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]

$\frac{\partial n }{\partial t} + \sum_i \frac{\partial(n \langle{v_i}\rangle)}{\partial x_i}=0,$

$\frac{\partial(n \langle{v_j}\rangle)}{\partial t} + n \frac{\partial \Phi}{\partial x_j} + \sum_i \frac{\partial(n \langle{v_i v_j}\rangle)}{\partial x_i}= 0 \qquad (j=1, 2, 3.)$

Here, the <…> notation means an average at a given point and time (x,t), so that, for example, $\langle{v_1}\rangle$ 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

$n \frac{\partial \langle{v_j}\rangle}{\partial t} + \sum_i n \langle{v_i}\rangle \frac{\partial{\langle{v_j}\rangle}}{\partial x_i} = -n \frac{\partial \Phi}{\partial x_j} - \sum_i \frac{\partial (n \sigma_{ij}^2)}{\partial x_i} \qquad (j=1, 2, 3.)$

where $\sigma_{ij}^2=\langle{v_i v_j}\rangle-\langle{v_i}\rangle \langle{v_j}\rangle$ 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]

References

1. pp. 195-197, §4.2, Galactic dynamics, James Binney, Scott Tremaine, Princeton University Press, 1988, ISBN 0-691-08445-9.
2. Merritt, David (2013). Dynamics and Evolution of Galactic Nuclei. Princeton, NJ: Princeton University Press.
3. p. 82, "On the theory of star-streaming and the structure of the universe", J. H. Jeans, Monthly Notices of the Royal Astronomical Society 76 (December 1915), pp. 70-84, Bibcode: 1915MNRAS..76...70J.