A day on Planet Sitnikov
Binary star systems are cool, and cooler still when they contain planets. The gravitational interaction of a planet with two central bodies leads to some weird paths — many unstable, a select few stable. A worldbuilder can use some creativity when designing the planetary system around a binary star. However, I must admit that many miss out on one of the more obscure orbital configurations.
Planetary systems are, on the whole, planar. This is because the original protoplanetary disk that formed each system was — well, disk-like: relatively flat. Thus, the objects that formed in it remained orbiting in pretty much the same plane. It takes significant disturbances to change this, although exotic things like the Kozai mechanism can temporarily lift an object out of the plane. Interactions with outside bodies are typically cited to explain planets orbiting outside this plane, but in reality, those interactions generally lead to all sorts of instabilities.
The setup that I have yet to see used is based on the Sitnikov problem. Here, two stars of approximately equal mass orbit around their center of mass, also called a barycenter. A third body of negligible mass oscillates along a line perpendicular to the stars’ plane of motion and centered on the barycenter.
Modeling the Sitnikov problem is an interesting one. In the case where the eccentricity of the orbits is 0, the problem is reduced to the MacMillan problem, and an analytical result may be obtained. For other eccentricities, though things are more difficult. Through mechanics, it can be found that the equation of motion of the central body is
where z is the height of the body above the plane, r is the distance between the body and one of the stars, and a dot denotes a derivative with respect to time. This is the differential equation that can be solved in the MacMillan problem.
In the general case, this can not be integrated. There are other options, including approximation by methods such as Taylor series. Computing these can take a while, but it’s not terrible. Besides, I would guess that most worldbuilders would be more interested in the MacMillan problem, just for simplicity.
The effects of this linear orbit are what make the Sitnikov problem so interesting:
- The equator and certain other regions of the planet will always be in starlight (this can be proved geometrically).
- There will be day-night cycles, as a point on the planet rotates away from the stars, and also as the planet moves away from and towards the plane.
There are, however, some issues:
- This arrangement is highly unlikely to occur in nature.
- While the orbit is stable for some small radial oscillations, more drastic ones can perturb it enough to destroy the orbit — and possibly the planet.
Still, it makes for quite an interesting setup. Extensions to the idea are possible, too, such as adding a circumbinary planet to the system. Imagine what it would be like for civilizations on these different planets to meet!
