Generation of random, stable n-body solar system

Question

I've been playing with solar system simulation lately using Barnes-Hut algorithm to speed things up. Now simulation works fine when feed with our solar system data, but I'd like to test it on something bigger. Now, I tried to generate 500+ random bodies, and even add initial orbital motion around center of gravity - but every time after short time while most of the bodies end up ejected far away into space. Are there any methods to generate random sets of planets/stars for simulations like this that will remain relatively stable ?

Answer 1

You should probably ask this question on the Physics or Mathematics stackexchange.

I think this is a very difficult question, to the point that great mathematicians have studied the stability of the solar system. Things are "easy" for the two body problem, but the three body problem is notorious for its chaotic behavior (Poincare studied it carefully and in the process laid out the fundament of the qualitative theory of dynamical systems). If I am not mistaken (feel free to check this online), instability of orbital dynamics of large number of bodies (large meaning three or more) is a condition, whose probability of occurrence is very high. Meanwhile, coming across stable configurations has a vary low probability.

Now, for so called integrable systems ("exactly solvable"), like n copies of decoupled sun-one-planed models of a solar/star system, small perturbations are more likely to yield stable dynamics, due to the Kolmogorov-Arnold-Moser's theorem. So I can say that it is more likely for you to come across stability, if you first set up the bodies in your simulation to be comparatively small gravity sources orbiting one significantly larger gravitational source. Each body has one dominating force from the large source and many much smaller perturbations from the rest of the bodies (or the averaged sources of your Barnes-Hut algorithm). If you consider only the dominating force, and turn off the perturbations, you would have a solar system with n decoupled two-body systems (each body following elliptical motion around a common gravitational center). If you turn on the perturbations, this dynamics changes, but it tends to deviate from the unperturbed one very slowly, and is more likely to be stable. So start with highly ordered dynamics and start changing slightly the body's masses and their positions and velocities. You could follow how the dynamics changes when you alter the parameters and the initial conditions.

One more thing, it is always a good idea to place the inertial coordinate system, with respect to which the positions and the velocities of the bodies are represented, in the center of mass of the group of bodies. This is more or less guaranteed when the initial momenta sum up to the zero vector. This set up yields the center of mass of the system is always fixed at some point in space, so a simple translation will move it to the origin of the coordinate system.