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I am coding a simulation of a globular cluster in python, starting with particles of the same mass and $0$ initial velocity in a random spherical distribution. I am trying to investigate the root-mean-square radius of the system over time and trying to see how long it takes for this $R_{rms}$ to stop contracting and reach an equilibrium state. However, I have the issue of stars being flung out of the system at very high velocities due to close encounters, which totally dominate my $R_{rms}$ calculations. What is the best way to ignore these particles in my calculation?

I have thought about ignoring them if:

  • the distance from the origin is outside of the original sphere of generation, however have the issue of the cluster as a whole moving until it is outside of the sphere of generation.
  • Ignoring the particles if they have a velocity > the escape velocity of the system. This seems like the most reasonable way, however, I am worried about ignoring a particle which has a velocity > the escape velocity but has a later encounter that keeps it in the system.

Here is a plot of the positions of the particles over time where you can see several particles have been flung out.

plots of orbits

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What you have found is that $R_{rms}$ isn't a good definition of "size" of such cluster: already one escaped particle will make it grow indefinitely and ruin its value as a measure of size of the cluster.

You can probably use some other measure of cluster's size, such as inverse of average of $\frac{1}{r_{ik}}$ where $r_{ik}$ is distance between particles $i,k$. This definition makes contribution of very distant particles' very small, and contribution of particles that are close to each other very big, which is maybe better for your purposes.

However, I suspect that in time, this average will also "diverge" towards zero, due to particles at the centre getting closer to each other. But at least you don't have to worry which particles are to be tossed out - you include all of them.

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Isn't a better metric just the radius that includes half of your original particles (or half the mass if the particles are of different masses) ?

If the evaporation rate is still so high that you feel even this is compromised, then you could instead work out the half-mass radius for particles that are within some arbitrarily large radius.

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