Editing Stellar dynamics (section)

=== A worked example for neutrinos in galaxies ===

For example, the phase space distribution function of non-relativistic neutrinos of mass m anywhere will not exceed the maximum value set by
<math display="block"> f(\mathbf{x},\mathbf{v},t) = {dN \over dx^3 dv^3} \le {6 \over (2\pi \hbar/m)^3},  ~~~  </math>
where the Fermi-Dirac statistics says there are at most 6 flavours of neutrinos within a volume <math> dx^3 </math> and a velocity volume <math display="block"> dv^3  = (dp/m)^3 = [(2\pi\hbar/dx)/m]^3,</math>.

Let's approximate the distribution is at maximum, i.e., 
<math display="block"> f(x, y, z, V_x, V_y, V_z) = {6 \over (2\pi \hbar/m)^3} q^{\alpha \over 2}, ~~ 0 \le q(E)={\Phi_{\max}- E \over V_0^2/2} \le 1, </math>
where <math> 0 \ge \Phi_{\max} \ge E= \Phi(x,y,z) + {V_x^2 + V_y^2 + V_z^2 \over 2} \ge \Phi_{\min} \equiv \Phi_{\max}- {V_0^2 \over 2} </math> such that <math> E_{\min}, E_{\max} </math>, respectively, is the potential energy of at the centre or the edge of the gravitational bound system.  The corresponding neutrino mass density, assume spherical, would be
<math display="block">\rho(r) = n(x,y,z) m = \int dV_x \int dV_y \int dV_z ~m~ f(x,y,z,V_x,V_y,V_z), </math>
which reduces to
<math display="block">\rho(r) = { C  (\Phi_{\max}-\Phi(r))^{3+\alpha \over 2} \over (\Phi_{\max}-\Phi_{\min} )^{\alpha \over 2} }, ~~~ C={6 m \pi 2^{5/2} B\left(1+{\alpha \over 2}, {3 \over 2}\right) \over (2\pi \hbar/m)^3} </math>

Take the simple case <math> \alpha \to 0 </math>, and estimate the density at the centre <math> r=0 </math> with an escape speed <math> V_0 </math>, we have
<math display="block"> \rho(r) \le \rho(0) \rightarrow { m^4 V_0^3 \over \pi^2 \hbar^3} \approx m_\mathrm{eV}^4 V_{200}^3 \times \text{[Cosmic Critical Density]}.</math>

Clearly eV-scale neutrinos with <math> m_{eV} \sim 0.1-1 </math> is too light to make up the 100–10000 over-density in galaxies with escape velocity <math> V_{200} \equiv V/(\mathrm{200km/s}) \sim 0.1-3.4 </math>, while 
neutrinos in clusters with <math> V \sim \mathrm{2000 km/s} </math> could make up <math> 100-1000 </math> times cosmic background density.

By the way the freeze-out cosmic neutrinos in your room have a non-thermal random momentum <math display="inline"> \sim {(\mathrm{2.7 K}) k \over c} \sim (1~\mathrm{eV}/c^2) (\mathrm{70 km/s}) </math>, and do not follow a Maxwell distribution, and are not in thermal equilibrium with the air molecules because of the extremely low cross-section of neutrino-baryon interactions.