Rotational modes of relativistic stars: Analytic results

Abstract

We study the r-modes and rotational "hybrid" modes of relativistic stars. As in Newtonian gravity, the spectrum of low-frequency rotational modes is highly sensitive to the stellar equation of state. If the star and its perturbations obey the same one-parameter equation of state (as with isentropic stars), there exist no pure r-modes at all -no modes whose limit, for a star with zero angular velocity, is an axial-parity perturbation. Rotating stars of this kind similarly have no pure g-modes, no modes whose spherical limit is a perturbation with polar parity and vanishing perturbed pressure and density.In spherical stars of this kind, the r-modes and g-modes form a degenerate zero-frequency subspace. We find that rotation splits the degeneracy to zeroth order in the star's angular velocity Ω, and the resulting modes are generically hybrids, whose limit as Ω → 0 is a stationary current with both axial and polar parts. Because each mode has definite parity, its axial and polar parts have alternating values of l. We show that each mode belongs to one of two classes, axial-led or polar-led, depending on whether the spherical harmonic with the lowest value of l that contributes to its velocity field is axial or polar. Newtonian isentropic stars retain a vestigial set of purely axial modes (those with l = m); however, for relativistic isentropic stars we show that these modes must also be replaced by axial-led hybrids. We compute the post-Newtonian corrections to the l = m modes for uniform density stars.On the other hand, if the star is non-isentropic (or, more broadly, if the perturbed star obeys an equation of state that differs from that of the unperturbed star) the r-modes alone span the degenerate zero-frequency subspace of the spherical star. In Newtonian stars, this degeneracy is split only by the order Ω 2 rotational corrections. However, when relativistic effects are included the degeneracy is again broken at zeroth order. We compute the r-modes of a non-isentropic, uniform density model to first post-Newtonian order. *

Eigenmode frequency distribution of rapidly rotating neutron stars

“…In the relativistic slow rotation approximation [10,13], their frequencies are found to be up to 15% lower than those of their Newtonian counterparts. Using the relativistic 'traditional' approximation a similar shift was found [14], but it applies to the frequencies in the corotating frame, rather than those in the asymptotic inertial frame.…”

“…Deviations from an isentropic model become important only if the angular spin frequency is comparable to or smaller than the Brunt-Väisälä frequency, which for neutron stars is of the order of 100Hz [10].…”

Differential rotation and r-modes in magnetized neutron stars

“…Furthermore, Lockitch, Andersson & Friedman (2000) demonstrate fully relativistic equations for r-modes in NSs with barotropic and nonbarotropic equations of state to be qualitatively different. For barotropic equation of state a relativistic analogue for Newtonian r-modes was found by Lockitch et al (2003) and they conclude that "unstable rmodes remain essentially unaltered when the problem is studied in full general relativity".…”

Slowly rotating general relativistic superfluid neutron stars

“…Another motivation for studying oscillations of slowly rotating superfluid configurations is provided by the gravitational-wave driven instability of the so-called r-modes [24,25]. A study of the r-modes in a relativistic superfluid star would be a natural generalisation of recent work of Lockitch, Andersson and Friedman [26] and would provide further insights into the details of such modes in realistic neutron stars. This is a crucial issue given the possibility that the gravitational waves from the r-modes are potentially detectable by LIGO II [27,14].…”