General relativistic polytropes in anisotropic stars

Spherically symmetric relativistic stars with a polytropic equation of state (EoS), which possess local pressure anisotropy, are considered within the framework of general relativity. The generalized Lane-Emden equations are derived for the arbitrary anisotropy parameter Δ=pt−pr (pt and pr being the transverse and radial pressure, respectively). They are then applied to some special ansatz for the anisotropy parameter in the form of a differential relation between the anisotropy parameter Δ and the metric function ν. The analytical solutions of the obtained equations are found for incompressible fluid stars and then used for getting their mass-radius relation, gravitational energy, and binding energy. Also, following the Chandrasekhar variational approach, the dynamical stability of incompressible anisotropic fluid stars with a polytropic EoS against radial oscillations is studied. It is shown that the local pressure anisotropy with pt>pr can make the incompressible fluid stars unstable with respect to radial oscillations, in contrast to incompressible isotropic fluid stars with a polytropic EoS which are dynamically stable.