Jeans Equations

A stellar system is a swarm of $10^{8}$ to $10^{14}$ stars that almost never collide, described completely by a distribution function $f(\mathbf{x},\mathbf{v},t)$ that flows through six-dimensional phase space obeying the collisionless Boltzmann equation. That is a beautiful object and a useless one for the observer: nobody measures $f$ . What a telescope records are projections of $f$ onto the sky — a surface brightness, a mean line-of-sight velocity, a velocity dispersion — each an integral of $f$ over the velocities we cannot resolve. The Jeans equations, written down by James Jeans in 1915, are exactly those integrals. They take the velocity moments of the Boltzmann equation and trade an unsolvable six-dimensional transport problem for a small tower of coarser, spatial, and above all observable equations. The construction is the stellar-dynamical twin of how kinetic theory distils fluid mechanics from Boltzmann's transport equation. Multiply the collisionless Boltzmann equation by $1$ and integrate over velocity: out comes a continuity equation for the star density $\nu$ . Multiply by $v_j$ and integrate: out comes a momentum equation — the Jeans equation proper — in which the gravitational potential drives the flow and a velocity-dispersion tensor $\sigma_{ij}^2=\overline{v_iv_j}-\bar v_i\bar v_j$ plays the part of pressure. The quantity $\nu\sigma_{ij}^2$ is a stress tensor, so the momentum equation reads like a steady Euler equation for a strange fluid. The strangeness is the whole point. A gas relaxes by collisions to a Maxwellian, its stress collapses to a scalar pressure, and an equation of state $P=P(\rho)$ closes the equations. Stars do not collide, their velocity ellipsoid can be wildly anisotropic and tilted, and no equation of state exists. Each moment leans on the next higher one, so the hierarchy never closes by itself — you must supply geometry, an anisotropy law, or Jeans-theorem constraints. That single caveat is why the Jeans equations both weigh every elliptical galaxy and dark halo we know, and hide a degeneracy that has haunted the field for fifty years.

Key takeaways

The Jeans equations are the velocity moments of the collisionless Boltzmann equation: multiply by powers of velocity, integrate over velocity space, and fluid-like conservation laws for the stellar "crowd" fall out.
The zeroth moment is a continuity equation for the star density $\nu$ ; the first moment is a stellar Euler equation in which the divergence of the stress tensor $\nu\sigma_{ij}^2$ replaces the pressure gradient.
Because stars are collisionless there is no equation of state — $\sigma_{ij}^2$ is a full, generally anisotropic tensor — so the hierarchy is unclosed and needs an assumed symmetry or anisotropy profile $\beta(r)$ to solve.
In spherical symmetry the equations reduce to one ODE linking the tracer density, radial dispersion $\sigma_r$ , anisotropy $\beta$ , and the enclosed mass $M(<r)$ — the workhorse for weighing ellipticals, clusters, and dark halos.
Axisymmetric forms give the vertical Jeans equation (the Oort local-density measurement) and the asymmetric drift, tying disk kinematics to the Milky Way's mass model.
Key idea: the $n$ -th velocity moment of the collisionless Boltzmann equation is a fluid-like law for the $n$ -th moment of $f$ , but it always contains the $(n{+}1)$ -th moment, so the Jeans hierarchy is unclosed — only symmetry or an anisotropy prescription lets you solve it.

The core insight

Take velocity moments of the collisionless Boltzmann equation: the $n$ -th moment is a fluid-like conservation law for the $n$ -th velocity moment of $f$ , but each equation contains the next higher moment, so the hierarchy is unclosed — symmetry or an anisotropy prescription is required to close and solve it.

Where this lives today

The Jeans equations are the everyday instrument of dynamical mass measurement. In dwarf spheroidal galaxies — the most dark-matter-dominated systems known — spherical Jeans modelling of a few thousand stellar line-of-sight velocities returns mass profiles that anchor the dark-matter interpretation of the Local Group; codes such as GravSphere and its higher-moment (virial) extensions were built specifically to fight the mass–anisotropy degeneracy by folding in the fourth velocity moment. In galaxy clusters, Jeans analysis of member-galaxy velocities complements X-ray hydrostatic and weak-lensing masses, and the three routes are routinely cross-checked against one another and against the virial theorem. For nearby ellipticals, axisymmetric Jeans Anisotropic Modelling (JAM) fits integral-field spectroscopy from surveys like ATLAS3D and MaNGA to map central black-hole masses and stellar mass-to-light ratios across whole galaxy populations. And in the Milky Way, Gaia's micro-arcsecond astrometry has turned the century-old Oort problem into a precision measurement: the vertical Jeans equation applied to millions of stars now pins the local total mass density and the local dark-matter density to a few percent, while the asymmetric drift and tilt of the velocity ellipsoid sharpen the Galactic rotation curve and mass model.

The physical picture

You cannot follow $10^{11}$ individual orbits, so you watch the crowd instead: how densely the stars are packed, which way they drift on average, and how widely their velocities scatter about that drift. The collisionless Boltzmann equation says the phase-space fluid is incompressible along orbits; the Jeans equations are the bookkeeping that turns that statement into forces on the crowd. The mean drift of a parcel of stars changes because two things push on it — the gradient of a pressure-like dispersion term, and the gravitational field of everything inside. In steady state those balance, and the first figure is that balance drawn out for a spherical system.

Figure

A stellar system in equilibrium: gravity pulls each parcel of stars inward while the outward gradient of the dispersion-pressure νσ_r² plus the anisotropy term 2βσ_r²/r hold it up — for β = 0 this is exactly hydrostatic support with pressure P = νσ_r²

In an ordinary gas, collisions relax the velocities to a Maxwellian and the "pressure" is a single scalar. Stars are effectively collisionless over the age of the Universe, so their pressure is a tensor: the spread of velocities along the radial direction can differ from the spread across it. That anisotropy is not a nuisance — it is a direct fingerprint of the orbits. A system built mostly of plunging radial orbits has a fat radial dispersion and a thin tangential one; a system of near-circular orbits is the reverse. The Jeans equation contains this shape as the anisotropy parameter $\beta$ , but — and this is the recurring theme — it can never tell you $\beta$ on its own. The second figure shows what $\beta$ looks like and why it is the free knob.

Figure

The velocity ellipsoid and the anisotropy parameter β = 1 − σ_θ²/σ_r²: tangential orbits give β = −1 with σ_θ/σ_r ≈ 1.41, isotropic orbits give β = 0 with σ_θ/σ_r = 1, radially biased orbits give β = +0.5 with σ_θ/σ_r ≈ 0.71 — the ellipse shape at each radius is precisely what the Jeans equation cannot determine by itself

The hierarchy behaves like trying to pin down a distribution by its mean, then its variance, then its skewness: each level demands the next. Astronomers close it not with an equation of state but with geometry — assume the system is spherical or axisymmetric — plus a guess for $\beta(r)$ , then check afterward that the answer is positive and physical. The mental picture for the spherical case is simple: gravity pulls stars in, the gradient of $\nu\sigma_r^2$ pushes out like a pressure, and anisotropy adds or removes a geometric slice of that support. If the orbits are radially biased, tangential support is weak and you need a steeper dispersion profile — or more enclosed mass — to hold the system up. That trade between mass and anisotropy is the central difficulty of the whole subject.

The core equations

Everything begins with the collisionless Boltzmann equation, the statement that $f$ is conserved along orbits in a smooth potential $\Phi$ . It is the exact microscopic law; the Jeans equations are what survive after we integrate out the velocities we cannot see.

\frac{\partial f}{\partial t} + \sum_i v_i\frac{\partial f}{\partial x_i} - \sum_i \frac{\partial \Phi}{\partial x_i}\frac{\partial f}{\partial v_i} = 0 \qquad (1)

where: $f$ — phase-space distribution function $f(\mathbf{x},\mathbf{v},t)$ ; $v_i$ — velocity components; $x_i$ — spatial coordinates; $\Phi$ — gravitational potential, with $-\partial\Phi/\partial x_i$ the force per unit mass. Reading: the total time derivative of $f$ along a phase-space orbit is zero — the density of stars in $(\mathbf{x},\mathbf{v})$ is carried unchanged, like an incompressible fluid. The observables are moments of $f$ over velocity. Three of them carry all the physics we can measure: the density, the mean streaming velocity, and the velocity-dispersion tensor about that mean.

\nu \equiv \int f\, \mathrm{d}^3v, \qquad \bar{v}_i \equiv \frac{1}{\nu}\int f\, v_i\, \mathrm{d}^3v, \qquad \sigma_{ij}^2 \equiv \overline{v_i v_j} - \bar{v}_i\bar{v}_j \qquad (2)

where: $\nu$ — number (or mass) density in configuration space; $\bar v_i$ — mean streaming velocity; $\sigma_{ij}^2$ — symmetric, generally anisotropic velocity-dispersion (stress) tensor; $\overline{v_iv_j}=\frac1\nu\int f v_iv_j\,\mathrm{d}^3v$ — the second raw moment. Reading: $\nu$ is what surface photometry traces, $\bar v_i$ the mean Doppler shift, $\sigma_{ij}^2$ the Doppler broadening of the spectral lines. Multiplying (1) by $1$ and integrating over all velocities kills the force term and leaves a continuity equation — the zeroth Jeans equation. It is mass conservation for the stellar crowd.

\frac{\partial \nu}{\partial t} + \sum_i \frac{\partial(\nu\bar{v}_i)}{\partial x_i} = 0 \qquad (3)

where: $\nu\bar v_i$ — the number-flux density of stars. Reading: the star density at a point changes only because the streaming flux $\nu\bar v_i$ converges or diverges there; in steady state the flux is divergence-free. Multiplying (1) by $v_j$ and integrating gives the first moment — the Jeans equation proper. It is the stellar-dynamical Euler equation, with the divergence of the stress tensor $\nu\sigma_{ij}^2$ standing where a fluid would have its pressure gradient.

\nu\frac{\partial \bar{v}_j}{\partial t} + \nu\sum_i \bar{v}_i\frac{\partial \bar{v}_j}{\partial x_i} = -\nu\frac{\partial \Phi}{\partial x_j} - \sum_i \frac{\partial(\nu\sigma_{ij}^2)}{\partial x_i} \qquad (4)

where: $\nu\sigma_{ij}^2$ — the stress (pressure) tensor of the stellar "fluid"; $-\nu\,\partial\Phi/\partial x_j$ — the gravitational body force per unit volume. Reading: compare term by term with Euler's equation — the only difference is that $\nu\sigma_{ij}^2$ is a full tensor, not the scalar $P\delta_{ij}$ , because nothing forces the velocity ellipsoid to be round. For a steady, non-rotating, spherical system with no net streaming, (4) collapses to a single ordinary differential equation. This is the form that weighs galaxies, and it introduces the anisotropy parameter $\beta$ .

\frac{1}{\nu}\frac{\mathrm{d}(\nu\sigma_r^2)}{\mathrm{d}r} + \frac{2\beta}{r}\sigma_r^2 = -\frac{\mathrm{d}\Phi}{\mathrm{d}r} = -\frac{GM(<r)}{r^2}, \qquad \beta(r) \equiv 1 - \frac{\sigma_\theta^2+\sigma_\phi^2}{2\sigma_r^2} \qquad (5)

where: $\sigma_r$ — radial velocity dispersion; $\beta$ — velocity anisotropy ( $\beta=0$ isotropic, $\beta\to1$ purely radial, $\beta\to-\infty$ purely circular); $M(<r)$ — total mass enclosed within $r$ ; $\nu$ — tracer density, which need not equal the mass density. Reading: three profiles on the left — the dispersion, its gradient, and the anisotropy — determine the gravitating mass on the right. Solving (5) for the mass turns it into an estimator: given the logarithmic slopes of the density and dispersion profiles and a value of $\beta$ , read off the enclosed mass at any radius.

M(<r) = -\frac{r\,\sigma_r^2}{G}\left(\frac{\mathrm{d}\ln\nu}{\mathrm{d}\ln r} + \frac{\mathrm{d}\ln\sigma_r^2}{\mathrm{d}\ln r} + 2\beta\right) \qquad (6)

where: $\mathrm{d}\ln\nu/\mathrm{d}\ln r<0$ — the (steep, negative) tracer slope; $\mathrm{d}\ln\sigma_r^2/\mathrm{d}\ln r$ — the dispersion slope, often near zero. Reading: the mass scale is set by $r\sigma_r^2/G$ , and the bracket — a sum of dimensionless slopes plus $2\beta$ — is an order-unity correction. Because $\beta$ appears additively with the density slope, a change in mass can be traded against a change in anisotropy: the notorious mass–anisotropy degeneracy. The axisymmetric disk has its own reductions. The vertical Jeans equation, paired with the local Poisson equation, is the tool Oort used to weigh the solar neighbourhood.

\frac{1}{\nu}\frac{\partial(\nu\sigma_z^2)}{\partial z} = -\frac{\partial\Phi}{\partial z}, \qquad \frac{\partial^2\Phi}{\partial z^2} \simeq 4\pi G\rho \qquad (7)

where: $\sigma_z$ — vertical velocity dispersion of a tracer population; $\rho$ — total (dynamical) mass density near the midplane; $z$ — height above the plane. Reading: measure how fast a tracer's density falls with height and how warm it is vertically, and the pair of equations returns the total mass density $\rho$ — stars, gas, and any dark matter — at the Sun's location.

Derivation AFrom the collisionless Boltzmann equation to the Jeans equations

The Jeans equations are nothing more than the velocity moments of the collisionless Boltzmann equation, obtained by the same recipe that kinetic theory uses to distil fluid dynamics from Boltzmann's transport equation. We multiply the collisionless Boltzmann equation by successive powers of velocity and integrate over all of velocity space, watching the force term collapse under integration by parts and the streaming terms reorganise into convective derivatives, until a fluid-like continuity equation and an Euler-like momentum equation emerge. The goal here is to see both moments fall out cleanly and to identify exactly where the closure problem enters.

Step 1. Integrate (1) over $\mathrm{d}^3v$ . The time term gives $\partial_t\!\int f\,\mathrm{d}^3v=\partial_t\nu$ ; the advection term gives $\partial_{x_i}\!\int v_i f\,\mathrm{d}^3v=\partial_{x_i}(\nu\bar v_i)$ because $v_i$ is an independent phase-space coordinate that passes outside the spatial derivative. Why: velocity and position are independent variables in phase space, so the spatial derivative and the velocity integral commute.

Step 2. The force term is $\int \partial_{x_i}\Phi\,\partial_{v_i} f\,\mathrm{d}^3v = \partial_{x_i}\Phi\int \partial_{v_i}f\,\mathrm{d}^3v = 0$ , a perfect divergence in velocity space whose surface integral vanishes at $|\mathbf v|\to\infty$ where $f\to0$ . Assembling gives the continuity equation $\partial_t\nu+\partial_{x_i}(\nu\bar v_i)=0$ . Why: no star has infinite speed, so the boundary term at infinity is exactly zero — this is what makes the zeroth moment close by itself.

Step 3. Now multiply (1) by $v_j$ and integrate. The time and advection terms give $\partial_t(\nu\bar v_j)+\partial_{x_i}(\nu\overline{v_iv_j})$ ; the force term integrates by parts as $\int v_j\,\partial_{v_i}f\,\mathrm{d}^3v=-\delta_{ij}\nu$ , leaving $+\nu\,\partial_{x_j}\Phi$ . Why: integration by parts converts the velocity derivative on $f$ into a Kronecker delta, which is how gravity re-enters the moment equation as a body force.

Step 4. Split the second raw moment with the definition $\overline{v_iv_j}=\sigma_{ij}^2+\bar v_i\bar v_j$ , then subtract $\bar v_j$ times the continuity equation (3). The $\bar v_i\bar v_j$ pieces regroup into a convective derivative of $\bar v_j$ , and the leftover is the Jeans equation (4). Why: subtracting the lower moment is the standard manoeuvre that isolates the genuine momentum flux $\nu\sigma_{ij}^2$ from the bulk-streaming flux.

Step 5. Observe the closure gap: the first-moment equation (4) contains $\sigma_{ij}^2$ , the second moment; its own evolution equation would contain the third moment, and so on forever. Why: no collision term relaxes the velocities to a fixed shape, so unlike a gas there is no equation of state to truncate the tower — closure must be imposed by hand through symmetry or an assumed $\beta$ . *Two quick checks: the force-term surface integral vanishes because $f\to0$ as $|\mathbf v|\to\infty$ , and dimensionally $[\partial_{x_i}(\nu\sigma_{ij}^2)/\nu]$ carries units of acceleration, matching $\partial_{x_j}\Phi$ .*

Go deeper · Derivation B · 1 minThe spherical Jeans equation and the enclosed-mass estimator

Derivation BThe spherical Jeans equation and the enclosed-mass estimator

The single most useful form of the Jeans equations follows by imposing spherical symmetry, steady state, and no mean streaming on the general first moment, then inverting for the mass. The aim is to reduce equation (4) to the one-dimensional ODE (5), package the raw second moments into the anisotropy parameter $\beta$ , and finally solve algebraically for the enclosed mass $M(<r)$ so that a measured density slope, dispersion profile, and anisotropy return a mass at every radius. This is the calculation behind every Jeans mass of an elliptical galaxy, galaxy cluster, or dwarf spheroidal.

Step 1. In spherical coordinates the radial component of (4), with $\partial_t=0$ and $\bar v_i=0$ , reads $\dfrac{\mathrm{d}(\nu\overline{v_r^2})}{\mathrm{d}r} + \dfrac{\nu}{r}\big(2\overline{v_r^2}-\overline{v_\theta^2}-\overline{v_\phi^2}\big) = -\nu\dfrac{\mathrm{d}\Phi}{\mathrm{d}r}$ . Why: the $1/r$ terms are geometric — they come from the divergence of a tensor in curved coordinates, not from any force, and they are what make the spherical equation richer than the Cartesian one.

Step 2. With no streaming the raw moments equal the dispersions, $\overline{v_r^2}=\sigma_r^2$ and $\overline{v_\theta^2}=\overline{v_\phi^2}=\sigma_t^2$ for a non-rotating system, so the bracket becomes $2\sigma_r^2-2\sigma_t^2 = 2\sigma_r^2\beta$ with $\beta=1-\sigma_t^2/\sigma_r^2$ . Why: this single substitution compresses the entire angular structure of the velocity ellipsoid into one dimensionless function $\beta(r)$ .

Step 3. Divide through by $\nu$ to reach the working form $\dfrac{1}{\nu}\dfrac{\mathrm{d}(\nu\sigma_r^2)}{\mathrm{d}r} + \dfrac{2\beta\sigma_r^2}{r} = -\dfrac{\mathrm{d}\Phi}{\mathrm{d}r} = -\dfrac{GM(<r)}{r^2}$ , using $\mathrm{d}\Phi/\mathrm{d}r=GM(<r)/r^2$ for a spherical mass distribution. Why: the right-hand side is now purely the enclosed mass, so the equation directly connects kinematics to gravity.

Step 4. Expand the derivative with logarithmic slopes, $\dfrac{1}{\nu}\dfrac{\mathrm{d}(\nu\sigma_r^2)}{\mathrm{d}r}=\dfrac{\sigma_r^2}{r}\Big(\dfrac{\mathrm{d}\ln\nu}{\mathrm{d}\ln r}+\dfrac{\mathrm{d}\ln\sigma_r^2}{\mathrm{d}\ln r}\Big)$ , and solve for the mass to obtain the estimator (6): $M(<r)=-\dfrac{r\sigma_r^2}{G}\Big(\dfrac{\mathrm{d}\ln\nu}{\mathrm{d}\ln r}+\dfrac{\mathrm{d}\ln\sigma_r^2}{\mathrm{d}\ln r}+2\beta\Big)$ . Why: writing everything in logarithmic slopes exposes the degeneracy — the density slope and $2\beta$ enter as a sum, so no purely radial dataset can separate them. *Dimensional check: $[\sigma_r^2/r]$ is an acceleration, consistent with $GM/r^2$ .*

Go deeper · Derivation C · 1 minThe self-gravitating isothermal sheet

Derivation CThe self-gravitating isothermal sheet

Combining the vertical Jeans equation with Poisson's equation for a plane-parallel disk yields the classic sech-squared vertical profile of an isothermal stellar sheet, the standard model for the vertical structure of a galactic disk. The goal is to take a population with a constant vertical dispersion $\sigma_z$ that also generates its own gravity, close the Jeans–Poisson pair, and integrate the resulting nonlinear ODE to a closed form whose scale height is set only by the dispersion and the midplane density. This is Spitzer's 1942 result and the physical basis of a disk's exponential thickness.

Step 1. For constant $\sigma_z$ the vertical Jeans equation (7) becomes $\sigma_z^2\,\dfrac{\mathrm{d}\ln\rho}{\mathrm{d}z}=-\dfrac{\partial\Phi}{\partial z}$ , taking $\nu\propto\rho$ so the tracer traces the mass. Why: a constant dispersion pulls out of the derivative, turning the balance into a first-order relation between the density gradient and the potential gradient.

Step 2. Differentiate once in $z$ and insert the thin-disk Poisson equation $\partial^2\Phi/\partial z^2=4\pi G\rho$ , valid near the midplane where radial derivatives are negligible: $\sigma_z^2\,\dfrac{\mathrm{d}}{\mathrm{d}z}\!\Big(\dfrac{1}{\rho}\dfrac{\mathrm{d}\rho}{\mathrm{d}z}\Big)=-4\pi G\rho$ . Why: differentiating eliminates $\Phi$ in favour of $\rho$ itself, closing the system into a single equation for the density.

Step 3. This nonlinear ODE $\dfrac{\mathrm{d}}{\mathrm{d}z}\!\big(\rho^{-1}\rho'\big)=-\dfrac{4\pi G}{\sigma_z^2}\rho$ integrates once using the midplane boundary condition $\rho'(0)=0$ , and its solution is $\rho(z)=\rho_0\,\mathrm{sech}^2(z/z_0)$ with $z_0=\sqrt{\sigma_z^2/(2\pi G\rho_0)}$ . Why: the $\mathrm{sech}^2$ is the exact self-consistent profile in which the population's own weight is supported by its constant vertical dispersion.

Step 4. Check the far-field behaviour: as $z\to\infty$ , $\mathrm{sech}^2(z/z_0)\to 4e^{-2z/z_0}$ , so the sheet has an exponential tail with scale height $z_0/2$ . Why: the exponential tail is what makes real disks look like exponential-in- $z$ layers far from the plane, while the $\mathrm{sech}^2$ core rounds off the density at the midplane. *Two checks: the far-field limit $\rho\to4\rho_0e^{-2z/z_0}$ confirms the exponential tail, and dimensionally $[z_0]=\sqrt{[\sigma^2]/[G\rho]}$ has units of length.*

Limiting cases

The full Jeans equations collapse to a handful of recognizable special cases once a symmetry and a closure are chosen:

Case	Result	Meaning
Isotropic sphere	$\beta=0\Rightarrow \frac{1}{\nu}\frac{\mathrm{d}(\nu\sigma_r^2)}{\mathrm{d}r}=-\frac{\mathrm{d}\Phi}{\mathrm{d}r}$	pure hydrostatic equilibrium: identical in structure to a self-gravitating gas with scalar pressure $P=\nu\sigma_r^2$
Fully radial orbits	$\beta\to1$	tangential support vanishes; the extra $2\nu\sigma_r^2/r$ term is maximal — the regime of infall-dominated halos
Circular orbits	$\beta\to-\infty$	$\sigma_r\to0$ ; support is entirely rotational/tangential, the cold-disk limit
Cold rotating disk	$\bar v_\phi^2\gg\sigma^2$	first moment reduces to centrifugal balance $\bar v_\phi^2/R\approx\partial\Phi/\partial R$ ; dispersion terms give the small asymmetric-drift correction
Thin isothermal layer	$\sigma_z^2\,\frac{\mathrm{d}\ln\nu}{\mathrm{d}z}=-\frac{\partial\Phi}{\partial z}$	vertical Jeans equation; with Poisson gives the $\mathrm{sech}^2$ profile and the Oort local-density measurement

Common trapThe dispersion tensor is not a scalar pressure, and a positive solution is not a physical one

Two traps catch newcomers. First, $\nu\sigma_{ij}^2$ is a full stress tensor, not an isotropic gas pressure — there is no equation of state, and treating $\nu\sigma^2$ as a scalar pressure quietly assumes $\beta=0$ , which is false for most hot stellar systems and biases every mass you derive. Second, and more subtle: the Jeans equations are only velocity moments, so a solution with everywhere-positive $\nu(r)$ and $\sigma_r^2(r)$ can still be unphysical. The underlying distribution function $f$ that would produce those moments may go negative somewhere in phase space, and a negative $f$ is meaningless. Passing the Jeans equations is therefore necessary but not sufficient — you must check that a non-negative $f$ actually exists, for example via an Eddington inversion, before trusting a Jeans model.

Worked examples

1 · Weighing an elliptical galaxy from stellar kinematics (fully worked)

At a projected radius $r=10\ \mathrm{kpc}$ in an elliptical galaxy, the stellar tracer density falls as $\nu\propto r^{-3}$ (so $\mathrm{d}\ln\nu/\mathrm{d}\ln r=-3$ ), the radial dispersion is measured flat at $\sigma_r=200\ \mathrm{km\,s^{-1}}$ (so $\mathrm{d}\ln\sigma_r^2/\mathrm{d}\ln r\approx0$ ), and the orbits are consistent with isotropy ( $\beta=0$ ). Estimate the enclosed mass.

Step 1. Insert the slopes into the mass estimator (6). The bracket is $\big(\mathrm{d}\ln\nu/\mathrm{d}\ln r + \mathrm{d}\ln\sigma_r^2/\mathrm{d}\ln r + 2\beta\big) = (-3 + 0 + 0) = -3$ , so $M(<r)=-\dfrac{r\sigma_r^2}{G}(-3)=\dfrac{3r\sigma_r^2}{G}$ . The minus signs cancel — a falling tracer density means a positive enclosed mass, as it must.

Step 2. Evaluate the mass scale with $G=4.301\times10^{-6}\ \mathrm{kpc\,(km/s)^2\,M_\odot^{-1}}$ : $\dfrac{r\sigma_r^2}{G}=\dfrac{(10)(200)^2}{4.301\times10^{-6}}=\dfrac{4.0\times10^5}{4.301\times10^{-6}}=9.30\times10^{10}\ \mathrm{M_\odot}$ .

Step 3. Multiply by the bracket magnitude: $M(<10\ \mathrm{kpc})=3\times9.30\times10^{10}=\mathbf{2.8\times10^{11}\ M_\odot}$ . For comparison, the stellar light inside 10 kpc of such a galaxy implies only $\sim10^{11}\ M_\odot$ — so the dynamical mass exceeds the stellar mass by a factor of a few, the direct Jeans-equation signature of a dark-matter halo.

Step 4. Note the fragility: had the orbits been radially biased at $\beta=+0.5$ , the bracket would become $(-3+0+1)=-2$ , dropping the mass to $1.9\times10^{11}\ M_\odot$ — a $33\%$ change from anisotropy alone at fixed data. This is the mass–anisotropy degeneracy in one number.

2 · The scale height of a self-gravitating disk (fully worked)

A stellar population in the solar neighbourhood has a vertical dispersion $\sigma_z=20\ \mathrm{km\,s^{-1}}$ and, treated as an isothermal self-gravitating sheet, a midplane density $\rho_0=0.1\ \mathrm{M_\odot\,pc^{-3}}$ . Find its scale height $z_0$ and the height at which the density halves.

Step 1. Use the isothermal-sheet result $z_0=\sqrt{\sigma_z^2/(2\pi G\rho_0)}$ . In disk-friendly units $G=4.301\times10^{-3}\ \mathrm{pc\,(km/s)^2\,M_\odot^{-1}}$ , so $2\pi G\rho_0 = 2\pi(4.301\times10^{-3})(0.1)=2.70\times10^{-3}\ \mathrm{(km/s)^2\,pc^{-2}}$ .

Step 2. Then $z_0=\sqrt{(20)^2/2.70\times10^{-3}}=\sqrt{1.48\times10^{5}}=\mathbf{385\ pc}$ . Far from the plane the profile decays as $\rho\to4\rho_0e^{-2z/z_0}$ , an exponential layer of scale height $z_0/2\approx\mathbf{192\ pc}$ — squarely in the observed range for the Milky Way's older thin disk.

Step 3. The density halves where $\mathrm{sech}^2(z/z_0)=\tfrac12$ , i.e. $z/z_0=\mathrm{arccosh}\sqrt2=0.881$ , so $z_{1/2}=0.881\times385\approx\mathbf{339\ pc}$ . A warmer population ( $\sigma_z\uparrow$ ) puffs up as $z_0\propto\sigma_z$ ; a denser midplane ( $\rho_0\uparrow$ ) squeezes it as $z_0\propto\rho_0^{-1/2}$ .

3 · The asymmetric drift of a warm disk population (estimate)

A disk population has radial dispersion $\sigma_R=40\ \mathrm{km\,s^{-1}}$ in a Galaxy with local circular speed $v_c=230\ \mathrm{km\,s^{-1}}$ . Estimate how far its mean rotation lags the circular speed.

Step 1. The axisymmetric radial Jeans equation gives the asymmetric-drift relation $v_a\equiv v_c-\bar v_\phi\approx\dfrac{\sigma_R^2}{2v_c}\Big[\dfrac{\sigma_\phi^2}{\sigma_R^2}-1-\dfrac{\partial\ln(\nu\sigma_R^2)}{\partial\ln R}\Big]$ , valid for $v_a\ll v_c$ . The prefactor is $\sigma_R^2/2v_c=(40)^2/(2\times230)=3.5\ \mathrm{km\,s^{-1}}$ .

Step 2. Estimate the bracket from typical solar-neighbourhood values: an epicyclic ratio $\sigma_\phi^2/\sigma_R^2\approx0.5$ , and a combined density-plus-dispersion decline $\partial\ln(\nu\sigma_R^2)/\partial\ln R\approx-4.8$ (a disk scale length of a couple of kpc). The bracket is $0.5-1+4.8=4.3$ .

Step 3. Then $v_a\approx3.5\times4.3\approx\mathbf{15\ km\,s^{-1}}$ . The key prediction is the scaling $v_a\propto\sigma_R^2$ : dynamically hotter, older populations lag rotation more, exactly the Strömberg relation seen when solar-neighbourhood stars are binned by age — and the intercept of that trend returns $v_c$ itself.

Physics of the portrait

The Jeans equations are one man's answer to a question kinetic theory had already faced for gases: if you cannot track every particle, what do the averages obey? The move from Boltzmann's collisional gas to a collisionless star system is the whole story of galactic dynamics, and it starts with James Jeans.

Origins · 1 minJames Jeans and how this was discovered

James Jeans

1877–1946 · collisionless stellar dynamics and the velocity-moment equations

James Hopwood Jeans was a Cambridge mathematical physicist who worked across an extraordinary range — the partition of energy in blackbody radiation (the Rayleigh–Jeans law), the stability and fragmentation of self-gravitating gas (the Jeans mass and Jeans length), and the dynamics of stellar systems. In 1915, in a paper on star-streaming, he adapted the moment method of gas kinetic theory to a system where the "molecules" are stars and collisions are utterly negligible over the age of the Universe. By multiplying the collisionless Boltzmann equation by powers of velocity and integrating, he produced the hierarchy of equations that now bear his name, recognizing that the second-moment term acts as an anisotropic pressure with no equation of state to close it. Jeans understood immediately that this open-endedness was structural, not a temporary gap — the very feature that makes stellar systems able to hold non-Maxwellian, anisotropic velocity distributions indefinitely is what denies the equations a clean closure. In later life he became one of the most successful popularizers of science of his era, but the moment equations remain his most-used technical legacy: every dynamical mass of a galaxy or cluster is, at bottom, a solution of a Jeans equation.

History

1877

— Boltzmann's transport equation establishes the moment method for gases

1915

— Jeans casts the velocity-moment equations for collisionless stellar systems (MNRAS 76, 70)

1932

— Oort infers the local dynamical density from vertical stellar motions, the Oort limit

1942

— Spitzer solves the self-gravitating isothermal sheet and its sech² profile

2018

— Gaia astrometry revives Jeans-based disk kinematics and the local dark-matter census

Going further

Texts. Binney & Tremaine, Galactic Dynamics, 2nd ed., §4.8–4.9 (the moment equations, spherical and axisymmetric Jeans, asymmetric drift) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 24 · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 5 and §11 (Jeans modelling of halos) · Ryden, Introduction to Cosmology, 2nd ed., Ch. 8 · Dodelson, Modern Cosmology, 2nd ed., Ch. 9 · Jeans, J. H. 1915, MNRAS 76, 70 (the original) · Spitzer, L. 1942, ApJ 95, 329 (the isothermal sheet). The zeroth and first moments follow by integration by parts, with the velocity surface term vanishing as $f\to0$ ; the $\mathrm{sech}^2$ profile satisfies the coupled Jeans–Poisson ODE with $z_0=\sqrt{\sigma_z^2/2\pi G\rho_0}$ . The two anchor numbers are $M(<10\,\mathrm{kpc})=3r\sigma_r^2/G=2.8\times10^{11}\,M_\odot$ for $\sigma_r=200\,\mathrm{km\,s^{-1}}$ , and $z_0=385\,\mathrm{pc}$ for $\sigma_z=20\,\mathrm{km\,s^{-1}}$ , $\rho_0=0.1\,M_\odot\,\mathrm{pc^{-3}}$ . No numerical constants beyond $G$ , $2\pi$ , and $4\pi$ are used.

Prerequisites. Collisionless Boltzmann Equation · Distribution Function and Phase Space · Galactic Potential Models · Jeans Theorem. Leads to. Virial Theorem in Stellar Systems · Orbits in Spherical Potentials · Epicyclic Approximation · Galactic Potential Models.

Ask the tutor: Ask me to run the spherical Jeans mass estimator on your own tracer slope, dispersion profile, and anisotropy, or to show how folding in the fourth velocity moment breaks the mass–anisotropy degeneracy.

Why take moments at all instead of solving the Boltzmann equation directly?

Because the collisionless Boltzmann equation lives in six-dimensional phase space and its unknown, $f(\mathbf x,\mathbf v)$ , is not something a telescope can measure. What we observe are quantities integrated over the velocities we cannot resolve: a projected density from photometry, a mean velocity and a dispersion from spectral lines. Taking moments is exactly the operation of integrating out those hidden velocities, so the Jeans equations are written in the same variables the data come in. You pay for this convenience with the closure problem — you have thrown away information about the shape of $f$ — but for equilibrium mass modelling that trade is almost always worth it.

What is the anisotropy parameter β really telling me?

It encodes the shape of the velocity ellipsoid — the relative width of the velocity distribution along the radial versus the tangential directions. Written $\beta=1-\sigma_t^2/\sigma_r^2$ , it runs from $\beta=0$ for isotropic orbits (a round ellipsoid), through $\beta\to1$ for purely radial, plunging orbits (a needle pointing outward), to $\beta\to-\infty$ for purely circular orbits (a pancake in the tangential plane). Physically $\beta$ is a compact summary of the orbital composition of the system. The catch is that the Jeans equations relate $\beta$ to the mass, so unless you constrain one you cannot cleanly measure the other — hence the mass–anisotropy degeneracy.

How is the mass–anisotropy degeneracy broken in practice?

The degeneracy arises because the projected line-of-sight dispersion depends on the combination of $M(<r)$ and $\beta(r)$ , so different pairs reproduce the same second-moment data. Several handles break it. Higher-order moments — the kurtosis of the line-of-sight velocity distribution, or the full velocity histogram — respond differently to mass and anisotropy, which is what virial-shape and distribution-function methods exploit. Independent mass probes such as gravitational lensing or X-ray gas fix $M(<r)$ from outside the kinematics. And multiple tracer populations sharing one potential, or proper motions that supply the tangential velocities directly (as Gaia now does for Milky Way satellites), can pin $\beta$ on its own.

Why do the Jeans equations look so much like the equations of fluid dynamics?

Because they are built by the identical mathematical operation. Fluid dynamics comes from taking velocity moments of Boltzmann's transport equation for a gas; the Jeans equations come from taking velocity moments of the collisionless Boltzmann equation for stars. The zeroth moment is continuity in both cases, and the first moment is a momentum (Euler) equation in both. The one structural difference is the stress term: a collisional gas relaxes to a Maxwellian so its stress collapses to a scalar pressure with an equation of state, while a collisionless star system keeps a full anisotropic stress tensor $\nu\sigma_{ij}^2$ and no equation of state. So the machinery is fluid-like, but the closure is not.

What does "collisionless" mean here, given stars clearly exert gravity on each other?

It means two-body encounters between individual stars are negligible over the system's age — the relaxation time is far longer than a Hubble time for a galaxy. Each star responds to the smooth, collective gravitational potential of all the others, not to close gravitational scattering off its neighbours, so its orbit is set by $\Phi(\mathbf x)$ and $f$ is conserved along it. That is precisely the content of the collisionless Boltzmann equation. Collisions do matter in dense systems like globular-cluster cores, where two-body relaxation drives core collapse; there the collisionless assumption breaks and a collision term must be restored.

Can the Jeans equations be wrong even when they fit the data?

Yes, in an important sense. They are only moment equations, so a set of profiles $\nu(r)$ and $\sigma_r^2(r)$ that satisfies them — and that looks perfectly reasonable, all positive — need not correspond to any real, non-negative distribution function. The physical requirement is that some $f\ge0$ generate those moments, and that is a stronger condition than solving the Jeans equations. A model can therefore pass the Jeans test and still be unphysical because its implied $f$ goes negative in part of phase space. Best practice is to check consistency with an explicit distribution-function model, for instance through an Eddington inversion in the spherical isotropic case.