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The black-hole sphere of influence is a characteristic radius where the central black hole appreciably changes surrounding stellar dynamics. Equating its Newtonian gravitational potential scale $G{M}_{\bullet }/r$ to the ambient squared velocity dispersion gives $r_{\inf }\simeq G{M}_{\bullet }/{\sigma }^2$. Dynamical mass measurements depend on angular resolution $r_{\inf }/D$, tracer brightness, and degeneracies in the orbital distribution, not solely on black hole mass.

The isotropic three-dimensional Cauchy velocity distribution $f_0={\rho }_{0}b/[{\pi }^2(v^2+b^2{)}^2]$, $b>0$, has density ${\rho }_0$ and one-dimensional marginal $F(u)={\rho }_{0}b/[\pi (u^2+b^2)]$. Its second velocity moment diverges, so $b$ is a scale rather than a finite velocity dispersion. At zero phase speed, $\int F^{\prime }(u)/udu=-{\rho }_0/b^2$, giving $k_J=\sqrt{4\pi G{\rho }_0}/b$. In the upper-half frequency plane, the marginal response is ${\rho }_0/(\omega /k+ib{)}^2$; the growing branch has $\omega =i(\sqrt{4\pi G{\rho }_0}-kb)$ for $0<k<k_J$.

The virial theorem balances a relaxed galaxy cluster's orbital kinetic energy, of order $M{\sigma }_v^2$, against its Newtonian gravitational potential energy, of order $-G{M}^2/R$. Their ratio gives the displayed mass scale. The numerical coefficient depends on the density profile, radius definition and conversion from line-of-sight velocity dispersion to three-dimensional orbital motion. Membership contamination, anisotropy and nonequilibrium dynamics can bias the inference.

An instantaneous kinetic suppression estimate replaces sound speed by the velocity dispersion of cold dark matter. Nonrelativistic particles in thermal contact with photons have $\sigma \propto a^{-1/2}$; after kinetic decoupling, their momenta redshift and $\sigma \propto a^{-1}$. A background dynamical estimate using total density is then constant during radiation domination and decreases as $a^{-1/2}$ during matter domination. Using only the species' own matter density instead makes the self-gravitating collisionless Jeans length decrease as $a^{-1/2}$ throughout the decoupled nonrelativistic era. Neither instantaneous estimate is the accumulated distance traveled by collisionless free streaming.

The vertical Jeans equation determines $K_z={\partial }_z\Phi$ from a tracer's density and velocity dispersion, with a correction for the tilted velocity ellipsoid. Integrating the axisymmetric Poisson equation for Newtonian gravity givesfor $z>0$ and reflection symmetry. Subtracting independently estimated stellar and gas surface densities constrains any additional gravitating component; a dark matter halo also contributes, so an excess is not automatically a dark disk.

For a stationary axisymmetric tracer with isotropic velocity dispersion, vanishing tilted stress, density $\nu \propto e^{-R/R_d}$, and ${\sigma }^2\propto e^{-R/R_{\sigma }}$, the radial Jeans equation gives the displayed expression. More generally its right side is $-{\sigma }^{2}d\ln (\nu {\sigma }^2)/d\ln R$. It requires a nonnegative streaming-speed square; the density and dispersion laws must describe the same tracer and the chosen height or consistent vertical average.

The Faber-Jackson relation is an empirical correlation between luminosity and stellar velocity dispersion for elliptical galaxies, approximately $L\propto {\sigma }^4$ in its classical form. Slope, scatter, sample and band matter. A heuristic combines $M\propto {\sigma }^{2}R$, common mass-to-light ratio, and common characteristic surface brightness $L/R^2$. The luminosity-dispersion correlation alone does not imply those auxiliary assumptions, and extrapolation to a dwarf spheroidal galaxy is not guaranteed.

For a pressure-supported scale-free tracer $\rho \propto (R^2+z^2/q^2{)}^{-\gamma /2}$ in a spherical Newtonian gravitational potential, the tensor virial theorem constrains integrated directional velocity dispersions. Modest oblate flattening and a decreasing radial mass density require greater in-plane random support per direction. The displayed expansion assumes finite global virial integrals; the corresponding ratio of force moments also has a cutoff-independent angular interpretation.

The fundamental plane relates effective radius, stellar velocity dispersion, and mean surface brightness of elliptical galaxies. Homologous virial equilibrium with constant mass-to-light ratio predicts $R_e\propto {\sigma }^2/I_e$; empirical exponents differ. The two-variable Faber-Jackson relation is a projection, not the entire plane.

Galactic disk flaring is an outward increase of vertical thickness. A constant vertical velocity dispersion in an approximately thin-sheet stellar Newtonian gravitational field gives $h\propto 1/{\Sigma }_{\ast }$. An exponential galactic disk then predicts $h\propto e^{R/R_d}$ for a gas tracer. Inside a thick stellar layer the potential is instead approximately harmonic, and the tracer width is ${\sigma }_z/{\nu }_z$ with vertical epicyclic frequency ${\nu }_z$.

Galactic disk heating increases the random velocities of stars through encounters with molecular clouds, nonaxisymmetric gravitational disturbances, and other perturbations. Older stellar populations usually have larger velocity dispersions and greater vertical thickness, although birth conditions and migration also affect the relation.

A galaxy of characteristic radius $R$ and velocity dispersion $\sigma$ has crossing time $R/\sigma$. Gravitational balance gives $GM/R\sim {\sigma }^2$, hence $t_{\mathrm{dyn}}\sim GM/{\sigma }^3$. A numerical coefficient depends on the radius definition, density profile, and whether one uses a crossing, orbital, or free-fall time; these are not identical clocks.

For identical point masses of total mass $M$, let $W=-G{M}^2/R_g$ exclude self-interactions. The virial theorem gives $⟨v_{\mathrm{eq}}^2⟩=GM/R_g$. Since the sum of individual potential energies counts each pair twice, the mass-weighted mean frozen-potential escape velocity squared is $-4W/M$. A dissipation-free collapse from zero total energy instead has kinetic energy $-W$, hence mean squared speed $-2W/M$. These are three-dimensional speeds relative to the centre of mass; a coherent collapse speed is not a local random velocity dispersion.

For a non-streaming spherical tracer population, the line of sight makes angle $\chi$ with the Galactocentric radial direction. The projected velocity dispersion iswhere ${\sigma }_t^2$ is one tangential variance and $\beta =1-{\sigma }_t^2/{\sigma }_r^2$ is the velocity-anisotropy parameter. At a fixed radius, averaging with the observational angular weights gives ${\sigma }_{\odot }={\sigma }_r\sqrt{1-\beta A}$, where $A=⟨{\sin }^2\chi ⟩$.

For an axisymmetric random velocity dispersion tensor with ${\Pi }_{11}={\Pi }_{22}=M{\sigma }_0^2$ and ${\Pi }_{33}=M{\sigma }_0^2(1-\delta )$, contracting with the line-of-sight unit vector $(\sin i,0,\cos i)$ gives the displayed mass-weighted velocity dispersion. A major-axis rotation amplitude projects as $V_r=V\sin i$. By contrast, an azimuthally averaged line-of-sight streaming root-mean-square is $V\sin i/\sqrt{2}$ when $V^2$ is the full in-plane streaming mean square. Aperture weights and the definition of the measured rotation statistic must match the intrinsic statistic.

In a spherical galactic distribution function with $F=L^{-1}{f}_E(E)$, write $L=r{v}_T$ and $d^{3}v=d{v}_r{v}_{T}d{v}_{T}d\psi$. The velocity Jacobian cancels the apparent singularity at $v_T=0$. Integration of Gaussian energy components then reduces to one full-line and one half-line Gaussian integral. The angular average makes each tangential velocity dispersion half the radial one, so the velocity-anisotropy parameter is $1/2$.

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 a spherical, nonstreaming stellar population with isotropic velocity dispersion, each component of the random pressure tensor equals $p=n⟨v_r^2⟩$. The observed second stellar velocity moment satisfies $N(R){\sigma }_{\mathrm{los}}^2(R)=2{\int }_R^{\infty }p(r)r/\sqrt{r^2-R^2}dr$. Composing this Abel transform with itself gives the displayed inverse under sufficient decay and regularity. The Spherical Jeans equation then gives ${\psi }^{\prime }=p^{\prime }/n$, with gravitational acceleration $\nabla \psi$. The Poisson equation for Newtonian gravity recovers total gravitating mass density, not just the tracer density. Without velocity isotropy the same projected moment also contains anisotropic stresses.