3 Oscillations and Stability3.4 Nonaxisymmetric Perturbations

3.5 Nonaxisymmetric Instabilities

3.5.1 Introduction

Rotating cold neutron stars, detected as pulsars, have a remarkably stable rotation period. But, at birth, or during accretion, rapidly rotating neutron stars can be subject to various nonaxisymmetric instabilities, which will affect the evolution of their rotation rate

If a protoneutron star has a sufficiently high rotation rate (larger than tex2html_wrap_inline2397 for uniformly rotating, constant density Maclaurin spheroids), it will be subject to a dynamical instability driven by hydrodynamics and gravity. Through the l =2 mode, the instability will deform the star into a bar shape. This highly nonaxisymmetric configuration will emit strong gravitational waves with frequencies in the kHz regime. The development of the instability and the resulting waveform have been computed numerically in the context of Newtonian gravity and hydrodynamics by Houser et al. [122].

At lower rotation rates, the star can become unstable to secular nonaxisymmetric instabilities, driven by gravitational radiation or viscosity. Gravitational radiation drives a nonaxisymmetric instability when a mode that is retrograde with respect to the star appears as prograde to a distant observer, via the Chandrasekhar-Friedman-Schutz (CFS) mechanism [123Jump To The Next Citation Point In The Article, 78Jump To The Next Citation Point In The Article]: A mode that is retrograde in the corotating frame has negative angular momentum, because the perturbed star has less angular momentum than the unperturbed one. If, to a distant observer, the mode appears prograde, it removes positive angular momentum from the star, and thus the angular momentum of the mode becomes increasingly negative.

The instability evolves on a secular timescale, during which the star loses angular momentum via the emitted gravitational waves. When the star rotates slowly enough, the mode becomes stable, and the instability proceeds on the longer timescale of the next unstable mode, unless it is suppressed by viscosity.

Neglecting viscosity, the CFS-instability is generic in rotating stars for both polar and axial modes. For polar modes, the instability occurs only above some critical angular velocity, where the frequency of the mode goes through zero in the inertial frame. The critical angular velocity is smaller for increasing mode number l . Thus, there will always be a high enough mode number l, for which a slowly rotating star will be unstable. Axial modes are generically unstable in all rotating stars, since the mode has zero frequency in the inertial frame when the star is nonrotating [139Jump To The Next Citation Point In The Article, 140Jump To The Next Citation Point In The Article].

The shear and bulk viscosity of neutron star matter is able to suppress the growth of the CFS-instability, except when the star passes through a certain temperature window. In Newtonian gravity, it appears that the polar mode CFS-instability can occur only in nascent neutron stars that rotate close to the mass-shedding limit [82Jump To The Next Citation Point In The Article, 83Jump To The Next Citation Point In The Article, 124Jump To The Next Citation Point In The Article, 125Jump To The Next Citation Point In The Article, 126], but the determination of neutral f -modes in full relativity [42Jump To The Next Citation Point In The Article, 92Jump To The Next Citation Point In The Article] shows that relativity enhances the instability, allowing it to occur in stars with smaller rotation rates than previously thought.

3.5.2 CFS-Instability of Polar Modes

The existence of the CFS-instability in rotating stars was first demonstrated by Chandrasekhar [123] in the case of the l =2 mode in uniformly rotating, constant density Maclaurin spheroids. Friedman and Schutz [78] show that this instability also appears in compressible stars and that all rotating self-gravitating perfect fluid configurations are generically unstable to the emission of gravitational waves. In addition, they find that a nonaxisymmetric mode becomes unstable when its frequency vanishes in the inertial frame. Thus, zero-frequency outgoing-modes in rotating stars are neutral (marginally stable).

In the Newtonian limit, neutral modes have been determined for several polytropic EOSs [85, 80, 81Jump To The Next Citation Point In The Article, 125Jump To The Next Citation Point In The Article]. The instability first sets in through l = m modes. Modes with larger l become unstable at lower rotation rates, but viscosity limits the interesting ones to tex2html_wrap_inline2413 . For an N =1 polytrope, the critical values of T / W for the l =3,4 and 5 modes are 0.079, 0.058 and 0.045 respectively; these values become smaller for softer polytropes.

The l = m =2 ``bar'' mode behaves considerably differently than the other modes. Its critical T / W ratio is 0.14, and it is almost independent of the polytropic index. Since soft EOSs cannot produce models with high T/W values, the bar mode instability appears only for stiff Newtonian polytropes of tex2html_wrap_inline2425 [129Jump To The Next Citation Point In The Article, 130Jump To The Next Citation Point In The Article]. In addition, the viscosity driven bar mode appears at the same critical T / W ratio as the bar mode driven by gravitational radiation (We will see later that this is no longer true in general relativity.).

The post-Newtonian computation of neutral modes by Cutler and Lindblom [114, 115Jump To The Next Citation Point In The Article] has shown that general relativity tends to strengthen the CFS-instability. Compared to their Newtonian counterparts, critical angular velocity ratios tex2html_wrap_inline2429 (where tex2html_wrap_inline2431 and tex2html_wrap_inline2433, tex2html_wrap_inline2435 are the mass and radius of the nonrotating star in the sequence), are lowered by as much as tex2html_wrap_inline2437 for stars obeying the N =1 polytropic EOS (for which the instability occurs only for tex2html_wrap_inline2441 modes in the post-Newtonian approximation).

In full general relativity, neutral modes have been determined for polytropic EOSs of tex2html_wrap_inline2443 by Stergioulas and Friedman [42, 92Jump To The Next Citation Point In The Article], using a new numerical scheme. The scheme completes the Eulerian formalism developed by Ipser and Lindblom in the Cowling approximation, (where tex2html_wrap_inline2213 was neglected) [124], by finding an appropriate gauge in which the time-independent perturbation equations can be solved numerically for tex2html_wrap_inline2213 . Because linear perturbations have a gauge freedom, four out of ten components of tex2html_wrap_inline2213 are fixed by the choice of gauge. In the Ipser and Lindblom scheme, the perturbed Euler equations are solved analytically. A complete neutral mode solution of the perturbation equations is then determined by setting the frequency in the inertial frame equal to zero and solving six perturbed field equations for tex2html_wrap_inline2213 and the perturbed equation of energy conservation for a scalar function tex2html_wrap_inline2453 .

The six perturbed field equations in the gauge of Stergioulas and Friedman are of different types. Three are second order ODEs, two are elliptic, and the other one is parabolic. Their solutions vanish at the center, at infinity and on the axis of symmetry, while they are either odd or even under reflection in the equatorial plane. The six equations, although of different type, are solved simultaneously on a two-dimensional grid, which extends to infinity by a redefinition of the radial variable. Solutions of the perturbed field equations are obtained for a set of trial functions tex2html_wrap_inline2455 . In the Newtonian limit, it was found that the real eigenfunctions can be expanded accurately in terms of these trial functions [81].

The remaining equation to be satisfied, the perturbed energy conservation equation, can be represented schematically as a linear operator L on the eigenfunction tex2html_wrap_inline2453 . Defining an inner product tex2html_wrap_inline2461, for the set of trial functions, the perturbed energy conservation equation is satisfied, when


Using this criterion, one starts with slowly rotating configurations and increases the angular velocity of the star until (25Popup Equation) is satisfied and a complete neutral mode solution is obtained.

The determination of neutral modes for N =1.0, 1.5 and 2.0 relativistic polytropes shows that relativity significantly strengthens the instability (which was already indicated in the post-Newtonian approximation). For the N =1.0 polytrope, the critical angular velocity ratio tex2html_wrap_inline2471, where tex2html_wrap_inline2473 is the angular velocity at the mass-shedding limit at same central energy density, drops by as much as tex2html_wrap_inline2385 for the most relativistic configuration. This is a large decrease compared to the Newtonian values, which significantly moves the onset of the instability away from the mass-shedding limit and which strengthens it with respect to the damping effect of viscosity.

A surprising result, which was not detected in the post-Newtonian approximation, is that the l = m =2 bar mode is unstable for relativistic polytropes of index N =1.0. The classical Newtonian result for the onset of the bar mode instability (tex2html_wrap_inline2481) is replaced by


in general relativity.

Also, in relativistic stars, the onset of the gravitational radiation driven bar mode is different from the onset of the viscosity driven bar mode. While in the Newtonian limit the two bar modes occur at the same critical rotation ratio [79], relativity strengthens the gravitational radiation instability, allowing softer configurations to become unstable, and suppresses the viscosity driven instability allowing it to occur only for very stiff EOSs [131Jump To The Next Citation Point In The Article].

An independent determination of the onset of the CFS-instability in the relativistic Cowling approximation by Yoshida and Eriguchi [121] agrees qualitatively with the conclusions in [92Jump To The Next Citation Point In The Article].

Morsink, Stergioulas and Blattning [132] extend the method presented in [92] to a wide range of realistic equations of state (which usually have a stiff high density region, corresponding to polytropes of index N =0.5-0.7) and find that the l = m =2 bar mode becomes unstable for stars with gravitational mass as low as tex2html_wrap_inline2487 . For tex2html_wrap_inline2117 neutron stars, the mode becomes unstable at tex2html_wrap_inline2491 of the maximum allowed rotation rate. For a wide range of equations of state, the l = m =2 f -mode becomes unstable at a ratio of rotational to gravitational energies tex2html_wrap_inline2497 for tex2html_wrap_inline2117 stars and tex2html_wrap_inline2501 for maximum mass stars. This is to be contrasted with the Newtonian value of tex2html_wrap_inline2503 . The empirical formula


where tex2html_wrap_inline2505 is the maximum mass for a spherical star allowed by a given equation of state, gives the critical value of T / W for the bar f -mode instability, with an accuracy of tex2html_wrap_inline2511 %, independent of the equation of state.

Conservation of angular momentum and the inferred initial period (assuming magnetic braking) of 6-9ms for the X-ray pulsar in the supernova remnant N157B [133Jump To The Next Citation Point In The Article] suggests that a fraction of neutron stars may be born with very large rotational energies. The f -mode bar CFS-instability thus appears as a promising source for the planned gravitational wave detectors [134Jump To The Next Citation Point In The Article]. It could also play a major role in the rotational evolution, through the emission of gravitational waves, of merged binary neutron stars, if their post-merger angular momentum exceeds the maximum allowed to form a Kerr black hole [135].

3.5.3 CFS-Instability of Axial Modes

In nonrotating stars, axial fluid modes are degenerate at zero-frequency, but in rotating stars they have nonzero frequency and are called r -modes in the Newtonian limit [136, 137]. To tex2html_wrap_inline2325, their frequency in the inertial frame is


Modes with different radial eigenfunctions can be computed at order tex2html_wrap_inline2521 [112, 138]. According to (28Popup Equation), r -modes with m >0 are prograde (tex2html_wrap_inline2527) with respect to a distant observer but retrograde (tex2html_wrap_inline2529) in the comoving frame for all values of the angular velocity. Thus, r -modes in relativistic stars are generically unstable to the emission of gravitational waves via the CFS-instability, which was first discovered by Andersson [139], in the case of slowly rotating, relativistic stars. This result is confirmed analytically by Friedman and Morsink [140], who show that the canonical energy of the modes is negative.

Two independent computations in the Newtonian Cowling approximation, by Andersson, Kokkotas and Schutz [127] and Lindblom, Owen and Morsink [141] show that viscosity is not able to damp the r -mode instability in rotating stars. In a temperature window of tex2html_wrap_inline2535 K tex2html_wrap_inline2537 K, the growth time of the l = m =2 mode becomes shorter than the shear or bulk viscosity damping time at a critical rotation rate that is roughly one tenth the maximum allowed angular velocity of uniformly rotating stars. The gravitational radiation is dominated by the current quadrupole term. These results suggest that a rapidly rotating proto-neutron star will spin down to Crab-like rotation rates within one year of its birth, because of the r -mode instability. The current uncertainties in the viscosity and superfluid mutual friction damping times make this scenario also consistent with somewhat higher initial spins, like the suggested initial spin of 6-9ms of the X-ray pulsar in the supernova remnant N157B [133]. Millisecond pulsars with periods less than tex2html_wrap_inline2545 ms can then only form after the accretion-induced spin-up of old pulsars and not in the accretion-induced collapse of a white dwarf.

The precise limit on the angular velocity of newly-born neutron stars will depend on several factors, such as the strength of the bulk viscosity, the cooling process, the superfluid mutual friction etc. In the uniform density approximation, the r -mode instability can be studied analytically to tex2html_wrap_inline2037 in the angular velocity of the star, and the resulting expressions for the timescales, given in Kokkotas and Stergioulas [142Jump To The Next Citation Point In The Article], can be used to study the effect of such factors on the instability. In [142] it is also shown that the minimum critical angular velocity for the onset of the r -mode instability is rather insensitive to the choice of equation of state.

A first study on the issue of detectability of gravitational waves from the r -mode instability is presented in [144Jump To The Next Citation Point In The Article] (See section 3.5.6 .), while Andersson, Kokkotas and Stergioulas [143] study the relevance of the r -mode instability in limiting the spin of recycled millisecond pulsars.

3.5.4 Effect of Viscosity on CFS-Instability

In the previous sections, we have discussed the growth of the CFS-instability driven by gravitational radiation in an otherwise nondissipative star. The effect of neutron star matter viscosity on the dynamical evolution of nonaxisymmetric perturbations can be considered separately, when the timescale of the viscosity is much longer than the oscillation timescale. If tex2html_wrap_inline2557 is the computed growth rate of the instability in the absence of viscosity, and tex2html_wrap_inline2559, tex2html_wrap_inline2561 are the timescales of shear and bulk viscosity, then the total timescale of the perturbation is


Since tex2html_wrap_inline2563 and tex2html_wrap_inline2561, tex2html_wrap_inline2567, a mode will grow only if tex2html_wrap_inline2557 is shorter than the viscous timescales, so that tex2html_wrap_inline2571 .

The shear and bulk viscosity are sensitive to several factors. We give here a summary of what is known to date from Newtonian and post-Newtonian computations:

In the neutrino transparent regime, the effect of bulk viscosity on the instability depends crucially on the proton fraction tex2html_wrap_inline2589 . If tex2html_wrap_inline2589 is lower than a critical value (tex2html_wrap_inline2593), only modified URCA processes are allowed, and bulk viscosity limits, but does not suppress completely, the instability [82, 83, 125]. For most modern EOSs, however, the proton fraction is larger than tex2html_wrap_inline2593 at sufficiently high densities [151], allowing direct URCA processes to take place. In this case, depending on the EOS and the central density of the star, the bulk viscosity could almost completely suppress the CFS-instability in the neutrino transparent regime [152], (but it will probably still not affect it for temperatures tex2html_wrap_inline2587 K).

In conclusion, the available Newtonian computations indicate that the CFS-instability in f -modes is effective in nascent neutron stars for temperatures between tex2html_wrap_inline1969 K and tex2html_wrap_inline2603 K and possibly also above tex2html_wrap_inline2603 K, if the star is opaque to neutrinos and the bulk viscosity is weak. If direct URCA reactions do participate in the cooling process, it appears that the instability can grow only for temperatures for which the star is opaque to neutrinos. Since the neutral mode computations in fully relativistic stars show that relativity strengthens the instability, the above conclusion should also hold in relativistic stars.

The uncertainties regarding the effect of viscosity on the CFS-instability in realistic neutron stars will be greatly reduced by the construction of mode eigenfunctions for fully relativistic, rotating stars.

3.5.5 Viscosity-Driven Instability

A different type of nonaxisymmetric instability in rotating stars is that driven by viscosity, which breaks the circulation of the fluid [153, 129Jump To The Next Citation Point In The Article]. The instability is suppressed by gravitational radiation, so it can act only in cold neutron stars that become rapidly rotating by accretion-induced spin-up. The instability sets in when the frequency of an l =- m mode goes through zero in the rotating frame. In contrast to the CFS-instability, the viscosity-driven instability is not generic in rotating stars. The m =2 mode becomes unstable at a high rotation rate for very stiff stars, and higher m -modes become unstable at larger rotation rates.

In Newtonian polytropes, the instability occurs only for stiff polytropes of index N <0.808 [129, 130]. For relativistic models, the situation for the instability becomes worse, since relativistic effects tend to suppress the viscosity instability (while they strengthen the CFS-instability). According to recent results by Bonazzola et al. [131Jump To The Next Citation Point In The Article], for the most relativistic stars, the viscosity driven bar mode can become unstable only if N <0.55. For tex2html_wrap_inline2117 stars, the instability is present for N <0.67.

These results are based on an approximate computation of the instability in which one perturbs an axisymmetric and stationary configuration and studies its evolution by constructing a series of triaxial quasi-equilibrium configurations. During the evolution only the dominant nonaxisymmetric terms are taken into account.

The method presented in [131] is an improvement (taking into account nonaxisymmetric terms of higher order) on an earlier method by the same authors [148]. Although the method is approximate, its results indicate that the viscosity-driven instability is likely to be absent in most relativistic stars, unless the EOS turns out to be unexpectedly stiff.

An investigation of the viscosity-driven bar mode instability, using incompressible, uniformly rotating triaxial ellipsoids in the post-Newtonian approximation, by Shapiro and Zane [149], also finds that the relativistic effects weaken the instability.

3.5.6 Gravitational Radiation from CFS-Instability 

The CFS-instability can limit the maximum angular velocity of nascent neutron stars, but it is also a mechanism for the generation of gravitational waves that could be strong enough to be detected by the planned gravitational wave detectors.

Lai and Shapiro [134] have studied the development of the f -mode instability using Newtonian ellipsoidal rotating models [154, 155]. They consider the case where a rapidly rotating neutron star is created in a core collapse. After a brief dynamical phase, the protoneutron star becomes axisymmetric but secularly unstable. The instability deforms the star into a nonaxisymmetric configuration via the l =2 bar mode. Since the star loses angular momentum via the emission of gravitational waves, it spins-down until it becomes secularly stable.

The frequency of the waves sweeps downward from a few hundred Hz to zero, passing through LIGO's ideal sensitivity band. A rough estimate of the wave amplitude shows that, at tex2html_wrap_inline2625 Hz, the gravitational waves from the CFS-instability could be detected out to the distance of 140Mpc by the advanced LIGO detector. This result is very promising, especially since for relativistic stars the instability will be stronger than the present Newtonian estimate.

The recently discovered CFS-instability in r -modes is also an important source of gravitational waves. Owen et al. [144] model the development of the instability and the evolution of the neutron star during its spin-down phase. The evolution suggests that a neutron star formed in the Virgo cluster could be detected by the advanced LIGO and VIRGO gravitational wave detectors, with an amplitude signal-to-noise ratio that could be as large as about 8, if near-optimal data analysis techniques are developed. Assuming a substantial fraction of neutron stars are born with spin frequencies near their maximum values, the stochastic background of gravitational waves produced by the r -mode radiation from neutron star formation throughout the universe is shown to have an energy density of about tex2html_wrap_inline2631 of the cosmological closure density, in the range 20 Hz to 1 kHz. This radiation is potentially detectable by the advanced LIGO as well.

In newly born stars or in the post-merger objects in binary neutron star mergers, rotating close to the Kepler limit, both the f and r modes will be unstable. Relativistic computations of growth times in rapidly rotating stars or even nonlinear evolutions, are needed to determine which mode will be strongest.


I am grateful to K. Kokkotas and J. L. Friedman for critical reading of the manuscript and for helpful suggestions. I would also like to thank the second anonymous referee for his many useful comments. I am indebted to the Departments of Physics of the Aristotle University of Thessaloniki, Greece and of the University of Wisconsin-Milwaukee, USA, for their support during my military service. I am also grateful for the generous hospitality of the Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute) in Potsdam, Germany, where part of this paper was completed.

3 Oscillations and Stability3.4 Nonaxisymmetric Perturbations

image Rotating Stars in Relativity
Nikolaos Stergioulas
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