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QUESTION: Philip,

I often hear of Phobos and Deimos being pulled in opposite directions by tidal forces, due to their individual orbital periods related to Mars' rotation period (Phobos faster, Deimos slower).

This suggests to me that if a moon were to form (or be captured) just at a planet's stationary orbit height, such as 42,000km for the earth, that it's orbit would remain stable indefinitely. Is this correct?

This also suggests to me that all planets orbiting the sun should be sped up, raising their orbits, at least to some limit in distance for the sun's tidal effects. Is this correct, and what is that limit?

Thanks,

Paul

ANSWER: Hello,

You write that you've "often heard" of Deimos and Phobos "being pulled in opposite directions by tidal forces, due to their individual orbital periods". But which sources can you cite for this, because I've found none?

It is of course true that some analogous gravitational tidal forces would be elicited by the Moons - but relative to Mars (i.e. acting on Mars' surface not the moons' surfaces) but given the masses of Mars' moons are so small, i.e. 0.000000015 x Mars for Phobos and 0.0000000031 x Mars for Deimos these forces would have to be extremely small.

Re: the hypothesis of a moon "to form" (as in be captured)- just at the geostationary orbital altitude (20,440 km for Mars) such that "it's orbit would remain stable indefinitely" would depend on the particular solution to a 3-body problem (technically 4) as applied to what we call "the Lagrangian points". In general there are 5 such points in all which refer to the five positions for a hypothetical orbit for which a small object (i.e. tiny moon as you suggested) can be incorporated without altering the overall Lagrangian pattern (which defines 5 different points at high and low gravitational potential, i.e. V = -GM/r)

Hence, a definitive answer to your question would hinge on: 1) the exact mass of the object captured, 2) the Lagrange point solutions based on this mass, and most importantly, 3) the assumptions used in considering the nature of the problem, i.e. two-body (ignoring Mars' small moons Deimos and Phobos entirely), or restricted 3-body, i.e. considering only the more massive, say Phobos. In any case, such a computation is beyond the framework for an All Experts answer, though given enough time I could likely crank it out using my Mathcad program. (Alas, I don't have this time, as I am finishing up two projects entailing book editing).

My one caution here would be the requirement that the orbit "remain stable indefinitely" which is a tall order if one's numerical precision is off even a tiny bit. Chaos, intruding via real number extra digits (arising from lack of total precision in the initial coordinates), would play havoc with the result. At minimum, it would mean a stable orbit over indefinite time would be difficult to find.

Re: your last query, this is a pretty decent conjecture on your part. For sure, any object orbiting the Sun at semi-major axis or mean distance closer than the Earth would normally have a shorter orbital period than the Earth. In other words, exhibit an observed "speed up", i.e. according to Kepler's 3rd or harmonic law.

However, this ignores the effect of the Earth's own gravitational pull. Therefore, say the object is directly between the Earth and the Sun, then the Earth's gravity weakens the force of attraction of the object to the Sun, hence **increasing** the orbital period of the object. The closer to Earth the object is (say a comet), the greater the effect. (Ditto for the Sun, i.e. planets at given distances from it, from its perspective!)

In regard to the last parenthetical, and given the significant distances of the planets to the Sun, if we try to apply the same effect to them and integrate in the Sun's tidal effects, we won't see much of a difference in orbital speed at all. The reason? The Sun's tidal effects aren't that much to speak of. To fix ideas, the solar tidal acceleration at the Earth's surface along the Sun-Earth axis is about 0.52 × 10^−7 g, where g is the gravitational acceleration at the Earth's surface. In other words, this would amount to just under 5.1 × 10^−7 m/sec/sec. By comparison, the centripetal acceleration for the Earth is more than four orders of magnitude larger.

---------- FOLLOW-UP ----------

QUESTION: Philip,

Thanks for the quick and involved answer. I'll try to explain my question more clearly.

I saw the reference to the moons of Mars last night while watching an episode of The Universe called "Alien Moons," about various moons in the solar system.

Phobos was described as orbiting faster than Mars is rotating, and therefore being slowed by tidal forces, always lowering its orbit. The Wikipedia article on Phobos agrees with this.

Deimos was described as orbiting slower than Mars is rotating, and therefore always being sped up by tidal forces, increasing its orbit.

I also understand that Earth's moon, orbiting slower than the Earth is rotating, is being sped up by tidal forces, and always increasing its orbit.

This led me to speculate that moons in general might always be either being sped up or slowed down by tidal forces, either pulling them in to the planet or throwing them out. Unless, if they just happened to be formed (or captured) right at geostationary orbit, then there would be no tidal effect, either way.

In other words, if the Earth's moon had just happened to have formed at 42,000km, orbiting once a day, wouldn't it still be doing that today?

I do not believe that this has anything to do with Lagrangian points, since what I'm asking about is all happening within a particular planet's

Hill sphere.

And a simpler question about the Sun would be: Do tidal forces associated with orbiting the Sun cause any of the planets to speed up and increase their orbit(s)?

Since this is all thought experiment for me, I think I'm more interested in the general principles involved, than the specifics of the math.

Thanks, Paul

Thanks for clarifying your question! It now becomes easier to address in terms of known physics laws. (N.B. In the context here, your proposal or conjecture rests soundly on a hypothetical **positional** basis - as I will show- but not on a dynamic basis - for which the L-points would have to be reckoned in via appropriate computations of overall mass interactions. In other words, it's a nice abstract, hypothetical proposal but one which, alas, we'd not see manifested in reality by an actual 'capture' of a planetoid or moon at the precise geostationary position)

Anyway, I attached a simple diagram to try to make understanding of the principle easier. Basically, we are just considering an elaborate case for the conservation of angular system - whether looking at Earth-Moon or Mars-Phobos. Note the direction of rotation shown for Earth- and that you're looking down from the N. pole as it were. Note also the Moon's direction of orbital motion.)

Take the Earth-Moon system which is the one illustrated in the diagram. Here, I denote two tidal bulges on the Earth's surface by x and x' at essentially opposite ends of the line through the center of Earth. As you probably know already, two tidal bulges occur as a result of a differential gravitational force. (The general origin for a given tidal force on Earth due to the Moon is from the variable values for the Moon's gravitational attraction at differing locations inside Earth.)

In the sketch I've shown, note that bulge x is closer to the Moon than bulge x' on the other side of Earth. This difference results in a net torque exerted on Earth (torque is the product of the moment of inertia, I, and the angular acceleration, a)

Specifically for our diagram, bulge x leads the Moon, and because it's also nearer the Moon than bulge x' the force exerted by the Moon on x is greater yielding a net torque. This net torque slows the Earth's rotation, in effect, slows its angular momentum. Now, at the same time, the bulge x is pulling the Moon forward in its orbital path, effectively speeding it up and causing it to move farther out. This is just a consequence of the conservation of angular momentum.

Thus consider two bodies m and M connected as shown with c the center of mass:

O---r1------c ---------r2----------o m

M

Both M and m rotate about c at distances r1 and r2 respectively. Since the angular momentum, e.g. L = Iw must be constant (where w = v/r)then if m somehow speeds up (higher v) then to compensate, r2 must increase. (w ~ 1/r and w ~ v)

Thus, the Moon constantly moves further out, and by about 3-4 cm a year.

Now, consider your case of Phobos - which as the Wiki article noted - is the converse of the case for our Moon. Thus, Phobos (orbiting faster then Mars is rotating) experiences the opposite of what our Moon does, i.e. a slowing down of its orbit. Again, by the law of conservation of angular momentum (and using the simple model shown above), if v is lessened or decreased then r must decrease. Deimos meanwhile, as you pointed out, moves in its orbit faster than Mars is rotating and therefore speeds up like our own Moon, and hence increases in its orbital distance.

Therefore, your conjecture that if our own Moon "just happened to form" at the geosynchronous distance, is technically correct. Since its orbital velocity is in synch with Earth's rotational velocity there is no net torque acting on Earth and hence, the Moon could still be orbiting. (The downside, as I pointed out, is that such a fortuitous happenstance has about as much probability as happening as a blind man playing eight -ball pool and getting all of his balls in before his opponent can!) This, the dynamic computations for such 'capture' - are basically what I was all about in the previous reply.

Now, what about the tidal forces in terms of planets orbiting the Sun? First off, the Sun's mass is roughly 99.9% of the mass of the solar system. Its rotational period is 26.5 days. We know that no planet shares this in terms of its own revolution time. (Mercury goes once around every 88 days).

So what effect would any planet have on the Sun in terms of tidal bulge? Actually, effectively zero. To what extent would all the planets contribute to a solar tidal bulge? I believe it was one of John Gribbin's antagonists in the mid-1980s (after he penned 'The Jupiter Effect') and he computed 1-2 mm. Nothing to write home about. Since even all the planets together exert zero net torque on the Sun, then we conclude the effect is roughly as if all the planets were in a 'geosynchronous' orbit. (Again, this is not because the planets are actually in any such 'solar -stationary' orbits but because the Sun's mass is so large, at 2 x 10^30 kg, that it dominates the angular momentum in the solar system! Hence, a net torque sufficient to alter a planet's distance would have to come from an external agent, a really massive agent - in collision with a planet).

In other words the Sun would not cause any of the planets to increase or decrease their orbital semi-major axes analogous to what happens with the Moon and Phobos.

Precise astrometric measurements confirm this to be the case to within maybe +/- 1 cm.

Hopefully, this answer provides some insight into the workings of these planet-moon systems.

Astronomy

I have more than forty years of experience in Astronomy, specifically solar and space physics. My specialties include the physics of solar flares, sunspots, including their effects on Earth and statistics pertaining to sunspot morphology and flare geo-effectiveness.

Astronomy: Worked at university observatory in college, doing astrographic measurements. Developed first ever astronomy curriculum for secondary schools in Caribbean. Gave workshops in astrophysics and astronomical measurements at Harry Bayley Observatory, Barbados. M.Phil. degree in Physics/Solar Physics and more than twenty years as researcher with discovery of SID flares. Developed of first ever consistent magnetic arcade model for solar flares incorporating energy dissipation and accumulation. Develop first ever loop solar flare model using double layers and incorporating cavity resonators.**Organizations**

American Astronomical Society (Solar Physics and Dynamical Astronomy divisions), American Mathematical Society, American Geophysical Union.**Publications**

Solar Physics (journal), The Journal of the Royal Astronomical Society of Canada, The Proceedings of the Meudon Solar Flare Workshop (1986), The Proceedings of the Caribbean Physics Conference (1985). Books: 'Selected Analyses in Solar Flare Plasma Dynamics', 'Physics Notes for Advanced Level'. 'Astronomy and Astrophysics: Notes, Problems and Solutions'. **Education/Credentials**

B.A. Astronomy, M. Phil. Physics **Awards and Honors**

American Astronomical Society Studentship Award (1984), Barbados Government Award for Solar Research (1980), Barbados Astronomical Society Award for Service as Journal Editor (1977-91)**Past/Present Clients**

Caribbean Examinations Council, Barbados Astronomical Society, Trinidad & Tobago Astronomical Society.