Jeans equations

The Jeans equations are moment equations of the collisionless Boltzmann equation. They relate a stellar system’s density, mean velocity, velocity dispersion, and gravitational potential.

Collisionless Boltzmann equation: A kinetic equation describing how a distribution of collisionless particles evolves through phase space under gravity. Taking its velocity moments produces the Jeans equations.
Dynamical mass: Mass inferred from the motions of matter responding to a gravitational field. Jeans modeling estimates this mass from observed stellar kinematics.
Phase space: A space whose coordinates specify both the position and momentum or velocity of a system. The stellar distribution function evolves over this space before its moments are taken.
Schwarzschild orbit superposition: A method that fits stellar systems by combining weighted orbits in a gravitational potential. It models orbit structure directly rather than relying only on Jeans moment equations.
Projection effect: The loss of depth information when a three-dimensional system is observed on a two-dimensional sky. Projected stellar data conceal velocity components needed to constrain Jeans models.
Velocity moment: An integral of a distribution function weighted by a power of velocity, such as density or velocity dispersion. The equations arise by taking successive velocity moments of the stellar distribution function.
Galaxy rotation curve: A plot of orbital speed in a galaxy against distance from its center. Jeans analyses relate observed motions to the gravitational field underlying the curve.
Distribution function (physics): A function describing the density of particles across position and velocity in phase space. Its velocity integrals define the density, mean motions, and dispersions in the equations.
Eddington inversion: A method for recovering an isotropic distribution function from a spherical density profile and potential. It imposes isotropy and spherical symmetry, stronger constraints than general Jeans modeling.
Proper motion: The apparent angular displacement of an object across the sky over time. Proper motions add transverse velocities that can help break dynamical modeling degeneracies.

Linked from 1 page

Stellar dynamicsBroader topic: They connect stellar velocity distributions to the gravitational structure of galaxies.