Black hole tidal problem in the Fermi normal coordinates

TL;DR

This paper derives a detailed tidal potential for stars orbiting Kerr black holes using Fermi-normal coordinates, extending previous models to higher-order terms and analyzing their impact on tidal disruption limits.

Contribution

It introduces a third and fourth-order tidal potential calculation in Fermi-normal coordinates and applies it to determine Roche limits for stars near Kerr black holes.

Findings

Higher-order terms significantly affect Roche limit calculations for close orbits.

The Roche limit for neutron stars depends on the star's equation of state.

The formulation improves understanding of tidal disruption in strong gravity regimes.

Abstract

We derive a tidal potential for a self-gravitating fluid star orbiting Kerr black hole along a timelike geodesic extending previous works by Fishbone and Marck. In this paper, the tidal potential is calculated up to the third and fourth-order terms in $R/r$, where $R$ is the stellar radius and $r$ the orbital separation, in the Fermi-normal coordinate system following the framework developed by Manasse and Misner. The new formulation is applied for determining the tidal disruption limit (Roche limit) of corotating Newtonian stars in circular orbits moving on the equatorial plane of Kerr black holes. It is demonstrated that the third and fourth-order terms quantitatively play an important role in the Roche limit for close orbits with $R/r \agt 0.1$. It is also indicated that the Roche limit of neutron stars orbiting a stellar-mass black hole near the innermost stable circular orbit may depend sensitively on the equation of state of the neutron star.