# Calculating RMS Radius of a Globular Cluster

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.

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.