Jeans equations

The Jeans equations describe the motion of a collection of stars in a gravitational field.

If n = n(x,t) is the density of stars in space, as a function of position x = (x₁, x₂, x₃) and time t, v = (v₁, v₂, v₃) is the velocity, and Φ = Φ(x,t) is the gravitational potential, the Jeans equations may be written as[1][2]

${\displaystyle {\frac {\partial n}{\partial t}}+\sum _{i}{\frac {\partial (n\langle {v_{i}}\rangle )}{\partial x_{i}}}=0,}$

${\displaystyle {\frac {\partial (n\langle {v_{j}}\rangle )}{\partial t}}+n{\frac {\partial \Phi }{\partial x_{j}}}+\sum _{i}{\frac {\partial (n\langle {v_{i}v_{j}}\rangle )}{\partial x_{i}}}=0\qquad (j=1,2,3.)}$

Here, the <...> notation means an average at a given point and time (x,t), so that, for example, ${\displaystyle \langle {v_{1}}\rangle }$ is the average of component 1 of the velocity of the stars at a given point and time. The second set of equations may alternately be written as

${\displaystyle n{\frac {\partial \langle {v_{j}}\rangle }{\partial t}}+\sum _{i}n\langle {v_{i}}\rangle {\frac {\partial {\langle {v_{j}}\rangle }}{\partial x_{i}}}=-n{\frac {\partial \Phi }{\partial x_{j}}}-\sum _{i}{\frac {\partial (n\sigma _{ij}^{2})}{\partial x_{i}}}\qquad (j=1,2,3.)}$

where ${\displaystyle \sigma _{ij}^{2}=\langle {v_{i}v_{j}}\rangle -\langle {v_{i}}\rangle \langle {v_{j}}\rangle }$ measures the velocity dispersion in components i and j at a given point.

The Jeans equations are analogous to the Euler equations for fluid flow and may be derived from the collisionless Boltzmann equation. They were originally derived by James Clerk Maxwell but were first applied to stellar dynamics by James Jeans.[3]