Lecture 11 Jeans equations and Virial Theorem from Boltzmann Equation II.



Physics 141/241

Winter 2020




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 Equations

Moments of the CBE contain important physics about time averages of the dynamical motion. Consider first the $0^{th}$ velocity moment of the CBE:

MATH

The following relation can be used for the mass density $\rho $ MATH

and the relation

MATH

is valid, if MATH for asymptotically large MATH. We introduce the average velocity,

MATH

and Eq. (5) becomes the continuity equation,

MATH

Consider now first moments in the velocity components:

MATH

We use the identity

MATH

Define

MATH

which gives the Jeans equation for the first velocity moments:

MATH

Subtract $\bar{v}_{j}$ times Eq.(9) from Eq.(13) :

MATH

and define

MATH

which describes the non-streaming motion locally. Eq.(14) becomes

MATH

where MATH is defined as the stress tensor. Eqs.(9,13,15) are known as the Jeans equations.

Virial Theorems

Multiply Eq.(13) by $x_{k}$ and integrate over the spatial variables:

MATH

We introduce the potential energy tensor $W_{jk}$:

MATH

symmetric in the $jk$ indices. The total gravitational potential energy is given by

MATH

The following definition of the kinetic energy tensor $K$ will be used:

MATH

By averaging $(k,j)$ and $(j,k)$ in Eq.(16) and using the symmetry properties of the tensors, we get

MATH

The moment of inertia tensor is defined as

MATH

Using the continuity equation, we find

MATH

Combining Eqs.(20,22b) we obtain the tensor virial theorem:

MATH

Since MATH

in steady state, $\ddot{I}=0$, we get the scalar virial theorem:

MATH

where M is the total mass of the system,

MATH

For the total energy, we find

MATH

and $K+W=E=0$ if stars are at rest at infinity; $E_{b}=-E$ binding energy.

Jeans Theorem

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