Editing Stellar dynamics (section)

=== Convention and notation in case of a thermal distribution ===
In most of stellar dynamics literature, it is convenient to adopt the convention that the particle mass is unity in solar mass unit <math> M_\odot</math>, hence a particle's momentum and velocity are identical, i.e.,
<math display="block"> \mathbf{p} = m \mathbf{v} = \mathbf{v}, ~ m=1, ~ N_\text{total} = M_\text{total},</math>

<math display="block"> {dM \over dx^3 dv^3} = f(\mathbf{x},\mathbf{v},t) =  f(\mathbf{x},\mathbf{p},t) \equiv {dN \over dx^3 dp^3} </math>

For example, the thermal velocity distribution of air molecules (of typically 15 times the proton mass per molecule) in a room of constant temperature <math> T_0 \sim \mathrm{300K} </math> would have a [[Maxwell distribution]]
<math display="block"> f^\text{Max}(x,y,z,m V_x,m V_y,m V_z) = {1 \over (2\pi \hbar)^3} {1 \over \exp\left({E(x,y,z,p_x,p_y,p_z) - \mu \over kT_0}\right) + 1 } </math>
<math display="block"> f^\text{Max} \sim {1 \over (2\pi \hbar/m)^3} e^{\mu \over kT_0 } e^ {-E \over m\sigma_1^2}, </math>

where the energy per unit mass <math display="block"> E/m = \Phi(x,y,z) + (V_x^2 + V_y^2 + V_z^2)/2, </math> where <math>\Phi(x,y,z) \equiv g_0 z = 0</math>

and <math display="inline"> \sigma_1 =\sqrt{k T_0/m} \sim \mathrm{0.3km/s}</math> is the width of the velocity Maxwell distribution, identical in each direction and everywhere in the room, and the normalisation constant <math> e^{\mu \over kT_0} </math> (assume the chemical potential <math display="inline">\mu \sim  (m\sigma_1^2) \ln\left[n_0 \left({\sqrt{2\pi}\hbar \over m \sigma_1}\right)^3\right] \ll 0 </math> such that the Fermi-Dirac distribution reduces to a Maxwell velocity distribution) is fixed by the constant gas number density <math>n_0 = n(x,y,0) </math> at the floor level, where 
<math display="block">
n(x,y,0) = \!\! \int_{-\infty}^\infty m dV_x \!\! \int_{-\infty}^\infty m dV_y \!\! \int_{-\infty}^\infty m dV_z f(x,y,0,mV_x,mV_y,mV_z) </math>
<math display="block"> n \approx {(2\pi)^{3/2} (m\sigma_1)^3 \over (2\pi \hbar)^3} e^{\mu \over m \sigma_1^2}.
</math>