Is this planetary system stable?

Question

The host star has a mass of 0.90 M$_\odot$ I'll refer to this star as Helios. I'll refer to the planets as Helios One, Helios Two, etc.

Helios One is 0.55 M$_{\oplus}$ and orbits at 0.75 AU. It's eccentricity is 0.012.

Helios Two is 0.94 M$_{\oplus}$ and orbits at 1.08 AU. Its eccentricity is 0.015

Helios Three is 0.32 M$_{\oplus}$ and orbits at 1.55 AU. It's eccentricity is 0.023

Helios Four is 0.42 times the mass of Jupiter and orbits at 6.02 AU. Its eccentricity is 0.044.

Helios Five is 0.17 M$_{\oplus}$ and orbits at 10 AU. Its eccentricity is 0.060.

Does this system seem realistic? And, of course, the main question is, is it stable?

I created the orbits randomly myself for a hypothetical alien solar system I created. I just need to know if they are stable orbits — Stephanie, Mar 01 '16 at 22:06
a side note - I didn't calculate, but it feels like there is no habitable planet in your system... well, no earth like planet. I do not have these formulas ready, but Helios one (which should be in the habitable zone) is to small to hold its atmosphere and Helios Two could be right outside the habitable zone (imagine Hoth from StarWars). As long as you do value this - non humans might be okay with Helios two... but to be sure, we need this "how big is that habitable zone" formula I do not find right now. edit: HDE 226868 seems to know more about this. — Confused Merlin, Mar 02 '16 at 10:21
A planet with half the mass of Earth is big enough to hold onto a thick enough atmosphere to support life, Mars is only 10% Earth's mass — Stephanie, Mar 02 '16 at 13:35

score 6 · Accepted Answer · edited Apr 13 '17 at 12:52

This is an exceedingly tricky question. Determining the stability of this system is akin to determining the stability of the Solar System, a difficult - and currently unfinished, as far as I know - task. Planetary systems with more than a couple planets are chaotic, meaning that on timescales of ~10⁷ years, they become chaotic and impossible to predict to any degree of accuracy.

If there were simply two planets, I could look for any orbital resonances that would destabilize the system, something that I've written about before. In the cases where $[n:1](2)$ resonances dominate - that is, when the satellites (here, planets) fall into a certain range of masses - then determining stability is (relatively) easy. Here, it isn't.

We can't determine analytically here whether or not the system is stable. I can give you a rough prediction, though, which is that it is stable. Why?

The planets have relatively low masses (all but one less than the mass of Earth).
The planets are pretty well spaced apart.

My one concern is that the third and first planets are nearly in a 3:1 resonance, which has the potential to cause instability. But this might not happen. In fact, there's a chance it could stabilize the system even more.

Is the system realistic? I would argue yes. You have three terrestrial planets within about 1.5 AU, which is nothing special. You then have a small gas giant at about 6 AU, a bit further out than Jupiter. Again, this is fine.

What concerns me is that planet way out at 10 AU. It's the least massive of them all - although not by too much - but may still be a terrestrial planet. You have to explain how it came to be way out there. If it formed there, then you need a good explanation; it seems oddly far out. If it was moved back there, you need to explain what caused it to go there. I see one body capable of severely disrupting an orbit, namely, the gas giant. Interactions with the protoplanetary disk and/or the other terrestrial planets could also have had an effect, but those are less likely.

Overall, I'd say you're fine.

I created two alien races to inhabit the first and second planet, Do you think the third planet would be too cold to support life? — Stephanie, Mar 02 '16 at 00:09
@Stephanie At 1.55 AU? It would be close to the edge of the habitable zone, which in this case would be a bit smaller than the Sun's. I would bet against it, but the planet won't be exceptionally cold. — HDE 226868, Mar 02 '16 at 00:16

score 2 · Answer 2 · answered Mar 01 '16 at 22:42

There is one empirical law that fits many of the stable systems we can see, both planets and moons, and that is Bode's Law. It gives a very close fit to the locations of the first seven planets, with a gap at the Asteroid belts.

Body    Actual distance (A.U.)  Bode's Law
Mercury         0.39               0.4
Venus           0.72               0.7
Earth           1.00               1.0
Mars            1.52               1.6
                                   2.8
Jupiter         5.20               5.2
Saturn          9.54              10.0
Uranus         19.19              19.6

Your system fits the Venus/Earth/Mars slots well. Helios 4 is a bit far out, but there's nothing else out there but a small planet in the Saturn slot. This might be an issue, but in our Solar System Neptune and Pluto don't fit Bode's Law particularly well.

Is this planetary system stable?

2 Answers2