Is this a potential planetary setup for the Alpha Centauri System?

Question

I originally posted this on the World Building SE, but I was suggested to post it here for a better answer. I am attempting to create a [semi]plausible star system in Alpha Centauri for a series. The system was originally intended to have 1 habitable planet per star, but after my first couple of attempts I noticed that it might be possible to get at least 3 planets to be habitable. The latest attempt actually shows 5 potentially habitable planets (2 of which are a binary[That is another SE question]). Ultimately, I would like this to be the case as it allows a much more diverse universe for the series.

Attempt:

• aCen A Planets
• aCen B Planets

I don't have access to, or knowledge how to use, any form of software like Universe Sandbox. I have read several reports on planet orbits and I think I did an "ok" job with this.

Here are some papers that I have looked at for this: http://adsabs.harvard.edu/full/1997AJ....113.1445W https://arxiv.org/pdf/1801.06131 https://core.ac.uk/download/pdf/25201586.pdf

My main concern is that the planets might orbit too closely. I have attempted to find a formula (that I could understand) that could aid in spacing the planets. The nearest I could manage is using the Mutual Hill Radii. There are conflicting reports where one says that 10 - 12 MHR (Delta-H) is good or a tightly packed system. Earth and Venus have around a 25 MHR value. One of the reports that I linked mentioned up to 25 MHR for aCenA, but it also shows several other numbers and after trying to comprehend everything my brain reached orbital velocity.

Question: Is the planetary spacing stable enough to host planets on Gigayear timescales? They don't necessarily need to be able to have spawned life, but they should be able to support life with little to no human intervention.

Note and Bonus, aCen V is a binary planet that orbits with a Semi-Major axis of 750,589km with an eccentricity of 0.01204. (There is an error in the image in Yellow that shows 148.623 and Eccentricity of 0.0910). The inclination should be 0 as both planets should be on the same plane. Last note: The Semi-Major Axis distance is in Megameters (1 million meters).

Answer 1

The only way to answer the question about stability is to do the integration, because this problem does not have an analytic solution. There are approximate solutions for the stability of two-planet systems (although these are based on a somewhat weaker constraint that allows the outermost object to escape to infinity) but they do not necessarily generalise to more planets. Furthermore, planetary systems tend to exhibit chaotic behaviour so you aren't going to get definitive answers even with an integration, because errors get introduced by the limited precision at which the calculations can be done.

You also need to bear in mind that the parameters you've listed in the table do not give sufficient information to set up an integration:

• You need the argument of pericentre ($\omega$) and the longitude of the ascending node ($\Omega$), which together with the inclination form a set of Euler angles describing the orbit's orientation in 3D space.
• You need the mean anomaly ($M$) or equivalent (e.g. mean longitude, $\lambda$) to describe where the objects are along their orbits.
• You need to specify at what epoch these parameters are specified for.

The integration is going to be further complicated by the fact that it isn't sufficient to just simulate Alpha Centauri AB, you also have to take into account Proxima, which may perturb the orbits of the AB pair on gigayear timescales, and is also on such a wide orbit that you probably need to take the galactic tide into account and the unknown history of stellar encounters along the system's path through the galaxy.

Answer 2

I think you may have given each star too many planets. Alpha Centauri AB is a fairly close system, with a semimajor axis of 23 AU and an eccentricity of 0.52, which means the stars approach to within 11 AU of each other. Planets aren't stable unless the binary star surrounding them is more than 3-4 times more distant, so I think your outer planets may be in trouble. If these systems are space roughly like our own, I'd think each star could have no more than about five planets, and that would be stretching it.