Black hole tidal problem in the Fermi normal coordinates

We derive a tidal potential for a self-gravitating fluid star orbiting a 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 $R/r\gtrsim 0.1$. 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.