The phase space density of mass, or Distribution Function (DF) of the N-body system was defined in Lecture 5 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
which gives the Jeans equation for the first velocity moments:
Subtract times Eq.(9) from Eq.(13) :
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:
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.