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    Re: Oblateness calculation given observer's latitude
    From: Frank Reed
    Date: 2018 Jan 31, 22:20 -0800

    Peter Monta, you wrote:
    "What analysis do you think might be artifacty or marginal---direct images of the lunar optical figure from Earth (which would be relatively low-quality since they're seeing-limited)? "

    What I'm getting at is that there is a certain point where the low level components of the expansion, whether it's spherical harmonics or any of a variety of other representations, become obscure when the object itself is less symmetrical. Naturally the expansions work, but rather than seeing a rapidly converging sequence of terms, you end up needing a very large number of terms none of which do much justice to the thing being modelled, individually. And you'll always get values in the lowest lying members of the sequence (like J2) in a lumpy body like the Moon, even if it's not by itself much use in describing the Moon's shape. 

    Maybe a simple idealized case would help explain what I'm talking about. Suppose we start out with a nearly perfect sphere: "That's no moon -- it's a space sta... oh wait, no, it's an idealized model of a moon". A perfect sphere has zero oblateness, of course. Now suppose I bang a couple of dents in it. One dent will extend from about the equator to about the pole in one hemiphere centered on longitude 150 (measured from some prime meridian), and another will be half that size centered in mid-latitudes centered at longitude -90. Now you take that shape and run it through the grinder to get a series representation of it. Naturally it will have an oblateness term, a J2 term, and it's completely legitimate, but it to say that this body is well-approximated by an oblate spheroid woud be misleading. The expansion in a series like this would necessarily have many significant terms that would have to be included with the mathematical oblateness to get a realistic picture of this idealized moon's shape. And note that from a great many orientations, its outline would appear perfectly circular, despite having a calculabe oblateness. Is our real moon that different?

    Another example that might help illustrate what I'm saying... Do you remember hearing at some point, maybe in fact when you were a kid, that the "earth is pear-shaped"? This emerged in early analyses of satellite orbits. It was one of the first relatable discoveries of the space age. Geographers had long known that the earth is an obate spheroid to excellent approximation (or more accurately, its "geoid" --its gravitational field at the surface is well-matched by an oblate spheroid), and space scientists had gone one better and found that it was a slightly pear-shaped oblate spheroid. Satellite, just by orbiting, had shown their worth... But go looking for it in the geoid as we know it today. Yes, there is a term in the expansion that corresponds to the "pear-shape" but by itself it tells us next to nothing about the hills and valleys in the geoid. The real geoid is a complicated shape, and it's nothing much like "pear-shaped". That early space age description is a mathematical artifact with little physical meaning.

    I know I'm not explaining this clearly enough, but I wanted to type something up tonight before I forgot about it. I'm happy to kick it around some more...

    A last thought: needless to say, the "real" low-level adjustment to the moon's shape makes it slightly elongated along the axis towards the Earth. If it were a mostly fluid body, it would tend to settle into synchronous rotation with a tri-axial ellipsoid shape having a long axis aligned with the earth, a short axis on the axis of rotation, and the perpendicular axis intermediate. Of course, as earth-based observers, the oblateness of the Moon in this case is just the projected apparent oblateness.

    Frank Reed

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