Nonaxisymmetric instabilities

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 proto-neutron star has a sufficiently high rotation rate (so that, e.g., $T ∕W > 0.27$ in the case of 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 by Houser et al. [155 ] and in full general relativity by Shibata et al. [273 ] (see Section 4.1.3).

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 in a frame corotating with the star appears as prograde to a distant inertial observer, via the Chandrasekhar-Friedman-Schutz (CFS) mechanism [60 , 118 ]: 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, for a distant observer, the mode is 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 more slowly than a critical value, 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. Many of the hybrid inertial modes (and in particular the relativistic r-mode) are generically unstable in all rotating stars, since the mode has zero frequency in the inertial frame when the star is nonrotating [6 , 117 ].

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 [160 , 159 , 161 , 326 , 203 ], but the computation of neutral f -modes in full relativity [292 , 295 ] shows that relativity enhances the instability, allowing it to occur in stars with smaller rotation rates than previously thought.

Going further A numerical method for the analysis of the ergosphere instability in relativistic stars, which could be extended to nonaxisymmetric instabilities of fluid modes, is presented by Yoshida and Eriguchi in [327 ].

3.5.2 CFS instability of polar modes

The existence of the CFS instability in rotating stars was first demonstrated by Chandrasekhar [60 ] in the case of the $l = 2$ mode in uniformly rotating, uniform density Maclaurin spheroids. Friedman and Schutz [118 ] 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).

Figure 7: The $l = m$ neutral f -mode sequences for EOS A. Shown are the ratio of rotational to gravitational energy $T ∕W$ (upper panel) and the ratio of the critical angular velocity $Ωc$ to the angular velocity at the mass-shedding limit for uniform rotation (lower panel) as a function of gravitational mass. The solid curves are the neutral mode sequences for $l = m = 2,3,4$ , and $5$ (from top to bottom), while the dashed curve in the upper panel corresponds to the mass-shedding limit for uniform rotation. The $l = m = 2$ f -mode becomes CFS-unstable even at 85% of the mass-shedding limit, for $1.4M ⊙$ models constructed with this EOS. (Figure 2 of Morsink, Stergioulas, and Blattning [230

].)

In the Newtonian limit, neutral modes have been determined for several polytropic EOSs [156 , 217 , 158 , 326 ]. 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 $l ≤ 5$ . 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, and these values become smaller for softer polytropes. The $l = m = 2$ “bar” mode has a critical $T∕W$ ratio of 0.14 that 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 $N ≤ 0.808$ [164 , 282 ]. In addition, the viscosity-driven bar mode appears at the same critical $T ∕W$ ratio as the bar mode driven by gravitational radiation [162 ] (we will see later that this is no longer true in general relativity).

The post-Newtonian computation of neutral modes by Cutler and Lindblom [79 , 198 ] has shown that general relativity tends to strengthen the CFS instability. Compared to their Newtonian counterparts, critical angular velocity ratios $Ωc ∕Ω0$ (where $3 1∕2 Ω0 = (3M0 ∕4R 0)$ , and $M0$ , $R0$ are the mass and radius of the nonrotating star in the sequence) are lowered by as much as 10% for stars obeying the $N = 1$ polytropic EOS (for which the instability occurs only for $l = m ≥ 3$ modes in the post-Newtonian approximation).

In full general relativity, neutral modes have been determined for polytropic EOSs of $N ≥ 1.0$ by Stergioulas and Friedman [292 , 295 ], using a new numerical scheme. The scheme completes the Eulerian formalism developed by Ipser and Lindblom in the Cowling approximation (where $δg ab$ was neglected) [161 ], by finding an appropriate gauge in which the time independent perturbation equations can be solved numerically for $δgab$ . The computation of neutral modes for polytropes of $N = 1.0$ , 1.5, and 2.0 shows that relativity significantly strengthens the instability. For the $N = 1.0$ polytrope, the critical angular velocity ratio $Ωc∕ΩK$ , where $ΩK$ is the angular velocity at the mass-shedding limit at same central energy density, is reduced by as much as 15% for the most relativistic configuration (see Figure 7 ). A surprising result (which was not found in computations that used the post-Newtonian approximation) is that the $l = m = 2$ bar mode is unstable even for relativistic polytropes of index $N = 1.0$ . The classical Newtonian result for the onset of the bar mode instability ( $Ncrit < 0.808$ ) is replaced by

in general relativity. For relativistic stars, it is evident that the onset of the gravitational-radiation-driven bar mode does not coincide with the onset of the viscosity-driven bar mode, which occurs at larger $T∕W$ [41

]. The computation of the onset of the CFS instability in the relativistic Cowling approximation by Yoshida and Eriguchi [328 ] agrees qualitatively with the conclusions in [292 , 295

Morsink, Stergioulas, and Blattning [230 ] extend the method presented in [295 ] 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 $1.0– 1.2 M ⊙$ . For $1.4M ⊙$ neutron stars, the mode becomes unstable at 80 – 95% 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 $T∕W ∼ 0.08$ for $1.4 M ⊙$ stars and $T∕W ∼ 0.06$ for maximum mass stars. This is to be contrasted with the Newtonian value of $T ∕W ∼ 0.14$ . The empirical formula

where $M smpahx$ 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 4 – 6%, for a wide range of realistic EOSs.

Figure 8: Eigenfrequencies (in the Cowling approximation) of the $m = 2$ mode as a function of the parameter $β = T ∕|W |$ for three different sequences of differentially rotating neutron stars (the $A −r1 = 0.0$ line corresponding to uniform rotation). The filled circle indicates the neutral stability point of a uniformly rotating star computed in full general relativity (Stergioulas and Friedman [295

]). Differential rotation shifts the neutral point to higher rotation rates. (Figure 1 of Yoshida, Rezzolla, Karino, and Eriguchi [333

]; used with permission.)

In newly-born neutron stars the CFS instability could develop while the background equilibrium star is still differentially rotating. In that case, the critical value of $T ∕W$ , required for the instability in the f -mode to set in, is larger than the corresponding value in the case of uniform rotation [333 ] (Figure 8 ). The mass-shedding limit for differentially rotating stars also appears at considerably larger $T∕W$ than the mass-shedding limit for uniform rotation. Thus, Yoshida et al. [333 ] suggest that differential rotation favours the instability, since the ratio $(T ∕W )critical∕(T ∕W )shedding$ decreases with increasing degree of differential rotation.

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 [241 , 259 ]. To order $𝒪 (Ω )$ , their frequency in the inertial frame is

while the radial eigenfunction of the perturbation in the velocity can be determined at order $Ω2$ [175 ]. According to Equation (55

), r-modes with $m > 0$ are prograde ( $ωi < 0$ ) with respect to a distant observer but retrograde ( $ω = ω + m Ω > 0 r i$ ) 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, as was first discovered by Andersson [6 ] for the case of slowly rotating, relativistic stars. This result was proved rigorously by Friedman and Morsink [117 ], who showed that the canonical energy of the modes is negative.

Figure 9: The r-mode instability window for a strange star of $M = 1.4M ⊙$ and R =10 km (solid line). Dashed curves show the corresponding instability windows for normal npe fluid and neutron stars with a crust. The instability window is compared to i) the inferred spin-periods for accreting stars in LMBXs [shaded box], and ii) the fastest known millisecond pulsars (for which observational upper limits on the temperature are available) [horizontal lines]. (Figure 1 of Andersson, Jones, and Kokkotas [11

]; used with permission.)

Two independent computations in the Newtonian Cowling approximation [207 , 16 ] showed that the usual shear and bulk viscosity assumed to exist for neutron star matter is not able to damp the r-mode instability, even in slowly rotating stars. In a temperature window of $105 K < T < 1010 K$ , 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 mass current quadrupole term. These results suggested 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. Due to uncertainties in the actual viscous damping times and because of other dissipative mechanisms, this scenario also is consistent with somewhat higher initial spins, such as the suggested initial spin period of several milliseconds for the X-ray pulsar in the supernova remnant N157B [223 ]. Millisecond pulsars with periods less than a few milliseconds can then only form after the accretion-induced spin-up of old pulsars and not in the accretion-induced collapse of a white dwarf.

Figure 10: Relativistic r-mode frequencies for a range of the compactness ratio $M ∕R$ . The coupling of polar and axial terms, even in the order $𝒪 (Ω)$ slow rotation approximation has a dramatic impact on the continuous frequency bands (shaded areas), allowing the r-mode to exist even in highly compact stars. The Newtonian value of the r-mode frequency is plotted as a dashed-dotted line. (Figure 3 of Ruoff, Stavridis, and Kokkotas [257

]; used with permission.)

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, superfluidity, the presence of hyperons, and the influence of a solid crust. In the uniform density approximation, the r-mode instability can be studied analytically to $𝒪 (Ω2)$ in the angular velocity of the star [182 ]. A study on the issue of detectability of gravitational waves from the r-mode instability was presented in [237 ] (see Section 3.5.5), while Andersson, Kokkotas, and Stergioulas [17 ] and Bildsten [35 ] proposed that the r-mode instability is limiting the spin of millisecond pulsars spun-up in LMXBs and it could even set the minimum observed spin period of $∼$ 1.5 ms (see [12 ]). This scenario is also compatible with observational data, if one considers strange stars instead of neutron stars [11 ] (see Figure 9 ).

Figure 11: Projected trajectories of several fiducial fluid elements (as seen in the corotating frame) for an $l = m = 2$ Newtonian r-mode. All of the fluid elements are initially positioned on the $ϕ0 = 0$ meridian at different latitudes (indicated with stars). Blue dots indicate the position of the fluid elements after each full oscillation period. The r-mode induces a kinematical, differential drift. (Figure 2c of Rezzolla, Lamb, Marković, and Shapiro [252

]; used with permission.)

Since the discovery of the r-mode instability, a large number of authors have studied in more detail the development of the instability and its astrophysical consequences. Unlike in the case of the f -mode instability, many different aspects and interactions have been considered. This intense focus on the detailed physics has been very fruitful and we now have a much more complete understanding of the various physical processes that are associated with pulsations in rapidly rotating relativistic stars. The latest understanding of the r-mode instability is that it may not be a very promising gravitational wave source (as originally thought), but the important astrophysical consequences, such as the limits of the spin of young and of recycled neutron stars are still considered plausible. The most crucial factors affecting the instability are magnetic fields [286 , 253 , 252 , 254 ], possible hyperon bulk viscosity [166 , 206 , 140 ] and nonlinear saturation [293 , 209 , 210 , 21 ]. The question of the possible existence of a continuous spectrum has also been discussed by several authors, but the most recent analysis suggests that higher order rotational effects still allow for discrete r-modes in relativistic stars [332 , 257 ] (see Figure 10 ).

Magnetic fields can affect the r-mode instability, as the r-mode velocity field creates differential rotation, which is both kinematical and due to gravitational radiation reaction (see Figure 11 ). Under differential rotation, an initially weak poloidal magnetic field is wound-up, creating a strong toroidal field, which causes the r-mode amplitude to saturate. If neutron stars have hyperons in their cores, the associated bulk viscosity is so strong that it could completely prevent the growth of the r-mode instability. However, hyperons are predicted only by certain equations of state and the relativistic mean field theory is not universally accepted. Thus, our ignorance of the true equation of state still leaves a lot of room for the r-mode instability to be considered viable.

Figure 12: Evolution of the axial velocity in the equatorial plane for a relativistic r-mode in a rapidly rotating $N = 1.0$ polytrope (in the Cowling approximation). Since the initial data used to excite the mode are not exact, the evolution is a superposition of (mainly) the $l = m = 2$ r-mode and several inertial modes. The amplitude of the oscillation decreases due to numerical (finite-differencing) viscosity of the code. A beating between the $l = m = 2$ r-mode and another inertial mode can also be seen. (Figure 2 of Stergioulas and Font [293

].)

The detection of gravitational waves from r-modes depends crucially on the nonlinear saturation amplitude. A first study by Stergioulas and Font [293 ] suggests that r-modes can exist at large amplitudes of order unity for dozens of rotational periods in rapidly rotating relativistic stars (Figure 12 ). The study used 3D relativistic hydrodynamical evolutions in the Cowling approximation. This result was confirmed by Newtonian 3D simulations of nonlinear r-modes by Lindblom, Tohline, and Vallisneri [206 , 209 ]. Lindblom et al. went further, using an accelerated radiation reaction force to artificially grow the r-mode amplitude on a hydrodynamical (instead of the secular) timescale. At the end of the simulations, the r-mode grew so large that large shock waves appeared on the surface of the star, while the amplitude of the mode subsequently collapsed. Lindblom et al. suggested that shock heating may be the mechanism that saturates the r-modes at a dimensionless amplitude of $α ∼ 3$ .

More recent studies of nonlinear couplings between the r-mode and higher order inertial modes [21 ] and new 3D nonlinear Newtonian simulations [136 ] seem to suggest a different picture. The r-mode could be saturated due to mode couplings or due to a hydrodynamical instability at amplitudes much smaller than the amplitude at which shock waves appeared in the simulations by Lindblom et al. Such a low amplitude, on the other hand, modifies the properties of the r-mode instability as a gravitational wave source, but is not necessarily bad news for gravitational wave detection, as a lower spin-down rate also implies a higher event rate for the r-mode instability in LMXBs in our own Galaxy [11 , 154 ]. The 3D simulations need to achieve significantly higher resolutions before definite conclusions can be reached, while the Arras et al. work could be extended to rapidly rotating relativistic stars (in which case the mode frequencies and eigenfunctions could change significantly, compared to the slowly rotating Newtonian case, which could affect the nonlinear coupling coefficients). Spectral methods can be used for achieving high accuracy in mode calculations; first results have been obtained by Villain and Bonazzolla [316 ] for inertial modes of slowly rotating stars in the relativistic Cowling approximation.

For a more extensive coverage of the numerous articles on the r-mode instability that appeared in recent years, the reader is referred to several excellent recent review articles [14 , 116 , 200 , 178 , 7 ].

Going further If rotating stars with very high compactness exist, then w-modes can also become unstable, as was recently found by Kokkotas, Ruoff, and Andersson [179 ]. The possible astrophysical implications are still under investigation.

3.5.4 Effect of viscosity on the 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 $τgr$ is the computed growth rate of the instability in the absence of viscosity, and $τ s$ , $τ b$ are the timescales of shear and bulk viscosity, then the total timescale of the perturbation is

Since $τgr < 0$ and $τb$ , $τs > 0$ , a mode will grow only if $τgr$ is shorter than the viscous timescales, so that $1∕τ < 0$ .

In normal neutron star matter, shear viscosity is dominated by neutron–neutron scattering with a temperature dependence of $T −2$ [101 ], and computations in the Newtonian limit and post-Newtonian approximation show that the CFS instability is suppressed for $T < 106 K$ – $107 K$ [160 , 159 , 326 , 198 ]. If neutrons become a superfluid below a transition temperature $Ts$ , then mutual friction, which is caused by the scattering of electrons off the cores of neutron vortices could significantly suppress the f -mode instability for $T < Ts$ [203 ], but the r-mode instability remains unaffected [204 ]. The superfluid transition temperature depends on the theoretical model for superfluidity and lies in the range $108 K$ – $6 × 109 K$ [239 ].

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 [262 ]. The bulk viscosity due to this effect can suppress the CFS instability only for temperatures for which matter has become transparent to neutrinos [190 , 40 ]. It has been proposed that for $T > 5 × 109 K$ , matter will be opaque to neutrinos and the neutrino phase space could be blocked ([190 ]; see also [40 ]). In this case, bulk viscosity will be too weak to suppress the instability, but a more detailed study is needed.

In the neutrino transparent regime, the effect of bulk viscosity on the instability depends crucially on the proton fraction $xp$ . If $xp$ is lower than a critical value ( $∼ 1∕9$ ), only modified URCA processes are allowed. In this case bulk viscosity limits, but does not completely suppress, the instability [160 , 159 , 326 ]. For most modern EOSs, however, the proton fraction is larger than $∼ 1∕9$ at sufficiently high densities [194 ], 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 [337 ]. At high temperatures, $9 T > 5 × 10 K$ , even if the star is opaque to neutrinos, the direct URCA cooling timescale to $T ∼ 5 × 109 K$ could be shorter than the growth timescale of the CFS instability.

3.5.5 Gravitational radiation from CFS instability

Conservation of angular momentum and the inferred initial period (assuming magnetic braking) of a few milliseconds for the X-ray pulsar in the supernova remnant N157B [223 ] 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 [190 ]. It could also play a role in the rotational evolution of merged binary neutron stars, if the post-merger angular momentum exceeds the maximum allowed to form a Kerr black hole [29 ] or if differential rotation temporarily stabilizes the merged object.

Lai and Shapiro [190 ] have studied the development of the f -mode instability using Newtonian ellipsoidal models [188 , 189 ]. They consider the case when a rapidly rotating neutron star is created in a core collapse. After a brief dynamical phase, the proto-neutron star becomes 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 $∼$ 100 Hz, the gravitational waves from the CFS instability could be detected out to the distance of 140 Mpc by the advanced LIGO detector. This result is very promising, especially since for relativistic stars the instability will be stronger than the Newtonian estimate [295 ]. Whether r-modes should also be considered a promising gravitational wave source depends crucially on their nonlinear saturation amplitude (see Section 3.5.3).

Going further The possible ways for neutron stars to emit gravitational waves and their detectability are reviewed in [44 , 45 , 121 , 100 , 306 , 265 , 80 ].

3.5.6 Viscosity-driven instability

A different type of nonaxisymmetric instability in rotating stars is the instability driven by viscosity, which breaks the circulation of the fluid [255 , 164 ]. The instability is suppressed by gravitational radiation, so it cannot act in the temperature window in which the CFS instability is active. 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 $N < 0.808$ [164 , 282 ]. For relativistic models, the situation for the instability becomes worse, since relativistic effects tend to suppress the viscosity-driven instability (while the CFS instability becomes stronger). According to recent results by Bonazzola et al. [41 ], for the most relativistic stars, the viscosity-driven bar mode can become unstable only if $N < 0.55$ . For $1.4M ⊙$ 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 [41 ] is an improvement (taking into account nonaxisymmetric terms of higher order) of an earlier method by the same authors [40 ]. 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 by Shapiro and Zane [268 ] of the viscosity-driven bar mode instability, using incompressible, uniformly rotating triaxial ellipsoids in the post-Newtonian approximation, finds that the relativistic effects increase the critical $T∕W$ ratio for the onset of the instability significantly. More recently, new post-Newtonian [88 ] and fully relativistic calculations for uniform density stars [129 ] show that the viscosity-driven instability is not as strongly suppressed by relativistic effects as suggested in [268 ]. The most promising case for the onset of the viscosity-driven instability (in terms of the critical rotation rate) would be rapidly rotating strange stars [130 ], but the instability can only appear if its growth rate is larger than the damping rate due to the emission of gravitational radiation – a corresponding detailed comparison is still missing.