Galaxy dynamics

Galaxy dynamics studies how gravity organizes the motions of stars, gas and dark matter in galaxies. Doppler shifts of spectral lines measure motion along the line of sight; proper motions, where measurable, supply transverse motion. A rotation curve constrains the radial gravitational force, while stellar random motions constrain support in a galactic bulge or elliptical galaxy. The interpretation depends on geometry and on whether the observed material follows nearly circular orbits. Stellar bars and spiral arms exert torques, exchange angular momentum with the surrounding galactic disk and can drive gas inward. Thus the dynamics both measures the mass distribution and explains how that distribution changes.^[ref]

An elongated nonaxisymmetric stellar structure in a galaxy, often approximated as a rotating perturbation of an axisymmetric or spherical gravitational potential. Its pattern angular velocity need not equal an individual star's orbital angular velocity. The Galactic bar is the particular example in the Milky Way.

Exchange of angular momentum between a stellar bar and a dark matter halo through orbital resonances. If the unperturbed halo distribution function depends only on energy and decreases with it, the resonant torque from a rotating perturbation is nonnegative for positive pattern angular velocity. Strict transfer requires nonzero resonant overlap. The reaction torque slows a rigid bar of constant positive moment of inertia; evolving bar structure and halo redistribution require updating the response.

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.

Extended stellar structures drawn out by time-dependent gravitational interactions between galaxies. Close, prograde encounters can coherently perturb rotating discs. Restricted particle calculations demonstrate that the structures need not be evidence of direct star-star impacts.^[ref]

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.

Let a weak tangential deceleration $D>0$ act on a test body moving along a slowly evolving family of circular orbits. Write its circular speed as $V_c(r)>0$ and its specific angular momentum as $j=r{V}_c(r)$. The specific torque is $\dot{j}=-rD$. Defining $q=d\log V_c/d\log r$ givesThis derivation follows angular-momentum balance; simply setting ${\dot{V}}_c=-D$ would omit the angular momentum changed by radial migration.

An adiabatic inward drift along radially stable circular orbits requires $1+q>0$. Indeed, with $\Omega =V_c/r$, the radial epicyclic frequency satisfiesFor this stable branch, the positive local timescale defined as $r/(2|\dot{r}|)$ is $(1+q)V_c/(2D)$. It is $V_c/D$ in a uniform-mass density core with $q=1$, and $V_c/(2D)$ for a flat galaxy rotation curve with $q=0$.

At $q=-1$, the circular-orbit angular momentum is locally independent of radius, so division by $1+q$ fails. For $q<-1$, the algebraic drift formula points outward, but those circular orbits are radially unstable; it is not a stable slow-inspiral solution. The quasistatic approximation also requires the drift time to exceed an orbital time. Finally, this local timescale is not automatically the remaining time to the center, which requires integrating the radius-dependent drift.

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.

A point-mass host creates radial differential acceleration $2GMx/D^3$, while a uniform satellite restores with $4\pi G\rho x/3$. Equality gives $D=[3M/(2\pi \rho ){]}^{1/3}$. The displacement cancels; the robust scaling is $(M/\rho {)}^{1/3}$. A rotating circular-orbit Jacobi calculation changes the radial coefficient from two to three.

The radial differential gravity of a point-mass host contributes $2GMx/D^3=2{\Omega }^{2}x$ for nearby separation $x$ at orbital radius $D$. A Jacobi tidal radius describes a different calculation: the satellite is on a circular orbit and the equilibrium is evaluated in its rotating frame.

The centrifugal contribution adds another ${\Omega }^{2}x$, making the radial outward coefficient $3{\Omega }^2$ at a stationary point in that frame. Thus comparing self-gravity with $2{\Omega }^{2}x$ and solving the rotating-frame equilibrium with $3{\Omega }^{2}x$ are distinct approximations. The discrepancy is not an inconsistent derivative of the same gravitational field. Orbital geometry and the chosen frame must accompany any quoted tidal-disruption coefficient.

A Jacobi tidal radius estimates the radial saddle distance from a low-mass satellite on a circular orbit in a smooth spherical host. Let the orbital radius be $r_o$, the host potential be ${\Phi }_h$, andIn the satellite's rotating frame, linearizing the combined host and centrifugal acceleration along the host-satellite line gives $[{\Omega }^2-{\Phi }_h^{\prime \prime }(r_o)]x$. Balancing this outward acceleration against the satellite's inward gravity givesFor an extended satellite, $M_s$ is its enclosed mass at $r_t$, so the equation can be implicit.

Differentiate ${\Phi }_h^{\prime }=G{M}_h/r^2$ and write $\alpha =d\log M_h/d\log r$ at $r_o$. Then ${\Phi }_h^{\prime \prime }=(\alpha -2){\Omega }^2$ andA finite radial saddle in this leading approximation requires $3-\alpha >0$, as well as $r_t\ll r_o$. For a point-mass host, $\alpha =0$ gives the familiar denominator $3M_h$. A flat galaxy rotation curve has $\alpha =1$ and instead gives $2M_h$.

Inside an exactly constant-mass density core, $\alpha =3$ makes the linear radial stretching coefficient vanish. Dividing by zero is not a prediction of a finite tidal boundary there; higher-order structure or a different model is needed. The Jacobi tidal radius describes a rotating-frame equilibrium scale, not a spherical material wall or an exact stripping rule for eccentric, evolving orbits. The distinction from the inertial differential gravitational field is explained by the tidal-gravity coefficient versus rotating Jacobi coefficient.

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 treats stars as gravitating bodies whose collective motions reveal and reshape their environment. In a collisionless stellar system, stars mainly follow the smooth collective gravitational potential; in a sufficiently dense star cluster, accumulated gravitational encounters produce two-body relaxation. These encounters need not involve physical contact between stars. Observers combine astrometry with radial velocities to constrain orbits and infer unseen mass. The stellar orbits at the Galactic centre provide a particularly direct example: following a short-period star around Sagittarius A* measures the compact central mass, rather than estimating it from the light emitted by its surroundings.^[ref]

Directional inequality of velocity dispersions. Inferring a three-dimensional dynamical mass from one line-of-sight dispersion requires an assumption about this anisotropy.

The leading second-order specific torque on a collisionless distribution from a weak steadily rotating perturbation with ${\Phi }_{\mathbf{k}}={\Psi }_{\mathbf{k}}{e}^{-im{\Omega }_{p}t}$ and $m=k_{\varphi }$. The sum includes conjugate harmonics. Linear response in action-angle variables gives the finite-time kernel $\sin (\Delta t)/\Delta$; its integrated long-time resonance kernel selects orbital resonances. This limit assumes a smooth continuum, regular resonant surfaces and negligible secular redistribution during the response calculation.

For a homogeneous collisionless stellar model with its mean field subtracted, the response to an imposed Newtonian gravitational potential is ${\tilde{\Phi }}_{\mathbf{k}}^{\mathrm{tot}}={\tilde{\Phi }}_{\mathbf{k}}^{\mathrm{ext}}/{ϵ}_{\mathbf{k}}$, whereThe phase-space distribution function $f_0$ is mass-normalized, and $\mathbf{k}\ne 0$. Causal analytic continuation uses the Landau contour wherever the equilibrium permits it. Zeros describe collective Landau modes. This response function is a gravitational analogue of the electrostatic dielectric function; its susceptibility sign reflects attraction.

A change in a star's mean orbital radius through exchange of angular momentum, particularly at a corotation resonance with a rotating perturbation. For the $k_R=0$ harmonic in action-angle variables, the leading change of radial action vanishes even while azimuthal angular momentum changes. Migration can therefore rearrange a galactic disk with much less heating than exchange at a Lindblad resonance.

The second-order, angle-averaged evolution of a galactic distribution function caused by weak fluctuating gravitational perturbations. In action-angle variables, the first-order response to a Fourier series harmonic feeds a second-order flux proportional to $\mathbf{k}(\mathbf{k}\cdot {\partial }_{\mathbf{J}}{f}_0)$ and to the perturbation's power. Orbital resonances select where transport is strongest; a finite pulse produces Gaussian resonance broadening when its time envelope is a Gaussian function. The general flux retains time memory until a further short-memory or pulse-integrated approximation is made.

For a perturbation with amplitude envelope $e^{-(t-t_0{)}^2/(2{\tau }^2)}$, the squared Fourier transform is proportional to $e^{-{\omega }^2{\tau }^2}$. Hence pulse-integrated transport weights orbital detuning with a normalized Gaussian function of width $\Gamma =(\sqrt{2}\tau {)}^{-1}$. A longer pulse selects a narrower band of orbital resonances. This spectral broadening does not by itself assume stochastic temporal forcing.

For two approximately spherical stars approaching with speed $v_{\infty }$, contact at separation $R_1+R_2$ has gravitationally focused cross section $\pi (R_1+R_2{)}^2[1+2G(m_1+m_2)/((R_1+R_2)v_{\infty }^2)]$. This measures direct stellar encounters, not the probability of a collective encounter between their host galaxies.

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 group of stars formed in a relatively compact region. Open clusters and globular clusters provide populations whose members have broadly related formation histories, making them useful laboratories for stellar evolution. Membership is tested using parallax, proper motion and, where available, radial velocity, rather than apparent proximity on the sky alone. A cluster's Hertzsprung-Russell diagram then compares stars of different masses at a shared distance; its main-sequence turnoff helps estimate stellar age. Space-based astrometry sharpens both membership selection and these evolutionary sequences, although some clusters contain multiple stellar populations.^[ref]

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.