Epicyclic oscillations in the Hartle-Thorne external geometry

The external Hartle-Thorne geometry, which describes the space-time outside a slowly-rotating compact star, is characterized by the gravitational mass $M$, angular momentum $J$ and quadrupole moment $Q$ of the star and gives a convenient description which, for the rotation frequencies of more than 95 % of known pulsars, is sufficiently accurate for most purposes. We focus here on the motion of particles in these space-times, presenting a detailed systematic analysis of the frequency properties of radial and vertical epicyclic motion and of orbital motion. Our investigation is motivated by X-ray observations of binary systems containing a rotating neutron star which is accreting matter from its binary companion. In these systems, twin high-frequency quasi-periodic oscillations are sometimes observed with a frequency ratio approaching $3:2$ or $5:4$ and these may be explained by models involving the orbital and epicyclic frequencies of quasi-circular geodesic motion. In our analysis, we use realistic equations of state for the stellar matter and proceed in a self-consistent way, following the Hartle-Thorne approach in calculating both the corresponding values of $Q$, $M$ and $J$ for the stellar model and the properties of the surrounding spacetime. Our results are then applied to a range of geodetical models for QPOs. A key feature of our study is that it implements the recently-discovered universal relations among neutron star parameters so that the results can be directly used for models with different masses $M$, radii $R$ and rotational frequencies $f_\mathrm{rot}$.