Polytropic spheres containing regions of trapped null geodesics

We demonstrate that in the framework of standard general relativity polytropic spheres with properly fixed polytropic index $n$ and relativistic parameter $\sigma$, giving ratio of the central pressure $p_\mathrm{c}$ to the central energy density $\rho_\mathrm{c}$, can contain region of trapped null geodesics. Such trapping polytropes can exist for $n > 2.138$ and they are generally much more extended and massive than the observed neutron stars. We show that in the $n$--$\sigma$ parameter space the region of allowed trapping increase with polytropic index for interval of physical interest $2.138 < n < 4$. Space extension of the region of trapped null geodesics increases with both increasing $n$ and $\sigma > 0.677$ from the allowed region. In order to relate the trapping phenomenon to astrophysically relevant situations, we restrict validity of the polytropic configurations to their extension $r_\mathrm{extr}$ corresponding to the gravitational mass $M \sim 2M_{\odot}$ of the most massive observed neutron stars. Then for the central density $\rho_\mathrm{c} \sim 10^{15}$~g\,cm$^{-3}$ the trapped regions are outside $r_\mathrm{extr}$ for all values of $2.138 < n < 4$, for the central density $\rho_\mathrm{c} \sim 5 \times 10^{15}$~g\,cm$^{-3}$ the whole trapped regions are located inside of $r_\mathrm{extr}$ for $2.138 < n < 3.1$, while for $\rho_\mathrm{c} \sim 10^{16}$~g\,cm$^{-3}$ the whole trapped regions are inside of $r_\mathrm{extr}$ for all values of $2.138 < n < 4$, guaranteeing astrophysically plausible trapping for all considered polytropes. The region of trapped null geodesics is located closely to the polytrope centre and could have relevant influence on cooling of such polytropes or for binding of gravitational waves in their interior.