Strange Coincidence

[nickcosmosonde]

I've just read about a most surprising fact, which may be pure coincidence, but if it is it's a truly flukey one. All the orbits of the planets are ellipses, with the sun sitting at one of the two focii, as Kepler saw and Newton proved. The other focal points of the planets are empty space, and seem to have no dynamical significance. In the case of the Moon's orbit, however, its empty focal point is just about 22,300 miles above the Earth. This happens to be the height at which geosynchronous satellites orbit us exactly once a day, so they only see one face of the Earth. At the focal point of the Moon, anyone located there would only ever see one face of the Moon too, as its rotation is in synch with its orbit. Due to the slight rocking motion of the Moon's synchronous orbit with us, over time we can see about 58% of its surface; but from its focal point this effect completely disappears, and you would only ever see precisely half of its surface, according to Prof Carl Murray, an expert on celestial mechanics at the Uni of London.

He says this can only be a complete coincidence. Can such a neat symmetry be down to pure chance? Any theories?

[carnyx]

@nick

Given that the moon is describing an elliptical orbit around both foci, and it's own rotation is exactly synchronised with the period of the orbit, it seems 'obvious' that the moon will show one face when viewed from either foci.

So I suspect the apparent 'rocking ' motion may be that an observer is on the surface of the earth and so is describing a circular orbit around the centre of the earth. In other words, an observer can get no closer than the earths radius of around 4k miles from one of the foci of the ellipse.

A way of visualising this effect would be get in an imaginary spaceship and manoeuvre to a place where the moon's elliptical orbit appears to be circular. From this position, both foci become overlaid and effectively appear to become one. Then, the moon will present only one face to this focus, if it's rotation is exactly synchronised with it's orbit.

And why the moon has become exactly synchronised in this way, is really interesting.

To digress and look elsewhere, are the millions of small moons that make up the rings of Saturn, all describing elliptical orbits? But if they are all describing the same circular orbit, is it the co-gravitation effects between these moonlets that keep them marching in step? And, how do the outer moonlets keep in step, yet have to travel faster to do so?

But back to the question. Compared to the earth .. the moon rotates one every 28 or so days and so has a very low angular momentum about it's own axis. But the moon has a high angular momentum about the earth-moon axis ( i.e. about either foci of the elliptical orbit)

Maybe the moon has somehow given up it's self-spin angular momentum to the earth somehow. Could sea tides have been the mechanism?

[nickcosmosonde]

May 3, 2011 8:14:11 GMT carnyx said:

Thanks Carnyx. I think I've not explained myself too well.

Given that the moon is describing an elliptical orbit around both foci, and it's own rotation is exactly synchronised with the period of the orbit, it seems 'obvious' that the moon will show one face when viewed from either foci.

Yes. But the coincidence is that the geonsynchronous orbit distance just happens to be at the same distance from the surface as the second focal point of the Moon. It's reminiscent of the other strange coincidence that the apparent size of the Moon almost exactly fits over the apparent size of the Sun during eclipses.

To digress and look elsewhere, are the millions of small moons that make up the rings of Saturn, all describing elliptical orbits?

I'll have to look that one up!

But if they are all describing the same circular orbit, is it the co-gravitation effects between these moonlets that keep them marching in step? And, how do the outer moonlets keep in step, yet have to travel faster to do so?

My understanding is the "in step" part is down to the phenomenon of resonance. The moons of Jupiter, and many if not all of the planets show the same thing. I'm not sure if it is known, exactly, what exactly is the nature of this resonance. I remember having a quite heated argument with a Cambridge Physics professor at a dinner party once, when this question came up for some reason. He adamantly insisted that no such phenomenon existed, or could exist, in the solar system. Very sadly he was murdered in a mugging incident in the street within a few days, so he never received the satisfaction of all the papers I'd assembled for him to prove him wrong.

But back to the question. Compared to the earth .. the moon rotates one every 28 or so days and so has a very low angular momentum about it's own axis. But the moon has a high angular momentum about the earth-moon axis ( i.e. about either foci of the elliptical orbit)

Maybe the moon has somehow given up it's self-spin angular momentum to the earth somehow. Could sea tides have been the mechanism?

Again, my understanding is yes, it's down to the sea tides - the bulk of water (and rock) being pulled out from the Earth by the Moon's gravity. This has a marginal but accumulative braking effect on the Moon, so that since its formation it has gradually achieved this synchrony.

[carnyx]

With regard to the coincidence, you may be on to something. AFAIK, the shape of the orbital ellipse can be arbitrary; anywhere between the limits of true circularity to the incipient hyperbolic.

This randomness must be a puzzle, and surely there must be some hidden factor at work.

Does this coincidence of the geosynchronous orbit with the 'empty' focus of the moon's orbit occur with any other single-moon planets?

[nickcosmosonde]

May 3, 2011 16:59:49 GMT carnyx said:

With regard to the coincidence, you may be on to something. AFAIK, the shape of the orbital ellipse can be arbitrary; anywhere between the limits of true circularity to the incipient hyperbolic.

Yes. Conic sections, aren't they? I spent some months back in the early nineties trying to build a model of the solar system using the GR equations, taking seriously the curved space-time idea, modelling the heliosphere as an inverted cone, seeing the planets' orbits as various slices through it, trying to fit Kepler's harmonices mundi calculations vis-avis the Bode's Law into that cone. I gave up in the end - being mainly an autodidact, I'm not a competent enough mathemtatician enough. (I find it hard enough just spelling it).

This randomness must be a puzzle, and surely there must be some hidden factor at work.

I have that suspicion, yes.

Does this coincidence of the geosynchronous orbit with the 'empty' focus of the moon's orbit occur with any other single-moon planets?

I don't know! I doubt it, somehow - it probably is just a coincidence. I just thought it was a startling one. Eamonn on the politically correct Science board has argued the coincidence is not as forceful as first suggested - though I'm not a competent enough mathemtatician to take the argument further (much as I'd like to.)

Mainly, I love the fact that the Moon fits over the Sun at times of maximum solar eclipses, don;t you?