# Would tidal recession significantly reduce the angular size of small moons?

Multiple large moons around a terrestrial planet is something that is seen in science fiction a lot, but is not really easily possible. An alternative is to use smaller moons but have them orbit closer.

With this in mind, I wanted to get the exotic look of multiple moons with large angular sizes but without the orbital instability. I decided to recreate the Galilean moons but to scale with Earth's mass as the host planet. I scaled down the mass of Io, Europa, Ganymede, and Callisto so my 4 moons would have an equal mass-ratio with Earth as the real moons do with Jupiter. This puts their masses in the range of very large asteroids like 4 Vesta and 2 Pallas, which depart from an equilibrium shape only due to rotation.

It's a bit of a handwave as to how a rocky planet could form multiple moons like this, as they are not thought to have planetary accretion discs like gas-giants but I'm assuming a collision happened like with the case of our moon, only it was more of a glancing collision that smeared the body into a large disc, out of which clumps formed into resonances in place.

I reduced the semi-major axis of my "Io" by 10, down to 42,170km and coincidentally that puts it just outside of geostationary orbit where material would be pulled down to Earth rather than recede. From there, another 2 moons follow the laplace resonance of the Galilean moons, so their semi-major axis is calculated from the orbital period for a 1:2:4 ratio. Finally, we have a Callisto, and I wasn't sure how that should be scaled to make it gravitationally equivalent, so I just made it the same ratio as Ganymede's orbit is to Callisto's.

I tested this in an N-body simulator, with these characteristics:

My goal is to get impressive sci-fi moon sizes so I thought copying(ish) the Galilean system would be a good starting point, and as a visual aid, I made this, using 4 Vesta as an angular size model: (If you sit at the right distance from your monitor that you can cover up the moon with your thumbnail, you'll get a reasonable picture of what it would be like to see such moons in the sky).

So ultimately, stability wise this doesn't seem to be a huge problem. Earth even has enough orbital real estate for 4 Ceres sized moons (so nearing twice the diameter of my case), when they are separated by 12 mutual hill radii.

The issue is tides. The moons are going to be pushed further and further away over time, just like our moon (in this case the resonance may or may not break). However, it is true that smaller moons would raise a smaller tidal bulge on Earth and so recede less rapidly than a larger moon would under equal conditions.

Here's my question given this information: how much angular size are the moons likely to lose after 4 billion years? As a ballpark, given their size, are they going to recede a very great distance or only a small amount?

• You know, i feel like this is the sort of question you cannot answer analytically. And simulating 4 billion years with any precision is not really going to work either. Just from having worked with GR and how orbits evolve, i would argue it is very unlikley this is a stable configuration. The non spherical nature of the moons will interduce issues as well. A point N-Body simulation is not really going to cut it here. Commented Jul 6, 2023 at 12:53
• @ErikHall - I'm not sure there's any reason to believe it would be especially more unstable than Jupiter's Galilean moons, since it's mostly just a mass scaled equivalent and the moons are a decent multiple of their mutual hill spheres. I was more wondering if there was ballpark answer to how quickly small moons should be repelled by tidal forces. If the moon is ~180+ times more massive than these moons, is the magnitude of the tidal force squared that value or something else. Does a moon half the mass (ceteris paribus) get repelled at half the rate? What is the relevant formula here? Commented Jul 14, 2023 at 16:47
• its a gut feeling as i said. You used Newtonian point simulations which will just not capture the gravitational effects the bodies exert on each other. Especially over such a long timescale. And there is some evidence for this. All planets came from debris clouds. Yet almost non of the rocky ones have small moons of your size. Could be evidence, could be coincidence. We would need to run an actual simulation to know the answer. Commented Jul 14, 2023 at 17:37
• Tidal forces are not equivalent and; especially with 4 bodies effecting tides (5 technically, but solar-linked tides are so minuscule in comparison that it's probably not worth calculating) Closer-in objects will have a greater tidal affect than those further out and larger objects more than smaller ones; so In General; if you have a large body that's close in; it will have massive tides vs a smaller body at the same distance Commented Aug 17, 2023 at 7:53