Would this solar system be stable?

[Junghalli]

OK, I'm developing a planetary system around Alpha Centauri B (yes, another fiction-related question), and I want it to have two "Earthlike" planets (a truly Earthlike planet and a Snowball Earth-like planet). The system I've got so far looks like this:

Alpha Centauri B - ~.94 Sol masses
Inner planet - ~.8 Earth masses - .73 AU orbit
with a moon like ours
Outer planet - ~.8 Earth masses - 1 AU orbit
also with a moon like ours.

The inner planet was taken from Solstation's estimate of where a planet orbit ACB would have Earthlike conditions, and the outer planet is just inside the outer radius of the point where CO2 clouds start to form in the atmosphere (I just don't want the headache of worrying about a climate with CO2 clouds).

The thing that worries me is that this is a little closer than Earth and Venus in the solar system - Venus orbits at .72 AU. Intuitively I think it shouldn't make that much of a difference - it's only a matter of .01 AU - but I don't know and that bothers me.

Is it possible to check the stability of this system? Does anyone know how?

Thanks.

PS, there's also a Venus-like .7 Earth mass planet in a .4 AU orbit and a Mars-like planet in a 1.5 AU orbit, but I didn't think that'd make much of a difference.

[Bottlestein]

^ How many periods do you want to check stability for?

There's obviously no closed form for an arbitrary 3 body problem, but you can check stability of planet 1 given IEO of planet 2, and then planet 2 given IEO of planet 1.

You aren't in a situation where deviations from the elliptic are going to compound very quickly - unless you want stability for an insanely large number of periods.

[someone_else]

What you need is called "orbital simulator".
This is a good one

Otherwise you can google "orbital simulator" and find cadres of others that may suit you.

Good luck with balancing your system

[Junghalli]

What you need is called "orbital simulator".
This is a good one

Yeah, I already have that.

Anyway, I have a set-up in mind for the system I'm pretty confident of, the main issue I have now is the planet's Earthlike moon. I tried it in gravity simulator and it gave me a very unstable-looking elleptical orbit (I'm guessing it's too close to the sun for an Earthlike moon to be stable). But according to this paper the radius of stable orbits should be around .48 the Hill sphere, which for my planet is around 1 million km (going by this), so an Earthlike moon should be fine. The paper focuses on gas giant satellites, but I don't see why it wouldn't apply to rocky planets.

I'm inclined to trust the paper over the simulator program, but if anybody has any advice here it'd be welcome.

[someone_else]

I can only point you to a solar system generatorfor Alternity, made by an astronomer that also happens to play. It is pretty self-contained, can work fine for any game really.

Copy-pasting from there. (page 50, Moons chapter, "Hill limit" paragraph)
There's a lot of somewhat complex mathematics involved, but thankfully we can use a short cut.
If a moon makes fewer than 9 orbits around its planet in the time that planet takes to orbit its
sun, then it's not going to remain stable and will be pulled away by the sun. If there are more
than nine “months” per “year”, the moon is safe.

[Junghalli]

Thanks, that's a nice link.

Huh, so that works out to 1/3 of the Hill sphere, which makes the farthest stable orbit around 330,000 km. Not good, as this is significantly closer than the orbit of our moon, and Alpha Centauri is 2 billion years older, so if anything a prograde moon should be farther out (plus I don't want a planet with huge tides).

I think the safest thing would probably be to go with a retrograde moon, as retrograde orbits are more stable (the numbers I usually see for them are .6-.98X Hill sphere radius, which works well for me .6 HS = 600,000 km for this planet). Such a moon would probably be a capture (also good; I have more freedom with the orbit) and rather than starting close to the planet and gradually spiralling out it would start far away and gradually spiral in.

One issue I see is how this would effect the rate of the orbit change. A retrograde moon is one of the scenarios in Neil F. Comins's What If the Earth Had Two Moons? and he has it captured 500 million years ago in an initially present Moon-like orbit and now be 9/10 of the moon distance. The real moon was more than 96% its present distance 650 million years ago. Does a retrograde moon spiral in faster than a prograde moon spirals out?

If it spirals in at the same rate our moon spirals out then the history of the system works very well. It passed 460,000 km a few billion years ago (the limit at which it becomes an effective obliquity stabilizer), by today it's maybe a little farther out than our moon, and over the next several billion years it will spiral in, hitting the Roche limit sometime after the planet has become a Venus-like world due to the brightening of the sun anyway. If the rate is faster though ... I suppose it's still workable since I could say it's a late capture, but it's less elegant. Blargh.

The only difference I can see offhand with a retrograde moon is it'd be up a little bit longer since it's moving west to east in the sky, which would raise the tides and slow down the planet's rotation faster - but I'd think the effect would be pretty small, as the orbital period is so long. Maybe I should start another thread for that.

[Simon_Jester]

Thanks, that's a nice link.

Huh, so that works out to 1/3 of the Hill sphere, which makes the farthest stable orbit around 330,000 km. Not good, as this is significantly closer than the orbit of our moon, and Alpha Centauri is 2 billion years older, so if anything a prograde moon should be farther out (plus I don't want a planet with huge tides).

Make the moon smaller and it reduces the tidal effects. Or can you do that?

[Junghalli]

Make the moon smaller and it reduces the tidal effects. Or can you do that?

I considered a small (1/4-1/5 LM) close-orbiting (~200,000 km) moon, but I read in a book I found on Google that a small moon (up to 1/2 lunar mass) wouldn't have the obliquity stabilization effect of our moon, so the planet would have a highly unstable climate. It didn't really give any more information than that, like whether this is a proper lower limit or whether they mean such a moon at present lunar distance, so I'd prefer to be safe and stick with a moon roughly the same size as ours.

If the moon is prograde, there's also still the problem that it should be farther away from our moon (maybe not since I have a source saying a half-sized moon should receede more slowly, but it doesn't quantify the difference).