Augmented stellar density ($\rho (r,\psi )$)

An augmented density $\rho (r,\psi )$ treats radius and relative potential as independent variables and reduces to the physical density on $\psi =\psi (r)$. Its extension away from that curve contains additional orbital-anisotropy information. Different extensions can give the same physical density; physical realizability additionally requires a nonnegative galactic distribution function.

With radial pressure zero at $\psi =0$, an augmented density supplies $\rho ⟨v_r^2⟩={\int }_0^{\psi }\rho (r,q)dq$ and $\rho ⟨v_{\theta }^2⟩=(2r{)}^{-1}{\int }_0^{\psi }{\partial }_r[r^2\rho (r,q)]dq$. The derivative holds $q$ fixed. Their chain-rule combination satisfies the Spherical Jeans equation, but the physical density alone does not choose an augmented extension.