The phase space density of mass, or Distribution Function (DF) of the N-body
system was defined in Lecture 6 by considering the quantity
where
designates the mass in the infinitesimal phase-space volume
around
at time
.
The DF
satisfies the collisionless Boltzmann Equation
(CBE),
The collisionless Boltzmann equation describes the evolution of the
distribution function
and it serves as the fundamental equation of collisionless N-body dynamics. In
components it is given
by
where the gravitational field
is determined self-consistently by Poisson's
equation,
Eqs. (3,4) may be viewed as a pair of coupled PDEs which together completely describe the evolution of a galaxy.
Moments of the CBE contain important physics about time averages of the
dynamical motion. Consider first the
velocity moment of the CBE:
The following relation can be used for the mass density
and the relation
is valid, if
for asymptotically large
.
We introduce the average velocity,
and Eq. (5) becomes the continuity equation,
Consider now first moments in the velocity components:
We use the identity
Define
which gives the Jeans equation for the first velocity moments:
Subtract
times Eq.(9) from Eq.(13) :
and define
which describes the non-streaming motion locally. Eq.(14) becomes
where
is defined as the stress tensor. Eqs.(9,13,15) are known as the
Jeans equations.
Multiply Eq.(13) by
and integrate over the spatial variables:
We introduce the potential energy tensor
:
symmetric in the
indices. The total gravitational potential energy is given by
The following definition of the kinetic energy tensor
will be used:
By averaging
and
in Eq.(16) and using the symmetry properties of the tensors, we get
The moment of inertia tensor is defined as
Using the continuity equation, we find
Combining Eqs.(20,22b) we obtain the tensor virial theorem:
Since
in steady state,
,
we get the scalar virial theorem:
where M is the total mass of the system,
For the total energy, we find
and
if stars are at rest at infinity;
binding energy.
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.