how to calculate the half-mass radius and tidal radius of a (simulated) globular cluster

I am exploring a dataset of a direct N-body simulation of a star cluster, such that all particles have the same mass and

G = M = −4E = 1

for the whole cluster. The data available is the positions and velocities of all stars (at certain snapshots of the simulation).

I'd like to be able to calculate some basic characteristics of the cluster. First I did the center of mass (which is just the mean of all coordinates since all masses are equal), but I am not sure how to proceed for these two:

• The half mass radius - I can't seem to find an established (i.e. most efficient) way to calculate this. My first idea is to use a bisection method with the center of mass and the total radius as starting points, and do window queries with a spherical envelope until it contains half the stars. Is there something more efficient than that?

• The tidal radius - what I've found so far always involves variables drawn from the galaxy the cluster is in (like the total mass of the galaxy and so on). However, this being a simulation, there is no such data. Yet the dataset states that some stars escape the cluster over the length of the simulation... I must be missing something here

Your first question is about computing, not astronomy. You find a cumulative histogram of distance from the centre of mass and where that reaches 50% of the stars.

As to your second point, there is no tidal radius; you need two bodies in order to have a defined tidal radius. The tidal radius of a cluster is directly related to the gravitational potential external to it.

Stars can escape from a cluster because sometimes they can accumulate enough kinetic energy in weak interactions with each other, that their speed exceeds the escape speed of the cluster. It is a classic result that the "evaporation" timescale is of order 100 times the cluster relaxation timescale, even in the absence of an external gravitational field (e.g. Gerhard 2000).

• Thanks, I checked out the paper. So in my case a star would (generally) escape the cluster if it's velocity is greater or equal than twice the root mean square of the velocities of all the stars ("Thus, on average, a particle with v≥2V=√12σ‖ will escape")
– KGS
Jan 7, 2020 at 15:41