Anisotropic compact stars in higher-order curvature theory

In this paper, we shall consider spherically symmetric spacetime solutions describing the interior of stellar compact objects, in the context of higher-order curvature theory of the f(R) type. We shall derive the non--vacuum field equations of the higher-order curvature theory, without assuming any specific form of the $\mathrm{f(R)}$ theory, specifying the analysis for a spherically symmetric spacetime with two unknown functions. We obtain a system of highly non-linear differential equations, which consists of four differential equations with six unknown functions. To solve such a system, we assume a specific form of metric potentials, using the Krori-Barua ansatz. We successfully solve the system of differential equations, and we derive all the components of the energy-momentum tensor. Moreover, we derive the non-trivial general form of $\mathrm{f(R)}$ that may generate such solutions and calculate the dynamic Ricci scalar of the anisotropic star. Accordingly, we calculate the asymptotic form of the function $\mathrm{f(R)}$, which is a polynomial function. We match the derived interior solution with the exterior one, which was derived in \cite{Nashed:2019tuk}, with the latter also resulting in a non-trivial form of the Ricci scalar. Notably but rather expected, the exterior solution differs from the Schwarzschild one in the context of general relativity. The matching procedure will eventually relate two constants with the mass and radius of the compact stellar object. We list the necessary conditions that any compact anisotropic star must satisfy and explain in detail that our model bypasses all of these conditions for a special compact star $\textit {Her X--1 }$, which has an estimated mass and radius \textit {(mass = 0.85 $\pm 0.15M_{\circledcirc}$\,\, and\, \,radius $= 8.1 \pm 0.41$km)}.