# How to distinguish primary hosts (stars) and orbiting satellites (planets) and tertiary bodies (moons) by their mass and trajectory?

I posted this question in the astronomy stackexchange. There are no responses, and it was suggested that I pose the question here. The "too long, didn't read" was taken from a comment, and my question is copy-pasted below.

TLDR:

You're asking about how some kind of classifier might work that accepts a group of trajectories or state vectors for 𝑁 bodies over an extended period of time, and identifies orbital hierarchies; i.e. "Who orbits who orbits who?"

Suppose one has run a gravitational simulation of N bodies (has the mass, vector positions, vector velocities, etc for each body), but knows nothing a priori about the system (ie, one does not know anything regarding composition, or which bodies are stars or planets). I would like to use the simulation results to categorize which bodies are primary hosts and which ones are satellites (and if possible, which "tertiary" bodies are orbiting the satellites). Is there a way to create a hierarchy that identifies whether a body is a primary host or satellite or tertiary? I would like to find a method of doing this such that the method can be applied to cases for N > 10^3 bodies.

For context, consider a 9-body system like a simplified example of our solar system (8 planets + 1 star). One can identify the Sun as the only primary host and the planets as the satellites because the ratio of the mass of the Sun to the mass of Jupiter (most massive planet of our solar system) is 1048. One could also use the approach of noticing that the center of mass would be inside of the Sun (or very close to in a point-mass simulation). But, the center of mass approach only works for nearly circular stable orbits around a single host; for unstable orbits, a body can deviate from its tangential path about the center of mass.

If we consider a binary star system for which both stars are of comparable mass, then I think that the ratio of the mass of any single host star to the mass of the most massive planet orbiting this binary star system can decrease; in other words, for each primary host body added to a system, the combined gravitational force at certain points in the orbit required to keep a satellite body in orbit should decrease if the system is stable, thereby decreasing the ratio of the mass of any host body to the mass of the most massive satellite body. (I suppose the same logic could be extended to host planets and satellite moons, though I suspect the ratio of masses would decrease; would this logic also apply to unstable orbits?)

Is this line of thinking correct? More importantly, what is the right way to think about this?

• What is the definition of primary host and satellite? In general the motion of 3+ body system can be very complex, not periodic, so you cannot tell what is orbiting what. May 22 at 14:21
• @MaximUmansky Say I have bodies A, B, and C. If A is orbiting B, then A is a satellite of B, and B is the host of A. Suppose B is orbiting C as well - then B is a satellite of C, and C is a host to B. In such a case, I would consider C to be the primary host because C does not orbit about another body (though it orbits the center of mass of the system). The simplest possible scenario for this could be A=moon, B=planet, C=star; though others can exist. For my purposes, this doesn't matter - I just want the hierarchy. Rogue bodies (for lack of better word) would have their own classification. May 22 at 14:32
• I think machine learning methods like classification, or possibly clustering are in order here. May 22 at 16:27
• @zeebeel If there is no significant separation of masses then there is no "regular" motion when something is orbiting something else, it will be chaotic motion like in a nonideal gas where there are attraction forces between molecules, for each object the trajectory would be something irregular around the common center of mass. Take a look at Wikipedia article en.wikipedia.org/wiki/Three-body_problem, they show an animation that illustrates it well. May 22 at 17:29
• @MaximUmansky I realize the orbits can be chaotic, so they may not follow nice keplerian trajectories. What I do not understand is why a body in a chaotic orbit (like the chaotic 3-body in the wiki animation) cannot have its own classification (ie, 3 primary hosts with no secondaries or tertiaries). May 23 at 0:30