Where are ther‐Modes of Isentropic Stars?

Abstract

Almost none of the r-modes ordinarily found in rotating stars exist, if the star and its perturbations obey the same one-parameter equation of state; and rotating relativistic stars with one-parameter equations of state have no pure r-modes at all, no modes whose limit, for a star with zero angular velocity, is a perturbation with axial parity. Similarly (as we show here) rotating stars of this kind have no pure g-modes, no modes whose spherical limit is a perturbation with polar parity and vanishing perturbed pressure and density. Where have these modes gone?In spherical stars of this kind, r-modes and g-modes form a degenerate zerofrequency 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 axial and polar parts. Lindblom and Ipser have recently found these hybrid modes in an analytic study of the Maclaurin spheroids. Since the hybrid modes have a rotational restoring force, they call them "rotation modes" or "generalized r-modes".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 lowest value of l that contributes to its velocity field is axial or polar. We numerically compute these modes for slowly rotating polytropes and for Maclaurin spheroids, using a straightforward method that appears to be novel and robust. Timescales for the gravitational-wave driven instability and for viscous damping are computed using assumptions appropriate to neutron stars. The instability to nonaxisymmetric modes is, as expected, dominated by the l = m r-modes with simplest radial dependence, the only modes which retain their axial character in isentropic models; for relativistic isentropic stars, these l = m modes must also be replaced by hybrids of the kind considered here.1 An appendix in this paper incorrectly claims that no l = m r-modes exist, based on an incorrect assumption about their radial behavior 2 Based on this equation, Kojima has argued that the spectrum is continuous, and his argument has been made precise in a recent paper of Beyer and Kokkotas (1999) (See also Kojima and Hosonuma 1999). Beyer and Kokkotas, however, also point out that the continuous spectrum they find may be an artifact of the fact that the imaginary part of the frequency vanishes in the slow-rotation limit.

Nonlinear coupling network to simulate the development of the r-mode instability in neutron stars. I. Construction

“…(w is related to the κ frequency parameter of Ipser and Lindblom [13] by w = κ/2.) Equation (14) for the gravitational potential is Laplace's equation with a discontinuity on the boundary. The delta function describes the surface of the Maclaurin spheroid.…”

“…With two elegant coordinate transformations, Bryan transformed equations (13) and (14) so that they become separable and allow the boundary conditions to be imposed. On the surface of the star these eigenfunctions correspond to waves that preserve the stellar volume.…”

“…We use the coefficient proposed by [14][19] for hot neutron star matter, and explore temperature ranges varying from 10 6 K to 10 9 K:…”

“…The rate of dissipation of energy due to bulk viscosity [14] can be written using equations (I14) and (I17) of [10] as follows 5 1 m=0 m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10 m=11 m=12 m=13 m=14 m=15 m=16 m=17 m=18 …”

Eigenmode frequency distribution of rapidly rotating neutron stars

“…Still, even in the relativistic case with rotation, it is possible to assign an overall parity to a mode by looking at the dominant term in an infinite series representation over the harmonic index ℓ. Lockitch & Friedman [8] call these modes axial-led or polar-led. If the spherical harmonic with lowest ℓ that contributes to the velocity field is axial, then the mode was refered to as 'generalized r-mode' [11].…”