Asking for some advice (orbital mechanics)

[Junghalli]

This is a follow up to my earlier question.

OK, I'm making up a fictional planet. The planet is supposed to orbit Alpha Centauri B at .72-.73 AU, be about .76 Earth masses, and have an Earthlike moon.

According to this paper the stability region for satellites of a planet is .48 times the Hill sphere for prograde orbits. This Hill sphere calculator gives a Hill sphere of roughly 1 million km for my planet, which should translate into a stability region of ~480,000 km. So going by that an Earthlike moon should be fine.

However, I tried simulating the arrangement with Gravity Simulator and I got a highly unstable orbit.

I'm inclined to trust the paper (it does focus on gas giant satellites but I don't see why this situation should essentially be any different), but my knowledge of orbital mechanics is nonexistant. Does anybody who knows more about the subject have any advice on this?

Thanks.

[Bottlestein]

^ The paper AFAIK talks about a restricted 3 body problem. (Will read the full paper after my group meeting - just skimmed abstract & conclusions now)

This means it's assuming an viable elliptic orbit for at least one of the bodies (probably the planet) - restricting the degrees of orbital freedom. This was what I mentioned in the previous thread you made - to solve the three body you need to assume either a form of reduced mass (i.e. CM system = CM star), or assume one planet has an elliptic orbit that is unaffected locally per time step by the third body.

To answer your question of paper vs simulator - in the absence of transparency of the simulator code - always paper. I think what's happening is the simulator is using a fixed order Runge Kutta in time (that's the easiest to code). To solve for the endpoints per Runge Kutta it's using some calculation that either
a) violates the CM (center of mass) assumptions - so you are violating angular momentum without realizing it
b) violates the restricted degrees of freedom assumption so you are getting accumulated error

In either case - paper trumps non-transparent code (Though I'll try to find the Gravity Sim source code...)

[Junghalli]

Thanks.

One possibility I'm thinking of is making the moon's orbit retrograde, as I've heard retrograde orbits are stable at greater distances (the number I remember is ~.9-1 Hill sphere vs ~.5 Hill sphere for prograde orbits).

I have Neil F. Comins's What If the Earth Had Two Moons in front of me now and one of the scenarios he explores is an Earth with a retrograde moon. One problem I see is that, rather than gradually having its orbit increased by tides like our moon, such a moon would have its orbit gradually decreased. I think that should be solveable by having the planet's rotation be retrograde as well (maybe the same impact that formed the moon knocked the planet into a retrograde rotation?)

IIRC, the greater stability of retrograde orbits has to do with that when they're between the planet and the sun they're moving against the sun relative to the planet, whereas a prograde orbit is moving parallel to the sun when seen from the planet in such a position. Do I have that right?

[Junghalli]

Hmm, looking around for information about retrograde orbits...

The asymmetry between the prograde and retrograde satellites can be explained very intuitively by the Coriolis acceleration in the frame rotating with the planet. For the prograde satellites the acceleration points outward and for the retrograde it points inward, stabilising the satellite.[7]

So I guess the effect is dependent on the planet's rotation after all?

[Bottlestein]

^ Yep - that is roughly why retrogade orbits are more stable.

Do you have the Gravity Sim source code? (I'm almost certain the simulator is using too low a Runge Kutta - and that's leading to the long term instability)

Essentially - I think the arrangement you have should be stable, at least to ~10^10 periods (which should be comparable to the lifetime of the star). It's probably just the gravity sim is not accounting for the restricted orbit of the star during each time step of the integration.

[Junghalli]

Do you have the Gravity Sim source code? (I'm almost certain the simulator is using too low a Runge Kutta - and that's leading to the long term instability)

It can be downloaded here, if that helps.

I tried emailing them, this is the reply I got:

I just created the simulation you describe. I get a stable orbit for the moon. I attached a screenshot A stable orbit does not mean that the orbit continues to trace the same exact path over and over again. The sun perturbs the orbit from a Keplerian ellipse. A stable orbit simply means that even over millions of years, the object will continue to orbit the planet, even if it is a sloppy looking orbit. I just tried a few extra values in the system you propose. I get 0.45 Hill Spheres as stable, and anything higher escapes within a few orbits. The fact that my value differs from the paper's value might mean that there's more to learn here. Perhaps its not purely a function of the Hill Sphere. Maybe a more massive planet can retain its satellites slightly farther out. Gravity Simulator is perfect for exploring this. Here's a Hill Sphere calculator you can use:

Thanks.