Stellar Dynamics: Only Isolating Integrals Should be used in Jeans' Theorem

Abstract

Jeans' theorem that the distribution function of an unrelaxed steady stellar system can be expressed as some (in general multivalued) † function of elementary time-independent integrals of the motion has often been used restrictively. Only obvious integrals such as energy and angular momentum have been considered while there are always five time-independent integrals in all. This paper considers the role of the neglected integrals ‡ and shows that certain classes of integrals should indeed be omitted. A criterion is developed for finding some classes of potentials in which further integrals are important.