Galaxy dynamics

Galaxy dynamics studies the gravitational motions and evolution of stars, gas, dark matter, and satellite systems in galaxies.

Stellar asymmetric drift is the lag of a pressure-supported stellar population's mean rotation behind the local circular speed. Neglecting the tilted velocity ellipsoid, radial Jeans equation balance givesA declining density and larger random velocities generally increase the lag; gravitational balance is supplied partly by random-motion stresses.

Reflection symmetry makes $\overline{v_R{v}_z}$ odd in height, so its value at the midplane vanishes while its derivative need not vanish. The radial Jeans equation includes this derivative in the stellar asymmetric drift. For a velocity ellipsoid locally aligned with spherical coordinates, $\overline{v_R{v}_z}\simeq ({\sigma }_R^2-{\sigma }_z^2)z/R$, and its midplane contribution is $-({\sigma }_R^2-{\sigma }_z^2)$. Dropping a mixed moment at the symmetry plane does not justify dropping its derivative.

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.

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.

Violent relaxation is rapid redistribution of the orbital energies of collisionless matter in a time-dependent collective gravitational potential. It can occur during collapse or a major galaxy merger. It differs from collisional two-body relaxation and from phase mixing in a stationary potential; it does not imply perfect final thermal equilibrium.

The Oort constants describe local differential galactic rotation: $A=-R{\Omega }^{\prime }/2$ and $B=-\Omega -R{\Omega }^{\prime }/2$. Thus $\Omega =A-B$, the radial epicyclic frequency obeys ${\kappa }^2=-4\Omega B$, and the epicyclic ellipse has axis ratio $\sqrt{-B/(A-B)}$.

This spherical gravitational potential, with $a,v_0>0$, has density $\rho =v_0^2(2a+r)/(4\pi Gr(a+r{)}^2)$ and circular speed $v_c=v_0\sqrt{r/(a+r)}$. It combines a central $1/r$ density cusp with an outer flat galaxy rotation curve, but its total mass is infinite. It differs from a logarithmic potential with a constant-density core.

The differential Newtonian gravitational field of a galaxy across a smaller bound system. Its tidal tensor can slowly change the angular momentum of distant comets, changing their periapses and their access to planetary encounters.

In a rapid encounter, the impulse approximation holds stellar positions fixed while integrating an external acceleration over time. A stellar velocity kick is $\Delta \mathbf{v}=\int {\mathbf{a}}_{\mathrm{tidal}}(t)dt$. If the encounter time $p/v$ is much shorter than the orbital time ${\Omega }^{-1}$, this gives reliable leading tidal heating for a weak, distant encounter. Slow perturbations require adiabatic shielding of tidal encounters instead.

Integrate the transverse force along a complete straight path: ${\int }_{-\infty }^{\infty }(1+s^2{)}^{-3/2}ds=2$. This gives the transverse kick. Conservation of relative speed gives longitudinal speed reduction $\Delta v_{\perp }^2/(2V)$ at second order. As an independent check, exact weak hyperbolic scattering has angle $2\arctan \eta$, where $\eta =GM/(b{V}^2)$, and reduction $V(1-\cos (2\arctan \eta ))=2V{\eta }^2/(1+{\eta }^2)$. The small-$\eta$ coefficient agrees. Keeping only one half of the encounter in the transverse kick makes this quadratic reduction four times too small.

For a straight shearing-sheet encounter, $V=S|x|$, and the outer and inner longitudinal changes have opposite signs. The complete-flyby gravitational impulse gives $\chi =2$. Some simplified prescribed kick models use a different coefficient; carrying it explicitly makes the corresponding one-sided impulse torque on a disk and migration rates unambiguous. This coefficient does not assert an exact solution of the full rotating Hill encounter.

Multiply encounter mass flux $\Sigma (x)Sxdx$ by specific angular-momentum gain $r_0\Delta v_y$. The net disk torque replaces $\Sigma (x)$ by $\Sigma (x)-\Sigma (-x)$, so a symmetric mass density gives zero net torque. For constant surface density, Keplerian shear and the full-flyby value $\chi =2$, the one-sided coefficient is $8/27$ in $q^2\Sigma r_0^4{\Omega }^2(r_0/H{)}^3$, matching the primary coplanar impulse normalization.

A local odd mass density slope $\Sigma (x)-\Sigma (-x)=2{\Sigma }_0\beta x/r_0$ changes the cancelling one-sided torques into this net torque. A local outer cutoff $L$ adds the factor $1-(H/L{)}^2$. Positive slope transfers net angular momentum from satellite to disk. An infinite linear mass density law is only a formal local extension, since it cannot remain nonnegative on both sides.

For a perturber following $(p,0,vt)$, the impulse approximation integrates its tidal tensor to $2G{m}_p\mathrm{diag}(1,-1,0)/(v{p}^2)$. A fixed target position therefore receives $\Delta \mathbf{v}=2G{m}_p(x,-y,0)/(v{p}^2)$. The longitudinal kick vanishes after integrating the symmetric trajectory.

If the initial stellar velocities are uncorrelated with the impulse approximation kicks, mean specific heating is $|\Delta \mathbf{v}{|}^2/2$. Spherical position averaging gives total heating $4G^2{m}_p^2{m}_g⟨r^2⟩/(3v^2{p}^4)$, where the mass-weighted mean square radius must be finite.

Adiabatic shielding suppresses the heating of an orbit when an external tidal tensor varies slowly compared with its orbital frequency. In a nonresonant slow encounter, adiabatic invariance of an orbital action lets the orbit adjust approximately reversibly, so a frozen-position impulse approximation overestimates the energy transfer. An orbital resonance can invalidate the simple slow-variation argument.

Chandrasekhar dynamical friction is the drag on a massive body produced by its gravitational wake in a background of lighter particles. Its magnitude is proportional to the body's mass and the background density.

For a slowly evolving circular orbit with tangential deceleration $D>0$, the specific angular momentum is $j=r{V}_c(r)$ and its torque is $\dot{j}=-rD$. Differentiating $j$ gives the displayed drift when $1+q\ne 0$. The positive local decay time $r/(2|\dot{r}|)$ is $(1+q)V_c/(2D)$. In a uniform-density core $q=1$, so it is $V_c/D$; for a flat galaxy rotation curve $q=0$, it is $V_c/(2D)$. Simply setting ${\dot{V}}_c=-D$ neglects the radial contribution to angular momentum change.

For an isotropic, integrable background phase-space distribution function, rotational symmetry makes the integral radial in velocity space. Its divergence with respect to $\mathbf{V}$ is $4\pi f(V)$ by the Dirac delta identity for the divergence of the inverse-square vector field. Integrating over a velocity-space sphere and using the divergence theorem proves the displayed identity. Exterior velocity shells have zero contribution, just as in the spherical shell theorem. This explains why the leading-log Chandrasekhar dynamical friction integral contains only particles slower than the test body.

For a test mass $M$ and isotropic background particles of mass $m$, the hyperbolic two-body problem uses the relative deflection scale $b_{90}=G(M+m)/w^2$. The test body's longitudinal velocity change in one encounter is $-2mw/[(M+m)(1+b^2/b_{90}^2)]$. Integrating the encounter rate over impact parameter gives a logarithm; replacing its velocity-dependent value by a common Coulomb logarithm in stellar dynamics gives the displayed systematic drag. The approximation $M+m\simeq M$ requires $M\gg m$, not merely $M>m$. Velocity diffusion and exact cutoff dependence are distinct effects omitted by this leading-log drag model.

The tidal radius bounds material that remains gravitationally attached to a satellite in the differential gravitational field of its host.

For a low-mass satellite of mass $M_s$ on a circular orbit of radius $r_o$ in a spherical host,This local approximation requires $r_t\ll r_o$.

For a low-mass singular isothermal sphere satellite on a circular orbit of radius $D$ in a singular isothermal sphere host, ${\Omega }^2=2{\sigma }_h^2/D^2$ and ${\Phi }_h^{\prime \prime }=-2{\sigma }_h^2/D^2$. The radial differential effective acceleration is $({\Omega }^2-{\Phi }_h^{\prime \prime })r$. Balancing it with satellite gravity $2{\sigma }_d^2/r$ gives the displayed Jacobi tidal radius and bound mass $m_t=\sqrt{2}D{\sigma }_d^3/(G{\sigma }_h)$. Equal mean density gives a different order-one approximation, $r_t=D{\sigma }_d/{\sigma }_h$; eccentric orbits require an orbital-history prescription.

Tidal stripping removes satellite material outside the tidal radius. The resulting mass loss weakens dynamical friction and can greatly delay or prevent a merger.

A tidal tail is an elongated stream of material removed by a differential gravitational field. A satellite generally produces one leading and one trailing tail as stripped material enters nearby orbits with slightly different energy and angular momentum.

Dynamical-friction stalling by tidal stripping occurs when inward motion shrinks a satellite's tidal radius so rapidly that its remaining mass and frictional drag tend to zero before it reaches the host center.

Stellar dynamics studies the collective gravitational motion of stars in galaxies and star clusters.

A loss cone consists of low-angular momentum stellar trajectories whose next pericentre enters a central absorbing region. For capture radius $r_c$ in a black hole-dominated potential, the Newtonian near-parabolic estimate is $J_{\mathrm{lc}}^2\simeq 2G{M}_H{r}_c$. At radius $r$ and speed $v$, its angular half-width is $J_{\mathrm{lc}}/(rv)$. Two-body relaxation can refill this depleted part of phase space.

For a homologous black hole-dominated cluster, write $E=-bG{M}_{H}Nm/(2R)$, and let a captured star's specific orbital energy be ${ϵ}_{\mathrm{cap}}=\xi E/(Nm)$. Removing negative orbital energy raises the remaining cluster energy, while growth ${\dot{M}}_H=m\Gamma$ deepens its potential. The energy budget gives the displayed radius response. Preferentially removing strongly bound stars can drive expansion; representative-energy removal instead gives the adiabatic contraction due to black-hole growth. The sign is not fixed by the loss cone angle alone.

Here ${\mathcal{R}}_{\mathrm{lc}}=J_{\mathrm{lc}}^2/J_c^2$ is the orbital loss cone fraction, $P$ an orbital period, and $t_{\mathrm{rel}}$ a stellar relaxation time. When angular diffusion per orbit is small relative to the cone, a steady two-dimensional diffusion solution is logarithmic and gives the empty-cone flux. Rapid refilling maintains an approximately isotropic full cone and gives the second rate. Numerical total rates need a stellar density and orbital-energy distribution, not merely a star count.

A stellar orbit with sensitive dependence on its initial phase-space point, often diagnosed by a positive Lyapunov exponent. It need not explore the whole energy hypersurface: regular islands and disconnected chaotic regions can coexist. If an orbit is dense in a connected energy-shell region, every continuous integral of motion is constant on its closure; that stronger condition excludes further independent continuous constraints within that region.

A spherical stellar model with total mass $M$, scale radius $a$, inner density proportional to $r^{-2}$ and outer density proportional to $r^{-4}$. Integrating its mass density gives $M(r)=Mr/(r+a)$, hence Newtonian gravitational potential $\Phi =(GM/a)\log [r/(r+a)]$, normalized to zero at infinity. Its circular speed satisfies $v_c^2=GM/(r+a)$: it tends to a constant inside the scale radius and becomes Keplerian outside. The central density cusp has vanishing enclosed mass as $r\to 0$ and does not imply a central point mass.

A collisional self-gravitating stellar core can lose heat to its halo, contract and become hotter because it has negative heat capacity of a virialized gravitational system. The increased temperature contrast drives further outward heat flow and further contraction. The classical gravothermal catastrophe is the loss of a finite-density thermodynamic equilibrium at a finite total energy along the confined three-dimensional Newtonian isothermal sequence. It concerns slow two-body relaxation, not simply a collisionless instability. The dimensional form of the gravitational interaction and the thermodynamic ensemble matter: a two-dimensional logarithmic gravitational gas need not have the same finite-energy instability.

For a mean-field gas of parallel infinite rods with total mass per unit length $M=Nm$ confined to radius $R_e$, put $q=GM\beta /2$, $\beta =m/(kT)$ and $F=1-q+q{R}^2/R_e^2$. The isothermal solution is $\psi =\psi (R_e)-2\log F/\beta$ with mass density $\rho =M(1-q)/(\pi R_e^2{F}^2)$ and enclosed line mass $M(<R)=M(R^2/R_e^2)/F$. Direct differentiation verifies the cylindrical Poisson equation for Newtonian gravity. Regularity requires $q<1$, or $T>T_{\min }$; equality is a singular core-concentration limit, not a regular equilibrium.

With $\Theta =T/T_{\min }>1$, the two-dimensional self-gravitating rod equilibrium has the displayed energy per unit length for the potential reference $\psi (R_e)=-GM\log (V/V_0)$. Integration of $-\rho \psi /2$ gives its gravitational energy; its two-dimensional Maxwell-Boltzmann velocity distribution has kinetic energy $NkT$. At fixed area, differentiation gives $C=Nk[2+2\Theta \log (1-{\Theta }^{-1})+\Theta /(\Theta -1)]$. Expanding the logarithm shows $C/(Nk)=1+{\sum }_{j\ge 3}(j-2)/(j{\Theta }^{j-1})>1$. The energy increases continuously from $-\infty$ to $+\infty$, so there is no finite-energy endpoint or turning point of the regular sequence. This excludes the classical gravothermal catastrophe in this model, while leaving a distinct canonical collapse at or below its minimum temperature.

A spherical mass density $\rho$ gives projected surface density $\Sigma (R)={\int }_{-\infty }^{\infty }\rho (\sqrt{R^2+z^2})dz$. On the positive half of the line of sight, substitute $r=\sqrt{R^2+z^2}$ to obtain the displayed Abel transform. Spherical symmetry is essential: a general three-dimensional density cannot be reconstructed uniquely from one projection.

The strip density is the mass per unit projected Cartesian coordinate $x$: $S(x)={\int }_{-\infty }^{\infty }\Sigma (\sqrt{x^2+y^2})dy$. It is even in $x$. For $x\ge 0$, changing variables to projected radius gives $S(x)=2{\int }_x^{\infty }\Sigma (R)R/\sqrt{R^2-x^2}dR$. Alternatively integrate the three-dimensional mass density over the plane of coordinates $y,z$ to obtain the displayed expression. Consequently $S^{\prime }(x)=-2\pi x\rho (x)$ for $x>0$.

For an isolated spherical distribution, shell assembly gives gravitational potential energy $W=-{\int }_0^{\infty }GM(r)r^{-1}dM(r)$. Use $S^{\prime }=-2\pi r\rho$ and $M^{\prime }=-2r{S}^{\prime }$ to write $W=2G\int M{S}^{\prime }dr=-2G\int S{M}^{\prime }dr=4G\int rS{S}^{\prime }dr$. A second integration by parts gives the displayed negative quadratic functional. The boundary terms $MS$ and $r{S}^2$ must vanish; finite-mass regular profiles with finite binding energy satisfy the usual sufficient decay conditions.

Put $F(u)=\Sigma (\sqrt{u})$ and $g(v)=\rho (\sqrt{v})$. Then $F(u)={\int }_u^{\infty }g(v)/\sqrt{v-u}dv$. Composing this Abel transform with itself and exchanging convergent integrals gives ${\int }_t^{\infty }F(u)/\sqrt{u-t}du=\pi {\int }_t^{\infty }g(v)dv$, because the inner integral from $t$ to $v$ is $\pi$. Differentiate after writing $u=t+s$ to obtain $g(t)=-{\pi }^{-1}{\int }_t^{\infty }{F}^{\prime }(u)/\sqrt{u-t}du$. Substituting $t=r^2,u=R^2$ yields the displayed reconstruction. Decay and regularity sufficient for the exchanges and derivatives are required.

For $A,b>0$ and $\alpha >1/2$, the projected profile $F(R)=A(R^2+b^2{)}^{-\alpha }$ has the displayed spherical Abel transform inverse. Indeed its composed transform is $A(r^2+b^2{)}^{1/2-\alpha }{I}_{\alpha }$, with $I_{\alpha }={\int }_1^{\infty }{s}^{-\alpha }(s-1{)}^{-1/2}ds=B(1/2,\alpha -1/2)$. Differentiate in $r$, divide by $-2\pi r$, and use the Gamma function recurrence to obtain the coefficient. The same kernel reconstructs tracer number density or isotropic random pressure from their respective projected profiles.

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.

The spherical Plummer model has $\psi =GM/\sqrt{a^2+r^2}$ and $\rho =3M{a}^2/[4\pi (a^2+r^2{)}^{5/2}]$. It is the $p=2$ hypervirial model. In unit scales its isotropic distribution is $24\sqrt{2}{E}^{7/2}/(7{\pi }^3)$ for positive relative energy.

The Hernquist model has relative potential $\psi =GM/(r+a)$ and density $\rho =Ma/[2\pi r(r+a{)}^3]$. It has enclosed mass $M{r}^2/(r+a{)}^2$, a central $r^{-1}$ cusp and an outer $r^{-4}$ tail. Since $r\rho =a{\psi }^3/(2\pi G^3{M}^2)$, the half-anisotropic distribution function is $f(E,L)=3a{E}^2/(4{\pi }^3{G}^3{M}^{2}L)$ for $0<E<GM/a$. Its moments are $⟨v_r^2⟩=\psi /4$ and each tangential moment $\psi /8$. Isotropic and other admissible anisotropic distributions can also support this same density; the gravitational potential does not select a unique velocity anisotropy.

An N-body simulation approximates a dynamical system by evolving $N$ interacting particles. In a collisionless galaxy calculation these particles sample a smooth mass distribution rather than represent individual physical stars. Their larger individual masses enhance artificial two-body relaxation; softening, force accuracy and convergence with particle number must be checked over the intended duration.

Gravitational softening regularizes a simulation particle's short-distance force, for example by replacing $-Gm/r$ by $-Gm/\sqrt{r^2+{ϵ}^2}$. It suppresses artificial close encounters and effectively raises the diffusion cutoff to order $\max (ϵ,Gm/u^2)$, but alters the resolved force below $ϵ$. Particle number, softening and duration must be chosen together; a point-particle relaxation estimate alone does not specify a simulation's accuracy.

Evaluating the potential of a razor-thin disk at height $ϵ$ gives a softened horizontal force. Its Fourier multiplier is $e^{-kϵ}$ times the razor-thin disk Poisson kernel. This suppresses short wavelengths while retaining the long-wave potential. A real finite-thickness profile generally has a different full reduction factor.

For a vertically separable perturbation with normalized height profile $h(z)$, the midplane gravity reduction factor is $\int h(z)e^{-k|z|}dz$. Its small-$k$ expansion is $1-k⟨|z|⟩+o(k)$ when this first moment is finite. Match it to $e^{-kϵ}$ to get the displayed choice. For $h=e^{-|z|/H}/(2H)$ the exact factor is $(1+kH{)}^{-1}$, demonstrating why no single exponential matches every wavelength.

The stellar crossing time is a characteristic time to traverse a system, $t_{\mathrm{dyn}}\sim R/v$. In a virialized self-gravitating system, $v^2\sim GM/R$, so $t_{\mathrm{dyn}}\sim \sqrt{R^3/(GM)}$. It is distinct from the much longer spherical stellar relaxation time. Radius versus diameter and characteristic-speed conventions change order-one factors.

A star cluster is a gravitationally associated population of stars formed in a compact region.

An open cluster is a relatively low-mass group of stars formed together, typically in a galactic disk. Two-body relaxation, stellar evaporation, and external tidal disturbances disperse many open clusters on much shorter times than the lifetimes of their low-mass member stars.

Two-body relaxation redistributes stellar energies and feeds stars above the escape threshold. The escaping population removes mass and generally nonnegative energy at infinity. Radius evolution and external tidal limits must be considered before inferring the change in cluster density.

For an isolated quasistatic cluster with near-zero escape energies, the bound negative energy is approximately fixed. The virial theorem and gravitational radius then imply $R_g\propto M^2$. A homologous mean density scales as $M^{-5}$ and increases as mass evaporates. Tidal limitation or finite positive escape energies modifies this simple model.

A globular cluster is an old, dense, approximately spheroidal star cluster whose long-term evolution is shaped by two-body relaxation and the tidal field of its host galaxy.

An oblate spheroid is the flattened case of a spheroid, with polar semiaxis no greater than its equatorial semiaxis.

An oblate spheroid is obtained by rotating an ellipse about its shorter axis. If its equatorial semiaxis is one and its polar semiaxis is $q\le 1$, then $q$ is its intrinsic axis ratio.

If the symmetry axis of an oblate spheroid makes angle $i$ with the line of sight, its projected axis ratio satisfiesThus a face-on spheroid appears round and an edge-on spheroid reveals its intrinsic axis ratio.

For an unoriented symmetry axis chosen uniformly on the sphere, $\cos i$ is uniform on $[0,1]$. At fixed intrinsic ratio $q$, the change-of-variables formula for a probability density gives

The spherical isochrone model has potentialIts radial period depends only on orbital energy, while its radial action and frequencies are elementary functions of the integrals of motion.

For specific energy $E<0$ and specific angular momentum $L$,Consequently the Hamiltonian in actions is

The radial and azimuthal frequencies obeyThe ratio tends to one half in the harmonic core and to one in the Keplerian exterior.

The hypervirial family has relative potentialand densityIts name comes from satisfying the scalar virial relation pointwise.

In unit scales $I=\int \rho \psi d^{3}x=(p+1)\Gamma (1+1/p{)}^2/[p\Gamma (2+2/p)]$. The local virial relation of the hypervirial model gives global energies $T=I/4$ and $W=-I/2$. Restoring physical scales multiplies each integral by $G{M}^2/a$.

For $p>0$, the relative potential $GM/(a^p+r^p{)}^{1/p}$ generates density $(p+1)M{a}^p{r}^{p-2}/[4\pi (a^p+r^p{)}^{2+1/p}]$. In unit scales the separable augmented density is $(p+1)r^{p-2}{\psi }^{2p+1}/(4\pi )$. Its Jeans moments from an augmented density give constant anisotropy $\beta =1-p/2$.

The hypervirial model is generated self-consistently by a constant-anisotropy distribution function proportional toIts velocity-anisotropy parameter is $\beta =1-p/2$.

In units $G=M=a=1$, velocity integration of $F=C{L}^{p-2}{E}^{(3p+1)/2}$ gives $C=(p+1)\Gamma (2p+2)/[2^{(p+5)/2}{\pi }^{5/2}\Gamma (p/2)\Gamma ((3p+3)/2)]$. The beta function separates the radial-speed and angular integrals. This constant is positive for every $p>0$.

For the hypervirial model, $⟨v^2⟩=\Psi /2=-\Phi /2$. The local kinetic- and potential-energy densitiestherefore obey $2K+W=0$ at every radius, a stronger statement than the integrated virial theorem.

A galactic distribution function gives stellar mass or number per phase space volume. Integrating over velocity gives spatial density, while velocity moments and projection along the line of sight predict kinematic observables.

A lowered Maxwellian velocity distribution subtracts the value of a Maxwell-Boltzmann velocity distribution at the escape velocity and sets the result to zero beyond escape. This gives a nonnegative phase-space distribution function vanishing continuously at the boundary. If $A=n_0/(2\pi {\sigma }^2{)}^{3/2}$ and $x_e=v_e/(\sqrt{2}\sigma )$, its actual number density is $n=n_0[\mathrm{erf}{x}_e-2x_e{e}^{-x_e^2}/\sqrt{\pi }-4x_e^3{e}^{-x_e^2}/(3\sqrt{\pi })]$. Thus $n_0$ is a normalization parameter, not generally the integrated density. Integrating only speeds below $V<v_e$ gives the same bracket with the first two terms at $X=V/(\sqrt{2}\sigma )$ and the last term $-4X^3{e}^{-x_e^2}/(3\sqrt{\pi })$.

For a galactic distribution function $f(\mathbf{x},\mathbf{v})$ with tracer number density $\nu =\int f{d}^{3}v$, a velocity moment is the weighted integral of a velocity monomial. The first moment gives the mean velocity ${\overline{v}}_i$; the second moment satisfies $\overline{v_i{v}_j}={\overline{v}}_i{\overline{v}}_j+{\sigma }_{ij}^2$, separating ordered motion from the random velocity dispersion tensor. Multiplying the Collisionless Boltzmann equation by $v_i$ and integrating yields the Jeans equation, with vanishing velocity-space boundary terms when the distribution decays sufficiently rapidly.

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: ${\sigma }_r^2={\sigma }^2$, ${\sigma }_{\theta }^2={\sigma }_{\varphi }^2={\sigma }^2/(1+r^2/r_a^2)$. 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.