If a protoneutron star has a sufficiently high rotation rate (larger than 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 [123, 78]: 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 [139, 140].
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 [82, 83, 124, 125, 126], but the determination of neutral f -modes in full relativity [42, 92] shows that relativity enhances the instability, allowing it to occur in stars with smaller rotation rates than previously thought.
In the Newtonian limit, neutral modes have been determined for several polytropic EOSs [85, 80, 81, 125]. 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 . 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 [129, 130]. 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, 115] has shown that general relativity tends to strengthen the CFS-instability. Compared to their Newtonian counterparts, critical angular velocity ratios (where and , are the mass and radius of the nonrotating star in the sequence), are lowered by as much as for stars obeying the N =1 polytropic EOS (for which the instability occurs only for modes in the post-Newtonian approximation).
In full general relativity, neutral modes have been determined for polytropic EOSs of by Stergioulas and Friedman [42, 92], using a new numerical scheme. The scheme completes the Eulerian formalism developed by Ipser and Lindblom in the Cowling approximation, (where was neglected) [124], by finding an appropriate gauge in which the time-independent perturbation equations can be solved numerically for . Because linear perturbations have a gauge freedom, four out of ten components of 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 and the perturbed equation of energy conservation for a scalar function .
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 . 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 . Defining an inner product , 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 (25) 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 , where is the angular velocity at the mass-shedding limit at same central energy density, drops by as much as 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 () 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 [131].
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 [92].
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 . For neutron stars, the mode becomes unstable at 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 for stars and for maximum mass stars. This is to be contrasted with the Newtonian value of . The empirical formula
where 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 %, 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 [133] 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 [134]. 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].
Modes with different radial eigenfunctions can be computed at order [112, 138]. According to (28), r -modes with m >0 are prograde () with respect to a distant observer but retrograde () 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 K 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 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 in the angular velocity of the star, and the resulting expressions for the timescales, given in Kokkotas and Stergioulas [142], 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 [144] (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.
Since and , , a mode will grow only if is shorter than the viscous timescales, so that .
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 normal neutron star matter, shear viscosity is dominated by neutron-neutron scattering with a temperature dependence of [145]. Computations in the Newtonian limit and post-Newtonian approximation show that the CFS-instability is suppressed for K - K [82, 83, 125, 115].
If neutrons become a superfluid below a transition temperature , then mutual friction, which is caused by the scattering of electrons off the cores of neutron vortices, can completely suppress the instability for . The superfluid transition temperature depends on the theoretical model for superfluidity and lies in the range K - K [146].
In a pulsating fluid that undergoes compression and expansion, the weak interaction requires a relatively long time to re-establish equilibrium. This creates a phase lag between density and pressure perturbations, which results in a large bulk viscosity [147]. The bulk viscosity due to this effect can suppress the CFS-instability only for temperatures for which matter has become transparent to neutrinos [134], [148]. It has been proposed that for K, matter will be opaque to neutrinos, and the neutrino phase space could be blocked ([134] See also [148].). In this case, bulk viscosity will be too weak to suppress the instability, but a more detailed study is needed.
In conclusion, the available Newtonian computations indicate that the CFS-instability in f -modes is effective in nascent neutron stars for temperatures between K and K and possibly also above 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.
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. [131], for the most relativistic stars, the viscosity driven bar mode can become unstable only if N <0.55. For 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.
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 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 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.
Acknowledgments
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.
Rotating Stars in Relativity
Nikolaos Stergioulas http://www.livingreviews.org/lrr-1998-8 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |