This is mostly a blackboard lecture and derivation today.

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Basic Physical Points (from Binney and Tremaine 1987:)

The structure of an isothermal self-gravitating sphere of gas is identical to a collionless systems of stars whose phase-space density can be described by a Maxwell-Boltzmann distribution.

We also make use of Jeans Theorem:

Any steady-state solution of the collisionless Boltzmann equation depends on the phase-space density coordinates only through integrals of motion in the galactic potential. Furthermore, any function of the integrals yields a steady-state solution of the collisionless Boltzmann equation.

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