Hyperbolic polytrope

In this work, we study self-gravitating objects that obey a polytropic equation of state in hyperbolic symmetry. Specifically, we describe in detail the steps to derive the Lane-Emden equation from the structure equations of the system. To integrate the equations numerically, we propose the Cosenza-Herrera-Esculpi-Witten anisotropy and study the cases $\gamma \ne 1$ and $\gamma = 1$ in the parameter space of the models. We find that the matter sector exhibits the usual and expected behavior for certain values in this parameter space: energy density (in absolute value) and radial pressure are decreasing functions and vanish at the surface, while the mass function is increasing toward the surface. We find that the anisotropy of the system is positive and decreasing, consistent with the behavior of the radial pressure, which reaches a local minimum at the surface (i.e., the pressure gradient is zero at the surface). We also study the compactness of the dense objects as a function of the polytropic index and obtain that it has an upper bound given by the maximum value it reaches for a certain $n$. Some extensions of the work and future proposals are discussed.