Derivation of the day: scale height

Background
The scale height, H, refers to the height above the plane of a disk of particles surrounding a larger body.  Due to eccentricities or inclinations of orbits from pressure within the system, these disks are not completely flat and each particle has an associated "random velocity" defined as u = iv_orb or u = ev_orb.  We find this height by balancing forces within the disk.

Units
u = random velocity
Ω = angular frequency
G = Newtonian gravitation constant
a = radius of orbit

Derivation
Let us consider a particle with an inclination given by i = arcsin(H/a).  Starting with the ideal gas law,

P = nkT ~ nmv² = mv²/H(Area)

where n is a volume density. Force = (Pressure)(Area) so we can write that the force pushing the particle up is approximately:

mv²/H

We also consider the downward force due to the gravitation of the star:

F_g = (GMm/a²)sin(i) = (GMm/a²)(H/a)

From Kepler's Law, we can substitute GM = Ω²a³ so

mv²/H = Ω²mH

Rearranging, we find an expression for H:

H = u/Ω