Paul Schlyter wrote:
Yes, there is an intuitive reason: the orbits are ellipses with the
Sun at one of the foci because gravity follows an inverse-square law.
If gravity had varied in some other way, then the orbits would (usually)
not have been ellipses.

That is not a use of the word "intuitive" with which I am familiar.
If I drop a crystal vase from the roof, it will fall with increasing
speed until it strikes the ground and smashes into a thousand pieces.
*That* is intuitive. The fact that it falls at a rate that increases
linearly with time, and so its distance from me increases quadratically
with time (neglecting air resistance for the moment)--well, it's likely
intuitive to you and me, but probably not to a quite a few people.

But to say that an inverse-square law leads to a conic section--*that*
is not intuitive. To paraphrase Tom from another thread, try to explain
that without any mathematics. I don't think you can. The fact that
other forces lead to different shapes doesn't explain to me, clearly
and convincingly, why this particular force leads to an ellipse (or a
parabola, or a hyperbola). You can't explain something like that in
the negative.

That it comes out as an ellipse, rather than some other oval, is
something deep, beautiful and unexpected about celestial mechanics. I
would like to internalize it more than I have previously, but as I say,
I'm not confident that it can be done.

--
Brian Tung
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