Make a simple (numerical)model of a spherical galaxy in which the dark matter particles moving in the (Newtonian) gravitational field they create behave like collisionless plasma particles moving in an electromagnetic field. Ignore the baryons.

(a) Adopt the fluid approximation, treat the pressure P as isotropic and equal to Kρ^4/3, and use the equation of hydrostatic equilibrium to solve for the mass and radius. Comment on the answer you get, and make a suitable approximation to define an effective mass and radius.

(b) Consider a line passing through the galaxy with impact parameter relative to the center given by b. Solve for the dark matter particles’ rms velocity along this line as a function of b.

(c) It is observed that the central densities of galaxies scale as the inverse square of the rms velocity for b = 0. How does the mass scale with the velocity?

(d) Solve for the distribution function of the dark matter particles assuming that it is just a function of the energy.