Galactic distribution function ($f(\mathbf{x},\mathbf{v},t)$)

A galactic distribution function gives stellar mass or number per phase space volume. Integrating over velocity gives spatial density, while velocity moments and projection along the line of sight predict kinematic observables.

An action-based galactic distribution function is a nonnegative function of orbital actions. Actions are integrals of motion in an integrable potential, so $f(\mathbf{J})$ automatically gives a steady collisionless model and can represent flattened or rotating systems.

A Schwarzschild orbit-superposition model integrates a library of orbits in a trial gravitational potential and chooses nonnegative orbit weights to reproduce observed density and kinematics.

A made-to-measure stellar-dynamical model evolves particles in a trial potential while adjusting their weights so that projected observables approach measured constraints.

Velocity dispersion is the standard deviation of velocities about their mean. Its tensor form records anisotropy, while a line-of-sight measurement combines intrinsic kinematics with projection and observational weighting.

For spherical radial dispersion ${\sigma }_r^2$ and total two-component tangential dispersion ${\sigma }_t^2$, the velocity-anisotropy parameter isPositive values are radially biased, zero is isotropic, and negative values are tangentially biased.

In a bounded stellar system, a relative potential shifts the gravitational potential so that $\Psi$ vanishes at a chosen escape boundary. A common sign convention is $\Psi ={\Phi }_0-\Phi$.

Relative energy is the binding energy per unit mass measured from the chosen escape level, commonly $\mathcal{E}=\Psi -v^2/2$. Bound phase-space points have $\mathcal{E}>0$ in this convention.

A spherical constant-anisotropy distribution function hasIts velocity moments have the radius-independent velocity-anisotropy parameter $\beta$.

For $f=f_0{L}^{-2\beta }{\mathcal{E}}^n$, velocity integration givesThus a scale-free monomial in radius and relative potential corresponds directly to a power-law constant-anisotropy distribution function.

Jeans theorem states that a steady Collisionless Boltzmann equation solution depends on phase-space coordinates only through integrals of motion, and that a nonnegative function of isolating integrals gives a steady collisionless distribution function on its domain.