Spherical Jeans equation

For a steady nonrotating spherical system,It relates radial pressure, velocity anisotropy, and the gravitational field.

For an isotropic self-gravitating collisionless stellar system with $k<3$ and no central point mass, $M(r)=4\pi A{r}^{3-k}/(3-k)$. Integrating the Spherical Jeans equation to an outer radius $b$ gives the displayed pressure, with $P_b=\rho (b){\sigma }_r^2(b)$. For $k<1$ the inner pressure is set by outer radii and ${\sigma }_r^2\sim C{r}^k$, with $C$ depending on the boundary. An unbounded pure power law has divergent pressure in this case. For $1<k<3$, inner radii dominate and ${\sigma }_r^2\sim 2\pi GA{r}^{2-k}/[(3-k)(k-1)]$. At $k=1$ a logarithm appears. For $k\ge 3$ the enclosed central mass diverges unless the density law is modified at small radii.

For a spherical nonrotating tracer with luminosity density $\lambda$, radial velocity dispersion ${\sigma }_r$, enclosed luminosity $L(r)$, and constant mass-to-light ratio ${\rm Y}$, let $q=\lambda {\sigma }_r^2$ and $\gamma =G{\rm Y}$. DefineThe Spherical Jeans equation and the projected second stellar velocity moment give $p=2A+R{A}^{\prime }$. To differentiate without a singular lower-end contribution, write $A(R)={\int }_0^{\infty }q(\sqrt{R^2+z^2})dz$. If $R^{2}A$ vanishes at both endpoints, ${\int }_0^{\infty }Rp(R)dR=0$. A finite outer aperture instead retains its boundary stress term.

For a scale-free tracer $\nu \propto r^{-k}$ in a point-mass gravitational potential, with $k>1$ and constant velocity-anisotropy parameter, the Spherical Jeans equation gives ${\sigma }_r^2=G{M}_{\bullet }/[(k+1-2\beta )r]$ when the outer pressure term vanishes. Project the tracer density and its second stellar velocity moment along the line of sight. The substitution $r=R\sec \chi$ reduces both integrals to powers of $\cos \chi$ and hence the beta function, yielding the displayed expression. The ratio of the two numerator angular integrals is $k/(k+1)$. This makes the mass-anisotropy ambiguity explicit. At $k=2$ the velocity-anisotropy parameter dependence cancels and ${\sigma }_{\mathrm{los}}^2=2G{M}_{\bullet }/(3\pi R)$; that special slope does not remove the degeneracy in general. Physical use also requires a nonnegative tracer distribution and control of outer departures from the power law.

For an isolated spherical stellar system in equilibrium with constant mass-to-light ratio, the virial theorem gives $2K+W=0$. Spherical averaging gives $K=3\pi {\rm Y}\int R\mu {\sigma }_p^{2}dR$, while $W=-4\pi G{{\rm Y}}^2\int r\lambda Ldr$. Dividing yields the displayed estimator, independent of the radial velocity-anisotropy parameter. It requires the full aperture and negligible boundary stress. A radially varying mass-to-light ratio or an independent dark matter halo prevents interpreting this expression as a general reconstruction of the mass profile.

A steady galactic distribution function depending only on relative energy $E=\psi -v^2/2$ is isotropic and has zero mean velocity. If its one-dimensional velocity dispersion is constant, the Spherical Jeans equation reduces to ${\sigma }_0^2{\rho }^{\prime }=\rho {\psi }^{\prime }$, yielding the displayed density. This is a tracer relation in a specified gravitational potential, not a requirement that the tracer generate that potential. With no escape-energy cutoff its local velocity distribution is Maxwell-Boltzmann velocity distribution.

Without an escape-energy cutoff, a galactic distribution function proportional to $\exp [(\Psi -v^2/2)/{\sigma }^2]$ gives $\rho \propto e^{\Psi /{\sigma }^2}$ and a constant one-dimensional velocity dispersion. Since $\Psi ={\Phi }_0-\Phi$, differentiation gives the displayed hydrostatic relation. An ideal gas at constant temperature has pressure $P=\rho k_{B}T/m_0$, so its hydrostatic equilibrium equation is identical at the displayed dispersion-temperature identification. Coupling either relation to the same Poisson equation for Newtonian gravity, with matched central density and boundary data, therefore gives the same density structure. Truncating the stellar velocities at escape changes this equivalence.

For an isothermal collisionless tracer without streaming or an escape cutoff, integrate its three-dimensional Maxwell-Boltzmann velocity distribution over the two transverse velocities. The remaining marginal is the displayed normal distribution. Its width and mean do not depend on position. Integrating along the line of sight therefore only multiplies the marginal by the projected surface density; the normalized observed velocity distribution remains the same Gaussian at every projected radius.

For tracer density $\nu \propto r^{-\alpha }$, circular speed $V_c\propto r^{\gamma }$, and constant anisotropy $\beta$, the boundary-free scale-free solution iswhich requires a positive denominator.

The mass--anisotropy--density degeneracy is the ability of different gravitational mass profiles, tracer-density profiles, and orbital-anisotropy profiles to predict similar projected tracer kinematics through the Jeans equation.