What can the alignments of the velocity moments tell us about the nature of the potential?

TL;DR

This paper demonstrates that specific symmetry and moment conditions in a stellar system's velocity distribution imply the potential must be separable, providing new criteria for understanding galactic potentials.

Contribution

It establishes new theoretical conditions linking velocity moment symmetries to the separability of the gravitational potential in stellar systems.

Findings

Symmetry of velocity distribution implies potential is separable.

Vanishing mixed second moments lead to potential separability.

Zero odd moments of radial velocity ensure a specific potential form.

Abstract

We prove that, if the time-independent distribution function $F(v;x)$ of a steady-state stellar system is symmetric under velocity inversion such that $F(-v_1,v_2,v_3;x)=F(v_1,v_2,v_3;x)$ and the same for $v_2$ and $v_3$, where $(v_1,v_2,v_3)$ is the velocity component projected onto an orthogonal frame, then the potential within which the system is in equilibrium must be separable (i.e. the Staeckel potential). Furthermore, we find that the Jeans equations imply that, if all mixed second moments of the velocity vanish, that is, $\langle v_iv_j\rangle=0$ for any $i\ne j$, in some Staeckel coordinate system and the only non-vanishing fourth moments in the same coordinate are those in the form of $\langle v_i^4\rangle$ or $\langle v_i^2v_j^2\rangle$, then the potential must be separable in the same coordinates. Finally we also show that all second and fourth velocity moments of tracers with an odd power to the radial component $v_r$ being zero is a sufficient condition to guarantee the potential to be of the form $\Phi=f(r)+r^{-2}g(\theta,\phi)$.