Example 4: Plummer model with circular orbits, external gravity

In this example, we consider a mass distribution that follows the Plummer model, also called the "softened point mass". It is a spherical system with a potential is very similar to the 1/r of the simple point mass, but it introduces a so-called "softening length" (usually denoted by the letter a or epsilon) that removes the singularity at the origin and, as its name suggests, softens the density profile.

For the velocity distribution, we take here the very simple case of purely circular orbits, that is, the magnitude of the velocity of each particle corresponds to the circular velocity at the particle's radius, for the Plummer potential. The direction of the velocity lies on its tangential plane, with random orientation. The point is to illustrate that this very simple setup is highly unstable under its own self-gravity.

First, we ignore self-gravity and impose an external potential on the particles. For this, SWIFT has to be compiled with the flag --with-ext-potential=point-mass-softened (set at configuration time). The parameters of the external potential are set in params.yml and should match those used to generate the initial conditions of the particles, set in makeIC.py. This is the case by default.

Running the example & visualizing the output

To generate the ICs using default values and run the simulation:

mkdir snap python makeIC.py bash run.sh

By default, the ICs contain 256k particles. This number can be changed with the -N argument.

To visualize the results, the film used in the lecture is made using

python ../visualize.py snap/snapshot_*.hdf5 -shift 10 -lim 5 -nbins 800 -frsp 0.4 -interp bilinear -fps 50 --savefilm

Plot the Velocity Dispersion

To see how the dispersion of the different components of the velocity evolves with time, the script dispersion_profile.py is provided. By default, when given a set of snapshots, the script will produce two plots corresponding to the radial and tangential components of the velocty dispersion against radius, for each snapshot. It achieves this by placing the particles in radial bins, projecting their velocities onto r and t components and computing the variance. Assuming default parameters were used, the following produces a comprehensive plot of the time-evolution:

python dispersion_profile.py snap/snapshot_0*00.hdf5 -Rmax 7

To save the plot, append --saveplot to the command.

Plot the Density profile

To quantify how the initially spherical, Plummer mass distribution evolves, the density_profile.py script is used to plot the mass density against radius, for a set of snapshot files (as above). To use the script, run e.g.

python density_profile.py snap/snapshot_0*00.hdf5

To save the plot, append --saveplot to the command.