## Orbits for Inner Planets of Binary Stars

What stable orbits are possible around binary stars? I poked at the problem some and found a couple interesting stable orbits.

This was started by the question on sci.astro, is it possible for a planet to be in a stable figure-8 orbit around the two stars in a binary system? As near as I can tell, the answer is no. But there are some interesting orbits to be had. I arbitrarily chose to work with a system with circular orbits and one star 4x heavier than the other. I think any weight ratio would have yielded the same types of orbits. I suspect having slightly elliptical orbits wouldn't make too much of a difference either, but I haven't checked.

Click on images to start them moving, or to stop them. Dragging changes viewpoint. It helps your CPU if you only have one running at a time.

 First, for reference, this is what a typical trajectory through a binary star system looks like. Stable orbits are few and far between. This is an inner planet (white) making three orbits per star system orbit. The star system is circular, with one star 4x the weight of the other. The 3::1 resonance again, except displayed once per orbit of the star system. The stars will have gone around once, the planet will have gone around twice, and they should all be in the same position every time. It's a strobe effect. The planet has been given nonnegligible mass and slight errors in the starting position, so it should wander a bit. The numbers in the corner count star system orbits (years) and show fractional energy misplaced by the simulation. See, it's stable. Same thing (3::1), no strobe, no trails. This is an inner planet (white) making four orbits per orbit of the star system (a 4::1 resonance). Strobe version of the 4::1 resonance. Again, the planet was given nonnegligible mass and some errors in the starting position. The lefthand number is years. Same thing (4::1 resonance), no strobe, no trails. This is an inner planet (white) making two orbits per star system orbit, using retrograde motion. (I couldn't find a 2::1 prograde orbit.) Strobe version of the 2::1 retrograde resonance. Nonnegligible mass and errors in the starting position were added. Same thing (2::1), no strobe, no trails. I tried finding a 2::1 prograde resonance. It would cross perpendicular to the heavier sun twice on the side away from the other sun, maybe at different distances. It would form an X directly opposite of that, between the two suns. The two lobes on either side of the X would be the same size. I have tried all reasonable positions and velocities for one of the perpendicular crossings. The X is never directly opposite the perpendicular position, it is always somewhat short of it. This bizarre orbit has the planet do 7 rounds for every 3 of the binary star system (7::3 resonance). See, the lobes are equal size, and the X falls short of the line between the two stars. The 7::3 resonance, strobe version. I didn't increase the mass or add errors to the starting position. It's stable for at least 170,000 years, that's as long as I ran the simulation. Same thing (7::3), no strobe, no trails.

More orbital simulations
http://www32.brinkster.com/snefru/space/ is another site that examines stable orbits around binary stars. Where I look at internal orbits, he mostly examines external orbits, and he chose his stars to be of equal mass.
Code for perfect hashing
Ye Olde Catalogue of Boy Scout Skits
Formulas for simulating orbits