Is this moon system stable?

The host star for this hypothetical system is 1.05 (M☉)

The gas giant is 20 percent more massive than Jupiter and orbits at 1.40 AU, now I need to know if these three orbits are stable around the planet

Moon One, 0.80 (M⊕) Orbits the host planet every 5 days

Moon Two, 0.56 (M⊕) Orbits the host planet every 11.8 days

Moon Three, 0.41 (M⊕) Orbits the host planet every 24 days

All three moons have eccentricities less than Earth.

Is life likely to exist on all of them? all three moons have large amounts of water on them.

• Hello Stephanie, it can be shown that there is no solution for the three-body problem, which is the answer to the stability question. All there is are approximations. – o.m. Mar 3 '16 at 19:05
• It seems like you're asking two questions here. The first one about stable orbits seems fine but might get a better answer on Physics SE; the second one about life is going to be too broad unless you give us more information about each of the moons, such as composition, atmosphere, and amount of radiation shielding. If you want to know the answer to both questions, I suggest you move one of them into a new question. – DaaaahWhoosh Mar 3 '16 at 19:11
• +1 on the suggestion to take the stability question to Physics SE – guildsbounty Mar 3 '16 at 19:15
• I'd question that even a large planet could hold onto hydrogen/helium at that distance from a sun, and thus would be considerably smaller. – Oldcat Mar 3 '16 at 19:36
• A Jupiter massed planet can hold onto its hydrogen & helium quite easily at that temperature. I'd have to pull up my spreadsheet to give you an exact answer. – Jim2B Mar 3 '16 at 19:44

So, your planet is fine...as has been mentioned in the comments, there are plenty of instances of Hot Jupiters being in close to stars and maintaining atmo...at 1.40 AU, you are plenty far enough away from your host star.

Alright, lets throw some math at this and see if it sticks. Given your three orbital periods, lets see how far they are from the 'center' of your gas giant.

Equation to determine orbital distance:

$$a=\sqrt[3]{\frac{GMT^2}{4\pi^2}}$$

Where G is the gravitational constant = $6.674×10^{-11}$, M is the mass of the larger body = $2.2776x10^{27}kg$ (120% Jupiter mass), and T is the orbital period (in seconds). We're solving for a, the semi-major axis (furthest distance the orbiting body moves from the central body)

Plug in numbers to give us some simpler math to do...

$$a=\sqrt[3]{\frac{6.674×10^{-11}*2.2776x10^{27}*T^2}{4\pi^2}}$$

Now, plugging in my three values for T and solving, and assuming circular orbits (for the sake of simplicity)...

Moon One: 895,688.787km

Moon Two: 1,587,687.917km

Moon Three: 2,548,695.561km

Now, there is no equation to simply derive the diameter of a planet based on its mass, because of composition. But if we assume it's basically Jupiter but bigger, I can spitball a very simple estimate here. Jupiter has a radius of 69,911km. Increasing mass by 20% isn't going to make it THAT much bigger, so we can safely say that your planets are orbiting well outside of its atmosphere, which is very good for their continued survival.

The next concern, as mentioned in the comments, is the Roche Limit, the distance at which a gravitational body will 'shred' another body. For large bodies (including Jupiter-sized ones) the rule-of-thumb is 2.5x the radius of the larger body. Unless your gas giant is more than 358,275km in radius (which, as mentioned above, it almost certainly is not) then your planets are safe in that way.

Now to the next concern: Tidal forces. If your Tidal Forces are too strong, you could cause lots of earthquakes, geothermal activity, horrific tides, and potentially shred the planet into tiny pieces if the Forces exceed the binding forces of the planet. The latter case should be a non-issue, as you are beyond the Roche Limit with all of these planets.

Here is our equation for computing the Tidal Force exerted on a body.

$$F_{tidal}=\frac{2GMmr}{d^3}$$

Where G is the gravitational constant, M is the mass of the larger body, m is the mass of the smaller body, d is the distance between them, and r is the radius of the smaller body. Again, we're going to have to guess a bit on that radius.

Planet 1 is at the biggest risk, due to proximity. At .8 Earth Masses, it's about the same size as Venus, so we'll just substitute in Venus' radius for simpler calculations.

$$F_{tidal}=\frac{2*6.674×10^{-11}*2.2776x10^{27}*4.777*10^{24}*6,052}{895,688,787^3}$$

So, this gives us a Delta-F of $2.56*10^{-9}$ Newtons. This is 2 orders of magnitude smaller than the Tidal Differential of the Moon on Earth. So...not a problem.

Now lets see what we can see about how the moons will interact with each other. The two innermost moons will see the most interaction, as they are the most massive and the closest together. Pulling out Newton's Law of Gravitation...

$$F=\frac{GMm}{r^2}$$

We can check out the max force between the two bodies when they are on their closest possible approach. Assuming circular orbits, the closest the two planets will get to one another is 691,999.13km. Plugging in our numbers...

$$F=\frac{6.674×10^{-11}*4.777*10^{24}*3.344*10^{24}}{691,999.13^2}$$

Solve, and we get a max influencing force of $2.223*10^{27}$ Newtons. Let's compare that to the Gravitational Force between the main planet and the inner moon.

$$F=\frac{6.674×10^{-11}*4.777*10^{24}*2.2776x10^{27}}{895,688.787^2}$$

The main planet is exerting $9.051*10^{23}$ Newtons on the inner moon. And here's where it all falls to shambles. At closest approach, the middle moon is exerting significantly more force on the inner moon than the planet they are orbiting is (distance is a really big deal when it comes to gravity). You don't even need to break out chaos theory and the three body problem for this one...this isn't going to work. I can't say precisely what will happen, because the three body problem is still a confusing complicated mess...but it will probably involve collisions or something getting ejected from the system.

So while your idea of having these moons in orbit around a gas giant is valid...they need to be much, much further apart. As long as you can verify that the Force applied to them by their parent planet is several orders of magnitude greater than the Force applied by any other object, then they should stay in orbit (you'll need to look at the Sun/Moon pair the same way I looked at the moon/moon pair in order to confirm this). So, this should be do-able, and I've given you all the equations you need to iron things out and ensure feasible stability for these moons.

EDIT: I say feasible because of the 3-body problem. you cannot be certain things won't go nuts, but you can at least present a layout where it cannot be definitively proven to be unstable.

• I'm a little confused on how to do the equations, Let's move the orbits out more a little, For Moon Two i'll move it out more so it orbits every 21 days, and Moon Three gets pushed out further so its orbit is now 47 days. They should still be in the gas giants hill sphere, even the third moon – Stephanie Mar 3 '16 at 21:38
• For their orbital radius, look at the second MathJax block I posted (the second one with a cube root in it)...convert your orbital time to seconds, and put it in for T. that will tell you how many km you are away from the gas giant. (MS Calculator in Scientific Mode has a cube root button). Then go to the last two MathJax blocks [continued] – guildsbounty Mar 3 '16 at 22:03
• In the first one, subtract Moon 1's orbital radius from Moon 2's orbital radius and replace "691,999.13" with the new number. In the second one, Moon 1 hasn't moved, so you don't need to recalculate that number. – guildsbounty Mar 3 '16 at 22:05
• You could have one Moon orbit another. After all, the Sun exerts a greater pull on the Earth's Moon than the Earth does. While @guildsbounty is generally true, special cases can be made to work. – Jim2B Mar 4 '16 at 2:18
• @Stephanie I calculate the giant's Hill Sphere to have about a 15 million kilometer radius. Sphere of Influence has a radius of around 13.6 million kilometers. A moon at 13.6 million kilometers would have a period of around 300 days. – HopDavid Mar 5 '16 at 0:28

It seems you asked several question about multiple-body configuration. As there is no solution to such problems, simulators could be an option to test your models. You can try for example :

• It would be valuable if you also posted some results from a simulation – SE - stop firing the good guys Mar 11 '16 at 9:23