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A moon is just a natural satellite - something orbiting a planet - so I'm interested in what kind of factors influence how many moons a planet can have, and how many moons a planet will actually have.

I assume the size and potentially mass of the planet primarily affect how many moons a planet can have. Anything else?

What sort of factors affect how many moons a planet will actually have? I'm thinking about, say, Jupiter compared with Earth - why so many more moons?

The simple answer is an infinite amount. Since you defined moon as "just a natural satellite" anything natural and in orbit counts. Be that a double planets partner (There are many definitions, yet I prefer the one with calculating the barycenter and if it is outside the bigger planet, it's a double planet. This makes Pluto and Charon double planets and Luna remains a moon.), a big moon in hydrostatic equilibrium, any kind of asteroid moon, small rocks, dust and arguably every hydrogen atom. If you want a better picture of this look up Saturn rings or better the rings of J1407b.

Something tells me that's not the answer you're looking for. So lets redefine moon to satellite in hydrostatic equilibrium orbiting a common barycenter within the planet. How many of these moons can we squeeze around a planet in optimal conditions?

In that case we need to calculate the Hill sphere.

$r_H = a(1-e)sqrt[3]fracm3M$

$r_H$ = radius Hill sphere

$a$ = semi major axis satellite

$e$ = orbital eccentricity satellite

$m$ = mass satellite (planet)

$M$ = mass central object (sun)

The inner 5th of the Hill sphere offers stable orbits to major moons that can last indefinitely. Orbits outside that limit can be occupied too, but the moons wont remain there forever. This gives us the outer boundary. For the inner boundary we take the Roche Limit, as any major satellite within it will be ripped apart.

$r_L=R_m(2fracM_MM_m)^frac13$

$r_L$ = Roche limit (from the center of the major object to the center of the minor one)

$R_m$ = radius satellite

$M_M$ = mass central object (planet)

$M_m$ = mass satellite (moon)

This needs to be calculated for each satellite individually. Furthermore satellites with high tensile strengths, i.e. primarily metallic ones, can survive within the Roche limit for a long time. I'm not sure if this can be extended to major moons, I only read about it in the context of metallic asteroid moons.

Now we need to figure out how close to an existing satellite we can place the next one. This paper claims that the influence of the of an object on another one becomes weak enough for it it have a stable orbit at 3.46 * $r_H$ (moon center to moon center). Calculate the Hill sphere of each moon, now using the planet's and the moon's mass instead of the sun's and moon's. The center of the next moon can be placed right at that boundary.

Finally setting up mean motion resonances like in the Jupiter (4:2:1) system or the Trappist-1 (24:15:9:6:4:3:2) system is likely needed to keep such a tight grouping of moons stable.

I'll leave the number-crunching to you. My suggestion would be to set up a spread sheet, as many factors are variables.

EDIT: Zeiss Ikon's answer about moon formation introduces a criteria my answer ignores. I offer a way to find out what the maximum amount of stable satellites can be. As Zeiss Ikon pointed out it is unlikely that so many satellites will form.

In terms of planet formation, there are esssentially three ways to get a moon.

It can form with the planet, by the same kind of particle accretion process, but with enough relative velocity not to get incorporated (this is how the larger "true" moons of the gas giants were formed -- Ganymede, Titan, etc.).

It can form by capture, where an already formed body (generally through a three-body interaction) manages to get captured into an orbit of some kind, which then circularizes and (generally) aligns with the equator due to tidal forces. This is believed to be how Mars got its moons, as well as the gas giants their smaller moons.

It can form by impact (the current leading theory about how Earth's moon formed), if enough ejecta goes high enough and fast enough for interactions between ejecta to put some of the pieces into orbits, where they eventually coalesce and the orbit regularizes like a captured planetesimal.

For the latter case, I'd be very surprised if you get more than one moon over geologic time. Smaller objects in orbit around our moon are unstable -- they'll either eventually intersect the surface, or they'll be ejected into cis-Lunar space where they'll be perturbed either into ejection from the Earth-Moon system, or into an impact. There's no limit to asteroid/planetesimal capture over long enough spans, if the gravity well is large enough. In between, there is co-formation -- which seems to be limited to around half a dozen moons for planets similar in size to our gas giants. A super-Jupiter could reasonably manage to form and hold a larger number, if it's not too close to its star.

Having a moon is really a balancing of two things: the Roche Limit and 'local' gravity.

The Roche limit determines the minimum distance from the planet the moon has to be, and anything that 'orbits' beneath that limit is either going to tear itself apart due to tidal forces, or crash into the planet itself.

It almost goes without saying that larger planets have a larger Roche limit and that smaller planets have a smaller one

I say 'local' gravity for the second parameter because the thing the moon is orbiting has to have enough of a force of gravity to hold onto that moon.

If the moon has a higher 'local' gravity, than what it's supposed to be orbiting, then the roles switch, and what was a planet is now the moon.

The 'local' part matters, since everything has gravity everywhere, it just diminishes according to The laws of Gravitation, where it gets weaker by the distance squared.

With those two parameters in place, we know that a moon has to fall within the gravity capture of a planet without being in the Roche Limit.

The theoretical minimum distance is about 2 1/2 times the radius of the stellar body, while the gravity capture is significantly more than that (I didn't look for the math on gravity wells, it's complicated stuff).

With all that- larger stellar bodies have a better chance of catching 'moons' because of the increased size of their gravity capture zone. While their Roche Limit is larger as well, it grows slower than the gravity well.

The limit for how many moons a planet can have has more to do with the likelihood of them bumping into each other, and their respective Roche limits.

The number of moons a planet is likely to have depends on positioning in the solar system, since larger bodies on the outside of a solar system will catch more moons before the inner planets get a chance

If you want more bodies to orbit a given one, you need to take into account that all of them will interact with each other.

In order to have these interactions allowing each body to orbit in the system, you need to have a certain distance between the bodies. This is why in the solar system we don't get a body orbiting at every km distance from the Sun: when they are too close the bodies either merge or kick each other out.

The problem that you get with increasing the distance is that it also lower the gravity, thus sooner or later the planet will no longer be the main attractor for your system.

As a consequence, the more massive is the planet the large is the zone in which it can impose its gravitational domain: as you correctly observed, Jupiter has more moons than Earth, because it's more massive and can control more moons, and these moons can be at a proper distance from each other.