# Could this three moons system be stable?

In my fictional world I have selected all parameters to maximize the Hills Sphere of my planet. The planet has three times Earth's mass and it is located in 2 AU from its star that's 1.4 times more massive than our Sun.
I assume that there are more planets on the system but in such way to have minimum interaction with my planet.

I want to have three moons on a 1:2:4 resonance. I also want:

• the first one to be as large as our moon to the sky
• the second 1.5 times larger
• and the third, half our moon's size.

To achieve this I have assumed:

• the first moon having 1.1 lunar radius,
• the second 2.3
• and the third 1.1.

Their orbits are 18, 36, 72 days.

I don't really care about their density and I assumed that their masses are

• 1 lunar mass (probably rocky)
• 3 lunar mass (water or ice world)
• 0.3 lunar mass (probably ice world)

I guess that to be more stable i should have eccentricities smaller than 0.05 and inclination equal to zero.
I am not sure about the spin of my moons, is there a chance being tidal locked?

This is my effort to achieve maximum stability of the system, what parameters should I reconsider to make it more stable? I am willing to change almost everything even the resonance, but I want them to be visible from my planet in similar way moon is to Earth.

Have in mind this question is about a fictional world for a fantasy story so even the minimum possibilities for that senario to happen is enough for me. So any ideas?

P.S I have asked a similar question on worldbuild.stackexchange but they directed me also here for a better answer.

• Howdy - this is probably better for the worldbuilding site or possibly the physics site. It is strange they sent you here :) – Fattie Jun 21 '16 at 12:49
• Similar in principle to this question so I guess it's applicable here. And a few people were prepared to have a go at that one. – Andy Jun 21 '16 at 13:39
• The 1:2:4 resonance is fine. The size of the Moons might be a problem. Moons that large could disrupt each other fairly significantly. Our Earth for example can't have two moons because our one moon is too big to have a long term stable region for another. Jupiter's so massive, so it's able to hold more moons in stable orbit. It would help if you made the planet more massive and further from it's sun so it has a larger hill sphere. That said, I'm kind of 50/50 on whether your specs could be stable. The math is over my pay-grade. – userLTK Jun 21 '16 at 18:17
• Cool idea for some moons. – wogsland Feb 12 '17 at 4:58

Nothing in your description sounds wildly implausible. I'll just go through and extend some of your properties to make sure they make sense though.

The planet has three times Earth's mass

I'll assume that your new planet is Earth-like in composition and density. That implies that it's radius should be about $R_p \approx 1.4 \: R_\oplus$. Let's put that aside to use for later.

Now let's look at your moons.

• the first one to be as large as our moon to the sky
• the second 1.5 times larger
• and the third, half our moon's size.

To achieve this I have assumed:

• the first moon having 1.1 lunar radius,
• the second 2.3
• and the third 1.1.

The first moon, you want to be as large as our current Moon in the sky and you say it has a physical radius of $R_{M1} = 1.1\:R_{M}$. Our current moon subtends about $30\:arcsec = 8.73\times10^{-3}\:rad$ on average. You can calculate the distance, $d_{M1}$, the moon must be from the planet to have the same apparent size as our moon with the following equation.

$$d = \frac{R}{tan(\delta/2)}$$

Using $R_{M1}$ for $R$ and $8.73\times10^{-3}\:rad$ for $\delta$, your moon must be $d \approx 4\times10^8\:m = 44.5\:R_p$. Your first moon must be about 44.5 planetary radii out. Compare this to the Moon's distance of about 60 planetary radii out.

Now, we want to know how long it would take for such a moon to orbit, given the planet's mass, the moon's mass, and orbital distance. You can get that from Kepler's third law.

$$P = \sqrt{\frac{4\pi^2}{G(M_p + M_{M1})}d^3}$$

We can say $M_p = 3\:M_\oplus$ (as you specified) and we calculated $d$ just now. We only need to specify the mass of the moon. You provided the desired masses for your moons, so we're all set. Note $G = 6.67\times10^{-11} m^3kg^{-1}s^{-2}$ is the gravitational constant. I find that $P_{M1} = 1.44\times10^6\:s=16\:days$.

Let's stop right here for a second now. We've reached a point where your numbers are in conflict. Given your planet and your first moon's apparent size, mass, and radius, we found that it would have to orbit the planet in 16 days. If you want to keep the 1:2:4 resonance, then you need your other moons to orbit in 32 and 64 days, respectively.

I won't go through the rest of the math, I'll leave that up to you. But what you can do, if you want to ensure everything is consistent with real physics, is say I know how long my remaining two moons need to orbit for (i.e., you know $P$ for them), then at what distance must they orbit (i.e., what is the value of $d$ for them)? Work Kepler's third law backwards to get that. Then, given your desired apparent sizes in the sky, determine how large must they be physically by working the angular size equation backwards. You'll get new radii for the moon. You might then want to verify that the new mass and radii for your moons correspond to densities that match the desired moon types. E.g., if your third moon is going to be an ice moon, it should have a density of $\sim1\:g/cm^3$. You can play with your numbers until you get a system that matches your requires and fits the equations.

An important note: All these equations use MKS units. That is, masses should be in kilograms ($kg$), distances and sizes in meters ($m$), time in seconds ($s$), and angles in radians ($rad$).

• Thank you for the explanation about the orbits it was more than useful .My main problem is if this planet can sustain three moons that large for a big amount of time. And second how fast do we want the moons to spin around themselves. I think that the must spin slower than the planet but not too slow cause they're gonna hit the planet eventually . Is that right? – teorf Jun 22 '16 at 20:01
• I don't see why the planet couldn't sustain the three moons. You might want to calculate the barycenter of the system and make sure the planet is still the dominant object, but I suspect it will be. I don't know what you mean by "how fast do we want the moons to spin around themselves." Are you referring to their rotation rate about their axes? If you're talking about the orbital rate, the that has to be governed by Kepler's third law if you remain true to physics. If you have them orbit at what that law prescribes, the won't fall in. – zephyr Jun 22 '16 at 20:05
• I am talking about the rotation rate about their axis. – teorf Jun 22 '16 at 20:07
• If you set your system to be in resonance and have all your moons tidally locked (i.e., the same face of the moon always faces the planet), which is certainly possible, then it will be completely stable. If your moons are not tidally locked (by the fact that the rotation time does not equal the orbital time) then your moons will want to move away from the planet and out of their resonance just as our moon does. – zephyr Jun 22 '16 at 20:12

I see no reason why this wouldn't work. The innermost Galilean moons are in a 1:2:4 resonance so it's clearly a stable orbital configuration. They could all be tidally locked if you want, but the planet itself can't be tidally locked with all of them. If the planet were tidally locked it would likely be locked with the innermost moon.

Also of note, the Galilean moons all have eccentricity less than 0.01 and inclinations less than 0.5 degrees.

• Is that a possibility? One moon to be tidal locked? I thought that if that happens the innermost moon is going to kick off the others eventually... – teorf Jun 22 '16 at 20:04