Runaway Heating By R-Modes of Neutron Stars in Low Mass X-ray Binaries.
arXiv:astro-ph/9810471v1 29 Oct 1998
Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125
–2– ABSTRACT 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 (0.13yrs in a particular model that we have considered) they viscously heat the star up to T ? 2.5 × 109 K; (iii) r-mode gravitational-radiation-reaction then spins the star down in tspindown ? 0.08(f?nal /130Hz)?6 yrs to a limiting rotational frequency f?nal , whose exact value depends on the not fully understood mechanisms of r-mode damping; (iv) the r-mode instability shuts o?; (v) the neutron star slowly cools and is spun up by accretion for ? 5 × 106 yrs, 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, and unlikely that advanced LIGO interferometers will see gravitational waves from extragalactic 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 Tcore ? 4 × 108 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.
–3– In this case (as Bildsten and Andersson et. al. have shown) the neutron stars in LMXBs could be potential sources of periodic gravitational waves, detectable by enhanced LIGO interferometers.
–4– 1. Introduction
Most of the rapidly accreting neutron stars in Low Mass X-ray Binaries (LMXBs) are observed to rotate in a strikingly narrow range of frequencies—from 260Hz to 330Hz (see e.g. Van der Klis 1997). A natural explanation for this could be some mechanism which prevents further neutron-star spin-up once the rotational frequency is su?ciently high. Recently several such mechanisms were proposed: White and Zhang (1997) suggested that magnetic braking could be responsible for halting the spin-up; this idea will not be discussed here. Bildsten (1998) pointed out that, because gravitational radiation reaction is a sharply increasing function of rotational frequency, it might halt the spin-ip. In his original manuscript, Bildsten identi?ed one mechanism for triggering the necessary gravitational waves: lateral density variations caused by temperature dependence of electron capture reactions. While his manuscript was being refereed, Bildsten learned of the discovery that an r-mode instability, driven by gravitational radiation, can be very strong in spinning neutron stars (Andersson 1998, Friedman and Morsink 1998, Lindblom, Owen and Morsink 1998, Andersson, Kokkotas and Schutz 1998, Owen et al 1998); and the r-mode experts learned of Bildsten’s gravitational-wave idea for saturating LMXB spinup. Both groups independently saw the connection: Bildsten (1998) and Andersson, Kokkotas and Stergioulas (1998) proposed that the r-mode instability could provide enough gravitational-radiation reaction to halt the LMXB spinup. In this letter we examine the consequences of this proposal. Our conclusions depend crucially on whether the dissipation of the r-modes decreases with temperature (as is the case, e.g., when shear viscosity dominates the r-mode damping), or instead is temperature-independent (as is the case when, e.g., the mutual friction of proton and neutron super?uids dominates the damping). In the former case (Section 2 of this paper) we ?nd that the neutron star will undergo a spin-up—heating—spin-down—cooling
–5– cycle; in the latter case (Section 3) it will probably settle down to a stable equilibrium state with an internal neutron-star temperature of about 4 × 108 K.
“Viscous” r-mode damping
Let us consider ?rst the case when dissipation is a decreasing function of temperature. We show that, if some r-modes become unstable in a neutron star spun up by accretion, then they heat up the neutron star through shear viscousity. As the neutron star heats up, the r-modes become more unstable. A thermo-gravitational runaway takes place, in which the r-mode amplitude grows, as a result of this growth the star’s temperature rises, the dissipation becomes weaker and the instability becomes stronger. Within a fraction of a year the r-modes’ gravitational radiation reaction spins the star down to a rotation frequency which is close to the minimum of the critical stability curve (probably around 100 ? 150Hz, but the exact value depends on poorly understood dissipation mechanisms—see below), with a ?nal temperature of about 2 × 109 K. The instability then shuts o? and the star begins a several-million-year epoch of neutrino cooling and accretional spinup, leading back to the original instability point. Fig. 1 shows a typical evolutionary trajectory A → B → C → D → B of the neutron
? star in the log(T8 ) ? ? plane, where T8 is the temperature of the star’s core measured in √ ? units of 108 K, and ? = ?/ πG?. Here ? is the angular velocity of the neutron star and ρ
ρ is it’s mean density. The portion A → B of the curve represents the accretional spin-up ? of the neutron star to the critical angular frequency ?cr (T ); B → C represents the heating stage in which the r-modes become unstable, grow and heat up the neutron star; C → D shows the spindown stage in which the r-mode amplitude saturates because of poorly understood nonlinear e?ects, and the angular velocity decreases due to the emission of gravitational radiation; and D → B represents cooling back to the equilibrium temperature
–6– with simultaneous spin-up by accretion. All four stages are discussed in more detail below. The initial (steady-state) temperature T0 of the neutron-star core in steadily accreting LMXB’s is somewhat uncertain; according to Brown and Bildsten (1998), who analyzed heat transport during steady thermonuclear burning of the accreted material and nuclear reactions in the deep ocean, T0 = 1 ? 4 × 108 K. In Fig. 1 we assume T0 = 108 K. The curve K ? L ? M is the so-called r-mode “stability curve” (Lindblom, Owen and Morsink 1998). If the neutron star is represented by a point above the curve, then some r-modes in the star are unstable and grow. Otherwise, all r-modes decay. The portion K ? L of the stability curve is determined by the shear viscosity, or by mutual friction if part of the star is super?uid. Its exact location is uncertain precisely because the dissipation of the r-modes at the relevant temperatures is poorly understood. If shear viscosity dominates the dissipation, then the equation of the K ? L portion of the stability curve is given by ? ?cr = 0.1 η η0
where η is the shear viscosity of the neutron star material, and η0 is the shear viscosity due to electron-electron scattering in the neutron star [we have used Eqs (2.10), (2.14), (2.15) and Table I of Owen et. al. (1998) to work out Eq. (1)]. If only shear viscosity due to to electron-electron scattering were operating, with the shear viscosity given by η0 = 347ρ9/4 T ?2 , where all quantities are in cgs units (see Cutler and Lindblom 1987 and references therein), then the critical rotational frequency at T = 108 K would be 130Hz, which is much less than observed values (van der Klis 1998). However, the friction is probably larger than this (and therefore ?cr is also larger) because of interaction of the core ?uid with the crust and maybe mutual friction in a super?uid state. The emphasis of this paper is not to ?gure out whether the r-mode instability is relevant for LMXBs, but to investigate the consequences if it is relevant. For purpose of illustration, we assume that η = 244 × η0 ; this makes the critical rotational frequency 330Hz at T = 108 K, which
–7– is consistent with observations (van der Klis 1998). This choice of viscosity is a cheat since we don’t yet know the T and ρ-dependence of η. However, unless the damping is due to mutual friction, η is likely to decrease with increasing temperature, which is a su?cient condition for thermo-gravitational runaway. Our choice of viscosity possesses this feature; therefore we believe it has a good chance of representing the real physics. The portion L ? M of the stability curve is determined by bulk-viscosity dissipation; it’s exact location is also a subject of yet unsettled controversy [the heart of the problem is the calculation of Lagrangian perturbation in density (Lindblom, Owen and Morsink 1998, Andersson, Kokkotas and Schutz 1998), which, to our best knowledge, has not been reliably carried out by any of the groups]. Of the two current estimates of the bulk-viscosity contribution to damping of the r-modes, we have chosen the one which gives the higher ? values of ?cr , thus maximizing its e?ect (see Lindblom, Owen and Morsink 1998). The fact that, for the evolution curve shown in Figure 1 no part is in the region where the bulk viscosity dominates suggests that the details of the bulk viscosity will not be of particular importance. In this work for concreteness we specialize to a polytropic model of a neutron star with p ∝ ρ2 , and consider the r-mode with l = m = 2, which is expected to have the strongest instability in such polytropes (Friedman and Morsink 1998, Lindblom, Owen and Morsink √ ? ρ 1998). We assume that the time evolution of the normalized angular velocity ? = ?/ πG? of the star and the dimensionless amplitude α of the r-mode are given by phenomenological Equations (3.14), (3.15), (3.16) and (3.17) of Owen et al (1998): ? ? 2α2 Q ? d? =? + dt 1 + α2 Q τv ˙ 41M × p, ? 3I M (2) (3)
dα 1 1 ? α2 Q 1 α =? + dt τgrav τv 1 + α2 Q
–8– when α2 < k (the saturation value of α2 , which we assume to be k = 1), and by α2 = k, (4)
when α is saturated due to not yet understood non-linear e?ects. Here α is the dimensionless amplitude of the r-mode de?ned by Eq. (1) of Lindblom, Owen and Morsink (1998), and τv and τgrav are the viscous and gravitational timescales for the r-mode dissipation and are given by Eqs (2.14) and (2.15) of Owen et al (1998): ? τgrav = ?3.26??6 sec, 1 τv 1 = τs ? 108 K T
? ? d? 2? kQ = dt τgrav 1 ? kQ
1 + τB ?
T 108 K
? ?2 .
In the above equation the viscous damping rate is a sum of contributions from the shear and the bulk viscosities; the former is determined by τs which we took to be 1.03 × 104 sec ? in order to ?t the observed data; the latter is determined by τB which is taken to be ? 6.99 × 1014 sec, in agreement with Owen et. al. (1998). Note that τgrav is negative since gravitational radiation always ampli?es the r-mode. ˙ The second term in Eq. (2) represents the neutron-star spin-up by accretion; M and M are the mass of the neutron star and its accretion rate respectively, and p is a factor of order unity which depends on the accretion radius and the angular velocity of the neutron star; it’s exact value is not essential for the physics discussed here and we set p = 1 from ? here onwards. The numerical parameters Q and I are given by 0.094 and 0.261 respectively for a polytrope star of adiabatic index γ = 2 (Lindblom, Owen and Morsink 1998). For ˙ the evolution shown in Fig. 1 we took M = 1.4M⊙ and M = 10?8 M⊙ /yr, and we assumed a random initial perturbation of magnitude α = 10?8 when the neutron star reaches the stability curve K ? L.
–9– Now consider the star’s thermal evolution. The r-mode deposits heat into the star at the rate Wdiss = ? α2 ?2 MR2 J 2Er?mode = , τv τv (7)
where Er?mode is the energy in the r-mode [cf. Eq. (3.11) of Owen et al (1998)]. Here R is ? the radius of the neutron star taken to be 12.53km, and J = 1.635 × 10?2 for the polytropic model considered here. At the relevant temperatures the neutron star is expected to cool predominantly by the modi?ed URCA process (this is not entirely true, since close to 108 K neutrino bremstruhlung cooling from the crust and radiative cooling by photons might become signi?cant. However, their cooling rates are not signi?cantly larger than that of the modi?ed URCA process at 108 K, and they become negligible at higher temperatures. In this work for simplicity we assume that modi?ed URCA is the only cooling process; the inclusion of other processes would not change the general evolutionary picture). The modi?ed URCA cooling rate, reduced by heating from nuclear reactions in the deep crust, is given by (Shapiro and Teukolsky, 1983)
8 ?8 Lcool = 7 × 1031 (T8 ? T8 )erg/sec.
? Here the subscript 8 indicates that the temperature is measured in units of 108 K and T is the equilibrium temperature of the neutron star before the r-mode heating starts, taken to be 108 K for our calculation. The thermal evolution equation is then given by dT Wdiss ? Lcool , = dt Cv (9)
where Cv is the heat capacity of the neutron star, taken to be 1.4 × 1038 (erg/K)×T8 [from Shapiro and Teukolsky (1983), Eq. (11.8.2). However, the heat capacity of neutron star with a super?uid core is less]. Equations (2), (3), (5), (4) and (9) determine the time evolution of the angular velocity ? and temperature T . Figure 1 shows the predicted evolution, for the representative parameter values, introduced above. The evolution consists of four stages:
– 10 – The ?rst stage A → B is the spin-up of the neutron star, during which it’s angular velocity ? is increasing towards the critical one, and the r-mode instability is suppressed by ? viscosity; since we assume that the star begins at its equilibrium temperature T8 = T8 = 1, its temperature changes little during the spin-up. For an assumed accretion rate of 10?8 M⊙ /yr this stage takes ? 5 × 106 years. When the angular velocity reaches its critical value, the r-mode starts to grow and the second stage B → C begins. The neutron star gets heated up by the r-mode through viscosity, the r-mode becomes more unstable, and thermo-gravitational runaway follows. It takes 0.13 years for the r-mode’s amplitude to evolve from α = αW to α = 1, where αW ? 1.2 × 10?5 is the value of the r-mode amplitude at which the accretional torque is exactly compensated by the gravitational radiation reaction [see Wagoner (1984)]. For our intuition it is useful to de?ne two characteristic α-dependent timescales for stage B → C: the thermal timescale [cf Eq. (7)] tth = Cv T dt 2 τv ? 3.7 × 10?5 T8 2 = d log T Wdiss α dt 1 τv τv ? ? 5 × 2. d log ? 2Q α2 α (10)
and the timescale for the decrease of angular velocity [cf Eq. (2)] t? = (11)
Clearly, the neutron star heats up much faster than it spins down due to gravitational radiation. Therefore, during this stage the angular velocity of the star decreases by only a ? ? small amount, ?? = 0.0003. Physically, the reasons for such little change in ? are that the r-mode amplitude grows so quickly, and that in this phase the angular momentum loss is not manifested in a reduction of the angular velocity, but instead in the growth of the r-mode itself (the r-mode which is driven by gravitational radiation reaction carries a negative angular momentum). Eventually the r-mode amplitude saturates due to nonlinear e?ects. This initiates the third stage of the evolution, in which all of the angular momentum loss is manifested
– 11 – by reduction of angular velocity (since the r-mode cannot grow any more), and the star spins down C → D to the critical angular velocity. At point C1 the temperature of the neutron star is such that the neutrino cooling exactly compensates the dissipative heating from the r-modes. After that the temperature does not change much until the spin-down stage is terminated. The physical reason for this is that even though the thermal timescale at C1 → D is comparable or smaller than the spindown timescale, the rate of dissipative heating does not change much. If the heat capacity Cv of neutron star were zero, we would have Wdiss = Lcool at all points of C1 → D. This would imply T ∝ ?1/4 , so even then the temperature would not change signi?cantly over this last part of the spin-down. An analytical expression for the duration of this rapid spin-down stage can be derived from Eqs (5) and (6): ? tspindown ? 0.08(1/k)(?f /0.1)?6 yr, (12)
where ?f is the angular velocity at the end of the spin-down. In our simulations tspindown is about 0.14 years. After the neutron star reaches the stability curve, the r-mode is damped by viscosity stronger than it is driven by gravitational-radiation reaction; therefore its amplitude decreases and the neutron star cools back to its original equilibrium temperature, while being spun up by accretion. This part of the evolution is represented by D → B on Fig. 1; its timescale is the same as that for the original accretional spin-up, i.e. ? 5 × 106 years. After this the cycle is closed and can repeat itself as long as the accretion continues. We believe that the sharp kink at point C is not a real physical e?ect, but a result of our poor understanding of the non-linear saturation of the r-mode; however, this arti?cial feature of our simulations does not seem to a?ect the existence of the thermo-gravitational runaway and the subsequent rapid spin-down to a lower angular velocity. Despite a large number of uncertainties in the details of the evolution, we believe that this scenario is
– 12 – robust so long as the r-mode instability does occur in LMXBs, and the damping of the r-modes decreases with temperature. If the above described evolutionary scenario is generic, it is then clear that none of the currently observed LMXBs can possess an actively operating r-mode instability—otherwise we would observe a rapid spindown on a time-scale less than a year. However, it is conceivable that many of the neutron stars in these LMXBs have undergone the r-mode instability at some stage of their evolution, and are currently below the stability curve, evolving along leg D → B of Fig. 1. From Equations (12) and (2) we can estimate the fraction r of neutron stars in extragalactic LMXBs that are in the phase of active emission of gravitational waves: ? ?f tspindown ? 1.6(1/k) × 10?8 r= taccretion 0.1
? The quantity ?f is bounded from below by the rotational frequencies of young pulsars (this statement is true only if the r-mode damping is the same for young and old pulsars at the same temperatures). The rotational frequency of the recently discovered N157B (Marshall et al 1998) is 62.5Hz. Using the braking index theory one can project the initial ? rotational frequency of this pulsar to be no smaller then 100Hz, which implies ?f > 0.08. Therefore, only r < (6/k) × 10?8 of neutron stars in extragalactic LMXBs are in the phase of rapid gravitational wave emission, which implies that to catch one star in this phase, gravitational-wave detectors must reach out through a volume large enough to encompass ? 0.1 ? 0.01/r ? 106 galaxies like our own (this assumes that there are 10 ? 100 strongly accreting neutron stars in LMXBs in our galaxy). An analysis similar to that of Owen et al (1998) shows that even “advanced LIGO” detectors are unlikely to be able to see these sources at such great distances.
– 13 – 3. temperature-independent r-mode damping
There is a possible alternative evolutionary scenario which is similar to the one proposed by Bildsten (1998) and Andersson, Kokkotas and Stergioulas (1998) (we thank Lee Lindblom for pointing this out). It may be that the r-mode damping is dominated not by normal dissipative processes, but by mutual friction in the neutron-proton super?uid. Detailed calculations of the e?ect of such friction on the r-mode damping are in progress (Lindblom and Mendell); however for our analysis the essential feature of this dissipative process is already known: it is temperature independent. Therefore, if this process dominates, one would not expect a thermo-gravitational runaway; instead the neutron star will reach a state of three-fold equilibrium. The neutron star will “sit” on the stability curve [(1/τgrav ) + (1/τv ) = 0], the amplitude of the r-mode will adjust so that the accretional torque is compensated by the gravitational-radiation reaction torque (α = αW ? 1.2 × 10?5 for our model), and the temperature of the neutron star will adjust so that the cooling compensates the frictional heating from the r-mode: Wdiss = Lcool . From Eqs (2), (6), (7) and (8) one can work out the equilibrium temperature: Teq f = 4.2 × 10 K 330Hz
˙ M 10?8 M⊙ /y
1.4M⊙ p M
where f is the rotational frequency of the star. It is interesting to examine how (and whether) the star reaches this equilibrium point. For temperature-independent damping, Eqs (2) and (3) form a closed system with two ? independent variables, ? and α. To investigate the behavior of the star after it reaches the ? ? ? ? ? stability curve at ? = ?cr , we set ? = ?cr + ?1 and expand Eqs (2) and (3) to ?rst order in ? ?1 . After trivial algebraic manipulations, we can then reduce the system of two ?rst-order di?erential equations to a single second-order di?erential equation: dx ?V (x) d2 x + γ(x) =? . 2 dt dt ?x (15)
– 14 – Here x = ln α, and γ(x) and V (x) are given by γ(x) = and V (x) = 6 τgrav Q exp(2x) x . ? τv τacc (17) 2Q exp(2x) τv (16)
? ? ˙ In the above Equations τgrav is given by Eq. (6), and τacc = (1/p) (3/4)?cr IM/M is the timescale for the neutron star to be spun up by accretion to the angular frequency ?cr . Clearly Eq. (15) can be thought of as an equation of motion for a particle of unit mass in the potential well given by V (x) and with the damping γ(x). The bottom of the potential well corresponds to the equilibrium state described above, and the damping insures that the “particle” gets there (i.e. that the neutron star settles into the equilibrium state). However, the damping is small. To see this, consider damped oscillatory motion close to the bottom of the well. The complex angular frequency of this motion is given by ω= 12/(τacc τgrav ) ? i/(2τacc ). (18)
The period of these small oscillations is M P ? 230 1.4M⊙
10?8 M⊙ y?1 ˙ M
but the timescale on which they are damped (i.e. the timescale on which the equilibrium is reached) is τeq ? 2τacc ? 107 y. Since the damping is so small, ?uctuating disturbances may keep this nonlinear oscillator o? it’s equilibrium position. For example, in our evolutionary scenario we have assumed that there is a mechanism which gives α some non-zero initial value. Presumably, the same mechanism could keep the oscillator in an excited state. Then the amplitude of the r-mode, and hence the temperature of the star’s core, would vary on the timescales of hundreds of years. Detailed investigation of these issues is a subject for further work.
– 15 – However it is clear that the time-averaged temperature should be close to the equilibrium value given by Eq. (14). If the r-mode damping does not depend on temperature, we can expect r-modes to be excited in many of the rapidly rotating neutron stars in LMXBs. These presumably super?uid steady gravitational-wave emitters could be detected by enhanced LIGO gravitational wave detectors, as discussed in Bildsten (1998) and Andersson, Kokkotas and Stergioulas (1998). Recently, Brady and Creighton (1998) have considered the computational cost of such detection. Their conclusion was that with the enhanced LIGO sensitivity and available computational capabilities one could detect gravitational-wave emitters in LMXBs that are as bright in X-ray ?ux as SCO-X1. If the rotational frequency of the emitting neutron star is localized to within a few 10s of Hz using astronomical observations (by, e.g., QPO’s), one could narrow-band the interferometer response around the frequency of r-mode oscillations (see e.g. Meers 1988). This could allow LIGO to detect gravitational-wave emitters in LMXBs which are 10 ? 100 times dimmer in X-ray ?ux than SCO-X1. Positive detection of gravitational waves at the r-mode oscillation frequency would make a strong case for the super?uid nature of the r-mode damping.
In this paper we have investigated the recent proposal that the accretional spin-up of the neutron star in an LMXB is stopped by r-mode gravitational radiation reaction. There are two possible evolutionary scenarios. In the ?rst scenario, the neutron star goes through cycles such as that shown on Fig. 1. The necessary condition for this scenario to be relevant is that r-mode damping should decrease with increasing temperature. In this case, it is
– 16 – very unlikely that any of the currently observed neutron stars in LMXBs in our galaxy are in the r-mode excited phase of the cycle. The detection of gravitational radiation from extragalactic LMXBs in the r-mode excited phase is also not likely, even with advanced LIGO interferometers. In the second scenario, r-mode damping is temperature independent, and a steady-state equilibrium is probably reached, where both angular velocity and temperature stay constant, or are oscillating with periods of several hundreds of years. Equation (14) makes a robust prediction for the temperature of these objects to be ? 4 × 108 K; this temperature is on the high end of what is typically expected; and it might be possible to test this prediction by observations. In this case the neutron stars are emitters of periodic gravitational waves, which could be detected by interferometers like enhanced LIGO. I wish to thank Kip Thorne, Lars Bidsten and Lee Lindblom for making suggestions and comments which are crucially important for this work. I thank Teviet Creighton, Curt Cutler, Benjamin Owen, Sterl Phinney and Bernard Schutz for discussions, and AEI-Potsdam, where part of this paper was completed, for hospitality. This work was supported by NSF grants AST-9731698 and PHY-9424337.
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A This manuscript was prepared with the AAS L TEX macros v4.0.
– 19 – Fig. 1.— Cyclic evolution of a strongly accreting neutron star in a LMXB. R-mode damping is assumed to decrease with the temperature T of the neutron-star core. Line K ? L ? M represents the “stability curve”; when the neutron star gets above this line, r-modes grow due to gravitational radiation reaction. Leg A → B of the evolutionary track represents the accretional spin-up of neutron star to the critical angular frequency; B → C represents the heating stage in which the r-modes become unstable, grow and heat up the neutron star; C → D shows the spindown stage in which the angular velocity decreases due to the emission of gravitational radiation; and D → B represents the neutron-star cooling back to the equilibrium temperature with simultaneous spin-up by accretion, thus closing the cycle.
2 M 500
K 400 0.3 B 0.13yrs C 300 0.2
D L 100
0 0 1 2