Analytical families of two-component anisotropic polytropes and their relativistic extensions

In this paper, we study a family of two-component anisotropic polytropes which model a wide range of spherically symmetric astrophysical systems such as early-type bary-onic galaxies. This family is found to contain a large class of models such as the hy-pervirial family (which satisfy the virial theorem locally), the Plummer and Hernquist models and Navarro-Frenk-White (NFW)-like models. The potential–density pair for these models are derived, as well as their velocity dispersions and anisotropy parame-ters. The projected quantities are computed and found to reduce to analytical expres-sions in some cases. The first section of this paper presents an extension of the two-term anisotropic polytropes to encompass a very wide range of potential–density pairs. In the next section, we present the general relativistic extension of the potential–density pair, and calculate the stress–energy tensor, the relativistic anisotropy parameter, the velocity of circular orbits and the angular momentum. Remarkably, for the case of the hypervirial family, the relativistic pressure in the Newtonian limit and the relativistic anisotropy parameter are found to coincide with the corresponding Newtonian expres