r-mode instability: Analytical solution with gravitational radiation reaction (original) (raw)

Nonlinear Evolution of the r-Modes in Neutron Stars

Physical Review Letters, 2001

The evolution of a neutron-star r-mode driven unstable by gravitational radiation (GR) is studied here using numerical solutions of the full nonlinear fluid equations. The dimensionless amplitude of the mode grows to order unity before strong shocks develop which quickly damp the mode. In this simulation the star loses about 40% of its initial angular momentum and 50% of its rotational kinetic energy before the mode is damped. The nonlinear evolution causes the fluid to develop strong differential rotation which is concentrated near the surface and poles of the star.

On the r-mode spectrum of relativistic stars: the inclusion of the radiation reaction

Monthly Notices of the Royal Astronomical Society, 2002

We consider both mode calculations and time evolutions of axial r-modes for relativistic uniformly rotating non-barotropic neutron stars, using the slow-rotation formalism, in which rotational corrections are considered up to linear order in the angular velocity Ω. We study various stellar models, such as uniform density models, polytropic models with different polytropic indices n, and some models based on realistic equations of state. For weakly relativistic uniform density models, and polytropes with small values of n, we can recover the growth times predicted from Newtonian theory when standard multipole formulae for the gravitational radiation are used. However, for more compact models, we find that relativistic linear perturbation theory predicts a weakening of the instability compared to the Newtonian results. When turning to polytropic equations of state, we find that for certain ranges of the polytropic index n, the r-mode disappears, and instead of a growth, the time evolutions show a rapid decay of the amplitude. This is clearly at variance with the Newtonian predictions. It is, however, fully consistent with our previous results obtained in the low-frequency approximation.

Bar Mode Instability in Relativistic Rotating Stars: A Post‐Newtonian Treatment

The Astrophysical Journal Supplement Series, 1998

We construct analytic models of incompressible, uniformly rotating stars in post-Newtonian (PN) gravity and evaluate their stability against nonaxisymmetric bar modes. We model the PN configurations by homogeneous triaxial ellipsoids and employ an energy variational principle to determine their equilibrium shape and stability. The spacetime metric is obtained by solving Einstein's equations of general relativity in 3+1 ADM form. We use an approximate subset of these equations well-suited to numerical integration in the case of strong field, three dimensional configurations in quasi-equilibrium. However, the adopted equations are exact at PN order, where they admit an analytic solution for homogeneous ellipsoids. We obtain this solution for the metric, as well as analytic functionals for the conserved global quantities, M , M 0 and J.

Analytic description of the r-mode instability in uniform density stars

We present an analytic description of the rmode instability in newly-born neutron stars, using the approximation of uniform density. Our computation is consistently accurate to second order in the angular velocity of the star. We obtain formulae for the growth-time of the instability due to gravitational-wave emission, for both current and mass multipole radiation and for the damping timescale, due to viscosity. The l = m = 2 current-multipole radiation dominates the timescale of the instability. We estimate the deviation of the second order accurate results from the lowest order approximation and show that the uncertainty in the equation of state has only a small effect on the onset of the r-mode instability. The viscosity coefficients and the cooling process in newly-born neutron stars are, at present, uncertain and our analytic formaulae enables a quick check of such effects on the development of the instability.

Approximate equation relevant to axial oscillations on slowly rotating relativistic stars

Physical Review D, 2000

Axial oscillations relevant to the r-mode instability are studied with the slow rotation formalism in general relativity. The approximate equation governing the oscillations is derived with second-order rotational corrections. The equation contains an effective ''viscositylike'' term, which originates from coupling to the polar g-mode displacements. The term plays a crucial role on the resonance point, where the disturbance on the rotating stars satisfies a certain condition at the lowest order equation. The effect is significant for newly born hot neutron stars, which are expected to be subject to the gravitational radiation driven instability of the r mode.

Dynamical bar-mode instability in rotating and magnetized relativistic stars

We present three-dimensional simulations of the dynamical bar-mode instability in magnetized and differentially rotating stars in full general relativity. Our focus is on the effects that magnetic fields have on the dynamics and the onset of the instability. In particular, we perform ideal-magnetohydrodynamics simulations of neutron stars that are known to be either stable or unstable against the purely hydrodynamical instability, but to which a poloidal magnetic field in the range of 10^(14)-10^(16) G is superimposed initially. As expected, the differential rotation is responsible for the shearing of the poloidal field and the consequent linear growth in time of the toroidal magnetic field. The latter rapidly exceeds in strength the original poloidal one, leading to a magnetic-field amplification in the the stars. Weak initial magnetic fields, i.e. ≲1015 G, have negligible effects on the development of the dynamical bar-mode instability, simply braking the stellar configuration via magnetic-field shearing, and over a timescale for which we derived a simple algebraic expression. On the other hand, strong magnetic fields, i.e. ≳10^(16) G, can suppress the instability completely, with the precise threshold being dependent also on the amount of rotation. As a result, it is unlikely that very highly magnetized neutron stars can be considered as sources of gravitational waves via the dynamical bar-mode instability.

Gravity modes in rapidly rotating stars

Astronomy and Astrophysics, 2010

Context. CoRoT and Kepler missions are now providing high-quality asteroseismic data for a large number of stars. Among intermediate-mass and massive stars, fast rotators are common objects. Taking the rotation effects into account is needed to correctly understand, identify, and interpret the observed oscillation frequencies of these stars. A classical approach is to consider the rotation as a perturbation. Aims. In this paper, we focus on gravity modes, such as those occurring in γ Doradus, slowly pulsating B (SPB), or Be stars. We aim to define the suitability of perturbative methods. Methods. With the two-dimensional oscillation program (TOP), we performed complete computations of gravity modes-including the Coriolis force, the centrifugal distortion, and compressible effects-in 2D distorted polytropic models of stars. We started with the modes = 1, n = 1−14, and = 2−3, n = 1−5, 16−20 of a nonrotating star, and followed these modes by increasing the rotation rate up to 70% of the break-up rotation rate. We then derived perturbative coefficients and determined the domains of validity of the perturbative methods. Results. Second-order perturbative methods are suited to computing low-order, low-degree mode frequencies up to rotation speeds ∼100 km s −1 for typical γ Dor stars or ∼150 km s −1 for B stars. The domains of validity can be extended by a few tens of km s −1 thanks to the third-order terms. For higher order modes, the domains of validity are noticeably reduced. Moreover, perturbative methods are inefficient for modes with frequencies lower than the Coriolis frequency 2Ω. We interpret this failure as a consequence of a modification in the shape of the resonant cavity that is not taken into account in the perturbative approach.

Quasi‐normal Modes of Rotating Relativistic Stars: Neutral Modes for Realistic Equations of State

The Astrophysical Journal, 1999

We compute zero-frequency (neutral) quasi-normal f -modes of fully relativistic and rapidly rotating neutron stars, using several realistic equations of state (EOSs) for neutron star matter. The zero-frequency modes signal the onset of the gravitational radiation-driven instability. We find that the l = m = 2 (bar) f -mode is unstable for stars with gravitational mass as low as 1.0 − 1.2M ⊙ , depending on the EOS. For 1.4M ⊙ neutron stars, the bar mode becomes unstable at 83% − 93% of the maximum allowed rotation rate. For a wide range of EOSs, the bar mode becomes unstable at a ratio of rotational to gravitational energies T /W ∼ 0.07 − 0.09 for 1.4M ⊙ stars and T /W ∼ 0.06 for maximum mass stars. This is to be contrasted with the Newtonian value of T /W ∼ 0.14. We construct the following empirical formula for the critical value of T /W for the bar mode, (T /W ) 2 = 0.115 − 0.048 M/M sph max , which is insensitive to the EOS to within 4 − 6%. This formula yields an estimate for the neutral mode sequence of the bar mode as a function only of the star's mass, M, given the maximum allowed mass, M sph max , of a nonrotating neutron star. The recent discovery of the fast millisecond pulsar in the supernova remnant N157B, supports the suggestion that a fraction of proto-neutron stars are born in a supernova collapse with very large initial angular momentum. If some neutron stars are born in an accretion-induced-collapse of a white dwarf, then they will also have very large angular momentum at birth. Thus, in a fraction of newly born neutron stars the instability is a promising source of continuous gravitational waves. It could also play a major role in the rotational evolution (through the emission of angular momentum) of merged binary neutron stars, if their post-merger angular momentum exceeds the maximum allowed to form a Kerr black hole.

Gravitational-Wave Instabilities in Rotating Compact Stars

Galaxies

It is generally accepted that the limit on the stable rotation of neutron stars is set by gravitational-radiation reaction (GRR) driven instabilities, which cause the stars to emit gravitational waves that carry angular momentum away from them. The instability modes are moderated by the shear viscosity and the bulk viscosity of neutron star matter. Among the GRR instabilities, the f-mode instability plays a historically predominant role. In this work, we determine the instability periods of this mode for three different relativistic models for the nuclear equation of state (EoS) named DD2, ACB4, and GM1L. The ACB4 model for the EoS accounts for a strong first-order phase transition that predicts a new branch of compact objects known as mass-twin stars. DD2 and GM1L are relativistic mean field (RMF) models that describe the meson-baryon coupling constants to be dependent on the local baryon number density. Our results show that the f-mode instability associated with m=2 sets the limi...