Upper Limit on the Size of Galactic Halo via Hamiltonian Approach

Using the approach of autonomous Hamiltonian dynamical system, we attempt to estimate the as yet unknown upper limit on the size of the galactic halo surrounding galaxies (lenses). The key to determine the size of the halo is to determine the maximum radius up to which stable material circular orbits are admissible. We shall illustrate the approach by considering a solution of the Weyl gravity containing a halo parameter $\gamma$. The upper limit for several observed lenses are calculated for a typical value of $gamma$ for definiteness, with and without the cosmological constant $\Lambda$. These lenses (all having Einstein radius $R_{\text{E}}\approx 10^{23}$ cm) consistently yield an upper limit $R_{\text{max}}^{\text{stable}}$($\simeq 4.25\times 10^{27}$ cm) inside the de Sitter radius only when $\gamma$ is \textit{negative}, while a positive $\gamma$ yields R_{\text{max}}^{\text{% stable}} always exceeding the de Sitter radius.