# How long can a moon orbit a planet?

Today it was announced that Phobos was formed in a giant explosion, similar to how the Moon was formed. Phobos is actually coming closer to Mars every year, and has an expected 10 million year lifetime. It seems highly unlikely that Phobos could have been formed billions of years ago, and yet we first see it just a few million years before it's death.

My question is, is this even possible? Namely, if an object orbits just inside the geostationary belt of said object, which is the point at which it will come either closer or further away, could it survive in orbit for billions of years?

• Announced is maybe a too strong statement. It certainly looks improbable for Mars to have two temporary moons. In equatorial orbits. This is a mystery. Within a percent of their orbital life time existence. Now. Earth does not impact things that create new moons every other million years or so. I dare guess that this idea still needs some working to it. – LocalFluff Jul 5 '16 at 15:57
• Regarding the first link, a better title would be "Some scientists think Phobos and Deimos formed as a result of a giant impact". That said, that e-zine is at least five years behind the times. People have been leaning toward a giant impact formation for Phobos and Deimos for most of this decade. The arxiv paper that underlies your second link is completely bogus. Read it. Better estimates are 30 to 70 million years, depending on the values of Phobos' k2 Love number and Mars' tidal quality factor Q. – David Hammen Jul 5 '16 at 16:38
• @LocalFluff who knows, maybe we just missed a much "cooler" looking moon that would have been an even more interesting coincidence. Or maybe we're too early. – uhoh Jul 5 '16 at 16:39
• The statement of "is this even possible" doesn't really mention what "this" is- even though I assume you meant having a moon orbit forever? – Magic Octopus Urn Feb 4 '19 at 18:41

## 1 Answer

There is no generic answer to your question of how long a moon can orbit a planet.

If a moon is in a prograde orbit that is lower than the geosynchronous distance, it has negative work done on it due to the tidal bulge its gravity raises on the planet below. If it's farther than geosynchronous, it has positive work done on it. In the first case, its orbit decays to lower radii, and in the second case (which applies to earth's moon) its orbit expands.

The strength of the effect depend on the details of the planet's physical characteristics and on those of the moon. For the earth-moon system, the effect is enhanced because the earth has liquid oceans and because our moon is very large for a moon. The effect on the Mars-Phobos system will be weakened by the fact that Mars is a solid body, but strengthened because Phobos is so close to Mars.

• Tidal bulge is the deformation of the planet itself, caused by the gravity of the moon in question? How does the rate of change of the moon's orbit radius vary with moon mass? Is is independent, or would the rate of change increase, or decrease with decreasing moon mass? Those two little moons are quite small compared to the Moon of Earth. – uhoh Jul 5 '16 at 16:44
• @uhoh -- Ben is writing about the bulge in the planet that is induced by the moon. Another name for this is solid body tide. For example, see en.wikipedia.org/wiki/Earth_tide. The tidal acceleration on the moon depends on how far apart the two bodies are (torque is inversely proportional to distance to the fifth power); how big of an effect this is, characterized by the $k_2$ tidal Love number; and how dissipative this effect is, characterized by the tidal quality factor $Q$. – David Hammen Jul 5 '16 at 19:44
• @uhoh: I believe the dipole moment induced in the planet is proportional to the mass m of the moon, which means the anomalous acceleration induced in the moon's motion would also be proportional to the moon's own mass. So putting that together with David Hammen's comment, we'd have a torque proportional to mr^-5. The m factor is small for Phobos compared to our moon, but the r^-5 is bigger. – Ben Crowell Jul 5 '16 at 23:18