I am indebted for this story to Kevin Brown… or maybe Fred Olden… or whoever is behind the mysterious (and often inspiring) website www.mathpages.com, which has been running for about 20 years now. I am relaying it here for my friends, together with my additional considerations. The idea stems from a little Newtonian detail that many of us could have overlooked… I quote: “It’s an interesting fact that the path of our Moon (with respect to the inertial rest frame of the Sun) always curves toward the Sun. This might seem surprising at first, considering that the Moon revolves around the Earth, but […] the Moon always has a positive acceleration toward the Sun.” (Original reference at: www.mathpages.com/home/kmath405/kmath405.htm)

Imagine the Moon orbiting about the Earth, while at the same time the Earth orbits around the Sun. At each newmoon, the Moon is exactly aligned between the Sun and the Earth, and it feels the opposite centripetal accelerations from both, W2R the acceleration from the Sun at average distance R and w2r the acceleration from the Earth at average distance r. From Kepler’s Third Law, the orbital angular velocities, W of the Earth, and w of the Moon, are related to their masses and the approximately circular orbit radii, respectively as M=W2R3 and m=w2r3. We can therefore compare the strength of the attraction the Moon feels from the Earth and the Sun. From the previous formulas, it is easy to obtain that the Moon will be more attracted toward the Sun than to the Earth, if the radius of the Moon’s orbit exceeds the “critical” value  rc = R (m/M)1/2.

At exactly this value, the attraction from the planet and the central star are equal; if the radius is larger than rcthe star’s attraction prevails (the Moon “falls” towards the Sun).Now, the interesting fact is that if we calculate this critical radius for the Earth, we get a figure of about 260,000 km for rc , while the Moon is orbiting at the larger distance of 384,500 km; therefore, our satellite is well beyond the critical radius, and it seems to be always attracted to the Sun, although being carried around in her motion by the Earth. It’s a bit like having a passenger in your car that constantly looks outside the window and does not care for you…!

Moreover, if we calculate this same rc  for all the planets of our Solar system, we obtain the following values, in comparison of their satellites’ characteristics:

Mercury and Venus : rc = 23,700 and 170,000 km. No moons, no rings.

Earth : only one Moon, and the only planet whose satellite has an average orbit radius much larger than the critical radius (see above).

Mars : rc = 130,000 and two moons, the furthest one (Deimos) at 23,500 km.

Jupiter, Saturn, Uranus, Neptune, and even Pluto : r= 24.1 million, 24.2 million, 19 million, 32.5 million, and 493,000 km, respectively; in all cases, the farther is the planet away from the Sun, the more the number of moons, and always with orbital radii smaller than critical rc. The closest call is Jupiter’s large collection of “moons” (actually many of them are a bunch of big rocks of just a few km diameter) dispersed on orbits of size up to about 95% of the critical radius.

It may seem therefore, that this critical radius rcconstitutes a sort of natural limit to the extension of the satellite orbits around planets, at least as far as our Solar system goes. It could appear reasonable that stable moons span orbits at distances such that the gravity of the host planet is always stronger than the Sun’s gravity, because otherwise there would be a tendency for the moons to runaway. Jupiter’s large size and its proximity to the asteroid belt probably accounts for the fact that its orbit space is fully populated by many moons and mini-moons, right up to the threshold radius rc. But then, our Moon is quite an exception?

I noticed that in the above simple formula for rc, the two contributing terms have a weird numerical near-coincidence. In fact, both the planet orbital radius R (for an orbit assumed circular, for the sake of simplicity), and the square-root of the mass ratio m/M, with M the mass of the Sun, change by nearly the same factor, going from the smaller to the larger value: R increases from 57,9 Mkm (Mercury) to 4,5Gkm (Uranus), whereas Ö(m/M) goes from 0.004 (Mercury) to 0.03 (Jupiter). In both cases, the ratio maximum/minimum is equal to about 76 … Maybe this simply means that the two factors are equally important in determining the rc (despite each maximum corresponds to different planets). Looking at the role of R, that is, the average distance of the planet to the Sun, could it be that for some reason, the moons of planets closer to the Sun are destined to run away ? If this is plausible, maybe Mercury and Venus in the past had their own moons, which have already left for good…? And maybe our Moon is destined to runaway as well…?

Browsing the literature on such exotic subjects, one finds several – equally exotic – suggestions. A first interesting one is that our Moon could have been a Venusian moon, which the Earth stole at some point in its gravitational circling (www.space.com/22966-earths-moon-from-venus.html). Another original hypothesis, again related to the origin of our satellite, is based on the geological similarity of Mercury’s composition with Moon’s one (aside of many differences); in this case, the Moon would have formed by the primordial planetary ring around the Sun, from which also Mercury, Venus and the Earth formed https://www.space.com/23223-mercury-clues-moon-origin.html.

The subject remains interesting, since there are still few clues on the origins of the Moon. However, today’s discussion seems to shed a mysterious light also on the future evolution of the Moon, besides its equally mysterious past. The most current theory, according to which the Moon was ripped away from the Earth following a giant planetary impact, tells us that Moon and Earth were much closer in the past, and the orbit was much smaller than the critical rc above. According to this theory, the recession of Moon’s orbit is due to tidal fluctuations, which result in the observed increase of the average orbital radius by 3.8 cm/year. That makes more than 2 m since I was born…

Anyway, if our astrophysical models are correct, by the time the Moon could run away from the Earth, the Sun will have already expanded to a red giant of size such that it will have swallowed all the planets up to about the Earth’s distance. No problems for us humans: by that time the Sun’s heat would have vaporized all oceans, and life would be impossible on our beloved Earth since millions of years already. Whatever would be left of humanity at that time would have probably migrated away on distant stars… Battlestar Galactica!

No moons, many moons, one Moon…

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