A second look at the Kurth solution in galactic dynamics

TL;DR

This paper investigates the Kurth solution in galactic dynamics, demonstrating under certain conditions that it is essentially unique among steady states that can be transformed into time-dependent solutions.

Contribution

It shows that if a steady state can be transformed into a time-dependent solution with specific properties, then it must be the Kurth solution, highlighting its uniqueness.

Findings

Kurth solution is essentially unique under certain transformation conditions.

Steady states with specific invariance properties are characterized as Kurth solutions.

The paper provides conditions under which the Kurth solution is the only possible form.

Abstract

The Kurth solution is a particular non-isotropic steady state solution to the gravitational Vlasov-Poisson system. It has the property that by means of a suitable time-dependent transformation it can be turned into a family of time-dependent solutions. Therefore, for a general steady state $Q(x, v)=\tilde{Q}(e_Q, \beta)$, depending upon the particle energy $e_Q$ and $\beta=\ell^2=|x\wedge v|^2$, the question arises if solutions $f$ could be generated that are of the form \[ f(t)=\tilde{Q}\Big(e_Q(R(t), P(t), B(t)), B(t)\Big) \] for suitable functions $R$, $P$ and $B$, all depending on $(t, r, p_r, \beta)$ for $r=|x|$ and $p_r=\frac{x\cdot v}{|x|}$. We are going to show that, under some mild assumptions, basically if $R$ and $P$ are independent of $\beta$, and if $B=\beta$ is constant, then $Q$ already has to be the Kurth solution. This paper is dedicated to the memory of Professor Robert Glassey.