Kyle Barbary

Astro 120

Spring 2002

Homework 2

II) N-body simulations

A) Intuitively, a homogeneous sphere of constant density will be unstable in the same way that the atmosphere on earth would be unstable if it were homogeneous and of constant density. Because there is a force acting down (or inward) on the molecules (or stars), there must be some balancing force or the molecules (stars) will collapse downward. In an atmosphere, this balancing force is due to the pressure gradient. In a sphere of stars, it is due to the density gradient.

Quantitatively, we know from Gauss that the force acting on any one star will be dictated by the total mass of all stars at a smaller radius than it, and its radius. In a sphere of constant density, the inward force would be proportional to r³ times r ^-2 or just r. This would mean that the gravitational inward force would increase linearly with the radius, precipitating a collapse:

At first, the sphere doesn’t do very much, but one it starts collapsing, its motion begins to accelerate. A few stars that have outward velocities at the beginning do not collapse into the center, but most do. After it reaches a very small sphere, it “bounces back out.” I would assume that it would keep doing this, although a few stars will probably be lost each time due to the fact that they reach escape velocity. However, I tested my idea and this is not in fact true. The final state of equilibrium is an extremely diffuse one:

I guess I am a bit confused about what happens in the collapse itself: do the bodies collide, or just pass each other? It doesn’t seem like they would have enough initial potential energy to reach such a diffuse final state.

The final state doesn’t correspond to the final state of the Plummer model; it is much more diffuse. Also, it is more isotropic than any of the final states of other models.

The density near the center appears to depend linearly on radius:

The velocity distribution also seems to increase with radius, although it is not clear from the point-plot whether it varies as r or r².

B) The Plummer model appears to be much more stable than the homogeneous sphere from above. Using several different timescales (some very long), I generally got the same result:

The system appears to be in roughly the same state at the end that it started off with, except that perhaps a few more bodies have escaped from the central sphere (but I’m not sure about this). The central mass distribution is very steeply peaked towards the center (left graph below):

The velocity distribution shows a maximum velocity and then falls off. This is what we would expect for a spherical distribution. The velocity should increase with radius out to the extent of the sphere, and then once outside the sphere, fall off as 1/r². The velocity increases not linearly within the sphere due to the fact that the mass distribution is concentrated in the center.

C) The expdisk model is not stable but instead evolves into two separate clusters (left) while the premade galaxy seems to shrink down a bit, and assume a non-zero total velocity:

The corresponding velocity curves for the two systems are quite interesting. On the left we have the curve for the expdisk model. Since the final state is two separated clusters, we have a velocity distribution that has a flatter peak. The velocities for the premade galaxy are as we would expect: the velocity increases linearly within the cluster; once outside it falls off as what looks like 1/r^1/2.