Collisionless Boltzmann equation

The collisionless Boltzmann equation states that the phase-space distribution function is constant along trajectories in a smooth gravitational potential: $dF/dt=0$.

In physical polar velocity components, a planar rotating system has accelerations ${\dot{v}}_R=2{\Omega }_b{v}_{\varphi }+v_{\varphi }^2/R+{\psi }_{\mathrm{eff},R}$ and ${\dot{v}}_{\varphi }=-2{\Omega }_b{v}_R-v_R{v}_{\varphi }/R+{\psi }_{\mathrm{eff},\varphi }/R$. The Collisionless Boltzmann equation is constant distribution along these characteristics. Its physical phase-space measure includes the spatial Jacobian $R$.

The zeroth velocity moment gives ${\partial }_t\rho +R^{-1}{\partial }_R(R\rho ⟨v_R⟩)+R^{-1}{\partial }_{\varphi }(\rho ⟨v_{\varphi }⟩)=0$. In the rotating-frame collisionless Boltzmann equation, velocity integration by parts supplies the geometric $\rho ⟨v_R⟩/R$ term. This equation expresses conservation of planar mass.

Linearize the Collisionless Boltzmann equation and the perturbation Poisson equation about a homogeneous velocity distribution $f_0$. A plane wave gives $f_1={\varphi }_1\mathbf{k}\cdot {\nabla }_v{f}_0/(\mathbf{k}\cdot \mathbf{v}-\omega )$. Integrating this density and using $-k^2{\varphi }_1=4\pi G\int f_1{d}^{3}v$ yields $1+(4\pi G/k^2)\int [\mathbf{k}\cdot {\nabla }_v{f}_0/(\mathbf{k}\cdot \mathbf{v}-\omega )]d^{3}v=0$. Growing modes have $\mathrm{Im}\omega >0$ and no real-velocity pole; real and damped frequencies require the causal continuation prescription. The Newtonian homogeneous background requires the Jeans swindle.

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 Jeans equations are velocity moments of the Collisionless Boltzmann equation. They relate density, mean motion, velocity dispersion, and gravitational acceleration without requiring a closed fluid equation of state.

For a steady axisymmetric stellar population with no mean vertical motion, the vertical Jeans equation balances the gradient of the vertical random-motion stress and the radial gradient of the tilted velocity ellipsoid against the vertical Newtonian gravitational field. Neglecting the mixed moment gives ${\partial }_z(n{\sigma }_z^2)=-n{\partial }_z\Phi$. A tracer density $n$ need not equal the density sourcing the gravitational potential.

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.

The first moments of the rotating-frame collisionless Boltzmann equation express radial and azimuthal momentum balance. Coriolis terms enter as $-2{\Omega }_b\rho ⟨v_{\varphi }⟩$ and $+2{\Omega }_b\rho ⟨v_R⟩$ on the left. Geometric stress terms are $\rho (⟨v_R^2⟩-⟨v_{\varphi }^2⟩)/R$ and $2\rho ⟨v_R{v}_{\varphi }⟩/R$. Second moments include random stress and streaming flux.

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.

For an isolated collisionless system, the tensor virial theorem is $\frac{1}{2}{\ddot{I}}_{jk}=2T_{jk}+{\Pi }_{jk}+W_{jk}$, separating ordered kinetic, random kinetic, and gravitational potential-energy tensors.

Use the position second-moment tensor $I_{ij}=\int \rho x_i{x}_j{d}^{3}x$, not the engineering moment-of-inertia convention. If a density pattern rotates steadily about its third principal axis, $I(t)=R(\Omega t)I(0)R(\Omega t{)}^T$. Differentiating twice at a principal-axis instant gives the displayed tensor. The tensor virial theorem then determines the pattern speed from the difference of its first two diagonal components. This density-pattern speed is distinct from stellar mean streaming; in the axisymmetric limit $I_{11}=I_{22}$, this difference cannot determine it.

For a finite stationary axisymmetric self-gravitating system with similar oblate density surfaces, let $q=b/a$, $s=\sqrt{1-q^2}$, $A_1=q\arccos (q)/s^3-q^2/s^2$, and $A_3=2/s^2-2q\arccos (q)/s^3$. Its stellar potential-energy tensor has $Q=A_1/(q^2{A}_3)$; the common radial integral cancels. The spherical limit has $A_1=A_3=2/3$ and $Q=1$. With mass-weighted in-plane velocity dispersion ${\sigma }_x$ and constant azimuthal streaming $v$, the tensor virial theorem gives $v^2/{\sigma }_x^2=2[(1-\beta )Q-1]$, where ${\sigma }_z^2=(1-\beta ){\sigma }_x^2$. Neither $Q=1/q$ nor the omission of the Cartesian streaming factor is exact.

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.

For $h=({\sin }^2\theta +q^{-2}{\cos }^2\theta {)}^{-\gamma /2}$, separation in spherical coordinates makes both stellar potential-energy tensor integrals contain $\int r^{3-\gamma }{\Phi }^{\prime }(r)dr$. Their ratio is the displayed positive finite angular expression. If the radial integrals diverge, identical spherical cutoffs give this same ratio of force moments, but there is no finite global pressure-energy ratio without further hypotheses. A finite-aperture tensor virial theorem retains its surface tensor.

The planar trace of the stellar pressure-energy tensor is the sum of its two Cartesian in-plane components. For an axisymmetric nonstreaming population it equals $\int \rho (⟨v_R^2⟩+⟨v_{\varphi }^2⟩)d^{3}x$, not merely the cylindrical radial second moment. A spherical isotropic population has ${\Pi }_{\perp }=2{\Pi }_{zz}$.

This force-moment tensor enters the tensor virial theorem. For an isolated self-gravitating population it is symmetric by pairwise interchange of the Newtonian interaction; for a tracer in an external field use the force-moment definition directly, with the appropriate symmetric combination in a time-dependent inertia identity.

This tensor integrates the local pressure tensor, where ${\sigma }_{ij}=⟨(v_i-u_i)(v_j-u_j)⟩$ is the velocity covariance. Half its trace is the random kinetic energy. It enters the tensor virial theorem with coefficient one.

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.

This tensor records the ordered kinetic energy of the streaming velocity $u_i=⟨v_i⟩$. It excludes random motion; its vanishing is compatible with a nonzero stellar pressure-energy tensor.

For an isolated self-gravitating fluid obeying ideal magnetohydrodynamics, let $I_{ij}=\int \rho x_i{x}_{j}dV$ and $K_{ij}=\frac{1}{2}\int \rho u_i{u}_{j}dV$. Use the force-stress convention $\rho D_t{u}_i={\partial }_j{T}_{ij}$, with pressure, magnetic and negative Newtonian gravitational stress tensor contributions. Two time derivatives of $I$ give $4K_{ij}$ plus $\int (x_i{\partial }_k{T}_{jk}+x_j{\partial }_k{T}_{ik})dV$. Integration by parts and symmetry yield the theorem when surface terms vanish. This fluid version complements the collisionless formulation of the tensor virial theorem.

At an initially resting instant with negligible pressure, sum the horizontal components of the magnetized-fluid tensor virial theorem. The horizontal magnetic trace is $-B_z^2/{\mu }_0$ and the gravitational force-stress trace is $({\partial }_z\Phi {)}^2/(4\pi G)$. Their negatives give the displayed radial second-moment balance. Vacuum field stresses contribute too; being initially at rest does not imply zero acceleration.