# Lecture 4 Collisionless Boltzmann Equation

## Collisionless dynamics

A typical galaxy has stars but is only crossing times old, so the cumulative effects of encounters between stars are not significant. This justifies the next step, which is to idealize a galaxy as a continuous mass distribution. In this limit, each star moves in the smooth gravitational field of the galaxy. Thus instead of thinking about motion in a phase space of dimensions, we can think about motion in a phase space of just dimensions. This is a very important simplification!

## Distribution function

Rather than keeping track of individual stars, a galaxy may be described by the one-body distribution function; let

be the mass of stars in the phase-space volume at (,) and time . This provides a complete description if stars are uncorrelated, as assumed above.

## Collisionless Boltzmann equation

The motion of matter in phase space is governed by the phase space flow,

How does this affect the total amount of mass in the phase space volume ? The rate of change of the mass is just the inflow minus the outflow; that is, the flow obeys a continuity equation in 6 dimensions:

where the derivatives with respect to , and are understood to be partial derivatives. Using the expression for the phase-flow yields the collisionless Boltzmann equation:

The collisionless Boltzmann equation or CBE describes the evolution of the distribution function . It serves as the fundamental equation of galactic dynamics.

## Fluid continuity equation

(See explanation with added class note here)

## Gravity

The gravitational field is given self-consistently by Poisson's equation,

Eqs. (4,5) may be viewed as a pair of coupled PDEs which together completely describe the evolution of a galaxy.

## Conservation of phase space density

Let be the orbit of a star. What is the rate of change of along the star's orbit? The answer is zero,

where the first equality is just the definition of the convective derivative in phase-space, the second equality follows on substituting the phase-flow (Eq. 2), and the last equality follows from the CBE (Eq. 4). Thus, phase-space density is conserved along every orbit.

This fundamental and completely general result shows that the CBE has a much greater level of symmetry than the N-body equations of motion; whereas the latter conserves a fairly small set of parameters, the CBE conserves f(r,v,t) along an infinite number of stellar orbits. We can take advantage of this infinite array of conservation laws to obtain some important results even when we can't explicitly solve the CBE.

## Jeans Theorem

If the function is an integral which is conserved along any orbit: we can use the canonical equations to show that is a steady state soultion of the CBE:

Theorem: Any steady-state solution of the CBE depends on the phase-space coordinates only through integrals of motion in the galactic potential, and any function of the integrals yields a steady-state solution of the CBE.

Proof: Suppose is a steady-state solution of CBE. Then is an integral, so that first part of theorem is true. Conversely, if to are integrals, then

so that is an integral and a steady state solution of CBE.