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Mars has two moons: Phobos and Deimos. Both are irregular and are believed to have been captured from the nearby asteroid belt.

Phobos always shows the same face to Mars because of tidal forces exerted by the planet on its satellite. These same forces causes Phobos to drift increasingly closer to Mars, a situation that will cause their collision in about 50 to 100 million years.

How I can calculate, given appropriate data, the estimated time at which Phobos will collide with Mars?

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Do you understand the nature of the tidal transfer of angular momentum? Why it happens and therefore what factors play into it's strength? – dmckee♦ Sep 1 '11 at 3:06
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Cross-posted: on Astronomy.SE. – dmckee♦ Sep 1 '11 at 3:11
You might be able to do this easily by measuring the radiating temperature of Phobos, assuming it's all light non-radioactive elements (I don't think such a small object has significant internal heating). The temperature will tell you the energy lost to tidal friction. – Ron Maimon Sep 5 '11 at 20:56
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@Ron: It will tell you the energy that tides on Phobos lose to friction. I would think that would be zero, because Phobos always turns the same face to Mars, and so has no tides. The energy lost because of the tides Phobos induces on Mars is going to be much too small to be inferred by measuring Mars's temperature. – Peter Shor Nov 25 '11 at 4:57

3 Answers

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First, you state a few things that aren't quite right in your question. While the view that's generally talked about is that Phobos and Deimos are likely captured asteroids, dynamically it's a pretty difficult problem (you generally need a third (in this case fourth?) body to take away the extra energy, and it's hard to get a circular orbit around the equator). See for a bit more on that.

In terms of Phobos' demise, there are two things that make this problem very difficult to estimate. First, Phobos' orbit evolves as it orbits around Mars, so you can't just take a linear approach and say, "It's moving towards Mars at 18.3 cm/year so it's going to hit in about 50 million years." It's more complicated and non-linear.

But besides that, there's the Roche Limit to consider, whereby the moon will break up due to tidal forces before it would actually hit. The problem there is that Phobos is already within the Roche Limit, meaning that it's only being held together now by the physical strength of the rock it's made of. And since we don't really know what it's made of inside (though we can make educated guesses and I'm sure there are models out there for its strength), these unknowns make it somewhat difficult to estimate.

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Since Phobos is within the Roche Limit, that implies that there can't be any loose material on its surface, at least not near the points pointing directly towards and directly away from Mars. Any such material would be pulled off the surface by the tide. It also means that the local effective gravitational acceleration is negative. (But it appears to be covered with at least 100 meters of regolith, which is a bit of a mystery.) Reference. It would be fascinating to see a plot of the local acceleration over the surface. – Keith Thompson Oct 12 '11 at 2:05
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Dynamical models over a likely timescale (say $10^6$ to $10^9$ years) would have significant error bars as mentioned above, and therefore one off predictions about individual moons have little validity.

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Because Phobos is already within the Roche Limit, within which it should disintegrate due to tidal forces, "appropriate data" would have to include quite a bit of detail about Phobos's structure and composition (information which we currently lack), which would let you determine not so much the "time at which" it will collide, but the period over which bits of Phobos would.

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This answer(v1) is similar to Stuart Robbins' answer over at Astronomy.SE, cf. astronomy.stackexchange.com/questions/1134/… – Qmechanic♦ Oct 25 '11 at 21:16
@Qmechanic: Yes it looks like there's also much additional detail there. Should I incorporate his; link to it — what's the correct etiquette? – raxacoricofallapatorius Oct 25 '11 at 21:26

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