Circular stable orbits in $f(R)$ realistic static and spherically-symmetric spacetimes

TL;DR

This paper explores the geodesic structure around neutron stars in a specific $f(R)$ gravity model, revealing stable orbital bands and the absence of photon spheres, which could impact astrophysical observations.

Contribution

It provides a detailed analysis of stable and unstable orbits in $f(R)$ gravity neutron star spacetimes, including numerical solutions and junction conditions.

Findings

Stable circular orbits exist in discrete radial bands.

No photon spheres are found outside the neutron star.

Orbital stability depends on stellar parameters and the $f(R)$ parameter.

Abstract

We investigate the geodesic structure of realistic static and spherically symmetric spacetimes embedding neutron stars in metric $f(R)$ gravity, focusing on the quadratic Starobinsky model $f(R)=aR^2$ with $a<0$. Neutron-star solutions are obtained by numerically solving the modified Tolman-Oppenheimer-Volkoff system for several realistic equations of state. Such solutions are then matched consistently to the exterior vacuum geometry by enforcing the full set of junction conditions required in metric $f(R)$ theories. Using an effective potential approach, we show that stable circular orbits appear in discrete radial bands separated by forbidden regions, with a dominant principal band of stability that depends sensitively on the stellar central pressure, the equation of state, and the magnitude of the parameter $|a|$. Outside the stable bands, massive particles can have bound but unstable precessing trajectories as well as unbounded motions. On the other hand, for null geodesics, we find no evidence for photon spheres outside the neutron star within the parameter range studied.