Lecture 6 Collisionless Boltzmann Equation continued



Physics 141/241

Winter 2018




Collisionless Boltzmann Equation summary

The phase space density of mass, or Distribution Function (DF) of the N-body system was defined in Lecture 5 by considering the quantity MATH

where MATH designates the mass in the infinitesimal phase-space volume $d^{3}rd^{3}v$ around MATH at time $t$. The DF MATH satisfies the collisionless Boltzmann Equation (CBE),MATH

The collisionless Boltzmann equation describes the evolution of the distribution function MATH and it serves as the fundamental equation of collisionless N-body dynamics. In components it is given byMATH

where the gravitational field MATH is determined self-consistently by Poisson's equation,MATH

Eqs. (3,4) may be viewed as a pair of coupled PDEs which together completely describe the evolution of a galaxy.

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 solution of the CBE:

MATH

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 $f$ is a steady-state solution of CBE. Then $f$ is an integral, so that first part of theorem is true. Conversely, if $I_{1}$ to $I_{n}$ are $n$ integrals, then

MATH

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