing with the calculated geocentric distance. This will be changing in a highly non-linear fashion when the distance is small so the sort of simple interpolation that was normal for long distance lunars would not work. In a modern calculation, you can calculate the geocentric lunar distance either for arbitrarily short intervals (every three seconds instead of every three hours, e.g.) or you can calculate it iteratively.

    I mentioned previously somewhere that the "real" next step required 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. I've been playing around with this myself for the past two years.

    PS: Trying to work through my January message backlog before February arrives in Chicago.

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