= Gaussian angular-momentum suppression in a stellar distribution
{title2=$f\propto e^{E/\sigma^2-L^2/(2r_a^2\sigma^2)}$}
In a spherical <Newtonian gravitational potential>, use binding energy $E=\psi-v^2/2$ and <specific angular momentum> $L$. The displayed stationary <galactic distribution function> has locally Gaussian radial and transverse velocities: $\sigma_r^2=\sigma^2$, $\sigma_\theta^2=\sigma_\phi^2=\sigma^2/(1+r^2/r_a^2)$. Its <mass density> is $\rho=\rho_c e^{[\psi(r)-\psi(0)]/\sigma^2}/(1+r^2/r_a^2)$. The <velocity-anisotropy parameter> is $\beta_{\mathrm{an}}=r^2/(r^2+r_a^2)$, so the core is nearly isotropic and the halo nearly radial. These exact Gaussian integrals use the untruncated distribution over all velocities; imposing an escape-energy cutoff changes them.
Back to article page
