Galactic distribution function ($f(\mathbf{x},\mathbf{v},t)$)

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.

For a self-consistent Gaussian angular-momentum suppression in a stellar distribution, put $y=[\psi (r)-\psi (0)]/{\sigma }^2$ and $s=\log (r/r_a)$. At large radius the Poisson equation for Newtonian gravity reduces to $y_{ss}+y_s\sim -\lambda e^y$, where $\lambda =4\pi G{\rho }_c{r}_a^2/{\sigma }^2$. The asymptotic balance $y_s\sim -1/s$ gives $e^y\sim 1/(\lambda s)$, the displayed density, and circular speed $v_c^2\sim {\sigma }^2/\log (r/r_{\ast })$. A sufficiently small regular core can have an intermediate isotropic singular isothermal sphere envelope with $\rho \simeq {\sigma }^2/(2\pi G{r}^2)$ and $v_c^2\simeq 2{\sigma }^2$. At the actual regular centre, however, $\rho \to {\rho }_c$ and $v_c\propto r$; the intermediate power law must not be mistaken for a central limit.

A stationary axisymmetric galactic distribution function depending on the specific orbital energy and axial specific angular momentum is a two-integral distribution. These quantities are conserved in a time-independent axisymmetric Newtonian gravitational potential, so the Collisionless Boltzmann equation holds by the chain rule. Even dependence on $L_z$ removes net azimuthal streaming velocity; an odd part can supply rotation without changing the mass density.

The singular axisymmetric logarithmic gravitational potential admits this stationary galactic distribution function withThe zero of specific orbital energy is the printed logarithmic convention. Products of Gaussian integrals normalize its two spatial density terms. For $1/2\le q^2\le 1$ the two terms are separately nonnegative; mildly prolate models require a bound on their sum.

For the logarithmic-potential two-integral distribution, $q^2>1$ makes $A<0$. Accessible phase space satisfies $L_z^2{e}^{-2E/V^2}\le V^2/e$, attained by equatorial circular motion. Hence the galactic distribution function is nonnegative precisely when $B+A{V}^2/e\ge 0$, giving the displayed upper bound. A larger prolate flattening parameter produces negative phase-space density even though the spatial mass density remains positive.

At fixed position a two-integral distribution depends on $v_R,v_z$ only through $v_R^2+v_z^2$. Swapping these integration variables makes their second moments equal. Their odd moments and mixed second moments vanish by reflection symmetry. For an even function of $L_z$ the azimuthal velocity is also centered, so all mixed entries of the velocity ellipsoid vanish.

The streaming velocity is the local mean stellar velocity obtained by averaging with the galactic distribution function. In an axisymmetric disk it usually denotes $⟨v_{\varphi }⟩$. It differs from the circular speed, which is set by the force field, and from the velocity dispersion, which measures fluctuations about the mean.

For a nonrotating axisymmetric galactic distribution function even in $L_z$, reverse every retrograde azimuthal velocity. The resulting distribution is $F_{+}(E,L_z)=2F(E,L_z)\Theta (L_z)$: it preserves density and even velocity moments but makes the streaming velocity $⟨|v_{\varphi }|⟩$. Dependence on conserved $E,L_z$ preserves stationarity. A centered Gaussian distribution of width $\sigma$ becomes a half-normal distribution, with mean $\sqrt{2/\pi }\sigma$.

For a stationary collisionless stellar system confined to a plane, velocity isotropy makes the galactic distribution function depend on speed through $u=v^2/2$. The Collisionless Boltzmann equation implies ${\nabla }_{x}f=f_u{\nabla }_x\Phi$, hence $f=F(E)$ with specific orbital energy $E=u+\Phi$. This conclusion requires stationarity as well as isotropy.

The surface density of a disk associated with a planar isotropic distribution is $S(\Phi )=2\pi {\int }_{\Phi }^{\infty }F(E)dE$. Differentiation gives $F(E)=-(2\pi {)}^{-1}dS/d\Phi {|}_{\Phi =E}$. A vanishing upper-limit boundary value recovers the original integral; monotonic decrease of $S(\Phi )$ ensures a nonnegative distribution.

An Osipkov-Merritt spherical distribution has $F=f(Q)$ for $Q=E-L^2/(2r_a^2)>0$ and vanishes otherwise. Rescaling transverse velocities by $\sqrt{1+r^2/r_a^2}$ makes its velocity integral isotropic in the rescaled variables. Its velocity-anisotropy parameter is $\beta =r^2/(r^2+r_a^2)$.

In unit Plummer model scales, $F=3Q^{3/2}/(\sqrt{2}{\pi }^3)$ for $Q=E-L^2/2>0$ gives $\rho =3{\psi }^3/[4\pi (1+r^2)]=3{\psi }^5/(4\pi )$. It reproduces the isotropic Plummer density but has anisotropy $r^2/(1+r^2)$ and different second velocity moments.

The density and self-consistent potential of a spherical galaxy need not determine its galactic distribution function. Different angular-momentum dependence can reproduce the same density with different velocity-anisotropy parameters. An isotropic Plummer model and its positive anisotropic Plummer distribution with unit anisotropy radius provide an explicit example. A chosen full augmented density is additional information.

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.

A regular phase-mixed distribution $F(H,p_{\varphi },I)$ in an axisymmetric Stäckel potential is even separately in $v_{\lambda }$ and $v_{\mu }$, because these integrals depend on those components only through squares. Odd integration then gives all three mixed second moments zero, even if the $p_{\varphi }$ dependence produces azimuthal streaming. Bare stationarity alone is insufficient in degenerate resonant potentials with additional integrals. For example the isotropic harmonic Hamiltonian has the extra conserved $K=v_x{v}_y+{\omega }^{2}xy$; a positive Gaussian $\exp (-aH+ϵK)$ with $a>|ϵ|>0$ is a steady tracer and has nonzero $⟨v_x{v}_y⟩$. Its harmonic gravitational potential is a Stäckel potential in these spheroidal coordinates. This tracer need not itself source that harmonic gravitational potential. The alignment claim thus needs the phase-mixed three-integral hypothesis, not merely the weak form of Jeans theorem.

For a spherical isotropic bound distribution $f(E)$ with $E=\psi -v^2/2$, no unbound population and $\rho (0)=0$, velocity integration gives $\rho (\psi )=4\pi \sqrt{2}{\int }_0^{\psi }f(E)\sqrt{\psi -E}dE$. Differentiation followed by Abel transform inversion yields $f(E)=(\sqrt{8}{\pi }^2{)}^{-1}(d/dE){\int }_0^E{\rho }^{\prime }(\psi )/\sqrt{E-\psi }d\psi$. The factor is $1/(\sqrt{8}{\pi }^2)$, with ${\pi }^2$ outside the square root. Regularity sufficient for these operations and nonnegativity of the resulting distribution are separate requirements; an arbitrary density-potential pair need not give a physical distribution.

If the galactic distribution function is not restricted to positive relative energy, isotropic velocity integration gives $\rho (\psi )=4\pi \sqrt{2}{\int }_{-\infty }^{\psi }f(E)\sqrt{\psi -E}dE$. The same Abel transform argument as Eddington inversion yields the displayed formula, with lower limit $-\infty$. For $\rho ={\rho }_0{e}^{\psi /{\sigma }_0^2}$, substitution $\psi =E-{\sigma }_0^2{s}^2$ gives $f(E)={\rho }_0(2\pi {\sigma }_0^2{)}^{-3/2}{e}^{E/{\sigma }_0^2}$. An escape-energy truncation changes both the density law and the velocity tails.

An action-based galactic distribution function is a nonnegative function of orbital actions. Actions are integrals of motion in an integrable potential, so $f(\mathbf{J})$ automatically gives a steady collisionless model and can represent flattened or rotating systems.

A Schwarzschild orbit-superposition model integrates a library of orbits in a trial gravitational potential and chooses nonnegative orbit weights to reproduce observed density and kinematics.

A made-to-measure stellar-dynamical model evolves particles in a trial potential while adjusting their weights so that projected observables approach measured constraints.

Velocity dispersion is the standard deviation of velocities about their mean. Its tensor form records anisotropy, while a line-of-sight measurement combines intrinsic kinematics with projection and observational weighting.

The velocity ellipsoid represents the positive velocity covariance tensor by its orthogonal principal axes and eigenvalue square roots. The uncentered second-moment tensor differs from covariance when a population streams. Vanishing mixed second moments align the second-moment ellipsoid with the chosen frame; for pure azimuthal streaming in a phase-mixed axisymmetric model, the off-diagonal covariance also vanishes.

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 tracers sampled uniformly on a Galactocentric shell of radius $r$, with the Sun at radius $R_{\odot }$, the correction in the heliocentric projection of a spherical velocity dispersion isThe limiting value at $r=R_{\odot }$ is $1/2$; it tends to $2/3$ as $r/R_{\odot }\to 0$ and to $2R_{\odot }^2/(3r^2)$ at large radius. Uniform heliocentric solid angle at fixed $r>R_{\odot }$ instead gives $A=2R_{\odot }^2/(3r^2)$ exactly. Specify the selection measure when applying the correction.

For spherical radial dispersion ${\sigma }_r^2$ and total two-component tangential dispersion ${\sigma }_t^2$, the velocity-anisotropy parameter isPositive values are radially biased, zero is isotropic, and negative values are tangentially biased.

In a bounded stellar system, a relative potential shifts the gravitational potential so that $\Psi$ vanishes at a chosen escape boundary. A common sign convention is $\Psi ={\Phi }_0-\Phi$.

Relative energy is the binding energy per unit mass measured from the chosen escape level, commonly $\mathcal{E}=\Psi -v^2/2$. Bound phase-space points have $\mathcal{E}>0$ in this convention.

A spherical constant-anisotropy distribution function hasIts velocity moments have the radius-independent velocity-anisotropy parameter $\beta$.

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 $1/2$.

For $f(E,L)=f_E(E)/L$, with $L=rv\sin \eta$ and $E=\psi -v^2/2$, velocity integration gives $r\rho =2{\pi }^2{\int }_0^{\psi }{f}_E(E)dE$. Thus $f_E(E)=(2{\pi }^2{)}^{-1}d(r\rho )/d\psi$ at $\psi =E$. The factor $1/L$ is locally integrable under the velocity-volume measure. Its angular measure is uniform in $\eta$; averages of ${\cos }^2\eta$ and ${\sin }^2\eta$ are both $1/2$, so the total tangential second moment equals the radial second moment and the velocity-anisotropy parameter is $\beta =1/2$. At the largest allowed radius $\psi (r_E)=E$, the derivative becomes $[\rho +r(d\rho /dr)]/(d\psi /dr)$, not $[\rho +d\rho /d\psi ]/(d\psi /dr)$. Positivity requires $d(r\rho )/d\psi \ge 0$.

For $f=f_0{L}^{-2\beta }{\mathcal{E}}^n$, velocity integration givesThus a scale-free monomial in radius and relative potential corresponds directly to a power-law constant-anisotropy distribution function.

Jeans theorem states that a steady Collisionless Boltzmann equation solution depends on phase-space coordinates only through integrals of motion, and that a nonnegative function of isolating integrals gives a steady collisionless distribution function on its domain.

For a phase-mixed one-dimensional collisionless stellar system, $E=v_z^2/2+\psi (z)$ is an integral of motion and $f_z=F(E)$. Integrating both signs of velocity gives $\nu (\psi )=\sqrt{2}{\int }_{\psi }^{\infty }F(E)(E-\psi {)}^{-1/2}dE$. Applying the Abel transform a second time gives ${\int }_E^{\infty }\nu (\psi )(\psi -E{)}^{-1/2}d\psi =\pi \sqrt{2}{\int }_E^{\infty }F(t)dt$; differentiating proves the inverse. Sufficient convergence, regularity, and correct escape-energy boundary conditions are essential. An arbitrary tracer-density/potential pair is physical only if the reconstructed phase-space distribution function is nonnegative. Applying this inversion to a galactic disk requires a locally separable vertical dynamics or a justified approximation to it, and control of radial mixing and the tilted velocity ellipsoid.