Issue 
A&A
Volume 603, July 2017



Article Number  A9  
Number of page(s)  7  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201629610  
Published online  30 June 2017 
Evolution of newborn rapidly rotating magnetars: Effects of Rmode and fallback accretion
^{1} School of Astronomy and Space Science, Nanjing University, 210093 Nanjing, PR China
email: dzg@nju.edu.cn
^{2} Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, 210093 Nanjing, PR China
^{3} MaxPlanckInstitut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
Received: 30 August 2016
Accepted: 24 April 2017
In this paper we investigate effects of the rmode instability on a newborn rapidlyrotating magnetar with fallback accretion. Such a magnetar could usually occur in corecollapse supernovae and gammaray bursts. We find that the magnetar’s spin and rmode evolution are influenced by accretion. If the magnetar is sufficiently spun up to a few milliseconds, gravitational radiation leads to the growth of the rmode amplitude significantly. The maximum rmode amplitude reaches an order of ~0.001 when the damping due to the growth of a toroidal magnetic field balances the growth of the rmode amplitude. If such a sufficiently spunup magnetar was located at a distance less than 1 Mpc, then gravitational waves would be detectable by the Einstein Telescope but would have an extremely low event rate. However, if the spinup is insufficient, the growth of the rmode amplitude is mainly due to the accretion torque. In this case, the maximum rmode amplitude is of the order of ~10^{6}−10^{5}.
Key words: stars: neutron / gravitational waves / stars: rotation / stars: magnetars / stars: oscillations / accretion, accretion disks
© ESO, 2017
1. Introduction
The rmode instability is of great astrophysical importance, since it occurs in all rotating perfect fluid neutron stars (Andersson 1998; Friedman & Morsink 1998). This instability is driven by gravitational radiation via the ChandrasekharFriedmanSchutz mechanism (Chandrasekhar 1970; Friedman & Schutz 1978), which leads to rapid growth of the rmode amplitude, while viscosity acts to suppress the instability. Given the dependence of gravitational wave emission on the neutron star spin and the dependence of viscosity on the neutron star temperature, the critical angular velocitytemperature curve dividing the rmode stability from instability (hereafter rmode instability window) was obtained by Lindblom et al. (1998), who also found that gravitational radiation of the l = 2 current multipole is stronger than the other multipoles. Owen et al. (1998) carried out a detailed study of the rmode instability in rapidly rotating neutron stars and calculated the gravitational wave forms. Such gravitational wave signals might be detected at a distance of 20 Mpc by the advanced Laser Interferometer Gravitationalwave Observatory (aLIGO). Further studies were carried out to obtain the properties of the rmode instability especially for gravitational waves from rotating neutron stars (Andersson et al. 1999a, 2000, 2001; Kokkotas & Stergioulas 1999; Lindblom et al. 1999, 2000, 2001; Bildsten & Ushomirsky 2000; Lindblom & Mendell 2000; Arras et al. 2003; Bondarescu et al. 2007, 2009; Yang et al. 2010; Haskell et al. 2012; Alford & Schwenzer 2014a,b, 2015; Chugunov 2015; Haskell 2015; Mytidis et al. 2015) and rotating strange stars (Madsen 1998, 2000; Andersson et al. 2002; Zheng et al. 2006; Yu et al. 2009; Alford & Schwenzer 2014b; Dai et al. 2016).
Neutron stars form during supernova (SN) explosions. Fallback accretion could happen after the core collapse of massive stars (Michel 1988; Chevalier 1989). Owing to rotating progenitors, a disk is likely to form before fallback materials are accreted onto the neutron star surface (e.g., we refer to Piro & Ott 2011). Such a fallback disk is responsible for the jets of pulsars (Blackman & Perna 2004), and early afterglows of gammaray bursts (GRBs; Dai & Liu 2012). The rmode instability of neutron stars (with subEddington accretion) in lowmass Xray binary (LMXB) systems was investigated (Levin 1999; Andersson et al. 1999b; Brown & Ushomirsky 2000; Haskell et al. 2012; Mahmoodifar & Strohmayer 2013). The accreted materials in LMXBs spin up old neutron stars significantly and cause these stars to enter the rmode instability window again. Following this, the heating due to shear viscosity affects the stellar thermal evolution while gravitational radiation spins down the star. The effects of hypercritical advectiondominated accretion flows without magnetic fields in close binary systems were investigated by Yoshida & Eriguchi (2000), who found that hyperaccreting disks considerably increase the stellar angular momentum, which can balance the spindown due to gravitational radiation.
Newborn neutron stars are generally strongly magnetized. The average magnetic field of isolated radio pulsars is ~2 × 10^{12} G (Arzoumanian et al. 2002). However, some astrophysical phenomena were driven by magnetars with field strengths up to 10^{16} G, for example GRBs (Usov 1992; Dai & Lu 1998a,b; Zhang & Mészáros 2001; Thompson et al. 2004; Metzger et al. 2007; Dall’Osso et al. 2011), soft gammaray repeaters (SGRs, Thompson & Duncan 1995, 1996; Vasisht & Gotthelf 1997; Kouveliotou et al. 1998) and some superluminous supernovae (Kasen & Bildsten 2010; Woosley 2010; Wang et al. 2015). Based on the Australia Telescope National Facility (ATNF) catalog^{1} (Manchester et al. 2005), the highest magnetic field discovered so far is ~2 × 10^{15} G in SGR 180620. Such an ultrastrong magnetic field plays an important role in rmode evolution (Ho & Lai 2000; Rezzolla et al. 2000, 2001a,b; Lee 2005; Cuofano & Drago 2010; Cuofano et al. 2012; Abbassi et al. 2012; Asai et al. 2015; Chugunov 2015).
Fallback accretion of rapidly rotating magnetars could drive a subset of transients. For example, the propeller effect during fallback accretion of newborn millisecond magnetars could directly power some broadlined type Ib/c or IIP SNe (Piro & Ott 2011); the early bumps of some GRB afterglows also hint that their central engines are hyperaccreting millisecond magnetars (Dai & Liu 2012). On one hand, the rmode could be unstable in these magnetars and affect the spin evolution via gravitational radiation. On the other hand, both magnetic dipole radiation and fallback accretion can affect evolution of the stellar spin and rmode amplitude. In this paper, we explore the effects of the rmode instability on evolution of newborn hyperaccreting magnetars, and test whether or not gravitational waves due to the rmode are detectable. If detected, such gravitational waves could provide an important probe for central engines of some SNe and GRBs. It was found that magnetars could be gravitational wave sources due to toroidal magnetic fieldinduced distortion of the neutron star (Cutler 2002; Dall’Osso et al. 2009) and the barmode instability (Corsi & Mészáros 2009). Piro & Thrane (2012) studied gravitational radiation due to barmodelike oscillations in accreting magnetars. They found that for a magnetar with radius ~20 km gravitational radiation lasts between hundreds of and a few thousand seconds at typical frequencies of 1000−2000 Hz (Piro & Thrane 2012). However, the rmode instability could dominate gravitational radiation if the saturation amplitude is large enough. In this case, the frequencies of gravitational waves due to rmode instability are ~100–1000 Hz as is shown below, which are the most sensitive frequencies of aLIGO (Harry & LIGO Scientific Collaboration 2010) and the Einstein Telescope (ET; Hild et al. 2011). Thus future detections of gravitational waves could distinguish between these two kinds of oscillations.
In this paper, we investigate the effects of the rmode instability on a newborn rapidlyrotating magnetar with fallback accretion. In Sect. 2, we present some basic equations of the rmode instability and hyperaccreting disks in magnetars and then we obtain evolution equations of the stellar spin and rmode amplitude. The numerical and semianalytical results are presented in Sect. 3. In Sect. 4, we estimate the detectability of gravitational waves due to the rmode instability. Our conclusions and brief discussions are given in Sect. 5.
2. Basic equations of Rmode instability
2.1. Gravitational radiation and viscosities
The rmode instability is generically driven by gravitational radiation and dissipated by viscosities. The typical timescales of gravitational radiation (t_{gw}), shear viscosity (t_{sv}), and bulk viscosity (t_{bv}) depend on the stellar equation of state (EOS). A polytropic EOS of N = 0 or 1 is usually used, while the resulting timescales for these two polytropic EOSs differ only by roughly a factor of two (we refer to a review by Andersson et al. 2001 and references therein). The effects of more realistic EOSs have been tested by Papazoglou & Moustakidis (2016), whose results show only small differences from those of the N = 1 polytropic EOS. Following Alford et al. (2012a,b), we consider the AkmalPandharipandeRavenhall (APR) EOS (Akmal et al. 1998), since this EOS gives a better approximation of the equilibrium configuration than the polytropic EOSs for rmode oscillations.
The gravitational radiation timescale is given by (Lindblom et al. 1998) and Owen et al. (1998), (1)where G is the gravitational constant. The strongest gravitational radiation reaction comes from the l = m = 2 mode (Lindblom et al. 1998; Owen et al. 1998), so we obtain (2)where the radius of the magnetar is R = 11.5R_{11.5} km, the mass is M = 1.4 M_{⊙} × M_{1.4}, and ν = Ω/2π = 10^{3}ν_{3} Hz is the rotational frequency of the magnetar.
Usually very simple models of shear and bulk viscosities are adopted. However, these models are inconsistent with observations of LMXBs even if taking into account the effect of an Ekman layer (Alford & Schwenzer 2014a), and thus reevaluations of the relevant physics are needed (Ho et al. 2011; Haskell et al. 2012; Gusakov et al. 2014; Haskell 2015). Here we consider more recent and complex models to calculate the dissipation timescales of viscosities. The shear viscosity is found to be contributed by electron scattering (Shternin & Yakovlev 2008; Shternin et al. 2013). Following Alford et al. (2012a,b), we adopt this electron scattering model to calculate the dissipation timescale of shear viscosity in this paper, which is (3)where T = T_{9}10^{9} K is the core temperature of the neutron star.
The bulk viscosity is more important, since it is the dominant dissipation mechanism for newlyborn neutron stars. Most previous studies have focused on the effects of the bulk viscosity from the linear perturbation theory. However, as the rmode amplitude (α) grows large enough α ≳ O(1), the bulk viscosity is significantly enhanced due to the suprathermal effect (Alford et al. 2012b). This effect provides an alternative mechanism to saturate the rmode amplitude with the order O(1) to O(10) (Alford et al. 2012b). In this case, the dissipation timescale of the bulk viscosity (t_{bv}) also depends on the rmode amplitude α. For a neutron star dominated by the modifiedURCA processes, this timescale is expressed as (for more details, we refer to Alford et al. 2012b), (4)The first term on the righthand side, namely , is consistent with the results from the linear perturbation theory in previous studies (Lindblom et al. 1998; Owen et al. 1998).
2.2. Effect of magnetic field
In rotating stars, the Coriolis force (F_{c}) provides the restoring force. However, in stronglymagnetized neutron stars, the magnetic force (F_{B}) also acts on the fluid motion. The effects of a magnetic field have been studied widely (Ho & Lai 2000; Rezzolla et al. 2000, 2001a,b; Lee 2005; Cuofano & Drago 2010; Cuofano et al. 2012; Abbassi et al. 2012; Asai et al. 2015; Chugunov 2015). Differential rotation might be induced, and thus transfer energy to produce a toroidal magnetic field (Rezzolla et al. 2000, 2001a,b). We consider the damping timescale due to the generation of a toroidal magnetic field following Cuofano & Drago (2010) and Cuofano et al. (2012), that is, (5)where B = B_{15}10^{15} G is the surface magnetic field of the magnetar and B_{t} = B_{t,14}10^{14} G is the volumeaveraged toroidal magnetic field, which follows from (6)A toroidal magnetic field in the interior can deform the neutron star to an ellipsoid, which leads to gravitational radiation and meanwhile spins down the star by a torque of N_{B,gw} = −IΩ /t_{B,gw}, where t_{B,gw} is the timescale of gravitational radiation related to this torque, is the moment of inertia, and . It is a nontrivial problem that the gravitational radiation from such a magneticallydistorted neutron star depends on details of the magnetic field distribution in the stellar interior, as discussed by many authors (Cutler 2002; Dall’Osso et al. 2009, 2015; Corsi & Mészáros 2009; Cuofano et al. 2012). If the inclination angle between the magnetic dipole moment and the rotation axis is χ = π/ 2, the timevarying mass quadrupole of the neutron star and consequently the gravitational radiation efficiency are maximized; otherwise, the gravitational radiation efficiency should be smaller. If a neutron star has an initial temperature T_{0} = 10^{10} K and cools down via the modified URCA process, as found by Dall’Osso et al. (2009), then the inclination angle evolves following (7)where χ_{i} is the initial inclination angle, which is usually small, and . Thus the magnetic dipole moment will be orthogonal to the rotation axis in less than ~1 ms when χ_{i} = 1deg. For the APR equation of state, we obtain for a neutron star with mass 1.4 M_{⊙} and radius R = 11.5 km (Owen et al. 1998). The timescale t_{B,gw} with χ = π/ 2 reads (Cuofano et al. 2012) (8)The surface magnetic field also slows down the magnetar. For simplicity, the magnetic dipole radiation model is adopted. The torque due to magnetic dipole radiation and corresponding timescale are then given by (9)and (10)where t_{dip} is the typical timescale due to magnetic dipole radiation and we set . R_{L} is the radius of the light cylinder (11)where c is the speed of light. The matter outside R_{L} no longer interacts with the magnetar.
2.3. Hypercritical accretion of a magnetar
Previous studies of the rmode instability in accretion neutron stars have focussed on LMXB systems (Andersson et al. 1999b; Brown & Ushomirsky 2000; Yoshida & Eriguchi 2000; Bondarescu et al. 2007). Here we consider hypercritical fallback accretion of newborn rapidly rotating magnetars, where the accretion torque can be either positive or negative. Following MacFadyen et al. (2001), Zhang et al. (2008) and Piro & Thrane (2012), we parameterize the fallback accretion rate as (12)where (13)and (14)where η ≈ 0.01−10 is a factor which accounts for explosion energies, t is in units of seconds and t_{0} ~ 10^{2}−10^{3} s is the typical time at which the accretion rate starts to decrease (MacFadyen et al. 2001). At late times, the accretion rate decreases as Ṁ ∝ t^{− 5/3} (Chevalier 1989). We thus obtain the total baryonic mass of the neutron star at an arbitrary time t, (15)where M_{b,i} is the initial baryonic mass of the magnetar. A fraction of the accreted mass will turn into the binding energy that will be radiated away in the form of neutrinos. Hence, the gravitational mass (M) is more applicable (Lattimer & Prakash 2001), (16)We take R to be a constant in this paper. In reality, the radius R changes especially when the mass grows very large (Lattimer & Prakash 2001). However, a small change of R cannot affect our results significantly, since the timescales acting on the rmode mainly depend on the core temperature and the spin.
In order to parameterize the torques due to accretion and magnetic dipole radiation, we introduce two useful radii. The first one is the corotation radius where the Keplerian angular velocity of the disk equals the angular velocity of the neutron star Ω, (17)The second one is the magnetospheric (Alfvén) radius where the ram pressure of the fallback materials balances the magnetic pressure, (18)where μ = BR^{3} is the magnetic dipole moment and Ṁ_{3} = Ṁ_{b}/ (10^{3}M_{⊙}/s).
Usually the fastness parameter ω is defined as the ratio of the stellar angular velocity and the Keplerian angular velocity at the magnetospheric radius r_{m} (we refer to e.g., Elsner & Lamb 1977; Ghosh & Lamb 1979), (19)Using these parameters, we can parameterize the accretion torque and corresponding timescale through (20)and (21)If r_{m}<r_{c}, the accreted matter transfers its angular momentum to the magnetar, and thus provides a positive torque. Otherwise, the fallback matter acts as a propeller and offers a negative torque. A detailed picture of accretion onto magnetic stars has been elucidated by many authors, for example Pringle & Rees (1972), Illarionov & Sunyaev (1975), Ghosh & Lamb (1979), Aly (1980), Ekşi et al. (2005), and Dai & Liu (2012). The expression of n(ω) differs slightly in different models. For simplification, we here adopt n(ω) = 1−ω, as in Piro & Ott (2011). This form of n(ω) includes the main properties of an interaction between the accretion disk and the magnetar. Furthermore, it has two advantages; n(ω) is continuous for all ω and becomes zero when r_{m} = r_{c}.
2.4. Evolution of the rmode
We quantify the rmode model based on Owen et al. (1998). The total angular momentum J is given by (22)where J_{c} is the canonical angular momentum of the rmode (Owen et al. 1998), (23)where and for a neutron star with the APR EOS, whose mass is 1.4 M_{⊙} and radius is R = 11.5 km, we have (Owen et al. 1998). The canonical energy of the rmode is given by (Owen et al. 1998), (24)We obtain evolution of the rmode using the time derivative of J and E_{c} (Owen et al. 1998; Andersson et al. 2001; Cuofano et al. 2012), where the torques due to accretion (N_{acc}) and magnetic dipole radiation (N_{dip}) affect the total stellar angular momentum together.
Substituting Eqs. (22)–(24) into Eqs. (25) and (26), we obtain the evolution equations of Ω and α, (27)and (28)where . We see that accretion and magnetic dipole radiation affect the amplitude and spin evolution simultaneously. The magnetic braking increases the rmode amplitude while it spins down the magnetar. The effect of accretion is slightly complicated, since accretion can either spin up or down the magnetar while it can also either decrease or increase the rmode amplitude.
In order to solve the above evolution equations, we must also know how the interior temperature of the magnetar evolves. The fallback accretion will start ~100 s later after the stellar birth (MacFadyen et al. 2001; Dai & Liu 2012). Meanwhile, the protoneutron star will become a neutron star ~50 s later after its birth, and at this time the neutron star is transparent to neutrinos (Ferrari et al. 2003). Thus we consider a hot young isothermal neutron star with an initial temperature of T_{0} = 10^{10} K. The newborn neutron star is then cooled down by the modifiedURCA processes, whose luminosity is erg/s (Shapiro & Teukolsky 1986; Alford & Schwenzer 2014b). The direct URCA processes might set in when the star grows heavy enough. But since we only consider the time (the first ~10^{4} s) when gravitational radiation is significant, the direct URCA processes do not significantly affect our results. We also take into account the heating effect due to shear viscosity in this paper, namely, H_{v} = 2E_{c}/t_{sv}. We do not consider the heating effect of accreted materials, since these materials have been cooled down by neutrinos in the accretion disk (Kohri & Mineshige 2002; Zhang & Dai 2010) and the accretion column (Piro & Ott 2011). The temperature evolution is then given by (29)where C_{V} ≈ 7.0 × 10^{38}T_{9} erg K^{1} (Shapiro & Teukolsky 1986) is the heat capacity of the neutron star.
3. Results
Integrating Eq. (6), (27)–(29), we can obtain evolution of the rmode amplitude, spin, and temperature. The initial spin is of the order of one millisecond, but since accretion spins up the star near the massshedding limit, the effect of the initial spin is less important than that of accretion. Therefore, we mainly focus on effects of accretion and initialize the spin at P_{0} = 1 /ν_{0} = 3 ms. The initial baryonic mass and radius of the magnetar in our model are taken to be M_{b,i} = 1.4 M_{⊙} and R = 11.5 km, and thus the gravitational mass is M_{i} ≃ 1.26 M_{⊙}. The initial amplitude of the rmode is fixed at 10^{8}.
Figure 1 gives four examples of evolution of the rmode amplitude (α), spin (ν) and toroidal magnetic field (B_{t}). We study two specific accretion rates; η = 0.01 with t_{0} = 3000 s (red lines) and η = 0.1 with t_{0} = 300 s (blue lines). The magnetic field is typically chosen to be B = 2 × 10^{14} G (solid lines) and B = 5 × 10^{14} G (dashed lines). For comparison, we also study the case without accretion for B = 2 × 10^{14} G (black solid line). It is shown that the combination of accretion and magnetic fields can significantly affect the rmode amplitude evolution. If the accretion can spin up the neutron star significantly, the amplitude grows quickly to the maximum value of ~ 0.001. Otherwise, the growth speed might be considerably suppressed. In the following, we use a semianalytical method to analyze the amplitude and spin evolution.
To find which mechanism dominates the evolution, we compare the terms in the right hand side of Eqs. (27) or (28). We take the prefactor A_{±} ≈ 1, since the amplitude is very small in an early stage. Since the viscosity terms and the gravitational wave terms due to magnetic deformation are very small, we mainly consider the other terms. Firstly, we compare the accretion torque with the first term in Eq. (27), that is, (30)Generally, ν< 10^{3} Hz holds in most of the evolution time (Fig. 1) and n(ω) is in the range of − 0.5 to 0.5, so we treat with the accretion torque as the dominant term over the first term.
Fig. 1
rmode amplitude (α), spin (ν) and toroidal magnetic field (B_{t}) as functions of time. The black solid line is plotted for B = 2 × 10^{14} G and η = 0 . The red solid or dashed lines are obtained in the case of η = 0.01 and t_{0} = 3000 s for B = 2 × 10^{14} G or B = 5 × 10^{14} G. The blue solid or dashed lines are obtained in the case of η = 0.1 and t_{0} = 300 s for B = 2 × 10^{14} G or B = 5 × 10^{14} G. 
The torque due to magnetic dipole radiation is smaller than the accretion torque, since the magnetospheric radius is generally much smaller than the light cylinder radius and n(ω) ∈ [−0.5,0.5] in the amplitude growing stage so that (31)The toroidal magnetic field damping term is comparable to the accretion term, if the toroidal magnetic field grows strong enough, since we have (32)However, for the spin evolution (Eq. (27)), there is a prefactor . Considering in this paper (Fig. 1) that the amplitude is generally much smaller than 0.2, we can assume that the accretion term dominates the spin evolution.
Adopting Eqs. (13) and (14) and considering only the accretion term in the righthand side of Eq. (27), we obtain an approximate spin evolution, (33)where . Substituting Eq. (33) into Eq. (19), we find that n(ω) = 0 happens at s or s, where ν_{0,333} = ν_{0}/ 333 Hz, ν_{c,800} = ν_{c}/ 800 Hz and t_{0,100} = t_{0}/ 100 s. The second solution (t ~ 110η^{3/5}(t_{0,100})^{13/10} s) corresponds to the time when the maximum spin is reached, which hints that before ~t_{0} the spin is increasing. This analytical expression for ν is consistent with the result shown in Fig. 1.
The amplitude evolution begins to be affected by the gravitational wave emission when (34)This means that gravitational radiation dominates when Hz. Approximately, it is required that , which is roughly .
We also numerically show that the timescales − t_{gw} (purple lines), t_{B,t} (brown lines) and t_{acc} (orange lines) evolve with time in Fig. 2. The accretion rate is fixed to be η = 0.01 at t_{0} = 3000 s. The solid lines correspond to the case of B = 2 × 10^{14} G and dashed lines to B = 5 × 10^{14} G. The sharp transitions happen when ω → 1, which leads to t_{acc} → ∞.
Fig. 2
Evolution of timescales − t_{gw} (purple lines), t_{B,t} (brown lines) and t_{acc} (orange lines) in the cases with η = 0.01 and t_{0} = 3000 s. The solid lines correspond to the case of B = 2 × 10^{14} G and dashed lines to B = 5 × 10^{14} G. 
The amplitude is suppressed by the positive accretion torque before ~t_{0}. After that, if the spin of the neutron star increases significantly enough, which leads to gravitational radiation dominating the rmode amplitude ( t_{gw}  <  t_{acc} ), rapid growth of the amplitude occurs (Fig. 1). To calculate this rapid growth, we only consider the contribution of gravitational radiation in Eq. (28), and obtain (35)Substituting Eq. (33) into this equation, we find (36)The rapid growth happens only when ν_{0}< 3.7  n(ω)  ηt^{3/2}, in which case we can ignore the ν_{0} term in Eq. (36). Therefore, we get (37)which is strongly dependent on n, η and t, meaning the amplitude can grow quickly with time. To reach saturation, the amplitude should grow up to α_{s}/α_{0} = 10^{3}−10^{8}, which means γ = ln(α_{s}/α_{0}) ≈ 7−18. This implies that gravitational radiation can drive rmode to saturation in a few hundred seconds if the star accretes a significant amount of matter. In this case, the maximum amplitude is reached when t_{gw} + t_{B,t} ≈ 0, which is .
However, if the spin grows very slowly, the accretion torque will dominate the amplitude growth ( t_{gw}/t_{acc}  > 1). Considering only the accretion torque term in Eq. (28) and using Eq. (33), we approximately obtain the growth rate of the rmode amplitude due to accretion, (38)where n(ω) is negative and is the amplitude when the accretion torque becomes dominant. In this case, the amplitude grows very slowly with time. Then, the maximum amplitude is reached when t_{B,t} + t_{acc} ≈ 0, which is , where n(ω) < 0.
4. Detectability of gravitational waves
When the rmode is unstable in a magnetar, gravitational waves are emitted. The rmode became very interesting after gravitational waves from the merger of double black holes were detected by aLIGO (Abbott et al. 2016). We now test whether or not such a gravitational wave signal from a single source can be detectable by gravitational observatories such as ET. The gravitational wave strain amplitude for the rmode, which is averaged over polarizations and orientations, is given by Owen et al. (1998) as(39)where D is the distance of the source and (40)where c is the speed of light. We mainly focus on the frequency spectrum of the gravitational wave by using the Fouriertransform, (Owen et al. 1998), where f ≈ 2Ω/3π is the frequency of the gravitational wave for the l = m = 2 current multipoles. A characteristic amplitude is usually defined and we show it in Fig. 3 for different cases versus the rms strain noise of ET, where S_{h}(f) is the noise strain amplitude.
The red, blue, magenta, and orange solid curves in Fig. 3 are the characteristic amplitude h_{c} in the cases with different accretion rates and magnetic fields, when the sources locate at a distance of D = 1 Mpc. The green line is the ET noise rms strain amplitude (Hild et al. 2011) and the characteristic amplitudes are higher than the noise curves. Thus it is possible to detect such sources once ET works.
Fig. 3
Characteristic strain amplitude of the rmode instability for accreting magnetars locating at a distance D = 1 Mpc. The green curve corresponds to ET noise rms strain amplitude. The red, blue, magenta, and orange curves correspond to the cases: B = 2 × 10^{14} G, η = 0.01, and t_{0} = 3000 s; B = 2 × 10^{14} G, η = 0.1, and t_{0} = 300 s; B = 5 × 10^{14} G, η = 1, and t_{0} = 100 s; and B = 5 × 10^{14} G, η = 10, and t_{0} = 15 s, respectively. 
5. Discussion and conclusion
In this paper, we have explored the rmode instability in newborn rapidlyrotating hyperaccreting magnetars. Their spin evolution is influenced by the accretion torque. There are two possibilities. Case I: if the neutron star is spun up significantly by accretion, namely, the parameters must satisfy the condition of , the rmode amplitude will grow quickly due to the gravitational radiation ln [α(t) /α_{0}] = 12(nη)^{6}(t/ 100 s)^{10}. The saturation happens when the damping due to the generation of a toroidal magnetic field balances the gravitational radiation for . The generated magnetic field reaches a value that is slightly weaker than the poloidal field. The saturation amplitude is ~0.001.
However, if the spinup is not significant (Case II), the accretion torque will dominate the amplitude growth, which leads to a much slower growth (Eq. (38)). The saturation is then met when . The maximum saturation amplitude is of the order of ~10^{6} to 10^{5}. The generated toroidal field is much weaker than that in the case of a significant spinup.
If the magnetar is sufficiently spunup (i.e., Case I discussed here), the gravitational radiation from such a magnetar would be possibly detectable with the future mission ET, if the source were located at a distance <1 Mpc. Corecollapse SNe and gammaray bursts are electromagnetic counterparts of such gravitational wave sources. However, based on the recent estimates (Li et al. 2011; Strolger et al. 2015), the local corecollapse supernova rate is ~a few × 10^{2} Mpc^{3} per century. Thus, the detectable gravitational wave event rate with ET is as low as roughly one per millennium.
Acknowledgments
We thank the referee for valuable comments and constructive suggestions that have allowed us to improve our manuscript significantly. We also thank YiMing Hu, Yan Yan and YunWei Yu for their helpful discussions. This work is supported by the National Basic Research Program (“973” Program) of China (grant No. 2014CB845800) and the National Natural Science Foundation of China (grant No. 11573014).
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All Figures
Fig. 1
rmode amplitude (α), spin (ν) and toroidal magnetic field (B_{t}) as functions of time. The black solid line is plotted for B = 2 × 10^{14} G and η = 0 . The red solid or dashed lines are obtained in the case of η = 0.01 and t_{0} = 3000 s for B = 2 × 10^{14} G or B = 5 × 10^{14} G. The blue solid or dashed lines are obtained in the case of η = 0.1 and t_{0} = 300 s for B = 2 × 10^{14} G or B = 5 × 10^{14} G. 

In the text 
Fig. 2
Evolution of timescales − t_{gw} (purple lines), t_{B,t} (brown lines) and t_{acc} (orange lines) in the cases with η = 0.01 and t_{0} = 3000 s. The solid lines correspond to the case of B = 2 × 10^{14} G and dashed lines to B = 5 × 10^{14} G. 

In the text 
Fig. 3
Characteristic strain amplitude of the rmode instability for accreting magnetars locating at a distance D = 1 Mpc. The green curve corresponds to ET noise rms strain amplitude. The red, blue, magenta, and orange curves correspond to the cases: B = 2 × 10^{14} G, η = 0.01, and t_{0} = 3000 s; B = 2 × 10^{14} G, η = 0.1, and t_{0} = 300 s; B = 5 × 10^{14} G, η = 1, and t_{0} = 100 s; and B = 5 × 10^{14} G, η = 10, and t_{0} = 15 s, respectively. 

In the text 
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