Fixing a Parameter of the Galactic Halo: A Mathematical Modelling by Hamiltonian Method

Abstract

We illustrate how the mathematical modelling of the equations of motion in terms of autonomous Hamiltonian dynamical system can definitively fix a sign for an otherwise indefinite sign of a certain astrophysical parameter. To illustrate it, we shall consider the Mannheim-Kazanas-de Sitter solution of Weyl gravity containing the parameter {\gamma}, which is believed to be significant in the halo gravity. The strategy we adopt is to calculate the maximum radius up to which the halo supports stable material circular orbits. The maximum radius for several observed lenses are calculated for both signs of {\gamma}, and with the observed value of cosmological constant {\Lambda}. These lenses (all having approximately the Einstein radius R_{E}{\approx}10^23 cm) consistently yield a maximum radius R_{max}^{stable}({\simeq}4.25{\times}10^27 cm) inside the de Sitter radius of the universe only when {\gamma} is negative, while a positive {\gamma} yields R_{max}^{stable} always exceeding the de Sitter radius.