R-mode Instability Research Articles

Properties ofrmodes in rotating magnetic neutron stars. I. Kinematic secular effects and magnetic evolution equations

The instability of r-mode oscillations in rapidly rotating neutron stars has attracted attention as a potential mechanism for producing high frequency, almost periodic gravitational waves. The analyses carried so far have shown the existence of these modes and have considered damping by shear and bulk viscosity. However, the magnetohydrodynamic coupling of the modes with a stellar magnetic field and its role in the damping of the instability has not been fully investigated yet. Following our introductory paper (Rezzolla, Lamb and Shapiro 2000), we here discuss in more detail the existence of secular higher-order kinematical effects which will produce toroidal fluid drifts. We also define the sets of equations that account for the time evolution of the magnetic fields produced by these secular velocity fields and show that the magnetic fields produced can reach equipartition in less than a year. The full numerical calculations as well as the evaluation of the impact of strong magnetic fields on the onset and evolution of the r-mode instability will be presented in a companion paper.

Gravitational Wave Emission from Core Collapse of Massive Stars

We derive estimates for the characteristics of gravitational radiation from stellar collapse, using recent models of the core-collapse of Chandrasekhar mass white dwarfs (accretion induced collapse), core-collapse supernovae and collapsars, and the collapse of very massive stars (~> 300 Msun). We study gravitational-wave emission mechanisms using several estimation techniques, including two-dimensional numerical computation of quadrupole wave emission, estimates of bar-mode strength, estimates of r-mode emission, and estimates of waves from black hole ringing. We also review the rate at which the relevant collapses are believed to occur, which has a major impact on their relevance as astrophysical sources. Although the latest supernova progenitor simulations produce cores rotating much slower than those used in the past, we find that bar-mode and r-mode instabilities from core-collapse supernovae remain among the leading candidate sources for LIGO-II. Accretion induced collapse (AIC) of a white dwarf could produce gravitational-wave signals similar to those from core-collapse. In the models that we examine, such collapses are not unstable to bar modes; we note that models recently examined by Liu and Lindblom, which have slightly more angular momentum, are certainly unstable to bar formation. Because AIC events are probably 1,000 times less common than core-collapse supernovae, the typical AIC event will be much further away, and thus the observed waves will be much weaker. In the most optimistic circumstances, we find it may be possible to detect gravitational waves from the collapse of 300 Msun Population III stars.

Magnetic effects on the viscous boundary layer damping of ther-modes in neutron stars

This paper explores the effects that magnetic fields have on the viscous boundary layers (VBLs) that can form in neutron stars at the crust-core interface, and it investigates the VBL damping of the gravitational-radiation driven r-mode instability. Approximate solutions to the magnetohydrodynamic equations valid in the VBL are found for ordinary-fluid neutron stars. It is shown that magnetic fields above ${10}^{9}{(10}^{10} \mathrm{K}/T)\mathrm{G}$ significantly change the structure of the VBL, and that magnetic fields decrease the VBL damping time. Furthermore, VBL damping completely suppresses the r-mode instability for $B\ensuremath{\gtrsim}{10}^{12} \mathrm{G}$ (independent of the temperature). These bounds refer to the strength of the radial component of the equilibrium field at the location of the core radius. Magnetic effects on the VBL vanish wherever the radial component of the equilibrium field vanishes. Thus, magnetic fields will profoundly affect the VBL damping of the r-mode instability in hot young pulsars (that are cool enough to have formed a solid crust). One can speculate that magnetic fields can affect the VBL damping of this instability in low-mass x-ray binary systems and other cold old pulsars (if they have sufficiently large internal fields).

Crust-core coupling and r-mode damping in neutron stars: a toy model

r-modes in neutron stars with crusts are damped by viscous friction at the crust–core boundary. The magnitude of this damping, evaluated by Bildsten & Ushomirsky (BU) under the assumption of a perfectly rigid crust, sets the maximum spin frequency for neutron stars spun up by accretion in low-mass X-ray binaries (LMXBs). In this paper we explore the mechanical coupling between the core r-modes and the elastic crust, using a toy model of a constant-density neutron star having a crust with a constant shear modulus. We find that, at spin frequencies in excess of ≈50 Hz, the r-modes strongly penetrate the crust. This reduces the relative motion (slippage) between the crust and the core compared with the rigid-crust limit. We therefore revise down, by as much as a factor of 102–103, the damping rate computed by BU, significantly reducing the maximal possible spin frequency of neutron stars with solid crusts. The dependence of the crust–core slippage on the spin frequency is complicated, and is very sensitive to the physical thickness of the crust. If the crust is sufficiently thick, the curve of the critical spin frequency for the onset of the r-mode instability becomes multivalued for some temperatures; this is related to avoided crossings between the r-mode and higher-order torsional modes in the crust. The critical frequencies are comparable to the observed spins of neutron stars in LMXBs and millisecond pulsars.

Non-linear r-modes in a spherical shell: issues of principle

We use a simple physical model to study the nonlinear behaviour of the r-mode instability. We assume that r-modes (Rossby waves) are excited in a thin spherical shell of rotating incompressible fluid. For this case, exact Rossby wave solutions of arbitrary amplitude are known. We find that: (a) These nonlinear Rossby waves carry ZERO physical angular momentum and positive physical energy, which is contrary to the folklore belief that the r-mode angular momentum and energy are negative. (b) Within our model, we confirm the differential drift reported by Rezzolla, Lamb and Shapiro (1999). Radiation reaction is introduced into the model by assuming that the fluid is electrically charged; r-modes are coupled to electromagnetic radiation through current (magnetic) multipole moments. We find that: (c) To linear order in the mode amplitude, r-modes are subject to the CFS instability, as expected. (d) Radiation reaction decreases the angular velocity of the shell and causes differential rotation (which is distinct from but similar in magnitude to the differential drift reported by Rezzolla et al.) prior to saturation of the r-mode growth. This is contrary to the phenomenological treatments to date, which assume that the loss of stellar angular momentum is accounted for by the r-mode growth. We demonstrate, for the first time, that r-mode radiation reaction leads to differential rotation. (e) We show that for l=2 r-mode electromagnetic radiation reaction is equivalent to gravitational radiation reaction in the lowest post-Newtonian order.

Nonlinear r-modes in rapidly rotating relativistic stars.

The r-mode instability in rotating relativistic stars has been shown recently to have important astrophysical implications, provided that r-modes are not saturated at low amplitudes by nonlinear effects or by dissipative mechanisms. Here, we present the first study of nonlinear r-modes in isentropic, rapidly rotating relativistic stars, via 3D general-relativistic hydrodynamical evolutions. We find that (1) on dynamical time scales, there is no strong nonlinear coupling of r-modes to other modes at amplitudes of order one-the maximum r-mode amplitude is of order unity. (2) r-modes and inertial modes in isentropic stars are predominantly discrete modes. (3) The kinematical drift associated with r-modes appears to be present in our simulations, but confirmation requires more precise initial data.

R‐Modes of Neutron Stars with a Solid Crust

We investigate the properties of r-mode oscillations of a slowly rotating neutron star with a solid crust by taking account of the effects of the Coriolis force. For the modal analysis we employ three-component neutron star models that are composed of a fluid core, a solid crust, and a surface fluid ocean. For the three-component models, we find that there exist two kinds of r-modes, that is, those confined in the surface fluid ocean and those confined in the fluid core, which are most important for the r-mode instability. The r-modes do not have any appreciable amplitudes in the solid crust if rotation rate of the star is sufficiently small. We find that the core r-modes are strongly affected by mode coupling with the crustal torsional (toroidal) modes and lose their simple properties of the eigenfunction and eigenfrequency as functions of the angular rotation velocity Ω. This indicates that the extrapolation formula, which is obtained in the limit of Ω → 0, cannot be used to examine the r-mode instability of rapidly rotating neutron stars with a solid crust unless the effects of mode coupling with the crustal torsional modes are correctly taken into account.

Effect of a neutron-star crust on the r-mode instability

The presence of a viscous boundary layer under the solid crust of a neutron star dramatically increases the viscous damping rate of the fluid r-modes. We improve previous estimates of this damping rate by including the effect of the Coriolis force on the boundary-layer eigenfunction and by using more realistic neutron-star models. If the crust is assumed to be perfectly rigid, the gravitational radiation driven instability in the r-modes is completely suppressed in neutron stars colder than about $1.5\ifmmode\times\else\texttimes\fi{}{10}^{8}$ K. Energy generation in the boundary layer will heat the star, and will even melt the crust if the amplitude of the r-mode is large enough. We solve the heat equation explicitly (including the effects of thermal conduction and neutrino emission) and find that the r-mode amplitude needed to melt the crust is ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ for maximally rotating neutron stars. If the r- mode saturates at an amplitude larger than ${\ensuremath{\alpha}}_{c},$ the heat generated is sufficient to maintain the outer layers of the star in a mixed fluid-solid state analogous to the pack ice on the fringes of the Arctic Ocean. We argue that in young, rapidly rotating neutron stars this effect considerably delays the formation of the crust. By considering the dissipation in the ice flow, we show that the final spin frequency of stars with r-mode amplitude of order unity is close to the value estimated for fluid stars without a crust.

Influence of the r-mode instability on hypercritically accreting neutron stars

We have investigated an influence of the r-mode instability on hypercritically accreting ($\dot{M}\sim 1M_\odot {y}^{-1}$) neutron stars in close binary systems during their common envelope phases based on the scenario proposed by Bethe et al. \shortcite{bethe-brown-lee}. On the one hand neutron stars are heated by the accreted matter at the stellar surface, but on the other hand they are also cooled down by the neutrino radiation. At the same time, the accreted matter transports its angular momentum and mass to the star. We have studied the evolution of the stellar mass, temperature and rotational frequency. The gravitational-wave-driven instability of the r-mode oscillation strongly suppresses spinning-up of the star, whose final rotational frequency is well below the mass-shedding limit, typically as small as 10% of that of the mass-shedding state. On a very short time scale the rotational frequency tends to approach a certain constant value and saturates there as far as the amount of the accreted mass does not exceed a certain limit to collapse to a black hole. This implies that the similar mechanism of gravitational radiation as the so-called Wagoner star may work in this process. The star is spun up by accretion until the angular momentum loss by gravitational radiation balances the accretion torque. The time-integrated dimensionless strain of the radiated gravitational wave may be large enough to be detectable by the gravitational wave detectors such as LIGO II.

Approximate equation relevant to axial oscillations on slowly rotating relativistic stars

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.

CLC][ITAL]r[/ITAL][/CLC]-Mode Runaway and Rapidly Rotating Neutron Stars

We present a simple spin-evolution model that predicts that rapidly rotating accreting neutron stars will be confined mainly to a narrow range of spin frequencies: P=1.5-5 ms. This is in agreement with current observations of neutron stars in both the low-mass X-ray binaries and the millisecond radio pulsars. The main ingredients in the model are (1) the instability of r-modes above a critical spin rate, (2) the thermal runaway that is due to the heat released as viscous damping mechanisms counteract the r-mode growth, and (3) a revised estimate of the strength of the dissipation that is due to the presence of a viscous boundary layer at the base of the crust in an old and relatively cold neutron star. We discuss the gravitational waves that are radiated during the brief r-mode-driven spin-down phase. We also briefly touch on how the new estimates affect the predicted initial spin periods of hot young neutron stars.

Viscous Boundary-Layer Damping of r-Modes in Neutron Stars.

Recent work has raised the exciting possibility that r-modes (Rossby waves) in rotating neutron star cores might be strong gravitational-wave sources. We estimate the effect of a solid crust on their viscous damping rate and show that the dissipation rate in the viscous boundary layer between the oscillating fluid and the nearly static crust is more than 105 times higher than that from the shear throughout the interior. This increases the minimum frequency for the onset of the gravitational r-mode instability to at least 500 Hz when the core temperature is less than 1010 K. It eliminates the conflict between the r-mode instability and the accretion-driven spin-up scenario for millisecond radio pulsars and makes it unlikely that the r-mode instability is active in accreting neutron stars. For newborn neutron stars, the formation of a solid crust shortly after birth affects their gravitational-wave spin-down and hence their detectability by ground-based interferometric gravitational-wave detectors.

Stability of ther-modes in white dwarf stars

The stability of the r-modes in rapidly rotating white dwarf stars is investigated. Improved estimates of the growth times of the gravitational-radiation driven instability in the r-modes of the observed DQ Her objects are found to be longer (probably considerably longer) than $6\ifmmode\times\else\texttimes\fi{}{10}^{9} \mathrm{yr}.$ This rules out the possibility that the r-modes in these objects are emitting gravitational radiation at levels that could be detectable by LISA. More generally it is shown that the r-mode instability can only be excited in a very small subset of very hot $(T\ensuremath{\gtrsim}{10}^{6} \mathrm{K}),$ rather massive $(M\ensuremath{\gtrsim}{0.9M}_{\ensuremath{\bigodot}})$ and very rapidly rotating ${(P}_{\mathrm{min}}<~P\ensuremath{\lesssim}{1.2P}_{\mathrm{min}})$ white dwarf stars. Further, the growth times of this instability are so long that these conditions must persist for a very long time $(t\ensuremath{\gtrsim}{10}^{9} \mathrm{yr})$ to allow the amplitude to grow to a dynamically significant level. This makes it extremely unlikely that the r-mode instability plays a significant role in any real white dwarf stars.

Runaway Heating byr‐Modes of Neutron Stars in Low‐Mass X‐Ray Binaries

Recently Andersson et. al., and Bildsten have independently suggested that an r-mode instability might be responsible for stalling the neutron-star spin-up in strongly accreting, Low Mass X-ray Binaries (LMXBs). We show that if this does occur, then there are two possibilities for the resulting neutron-star evolution: If the r-mode damping is a decreasing function of temperature, then the star undergoes a cyclic evolution: (i) accretional spin-up triggers the instability near the observed maximum spin rate; (ii) the r-modes become highly excited through gravitational-radiation reaction, and in a fraction of a year they viscously heat the star; (iii) r-mode gravitational-radiation reaction then spins the star down in a fraction of a year to some limiting rotational frequency; (iv) the r-mode instability shuts off; (v) the neutron star slowly cools and is spun up by accretion, until it once again reaches the instability point, closing the cycle. The shortness of the epoch of r-mode activity makes it unlikely that r-modes are currently excited in the neutron star of any galactic LMXBs. Nevertheless, this cyclic evolution could be responsible for keeping the rotational frequencies within the observed LMXB frequency range. If, on the other hand, the r-mode damping is temperature independent, then a steady state with constant angular velocity and $T_{\rm core}\simeq 4\times 10^8$K is reached, in which r-mode viscous heating is balanced by neutrino cooling and accretional spin-up torque is balanced by gravitational-radiation-reaction spin-down torque. In this case the neutron stars in LMXBs could be potential sources of periodic gravitational waves, detectable by enhanced LIGO interferometers.

On the Relevance of ther‐Mode Instability for Accreting Neutron Stars and White Dwarfs

We present a case study for the relevance of the r-mode instability for accreting compact stars. Our estimates are based on approximations that facilitate back-of-the-envelope calculations. We discuss two different cases: 1) For recycled millisecond pulsars we argue that the r-mode instability may be active at rotation periods longer than the Kepler period (which provides the dynamical limit on rotation) as long as the core temperature is larger than about $2\times10^5$ K. Our estimates suggest that the instability may have played a role in the evolution of the fastest spinning pulsars, and that it may be presently active in the recently discovered 2.49 ms X-ray pulsar SAX J1808.4-3658 as well as the rapidly spinning neutron stars observed in low mass X-ray binaries. This provides a new explanation for the remarkably similar rotation periods inferred from kHz quasi-periodic oscillations in the LMXBs. The possibility that the rotation of recycled pulsars may be gravitational-radiation limited is interesting because the gravitational waves from a neutron star rotating at the instability limit may well be detectable with the new generation of interferometric detectors. 2) We also consider white dwarfs, and find that the r-mode instability may possibly be active in short period white dwarfs. Our order of magnitude estimates (for a white dwarf of $M=M_\odot$ and $R=0.01 R_\odot$ composed of $C^{12}$) show that the instability could be operating for rotational periods shorter than $P\approx 27-33$ s. This number is in interesting agreement with the observed periods ($>28$ s) of the rapidly spinning DQ Herculis stars. However, we find that the instability grows too slowly to affect the rotation of these stars significantly .

Stochastic background of gravitational waves generated by a cosmological population of young, rapidly rotating neutron stars

We estimate the spectral properties of the stochastic background of gravitational radiation emitted by a cosmological population of hot, young, rapidly rotating neutron stars. Their formation rate as a function of redshift is deduced from an observation-based determination of the star formation history in the Universe, and the gravitational energy is assumed to be radiated during the spin-down phase associated with the newly discovered r-mode instability. We calculate the overall signal produced by the ensemble of such neutron stars, assuming various cosmological backgrounds. We find that the spectral strain amplitude has a maximum ≈ (2–4)× 10−26Hz−1/2, at frequencies ≈ (30–60) Hz, while the corresponding closure density, h2ΩGW, has a maximum amplitude plateau of ≈ (2.2–3.3) × 10−8 in the frequency range (500–1700) Hz. We compare our results with a preliminary analysis done by Owen et al., and discuss the detectability of this background.

How to Identify a Strange Star

Contrary to young neutron stars, young strange stars are not subject to the r-mode instability which slows rapidly rotating, hot neutron stars to rotation periods near 10 ms via gravitational wave emission. Young millisecond pulsars are therefore likely to be strange stars rather than neutron stars, or at least to contain significant quantities of quark matter in the interior.

Resonant interactions between cometary ions and low frequency electromagnetic waves

We explore the conditions for resonance between cometary pick-up ions and parallel propagating electromagnetic waves. A model ring—beam distribution for the pick-up H 2O + ions is adopted which allows a direct comparison of the source of free energy for growth from either the beam or the gyrating ring in the limit near marginal stability. Under average solar wind conditions in the inner solar system, the gyrating ring provides the dominant contribution to wave growth. The presence of a field-aligned beam is only important to allow resonance with R-mode waves which occur in two distinct frequency bands either well above or below the pick-up ion gyrofrequency. The most unstable mode is the low frequency R-mode or fast MHD wave, though higher frequency whistlers or low frequency L-mode waves may also be excited by the same source of free energy. The nature of the unstable waves is strongly influenced by the inclination α of the interplanetary field. For α ≲ 3° the rate of the low frequency R-mode growth is dramatically reduced and resonant L-mode waves should experience net ion beam damping. Conversely for α ≳ 75°, the ion beam velocity will be insufficient to allow resonant R-mode instability; L-mode waves should therefore predominate. The low frequency fast MHD mode should experience the most rapid amplification for intermediate inclination; 30° ≲ α ≲ 75°. In the frame of the solar wind such waves must propagate along the field in the direction upstream towards the Sun with a phase speed lower than the beaming velocity of the pick-up ions. The waves are consequently blown back away from the Sun and would thus be detected with a left-hand polarization by an observer in the cometary frame. We consider this the most likely mechanism to account for the interior MHD waves observed by satellites over an extended spatial region surrounding comets Giacobini-Zinner and Halley.