# Is this circumbinary planet set up in a stable manner?

This is the ideal set up for my solar system: EDIT
It's been pointed out that this is not a stable set up, so I want to clarify that everything in this set up can be changed at will to fit my planet. The only thing that's essential is the size, gravity and temperature of Enkei as it needs to support the fauna in my world. Yaima is intended to be a planet but can be a moon.

Magnus (Main Star):
Mass: 2.1879*(10^30)kg (1.1 Sol)
Luminosity: 5.3688798*(10^26)W (1.396 Sol)
Diameter: 1530540km (1.073 Sol)
Habital Zone: 1.13-1.63 AU

Ignis (Circumbinary Star):
Mass: 5.1714*(10^29)kg (0.26 Sol)
Luminosity: 3.4467993*(10^24)W (0.00896 Sol)
Diameter: 513400km (0.369 Sol)
Semimajor Axis: ???

Enkei (Planet):
Mass: 5.63094*10^24kg (0.9428 Earth)
Volume: 1390160000000km^3 (1.2834 Earth)
Gravity: 7.84m/s^2 (0.8G)
Density: 4.05g/cm^3 (0.735 Earth)
Surface Temp: 292K 19°C
Semimajor axis: 260615093km 1.7421042951765742AU

Firstly, I have no idea what the most suitable orbit distance is for my binary (P-Type) Star but I found the minimum distance to be 0.11AU so 0.2 seemed a good guess. Obviously, I need a stable System before I can even begin to populate the world so this is the most important. Worst case scenario I can scrap the binary and just tweak our solar system accordingly.

292 kelvin is the maximum temperature I want for my planet as it effects climate and evolution. Since I couldn't find anything concrete for calculating the relationhip between temperature and atmospheric density, I plotted the disparity between expected temperature and actual temperature for Earth, Mars and Venus alongside the atmospheric pressure (1, 0.006 and 90, respectively). using this scale predicted my Planet's temperature (12x Earth Atmospheres) to be 60-80% of it's actual temperature and because the albedo would be similar to earth, this then gave me an orbit distance of roughly 1.74AU which is just outside of the habitable zone. That makes the expected temperature the same as mars but an atmosphere 12x stronger would provide a considerable buffer and an Earth-like landscape (i.e. Large Oceans) should help regulate temperature.

If that seems inaccurate, Is there an equation for temperature and atmospheric pressure? Or is anyone able to calculate a more accurate orbital distance that would keep the planet at around 19°C?

Bear in mind that distance affects year length. My current set-up allows 32-hour days for 539.751 days in a year, which is 540 days a year -1 every four years except for the start of new millenia.

• @JDługosz Ignis is the second star, so the planets outside Ignis are indeed circumbinary. Apr 20 '17 at 19:19
• Normally, I would write a Rebound script and run it this weekend to see if your setup is stable. However, it won't be. With the second star so close to the first, the tidal forces would prevent nearby planet formation, plus the conjunctions of the stars (so close in mass) would create relatively enormous gravitic variations. There is no way anything near two stars that close would remain stable through ~4 billion years of life evolution. Apr 20 '17 at 19:25
• Put it way out there. Basically, for a multi-body system to be stable, each interaction has to more or less 'act like' a 2-body interaction. The Sun's mass dominates the Solar System sufficiently, and the planets are far enough apart, that each planet's orbit can mostly be treated as a planet-Sun interaction. Multiple star systems (more than 2) are composed of binaries and single stars (Alpha Centauri is a binary with a very distant third member - Proxima is distant enough to treat Alpha A and B as a single source of gravity; Epsilon Lyrae is two binaries, far enough apart that each can treat Apr 21 '17 at 4:24
• Maybe you could use a stable Lagrange point for the two-star system as the centre-of-mass coordinate of your two-planet system. Of course, this setup screams intelligent design but I think it should work. Apr 30 '17 at 1:19
• It's a bit clunky, but one gravity simulator you might try is here: phet.colorado.edu/sims/my-solar-system/my-solar-system_en.html May 5 '17 at 21:17

# Setup

I tested your scenario using Rebound, an orbital simulation package with a nifty python interface. If you want to see the code I used, it is available on my github here.

Here are the numbers I used for my first run:

m_magnus = 1.1          #2.19e30 kg
m_ignis =  0.26         #5.17e29 kg
m_enkei =  2.83e-6      #5.63e24 kg

a_ignis = 0.2     # AU
a_enkei = 1.74    # AU

e_ignis = 0.01
e_enkei = 0.01


In addition to the numbers you provided, I just threw in some eccentricities to see if it worked.

# Results

The simulation ran for 1 million Earth years. I only included Enkei and the two stars in the simulation. At a distance of 0.2 AU, Ignis did not do much to disrupt the orbit of Enkei.

The maximum variation that I saw in the semi-major axis of Enkei in those million years was about 1.7%; this variation was somewhat random, I was unable to plot any patterns for it. There are definitely some interesting orbital harmonics going on, as you can see in the graph below:

I broke the semi-major axis graph out onto a longer time scale.

So these are some interesting orbital dynamics. The eccentricity changes will cause your planet to basically alternate hot and cold years with about the same variability we see on our planet year to year. The semi-major axis changes will be more significant, causing up to a 3.5% change in insolation on around a decade time span. That is enough insolation to produce noticeable, several degree C changes from warmest to coolest.

So, without accounting for any of the other planets, the setup listed above appears to be stable, and has some interesting orbital characteristics to boot. Just to be clear, numerical simulation, no matter how advanced, cannot prove stability for an n-body system, but it can suggest that it is likely.

If you want to add mass, semi-major axis, and eccentricity information for some of the other planets, I can add them into the simulation.

# Insolation

When I get some more time, I will update this question with insolation graphs, as I did in this question. Hopefully, this will help determine what the planet's seasons and average temperatures will be like, and shed some more light on the orbital harmonics.

• Wow this is useful! It says rebound doesn't support windows so I don't think I'll be able to test it myself. However this looks promising. From what you know, is there any reason to suspect the planet would not remain stable long enough to sustain life? Perhaps in excess of 100mya? May 15 '17 at 22:45
• @ZacWalton Well n-body systems (such as a solar system with 2 stars and n-2 planets) are inherently chaotic. That is, slight perturbations can have massive results on the outcomes. What I do with my simulation is to demostrate that there is a set of initial conditions that could work. I can't, with my little desktop computer, prove anything. however, if this system appears to be stable for 1 million + orbits, it is at the very least plausible, and even likely, that it will be stable for the lifetime of the primary star. May 15 '17 at 23:37
• "When I get some more time, I will update this question with insolation graphs" - so have you had more time? Dec 7 '17 at 13:47
• @Molot Funny, I think I have some free time coming up! Dec 7 '17 at 14:59
• @kingledion Apologies for the long hiatus, I've been using this as a framework so far but I'd like to delve into more detail with the planet's climate. You said you saw a 1.7% variance in the semimajor axis but the graph only shows a difference of 0.0005 AU, which is considerably less than 1.7%. Is there something I am missing or is the graph just a snapshot of the timeline? I understand it's been a while but if you could shed any light to this I would be extremely grateful! Sep 3 '20 at 16:00