NavList:

Richard, you wrote:
"Equipotential surfaces ARE smooth"

Very bumpy compared to the mathematically simple ellipsoid, but SMOOTH, yes, compared to arbitrary mathematical surfaces. Certainly all of the gravitational equipotentials are much smoother than the Earth's surface topography. There are no "craggy peaks" in the geoid. That gravitational hill at Bermuda, as I mentioned before, is only 20 feet high. And it's approximate width is a couple of dozen miles. So that's a negligible bump --until you start worrying about tilting surfaces by a minute of arc or thereabouts, which, of course is exactly the level of concern for accurate celestial navigation (or even more so for inertial navigation). Then suddenly it's a measurable bump. It doesn't have any sharp edges, but I think that much is obvious, isn't it? The geoid is a surface that is smooth in the mathematical sense of having continuous first derivatives in x and y at all points (and even this is only strictly true if we place certain limits on the "type" of gravitational physics that we allow). But the geoid is a smooth "bumpy" surface at every scale. It's all a matter of degree.

You also mentioned that the equipotential surface is everywhere convex. This is obviously true for a sphere so at sufficient distance from any gravitational source, it's necessarily true. And since the Earth has no major holes in it, no really substantial "lumps" in its mass distribution, it is also true for the Earth's geoid, the equipotential surface closest to mean sea level. But this doesn't have to be true generally. An equipotential surface on another planet or especially an asteroid CAN be concave. And that is exactly the limiting condition for the practice of celestial navigation on some planetary body. If there were any concave depressions in the Earth's geoid, that would imply that two different locations on the Earth's surface would see the exact same sky (same zenith, same altitudes for all stars) at the same instant. In other words, there would no longer be a one-to-one match between points on the Earth and points on the celestial sphere. Celestial navigation would fail to work even in principle. On small asteroids, this "celestial failure" would be a common occurrence. Of course, no one is proposing that we start sailing the seas of dust on the surfaces of small asteroids using sextants; it's merely interesting to know the physical circumstances that make the system of celestial navigation possible in the first place.

-FER

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