# On the Jeans Theorem and the “Tolman–Oppenheimer–Volkoff Equation” in ${R}^{2}$ Gravity

Corda, Christian  (Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran) ; Jain, Rishabh (Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA) ; Sidharth, Burra G.  (International Institute for Applicable Mathematics & Information Sciences (IIAMIS), Hyderabad, India) (International Institute for Applicable Mathematics & Information Sciences (IIAMIS), Udine, Italy) (G. P. Birla Observatory & Astronomical Research Centre, B. M. Birla Science Centre, Adarsh Nagar, Hyderabad 500 063, India)

13 October 2016

Abstract: Corda, Mosquera Cuesta, and Lorduy Gòmez have shown that spherically symmetric stationary states can be used as a model for galaxies in the framework of the linearized ${R}^{2}$ gravity. Those states could represent a partial solution to the Dark Matter Problem. Here, we discuss an improvement of this work. In fact, as the star density is a functional of the invariants of the associated Vlasov equation, we show that any of these invariants is in its turn a functional of the local energy and the angular momentum. As a consequence, the star density depends only on these two integrals of the Vlasov system. This result is known as the “Jeans theorem.” In addition, we find an analogy of the historical Tolman–Oppenheimer–Volkoff equation for the system considered in this paper. For the sake of completeness, in the final section of the paper, we consider two additional models which argue that Dark Matter could not be an essential element.

Published in: Advances in High Energy Physics 2016 (2016) 2601741