Jeans Equations

The Jeans equations are a coupled system of partial differential equations in stellar and galactic dynamics—derived by British physicist and astronomer Sir James Hopwood Jeans in 1915 from a hydrodynamic moment formulation originally introduced by James Clerk Maxwell in 1867—that relate the spatial number density ν(x, t), mean streaming velocity v̄(x, t), and velocity dispersion tensor σ_ij²(x, t) of a collisionless population of stars or dark matter particles to the gravitational potential Φ(x, t). Because the six-dimensional phase-space distribution function f(x, v, t) governed by the collisionless Boltzmann equation cannot be measured directly in external galaxies or distant globular clusters, observational astronomers instead measure sky-projected surface densities and line-of-sight Doppler velocity moments. The Jeans equations provide the dynamical bridge between these observable velocity moments and the total enclosed gravitational mass M(r), underpinning dynamical measurements of supermassive black hole masses, galactic disk surface densities, asymmetric drift, and dark-matter halo profiles.

For a collisionless stellar population governed by the phase-space distribution function f(x, v, t) moving in a smooth gravitational potential Φ(x, t), phase-space continuity requires f to obey the collisionless Boltzmann equation:

∂f/∂t + v_i(∂f/∂x_i) − (∂Φ/∂x_i)(∂f/∂v_i) = 0

where repeated indices i, j ∈ {1, 2, 3} are summed over all three Cartesian axes. Integrating f over velocity space (v ∈ ℝ³) defines the spatial number density ν(x, t) ≡ ∫ f d³v and the mean velocity v̄_i(x, t) ≡ (1/ν) ∫ v_if d³v:

1. Zeroth Velocity Moment (Continuity Equation): Integrating the collisionless Boltzmann equation over all velocities (where f → 0 as |v| → ∞) yields the configuration-space continuity equation, ∂ν/∂t + ∂(ν*v̄*_i)/∂x_i = 0.
2. First Velocity Moment (Momentum / Jeans Equations): Multiplying the collisionless Boltzmann equation by v_j and integrating over all velocities yields ∂(ν*v̄*_j)/∂t + ∂(ν⟨v_iv_j⟩)/∂x_i + ν(∂Φ/∂x_j) = 0.

Decomposing the second velocity moment tensor ⟨v_iv_j⟩ into ordered streaming motion v̄_iv̄_j and the symmetric velocity dispersion tensor σ_ij² ≡ ⟨(v_i − v̄_i)(v_j − v̄_j)⟩ = ⟨v_iv_j⟩ − v̄_iv̄_j, and subtracting the continuity equation multiplied by v̄_j, produces the Euler-like form of the Jeans equations:

ν(∂v̄_j/∂t) + ν*v̄*_i(∂v̄_j/∂x_i) = −ν(∂Φ/∂x_j) − ∂(νσ_ij²)/∂x_i

In this balance, the gravitational force −ν∇Φ is supported by the divergence of an anisotropic three-dimensional stress tensor Π_ij = νσ_ij² rather than an isotropic scalar fluid pressure. Because the equations for the first moments v̄_j depend on the second moments σ_ij², the moment hierarchy is not closed without supplying an orbital anisotropy model or solving for f via the Eddington inversion formula or Stäckel potential actions.

For a steady-state (∂/∂t = 0), non-rotating (v̄ = 0), spherically symmetric stellar system (such as a globular cluster, dwarf spheroidal galaxy, or elliptical galaxy modeled by a Plummer model, Hernquist profile, Jaffe profile, or Einasto profile), spherical symmetry requires σ_θ²(r) = σ_ϕ²(r). The radial component of the Jeans equations simplifies to a single ordinary differential equation—the spherical Jeans equation:

d(νσ_r²)/dr + (2β(r)/r)νσ_r² = −ν(r)(dΦ/dr) = −ν(r)[GM(r)/r²]

where M(r) is the total enclosed gravitational mass within radius r, and β(r) is the dimensionless Binney velocity anisotropy parameter (Binney, 1980):

β(r) ≡ 1 − σ_θ²(r) / σ_r²(r)

Here β = 0 corresponds to isotropic orbits (σ_θ = σ_r), 0 < β ≤ 1 corresponds to radially biased orbits (σ_r > σ_θ, typical of outer halos formed via incomplete violent relaxation), and β < 0 corresponds to tangentially biased orbits (with β → −∞ for circular orbits). Rearranging the spherical Jeans equation expresses the enclosed gravitational mass M(r) directly in terms of logarithmic radial gradients:

M(r) = −[rσ_r²(r)/G] [(d ln ν / d ln r) + (d ln σ_r² / d ln r) + 2β(r)]

Because spectroscopy of a distant spherical galaxy measures only the sky-projected surface brightness I(R) and the line-of-sight velocity dispersion profile σ_LOS²(R),

I(R)σ_LOS²(R) = 2 ∫_R^∞ [1 − β(r)(R²/r²)] [ν(r)σ_r²(r)r dr / (r² − R²)^1/2]

a single observed profile σ_LOS(R) cannot simultaneously determine both M(r) and β(r). This ambiguity is the mass–anisotropy degeneracy (Binney & Mamon, 1982): a peaked central velocity dispersion σ_LOS(0) can arise either from a central supermassive black hole with isotropic orbits (β = 0) or from radially biased stellar orbits (β > 0) without a black hole. Breaking this degeneracy requires measuring fourth-order Gauss–Hermite velocity moments (h₄), combining kinematics with gravitational lensing, or evaluating the enclosed mass at the three-dimensional half-light radius r_1/2 ≈ (4/3)R_e via the Wolf et al. (2010) estimator M(r_1/2) ≈ 3G⁻¹⟨σ_LOS²⟩r_1/2.

In cylindrical coordinates (R, ϕ, z) for axisymmetric disk galaxies like the Milky Way (modeled with a Miyamoto–Nagai potential) and flattened early-type galaxies (via Jeans Anisotropic Multi-Gaussian Expansion, or JAM modeling; Cappellari, 2008), the radial Jeans equation in the midplane (z = 0) yields the Strömberg asymmetric drift relation:

v_c² − v̄_ϕ² = −σ_R² [(∂ ln ν / ∂ ln R) + (∂ ln σ_R² / ∂ ln R) + 1 − (σ_ϕ²/σ_R²) + (R/σ_R²)(∂σ_Rz²/∂z)]

Because both ν(R) and σ_R²(R) follow an exp(−R/R_d) radial decay outward in a galactic disk, older stellar populations with higher velocity dispersion (driven by vertical disk heating and Lindblad resonance scattering) lag behind the local circular speed v_c by Δv_ϕ ∝ σ_R². Meanwhile, the vertical Jeans equation near the midplane, (1/ν)∂(νσ_z²)/∂z = −∂Φ/∂z ≡ K_z(z), allows astronomers (following Jan Oort in 1932) to measure the total surface mass density of the Galactic disk (Σ_tot(|z| < 1.1 kpc) ≈ 70 M_☉ pc⁻²) and the local dark-matter density (ρ_DM,☉ ≈ 0.010–0.013 M_☉ pc⁻³).

What is the difference between the Jeans equations and the collisionless Boltzmann equation?

The collisionless Boltzmann equation governs the full six-dimensional phase-space distribution function f(x, v, t) in position and velocity space. The Jeans equations are obtained by integrating the collisionless Boltzmann equation over velocity space to produce three-dimensional partial differential equations for the spatial density ν(x) and velocity moments (v̄_i, σ_ij²).

What is the mass–anisotropy degeneracy in spherical galaxies?

In a spherical stellar system, only the projected surface brightness I(R) and the single line-of-sight velocity dispersion profile σ_LOS(R) are observable on the sky, whereas the spherical Jeans equation contains two unknown functions of radius: enclosed mass M(r) and velocity anisotropy β(r) = 1 − σ_θ²/σ_r². Different combinations of M(r) and β(r) can reproduce the same σ_LOS(R) curve unless fourth-order velocity moments (h₄) or proper motions are measured.

Why is the enclosed mass at the half-light radius insensitive to velocity anisotropy?

As shown by Wolf et al. (2010), radial orbits (β > 0) raise σ_LOS(R) near the galaxy center (R ≪ R_e) and lower it in the outer halo (R ≫ R_e), whereas tangential orbits (β < 0) do the reverse. At the intermediate pivot radius r_1/2 ≈ (4/3)R_e, the σ_LOS(R) curves for different anisotropies cross one another, making M(r_1/2) ≈ 3G⁻¹⟨σ_LOS²⟩r_1/2 insensitive to β.

1. Jeans, J. H. (1915). "On the theory of star-streaming and the structure of the universe." Monthly Notices of the Royal Astronomical Society, 76(2), 70–84. https://doi.org/10.1093/mnras/76.2.70
2. Binney, J., & Mamon, G. A. (1982). "M/L and velocity anisotropy from observations of spherical galaxies." Monthly Notices of the Royal Astronomical Society, 200(2), 361–375. https://doi.org/10.1093/mnras/200.2.361
3. Cappellari, M. (2008). "Measuring the inclination and mass-to-light ratio of axisymmetric galaxies via anisotropic Jeans models of stellar kinematics." Monthly Notices of the Royal Astronomical Society, 390(1), 71–86. https://doi.org/10.1111/j.1365-2966.2008.13754.x
4. Wolf, J., Martinez, G. D., Bullock, J. S., Kaplinghat, M., Geha, M., Muñoz, R. R., Simon, J. D., & Avedo, F. F. (2010). "Accurate masses for dispersion-supported galaxies." Monthly Notices of the Royal Astronomical Society, 406(2), 1220–1237. https://doi.org/10.1111/j.1365-2966.2010.16753.x
5. Binney, J., & Tremaine, S. (2008). Galactic Dynamics (2nd ed.). Princeton University Press. https://doi.org/10.1515/9781400828722