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

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 $\Phi=f(r)+r^{-2}g(\theta,\phi)$.