Tuesday, October 9, 2018

The Origins of Moons

There are a lot of moons in our solar system, and it has been impossible to detect whether there are similar numbers in any of the distant solar systems which have been detected or even if there are any at all, with one possible exception. The existence of Earth's moon may have played a large role in the origin of life here, and so it is an interesting question to ask where they might come from. It is certainly not necessary to assume that all moons originate in the same way.

Let's try to imagine the various ways a moon could originate. It could originate in place, in other words, form as a binary planet. A rotating cloud of gas and dust might be spinning too fast to simply condense directly into a single planet, and, similar to the formation of a binary star, the condensation starts in two places and continues to draw in the gas cloud, winding up with a planet and a moon. This would leave both planet and moon spinning rapidly, as the angular momentum of the whole cloud gets collected in the planet and moon, which must spin faster and faster as they condense. Tidal effects take over at this point, and begin to slow down the rotation of the planet and moon, while moving them closer together. If there was differential motion in the gas cloud that they condensed from, so the remainder is not following the same orbit as the planet-moon system, they will move into other regions of cloud and then accrete more mass, which may also affect the rotation and orbital rates over very long periods of time. The moon is smaller, and so it would intercept and accrete directly less gas, but since it is orbiting the planet, it sweeps out a much larger volume that would be swept by the cross-section of the moon. It sweeps out a volume corresponding to the cross-section of the swept volume of the moon's orbit, which can be huge compared to the moon itself. So the moon would grow in mass faster than the planet in this situation. Possibly the Pluto-Charon pair might have this origin.

If not formed in place, there must be a capture event. If the planet, already existing, has a large atmosphere, a smaller object could approach the planet and penetrate the atmosphere, losing relative speed. If its relative velocity was not too different from the planet, this might be enough to put it into an orbit around the planet, and then more passes would tend to circularize the orbit enough to stop the repeated interceptions of the planet's atmosphere, and the usual tidal effects could start their slow process to further circularize the moon's orbit. Both atmospheric drag and tidal pull tend to reduce angular momentum of the orbiting moon relative to the planet, which increases the major axis of the orbit. Tidal pull is stronger at the peri-planet part of the moon's orbit than at the apo-planet, reducing the eccentricity of the orbit. Tidal pull from a rotating planet also serves to reduce the axial tilt of the orbit. Likewise it affects the axial rotation of the moon, meaning it would tend to slow down any large rotation, relative to the planet, that it might have started with.

The reduction of angular momentum by the planet's atmosphere is proportional to the cross-section of the moon, meaning it goes as the square of the moon's mean radius. Angular momentum, however, goes as the mass of the moon, which varies as the third power of the moon's mean radius. This means that the chance of slowing a larger moon is less than that of slowing a smaller one with approximately the same orbit. Some of the moons around our solar system's four large gas giants might have come from this mode of capture. None of the moons is large in mass compared to the planet it orbits.

The likeliness of capture also depends on the relative velocity at the time of atmospheric entry. Too much speed, and the smallest of the impacting objects will burn up. Larger ones will go through the atmosphere and leave with sufficient remaining velocity to stay uncaptured. Gas giants have very deep atmospheres, and so there is also the chance that the impacting object will enter at an angle more steep than just grazing, and go so deep that the drag will overcome all of its velocity. Then its mass would simply be absorbed by the planet. For any giant planet-moon combination, there is probably a very sharp difference between nearby angles of entry, where one leaves the planet forever, another leads to absorption, and the gap between them, capture.

For rocky planets with minimally thick atmospheres, the possibility of capture by atmospheric drag must be very small. The only analog is impacting trajectories. If an object comes in and impacts a rocky planet's surface, it might lose some angular momentum and become a moon. Again there is likely a small gap between an angle of impact where the impactor is simply absorbed by the planet, possibly with some shedding of debris, and an angle of impact where the impactor simply leaves the region of the planet, and there would be a small range of angles where it stays on as a moon. The size of this gap may be negligible for large impactors, with one exception. If the incoming relative velocity is not much more than the additional velocity caused by mutual gravitational pull, the gap might be large enough so that some probability of retention of the moon is possible.

How could this velocity match happen? The first thing to come to mind might be some variation of the gravitational slingshot idea very frequently used in the trajectories of probes heading either toward the sun or toward the outer planets. These are used to augment or decrement the velocity of the probe by using the gravity of the planet and sun. Regrettably, these do not lead to orbital capture, much less low impact velocity. Another possibility is for the planet and impactor to have the same velocity almost exactly, as they would if they were in the same orbit around the sun. If a ring of gas around the sun did not condense into simply one single planet, but into two, at co-Lagrangian points, they would have nearly identical velocities, and if a slow migration started bringing them closer together, their relative velocity at impact would be only that provided by mutual gravity.

The L4 and L5 Lagrangian points are stable, and could conceivably collect mass from a gas ring simultaneously. Currently, there are a few asteroids at these points, but nothing large compared to the planet owning the orbit. Over time, the effect of other planets would probably make the two planets drift out of mutual Lagrangian stability, and thus an impact at slow relative velocity might happen, leading to a moon around a rocky planet. Like the Earth and Luna, for example.

In exo-solar planetary systems, there are often smaller planets at larger distances from the star, so there is no reason to immediately suspect that there would be radius bands for moon capture, mimicing what we have in our own solar system. Here we can hypothesize an inner band where Lagrangian impact might happen, then a band where large gas giants can capture relatively small moons, and then a band where icy planets condense in binary fashion. These bands might not correspond to anything relevant in other systems. However, the same mechanisms might exist, and can potentially serve as a guide for where to search.