GRB afterglow plateaus and Gravitational Waves: multi-messenger signature of a millisecond magnetar?
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Preprint typeset using L A TEX style emulateapj v. 08/22/09
GRB AFTERGLOW PLATEAUS AND GRAVITATIONAL WAVES:MULTI-MESSENGER SIGNATURE OF A MILLISECOND MAGNETAR?
Alessandra Corsi and Peter M´esz´aros Draft version November 8, 2018
ABSTRACTThe existence of a shallow decay phase in the early X-ray afterglows of gamma-ray bursts is acommon feature. Here we investigate the possibility that this is connected to the formation of ahighly magnetized millisecond pulsar, pumping energy into the fireball on timescales longer thanthe prompt emission. In this scenario the nascent neutron star could undergo a secular bar-modeinstability, leading to gravitational wave losses which would affect the neutron star spin-down. Inthis case, nearby gamma-ray bursts with isotropic energies of the order of 10 ergs would produce adetectable gravitational wave signal emitted in association with an observed X-ray light-curve plateau,over relatively long timescales of minutes to about an hour. The peak amplitude of the gravitationalwave signal would be delayed with respect to the gamma-ray burst trigger, offering gravitationalwave interferometers such as the advanced LIGO and Virgo the challenging possibility of catching itssignature on the fly. Subject headings: gamma rays: bursts; radiation mechanisms: non-thermal; gravitational waves INTRODUCTION
Thanks to
Swift observations (e.g. Nousek et al.2006; Zhang et al. 2006), it has now become evidentthat the “normal” power-law behavior of long GRB X-ray light curves F ( T ) ∝ T α with α ∼ − . F ( T ) is the observed flux and T is the observer’s time),is often preceded at early times by an initial steep de-cay ( α ∼ − α & − .
5, see Fig. 1). The steep-to-shallow andshallow-to-normal decay transitions are separated by twocorresponding break times, 100 s . T break, .
500 sand 10 s . T break, . s. During the shallow-to-normal transition the spectral index does not change andthe decay slope after the break ( α ∼ − .
2) is gener-ally consistent with the standard afterglow model (e.g.M´esz´aros & Rees 1997; Sari et al. 1998), while the de-cay slope before the break is usually much shallower. Thelack of spectral changes suggests that the shallow phasemay be attributed to a continuous energy injection bya long-lived central engine, with progressively reducedactivity (for a review see Zhang et al. 2006, and refer-ences therein). Recently, Panaitescu & Vestrand (2008)have pointed out that the effects of a late-time energyinjection may also be evident in some optical afterglows,around 30 − s after the trigger. Although it is stillnot clear if a typical “steep-flat-steep” behavior doesexist also in short GRB X-ray afterglows, the case ofGRB 051221a does fit this scheme remarkably, with aplateau observed right in the middle of the afterglow de-cay (Soderberg et. al 2006). Electronic address: [email protected] address: [email protected] Universit`a di Roma “Sapienza” and INFN-Roma, PiazzaleAldo Moro 2, 00185 - Roma (Italy) Department of Astronomy & Astrophysics, Department ofPhysics, and Institute for Gravitation and the Cosmos, The Penn-sylvania State University, University Park, Pennsylvania 16802(USA) IASF-Roma/INAF, Via Fosso del Cavaliere 100, 00133 - Roma(Italy)
Newborn magnetars are among the progeni-tors proposed to account for shallow decays orplateaus observed in GRB light curves (Dai & Lu1998; Zhang & M´esz´aros 2001; Fan & Dong 2006;Yu & Huang 2007). Independent support for thisscenario comes from the observation of SN2006aj,associated with the nearby sub-energetic GRB 060218,suggesting that the supernova-GRB connection mayextend to a much broader range of stellar masses thanpreviously thought, possibly involving two differentmechanisms: a “collapsar” for the more massive starscollapsing to a black hole (BH), and a newborn neutronstar (NS) for the less massive ones (Mazzali et al.2006).Previous studies aimed at accounting for the after-glow plateaus by invoking a magnetar-like progenitorhave assumed that the magnetar’s slow-down is domi-nated by magnetic dipole losses, neglecting the contri-bution from the emission of gravitational waves (GWs,see Dai & Lu 1998; Fan & Dong 2006; Yu & Huang2007), or treating them separately as a limiting casefor a NS with sufficiently high, constant eccentricity(Zhang & M´esz´aros 2001). Such studies have shownhow magnetars dipole losses may indeed explain the flat-tening observed in GRB afterglows. In the simplest ver-sion of the magnetar scenario, the end of the shallowdecay is accompanied by an achromatic break, while sev-eral cases of chromatic breaks have also been observed(e.g. Panaitescu 2009). Additional mechanisms such asvariable micro-physical parameters in the fireball shockfront (e.g. Panaitescu et al. 2006) or a structured jetmodel (e.g. Racusin et al. 2008) can be invoked to ex-plain such chromatic breaks. Anyhow, a larger sampleof simultaneous optical-to-X-ray observations is neededto firmly asses the achromatic or chromatic behavior ofbreaks associated to the end of the shallow-decay phase.In this paper, we investigate in more detail the ef-fects of GW losses on the magnetar’s spin-down, andexplore quantitatively the signatures which could test A. Corsi and P. M´esz´aroswhether this is indeed the mechanism at work in theshallow X-ray light curves. Although the precise evolu-tion of a newborn magnetar from birth up to timescalesof ∼ − s is difficult to predict or to follow withnumerical simulations, here we point out that among thepossible evolutionary paths which one may reasonablyconsider, one plausible and particularly interesting pos-sibility to explore is that of a newborn NS left over aftera GRB explosion, which undergoes a secular bar-modeinstability. In this scenario, simple estimates accountingfor the most relevant energy loss processes can provideuseful insights into the viability of having efficient GWemission associated with a GRB X-ray afterglow plateau.Although these estimates are clearly approximate, sincepossible complications like viscosity effects or magneticfield driven instabilities are neglected, they nonethelessallow us to make a first statement on the relevance of theconsidered process. Moreover, while other scenarios arealso possible, the interesting aspect of this particular oneis that, on the one hand, GW observations would be facil-itated by the presence of an electromagnetic signature topinpoint the GW signal search, while on the other handthe detection of bar-mode like GWs in coincidence with aGRB X-ray plateau would be a smoking-gun signature ofa magnetar pumping energy into the fireball, thus identi-fying the much-debated plateau mechanism. Given thatseveral alternative scenarios have been invoked to explainthe afterglow flattening, which are not expected to be as-sociated with GW signals (see e.g. Panaitescu 2008),this would represent a significant step forward in our un-derstanding of GRB physics. Moreover, identifying thepresence of a magnetar would confirm that not all GRBexplosions necessarily lead to the prompt formation of aBH.A point of interest for current analyses that GW de-tectors are carrying out (see e.g. Acernese et al. 2008;Abbott et al. 2008a,b) is that the scenario describedhere involves a new class of GW signals, which shouldbe searched for in coincidence with GRBs. These wouldhave a longer duration (10 − s) and a different fre-quency evolution than the type of GW signals currentlyconsidered to be possibly associated with GRBs. More-over, being delayed by minutes to . γ -ray trigger, the GW signal associatedwith a GRB plateau would offer the challenging possibil-ity of an on-line detection. In light of the fact that theVirgo and LIGO interferometers are now progressingtoward their enhanced/advanced configurations, and get-ting prepared for performing on-line data analyses, thisprospect appears very appealing. It is worth noting thatdespite a GW signal in coincidence with a GRB plateaucould also be searched off-line by LIGO or Virgo, an on-line detection would be highly preferable, since it couldserve as a trigger for ground-based optical follow-ups,even if a GRB trigger alert is absent for any reason.The paper is organized as follows. In Sec. 2 we brieflydescribe how GRB afterglow plateaus are modeled in thecontext of the magnetar model. In Sec. 3 we reviewthe main processes that can lead to GW emission as-sociated with NS formation. The aim of this section isto show that, among the different mechanisms that can Fig. 1.—
Cartoon representation of the typical light curve behav-ior observed by
Swift
XRT. The “standard” power-law decay withindex α = − . − s,during which the decay index is α = − . come into play, the high efficiency of the secular bar-mode instability is conducive to producing GW signalswhich are detectable also from relatively nearby extra-galactic sources. Moreover, it develops on timescalescompatible with the observed durations of GRB plateaus.Sec. 4 describes the general idea and particular aspects ofthe scenario being explored here, and how it can explainGRB afterglow plateaus with the presence of a magne-tar whose spin-down includes both magnetic dipole andbar-mode GW losses. In Sec. 5 we present the results ofour calculations, and in Sec. 6 we discuss these results,summarizing our conclusions in Sec. 7. GRB PLATEAUS IN THE MAGNETAR SCENARIO
Although a wide range of GRB progenitors endin the formation of a BH-debris torus system, ithas been proposed that some progenitors may leadto a highly magnetized rapidly rotating pulsar (e.g.Usov 1992; Duncan & Thompson 1992; Thompson1994; Yi & Blackman 1998; Blackman & Yi 1998;Dai & Lu 1998; Kluzniak & Ruderman 1998;Nakamura 1998; Spruit 1999; Wheeler et al. 2000;Ruderman et al. 2000; Levan et al. 2006; Mereghetti2008; Bucciantini et al. 2009), with such possibilitybeing realized not only in the case of long GRBsassociated to collapsars, but eventually also in scenariosrelevant for short GRBs, such as NS binary mergers(Dai & Lu 1998, and references therein).Fast rotating highly magnetized pulsars, are amongthe class of progenitors that may be associated withsignificant energy input in the fireball for timescaleslonger than the γ -ray emission, thus being relevant forexplaining GRB afterglow plateaus. A detailed analy-sis of the observable effects linked to the presence of apulsar pumping energy into the fireball was performedby Zhang & M´esz´aros (2001), the results of which webriefly recall in what follows.Consider the general scenario where the GRB is pow-ered by a central engine that emits both an initial impul-sive energy input, E imp , as well as a continuous luminos-ity, the latter varying as a power-law with time, i.e. L = L (cid:16) TT (cid:17) q where T is the observer’s time. This could beRB afterglow plateaus and Gravitational Waves 3the case if the central engine is a pulsar and the initial im-pulsive GRB fireball is due to ν - ν annihilation or magne-tohydrodynamical processes (see e.g. MacFadyen et al.1999; Popham et al. 1999; Di Matteo et al. 2002; Lee2005; Oechslin & Janka 2006; Zhang & Dai 2009). Insuch a case, a self-similar blast wave is expected to format late times. In a GRB, the timescale T at whichthe self-similar solution applies is roughly equal to thetime for the external shock to start decelerating whilecollecting material from the interstellar medium (e.g.Sari & Piran 1999). At different times, the total en-ergy into the fireball may be dominated by either the ini-tial impulsive term, or by the continuous injection one,whose contribution will scale as E inj = L T q +1 (cid:16) TT (cid:17) q +1 .The continuous energy injection term can dominate onthe impulsive one for T & T c (where T c & T so as to as-sure that the self-similar solution has already developedwhen the continuous injection law dominates), if q > − E inj ( T c ) ∼ E imp . In the particular case in which L T ∼ E imp then T c ∼ T and the dynamics is dom-inated by the continuous injection as soon as the self-similar evolution begins. Generally speaking, one canwrite T c = max n T , T [( q + 1) E imp / ( L T )] / (1+ q ) o (Zhang & M´esz´aros 2001). Note that the continuousinjection may, in addition, have another characteristictimescale T f at which the continuous injection power-law index q > − q < −
1. Insuch a case, it is only for T c < T f that the continuousinjection has a noticeable effect on the afterglow lightcurve (Zhang & M´esz´aros 2001).During the energy-injection dominated phase, thepeak flux, peak frequency and cooling frequency ofthe synchrotron photons produced by the forwardshock (Sari et al. 1998) scale with time as F m ∝ T q , ν m ∝ T − (2 − q ) / , ν c ∝ T − ( q +2) / , respectively(Zhang & M´esz´aros 2001), that reduce to the standardscalings for q = − q ∼
0, one has F m ∝ T , ν m ∝ T − , ν c ∝ T − , respectively. Thesescalings allow one to compute the temporal indices ofthe afterglow light curve expected during the injectionphase. Supposing to be in slow cooling, these are F ν ∝ F m ν ( p − / m ∝ T α = T (3 − p ) / for ν m < ν < ν c and F ν ∝ F m ν ( p − / m ν / c ∝ T α = T (2 − p ) / for ν > ν c , where wehave indicated with p the power-law index of the electronenergy distribution in the shock front (Sari et al. 1998).For 2 < p <
4, one has 0 . > α > − . ν m < ν < ν c and 0 > α > − ν > ν c , to comparewith α & − . − / > α > − / ν m < ν < ν c , and − > α > − / ν > ν c , for thesame range of p values. Thus, the presence of a pul-sar pumping energy into the fireball at a nearly constantrate, is expected to cause a flattening in the typical decayof the afterglow light curve, with α & − .
5, in agreementwith
Swift observations (see Fig. 1). GWS BY NS FORMATION
Gravitational collapse leading to the formation of a NShas long been considered an observable source of GWs. During the core collapse, an initial signal is expected tobe emitted due to the changing axisymmetric quadrupolemoment. A second part of the GW signal is producedwhen gravitational collapse is halted by the stiffeningof the equation of state above nuclear densities and thecore bounces, driving an outwards moving shock, withthe rapidly rotating proto-neutron star (PNS) oscillat-ing in its axisymmetric normal modes. In a rotatingPNS, non-axisymmetric processes can also yield to theemission of GWs with high efficiency. Such processes areconvection inside the PNS and in its surrounding hot en-velope, anisotropic neutrino emission, dynamical insta-bilities, and secular gravitational-radiation driven insta-bilities, that we briefly recall in what follows. We referthe reader to e.g. Kokkotas (2008) for a recent, moredetailed review.-
Convection and neutrino emission -
2D simulationsof core collapse (M¨uller et al. 2004) have shown thatthe GW signal from convection significantly exceeds thecore bounce signal for slowly rotating progenitors, beingdetectable with advanced LIGO for galactic sources. Inmany simulations, the GW signature of anisotropic neu-trino emission has also been considered (Epstein 1978;Burrows & Hayes 1996; M¨uller & Janka 1997) and es-timated to be detectable by advanced LIGO for galacticsources.-
Dynamical instabilities -
They arise from non-axisymmetric perturbations and are of two differenttypes: the classical bar-mode instability and the morerecently discovered low-
T / | W | bar-mode and one-armedspiral instabilities. In Newtonian stars, the classical m = 2 bar-mode instability is excited when the ratio β = T / | W | of the rotational kinetic energy T to the grav-itational binding energy | W | is larger than β dyn = 0 . β becoming & β dyn ( β ∝ /R during contraction). The instability growson a dynamical timescale (the time that a sound waveneeds to travel across the star) which is about one ro-tational period, and may last from 1 to 100 rotationsdepending on the degree of differential rotation (e.g.Baiotti et al. 2007; Manca et al. 2007). If the bar per-sists for ∼ m = 1 one-armed spiralinstability has also been shown to become unstable inPNS, provided that the differential rotation is sufficientlystrong (with matter on the axis rotating at least tentimes faster than matter on the equator, Centrella et al.2001; Saijo et al. 2002). In recent simulations of rotat-ing core collapse to which differential rotation was added(Ott et al. 2005), the emitted GW signal reached a max-imum amplitude comparable to the core-bounce axisym-metric signal, after ∼
100 ms and at a frequency of ∼ Secular instabilities -
At lower rotation rates, a starcan become unstable to secular non-axisymmetric in-stabilities, driven by gravitational radiation or viscos-ity. Secular GW-driven instabilities are frame-dragginginstabilities usually called Chandrasekhar-Friedman-Schutz (CFS, Chandrasekhar 1970; Friedman 1978) in-stabilities. Neglecting viscosity, the CFS-instability is A. Corsi and P. M´esz´arosgeneric in rotating stars for both polar and axial modes.In the Newtonian limit, the l = m = 2 f -mode, whichhas the shortest growth time of all polar fluid modes(1 s . τ GW . × s for 0 . & β & .
15, seeLai & Shapiro 1995), becomes unstable when β & . f -mode instability, also referred to as the secu-lar bar-mode instability, is an excellent source of GWs.In the ellipsoidal approximation, Lai & Shapiro (1995)have shown that the mode can grow to a large nonlin-ear amplitude, modifying the star from an axisymmet-ric shape to a rotating ellipsoid, that becomes a strongemitter of GWs until the star is slowed-down towardsa stationary state. This stationary state is a Dedekindellipsoid, i.e. a non-axisymmetric ellipsoid with inter-nal flows but with a stationary (non-radiating) shapein the inertial frame. During the evolution, the non-axisymmetric pattern radiates GWs sweeping throughthe advanced LIGO/Virgo sensitivity window (from 1kHz down to about 100 Hz), which could become de-tectable out to a distance of more than 100 Mpc. Tworecent hydrodynamical simulations (Shibata & Karino2004; Ou et al. 2004, in the Newtonian limit and usinga post-Newtonian radiation-reaction, respectively) haveessentially confirmed this picture.Among axial modes, the l = m = 2 r -modeis an important member (see e.g. Andersson1998; Friedman & Morsink 1998; Lindblom et al.1998; Owen et al. 1998; Lindblom & Owen 2002;Andersson & Kokkotas 2001; Andersson 2003;Bondarescu et al. 2009). If the compact object isa strange star, such instability is predicted to persistfor a few hundred years (at a low amplitude) and,integrating data for a few weeks, could yield to aneffective amplitude h eff ∼ − for galactic signals, atfrequencies ∼ − Other magnetic-field related effects - Finally, mech-anisms different from rotational instabilities can be in-voked as GW sources in newborn magnetars. E.g.,in several scenarios the star’s shape may be domi-nated by the distortion caused by very high internalmagnetic fields (e.g. Palomba 2000; Cutler 2002;Arons 2003; Stella et al. 2005; Dall’Osso & Stella2007; Dall’Osso et al. 2008). GW signals produced bythese kind of processes are typically estimated to be de-tectable by the advanced interferometers up to the VirgoCluster (i.e. distances of the order of 20 Mpc). THE NS SPIN-DOWN
On the longer afterglow timescales that are of interestfor the present work, the energy injection into the fireballby a magnetar eventually surviving after the GRB explo-sion, is expected to be mainly through electromagneticdipolar emission (Zhang & M´esz´aros 2001). For whatconcerns GW losses, in this work we focus on the secu-lar bar-mode instability, given its high efficiency in theproduction of GWs, and being its characteristic timescale τ GW compatible with the one of GRB plateaus (see Sects.3 and 4).As discussed in the previous section, a collapsing corerotating sufficiently fast is expected to become non-axisymmetric when β is sufficiently large. Since a new-born NS can be secularly unstable but dynamically stableonly if the rotation rate of the pre-collapse core lies in anarrow range, and since during the collapse β increases proportionally to R − , Lai & Shapiro (1995) consideredmore likely that the core becomes dynamically unstable( β > β dyn ) following the collapse, provided the initial β i is not too small. On a short dynamical timescale,such NS will evolve toward a nearly axisymmetric equi-librium state, with β decreasing below β dyn , but possi-bly remaining above β sec (see Lai & Shapiro 1995, andreferences therein). Due to gravitational radiation, thenearly axisymmetric core (secularly unstable Maclaurinspheroid) will evolve into a non-axisymmetric configu-ration (Riemann-S ellipsoid), on a secular dissipationtimescale ∼ τ GW . While an initial dynamical unsta-ble phase would possibly produce a GW burst duringthe GRB, the secular evolution takes place on longertimescales, thus being relevant for the shallow phase(100 s . T . s) observed in GRB afterglows (seeFig. 1). For this reason, in what follows we focus onthe secular bar-mode instability. It is worth noting,however, that also the presence of a bar-like GW burstfrom a dynamical bar-mode instability, would provide ahint for a magnetar being formed in the GRB explosion.BH formation, in fact, is not expected to lead to strongquadrupole moments (except if it is argued for blobsforming in the infall, see e.g. Kobayashi & M´esz´aros2003), and in any case a dynamically unstable magne-tar would presumably give rise to a more regular signal.Fully general relativistic axisymmetric simulations ofrotating stellar core collapse in three spatial dimension,performed for a wide variety of initial conditions (rota-tional velocity profile, equations of state, total mass),indicate that the threshold β = 0 .
27 for the onset of theclassical dynamical instability is passed if the the progen-itor of the collapse is: (i) highly differentially rotating;(ii) moderately rapidly rotating with 0 . . β i . . β above the secular instabilitythreshold. Based on their results, Dimmelmeir et al.(2008) consider it unlikely that a PNS in nature de-velops a high- β dynamical instability at or early aftercore bounce. While many of the PNS could theoreti-cally reach β dyn during their subsequent cooling to thefinal condensed NS (if angular momentum is conserved),it is however considered more likely that the secular in-stability driven by dissipation or gravitational radiationback-reaction will set in first (Dimmelmeir et al. 2008).Still, three-dimensional simulations are necessary to pro-vide conclusive tests of these predictions.Under the hypothesis that a secular bar-mode insta-bility does indeed set in, in this work we follow the NSquasi-static evolution under the effect of gravitational ra-diation according to the analytical formulation given byLai & Shapiro (1995). Such evolution can in principlebe studied using the full dynamical equations of ellip-soidal figures (Chandrasekhar 1969), including gravita-tional radiation reaction. However, since τ GW is gener-ally much longer than the dynamical time of the star,RB afterglow plateaus and Gravitational Waves 5the evolution is quasi-static, i.e. the star evolves alongan equilibrium sequence of Riemann-S ellipsoids. Dif-ferently from what done by Lai & Shapiro (1995), herewe add in the energy losses the contribution of magneticdipole radiation, under the hypothesis that those will notsubstantially modify the dynamics, but will act speedingup the overall evolution of the bar along the same se-quence of Riemann-S ellipsoids that the NS would havefollowed in the absence of radiative losses. As we aregoing to show in the following section, dipole losses arenearly constant during the bar evolution so that, accord-ing to what discussed in Sec. 2 for the q = 0 case, theycan act as a source of continuous energy supply into thefireball, explaining the observed slope of GRB afterglowplateaus.It is worth noting that in a real situation wheremagnetic field instabilities and viscosity effects are alsopresent, the relevant timescales may be altered. A secu-larly evolving bar can last up to a timescale of the orderof 10 s, as far as viscosity or magnetic field induced in-stabilities do not substantially modify the dynamics. Vis-cosity may play a role on the secular evolution when thePNS has cooled to below ∼ T ∼ s). Magnetic effects are notoriouslydifficult to predict (see e.g. Shibata & Karino 2004), andin general require making heuristic assumptions. In thecontext of the secular r-mode instability, Rezzolla et al.(2001) have shown that the growth of an initial magneticfield associated with the secular kinematic effects emerg-ing during the evolution of the instability, possibly dampsthe growth of the instability itself. Despite the differentcontext (r-modes), these results do suggest that magneti-cally driven instabilities may complicate the scenario. Inwhat follows, we explore the quantitative consequences ofmaking the plausible assumption that magnetic instabil-ities are less efficient at spinning-down the bar than GWemission and magnetic dipole losses. Moreover, apartfrom magnetic braking (spin-down due to dipole losses)which we do consider here, the presence of a magneticfield can influence the secular evolution in other ways. Amagnetic field anchored on the star’s surface is in factperturbed by the instability itself, and this can lead toelectromagnetic losses which can enhance the CFS mech-anism. E.g. in the context of r-modes, Ho & Lai (2000)have considered the electromagnetic radiation associatedwith the shaking of magnetic field lines by the r-modeoscillations. This effect has been estimated to be negli-gible for NS with magnetic field strengths below 10 G.In our case, magnetic field lines anchored on the surfacewould be distorted by the bar-mode instability. In ourtreatment, we neglect this effect, and include only dipolelosses associated with a magnetic field whose flux is con-served on a sphere of radius equal to the mean radius ofthe ellipsoid. MODELING OF THE NS EVOLUTION
A general Riemann-S ellipsoid is characterized by anangular velocity Ω b e of the ellipsoidal figure (the patternspeed) about a principal axis b e , and by internal fluidmotions which are assumed to have uniform vorticity ζ b e along the same axis (in the frame co-rotating with thefigure). Labeling with a and a the principal axes ofthe ellipsoidal figure in the equatorial plane , and with x and x Cartesian coordinates in such a plane, it can beshown that the fluid velocity in the inertial frame reads ~u = ~u + Ω( b e × ~r ) (1)with ~u = a a Λ x b e − a a Λ x b e (2)the velocity in the frame co-rotating with the figure(Chandrasekhar 1969), where b e and b e are unit vectorsalong the Cartesian axes x and x ; ~r is the positionvector; × indicates the vector product; andΛ = − a a a + a ζ (3)is the angular frequency of the internal fluid motions,i.e. of the elliptical orbits that the particles span aroundthe rotational axis in addition to the pattern motion.The velocity ~u is contained in the plain perpendic-ular to the rotational axis b e , so indicating with ~r ⊥ the projection of the position vector ~r in such planewe can write ~r = ~r ⊥ + x b e , and ~u = d ( ~r ⊥ ) /dt +( dx /dt ) b e = d ( ~r ⊥ ) /dt (i.e. dx /dt = 0 since the compo-nent of ~u along b e is null). Further, we can write ~u =( dr ⊥ /dt ) b r ⊥ + r ⊥ d ( b r ⊥ ) /dt = ( dr ⊥ /dt ) b r ⊥ + r ⊥ ( ~ Ω × b r ⊥ ),where we have set ~ Ω = 1 r ⊥ ( b r ⊥ × ~u ) . (4)Note that ~ Ω is defined in such a way that r ⊥ Ω givesthe component of the particle velocity perpendicular tothe polar radius ~r ⊥ , measured in the inertial frame. How-ever, as underlined above, the motion of fluid particleson the surface can be viewed as the superposition of acircular motion with the pattern frequency Ω, plus anelliptical motion on paths contained on the pattern ellip-soid (resulting in maintaining the pattern fixed). Sincethe internal fluid motions are ellipses rather than circles,there is an additional component of the velocity parallelto ~r ⊥ . Using Eq. (1), one has ~ Ω = (cid:20) Ω − (cid:18) a x + a x r ⊥ a a (cid:19) Λ (cid:21) b e (5)In the frame co-rotating with the pattern, fluid par-ticles on the star’s surface move around the rotationalaxis, on ellipses contained in x = const planes. Thoseellipses are self-similar to the equatorial one and haveequation: In the ellipsoidal approximation, surfaces of constant densityare assumed to be self-similar ellipsoids, so the geometry of theconfiguration is completely specified by the three principal axes a , a and a , and the axis ratio a /a and a /a are the same forall interior isodensity surfaces (Lai, Rasio & Shapiro 1993). A. Corsi and P. M´esz´aros x a (cid:16) − x a (cid:17) + x a (cid:16) − x a (cid:17) = 1 (6)For such fluid particles, a x + a x = a a (cid:16) − x a (cid:17) ,and h r ⊥ i = r a a (cid:16) − x a (cid:17) , so that in one cycle D a x + a x r ⊥ a a E = 1. Thus, in the inertial frame, fluid parti-cles on the star’s surface are characterized by an angularfrequency: Ω eff = h Ω i = Ω − Λ (7)Since the gravitational radiation reaction acts like apotential force, the fluid circulation along the equator ofthe star, C = Z equator ~u · d~l = πa a ζ , (8)where d~l is taken along the star’s equator and ζ is thevorticity in the inertial frame, is conserved in the absenceof viscosity (Lai, Rasio & Shapiro 1993). Therefore, theNS will follow a sequence of Riemann-S ellipsoids withconstant circulation. Treating the NS as a polytrope ofindex n (Chandrasekhar 1939), total mass M , and in-dicating with R the radius of the non-rotating, spher-ical equilibrium polytrope with same mass M , one has(Lai & Shapiro 1995):¯ C = C√ GM R = − M k n C π √ GM R (9)where G is the gravitational constant; k n is a con-stant which depends on the index n of the consideredpolytrope (see e.g. Lai, Rasio & Shapiro 1993). Notethat ¯ C = C / √ GM R is an adimensional quantity, C = − ( k n M C ) / (5 π ) has the dimensions of an angular mo-mentum, and both are proportional to the conserved cir-culation C . It can be shown that (Lai & Shapiro 1995): C = I Λ − k n M a a Ω (10)where I = k n M ( a + a ) / dEdT = − B p R Ω eff c − GI ǫ Ω c = L dip + L GW (11)where E is the NS total energy, L GW = dE GW /dT ac-counts for GW losses, while L dip = dE dip /dT for mag-netic dipole ones. Here ǫ = ( a − a ) / ( a + a ) isthe ellipticity; B p is the dipolar field strength at thepoles; Ω is the pattern angular frequency of the ellip-soidal figure ; R is the mean stellar radius ; c is thelight speed; T is the time measured in an inertial framewhere the pulsar is at rest. L dip is computed conservingthe magnetic field flux over a sphere of radius equal tothe mean stellar radius (i.e. B p R = const = B p, R along the sequence, where R is the geometrical mean of the ellipsoid principal axes), and using the effective an-gular frequency Ω eff , which includes both the patternspeed and the effects of the internal fluid motions. Theuse of Ω eff = h Ω i = D r ⊥ | b r ⊥ × ~u | E accounts for thefact that in the frozen-in magnetic field approximation(see e.g. Goldreich & Julian 1969; Baym et al. 1969;Thompson & Duncan 1996; Morsink & Rezania 2002;Thompson et al. 2002) the magnetic field lines are ineffect tied to the fluid particles on the stellar surface.Note that Ω eff (and the corresponding dipole loss term)is measured in the inertial frame, which is where we com-pute dE/dT as well.Once ( ¯ C , n, M, R , B p, ) are assigned, each configura-tion along a constant- ¯ C sequence is completely deter-mined specifying the axis-ratio x = a /a in the ellip-soid equatorial plane. Thus, all relevant quantities canbe considered as functions of x only, and Eq. (11) canbe written as: dxdT = L dip ( x ) + L GW ( x ) dE/dx (12)We solve the above equation numerically, with its righthand side evaluated along a constant- ¯ C Riemann-S se-quence, and imposing an initial condition sufficientlynear to a uniformly rotating Maclaurin spheroid, ( x ( t i ) = x i →
1) of the given circulation ¯ C . RESULTS AND DISCUSSION
In Fig. 2 we compare the luminosity emitted in GWs,computed with (black-solid line) or without (black-dash-dotted line) the addition of the dipole loss term (red-dashed line) in Eq. (11), for a typical choice of pa-rameters, ( ¯ C , n, M, R , B p, ) = ( − . , , . ⊙ , 20 km,10 G). Note that ¯ C = − .
41 corresponds to a valueof β = 0 .
20 for the initial Maclaurin configuration, i.e.in the middle of the 0 . < β < .
27 range for thesecular instability. As evident from the lower panel ofFig. 2, as long as the circulation is conserved, Ω eff remains nearly constant during all the evolution, and | L dip | ∼ × ergs / s = L (upper panel, red-dashedline). As underlined in Sec. 2, energy pumped into thefireball at a constant rate is sufficient to explain theobserved temporal behavior of afterglow plateaus (i.e. α & − .
5, see Fig. 1). For what concerns the durationof the plateau, for a GRB with impulsive isotropic energyof the order of E imp ∼ ergs, the effect of the energyinjection in the light curve will become visible after atime T c ∼ E imp /L ∼ (10 ergs) / (3 × ergs / s) ∼ T break, ∼ −
500 s (see Fig. 1). Supposing the energyinjection ends or starts fading significantly when the starapproaches the final Dedekind state (see the discussionat the end of this section), the GRB light curve will re-turn to its standard behavior after T break, & , to becompared with the observed range of 10 − s. Thus,for a GRB with such impulsive energy, the properties ofthe plateau associated to the NS secular evolution are inagreement with those typically observed . Note that in our discussion we are neglecting redshift effects,since we are interested in nearby GRBs at z . . d L .
150 Mpc.
RB afterglow plateaus and Gravitational Waves 7
Fig. 2.—
NS evolution along a Riemann-S sequence with param-eters ( ¯ C , n, M, R , B p, ) = ( − . , , . ⊙ , 20 km, 10 G). Up-per panel: Rate of energy loss in units of 10 ergs/s, when bothmagnetic dipole losses (red-dashed line) and GW losses (black-solid line) are taken into account in the magnetar’s spin-downlaw. For reference, we also plot the rate of energy loss in thecase only GW emission is considered (black-dash-dotted line), asin Lai & Shapiro (1995). Lower panel: absolute value of the sur-face fluid particles angular frequency divided by a factor of π (i.e. | Ω eff | /π ), when both magnetic dipole and GW losses are consid-ered (black-solid line). For reference, we also plot the same quan-tity when only GW losses are taken into account in the magne-tar’s spin-down law (black-dash-dotted line), as in Lai & Shapiro(1995). Note that the vertical axis in the lower panel is a linearscale: between 10 s and ∼ s, Ω eff /π changes from ∼
800 Hzto ∼
750 Hz, i.e. less than ∼
10% of its initial value. Thus, be-tween 10 s and 10 s the power-law approximation to dipole lossesis L dip ∝ T − . , so that q ∼ T . s.(See the electronic version for colours). The waveform of the GW signal emitted in as-sociation with the afterglow plateau is computed as(Lai & Shapiro 1995): h + = − h ( t )2 cos Φ(1 + cos θ ) h × = − h ( t ) sin Φ cos θ (13)where θ is the angle between the line of sight and therotation axis of the star, Φ = 2 R tt Ω t is twice the orbitalphase, and h ( t ) = s(cid:18) c d Ω G (cid:19) − | L GW | = 4 G Ω c d Iǫ (14)where d is the distance to the source, L GW and Ω areshown in Fig. 2 (upper panel, black-solid line) and Fig. 3(lower panel, black solid line), respectively. The resultingGW signal is quasi-periodic, with frequency f = Ω /π .To estimate the GW signal detectability, we proceedas follows. For broad-band interferometers such as LIGOand VIRGO, the best signal-to-noise ratio is obtained byapplying a matched filtering technique to the data, whena waveform template is available. In such a case, ρ = 4 Z + ∞ ( F | e h + ( f, θ ) | + F × | e h × ( f, θ ) | ) S h ( f ) df (15) Fig. 3.—
Upper panel: characteristic GW amplitude h c at d = 100 Mpc, with dipole plus GW (black-solid line) and only GW(black-dash-dotted line, see also Lai & Shapiro 1995) losses beingconsidered. A typical fit to the sensitivity expected for advanceddetectors (purple-dashed line, see e.g. Cutler & Flanagan 1994;Owen et al. 1998), Virgo nominal sensitivity (blue-dotted line),and the advanced Virgo sensitivity optimized for binary searches(blue-dash-dot-dot-dotted line, Acernese et al. 2008), are alsoshown. Lower panel: evolution of the GW signal frequency, withdipole plus GW (black-solid line) and only GW (black-dash-dottedline) losses being considered in the NS spin down. (See the elec-tronic version for colours). where e h is a Fourier transform; S h ( f ) is the powerspectral density of the detector noise; F + , F × arethe beam pattern functions (0 < F , F × < ρ = Z + ∞ h ( t )( dt/df )( F (1 + cos θ ) / F × cos θ ) S h ( f ) df. (16)Since we expect to be observing the GRB on-axis, θ ≃
0. In case of optimal orientation, ρ max = Z + ∞ f h ( t )( dt/df ) f S h ( f ) d (ln f ) == Z + ∞ (cid:18) h c h rms (cid:19) d (ln f ) , (17)being h c = f h ( t ) p dt/df the characteristic ampli-tude, and h rms = p f S h ( f ). In the upper panel ofFig. 3, we compare h c computed for a GRB at d =100 Mpc, with the h rms expected for the advanced de-tectors (Acernese et al. 2008; Cutler & Flanagan 1994;Owen et al. 1998), for which ρ max & d .
100 Mpc, or d .
150 Mpc if we make the assump-tion that knowledge of the GRB trigger time reducesthe detection threshold, of a factor which as a rule-of-thumb we take equal to 1 . h ρ i sky = p / ρ max & h rms (Cutler & Flanagan 1994). With the help of the GRBtrigger time roughly compensating the factor of p / d .
100 Mpc.BATSE results show that about 3% of short GRBsare expected to be within 100 Mpc (Nakar et al. 2006),which translates into ∼ − ∼
10 short GRBs per year) plus GBM(GLAST - ∼ / ∼
200 GRBs per year) sample. Asfar as low-luminosity long GRBs, two of them (980425and 060218) were already observed at a d ∼
40 Mpcand d ∼
130 Mpc, and their local rate ( &
200 Gpc − yr − ) is expected to be much higher than that of nor-mal bursts (1 Gpc − yr − , e.g. Virgili et al. 2009).INTEGRAL has detected a large proportion of faintGRBs inferred to be local (Foley et al. 2008), a samplewhich may be increased by future missions such as Janus(Stamatikos et al. 2009) and EXIST . To compare withthe case discussed here, the standard progenitor scenariofor long GRBs predicts ρ ∼ short GRBs isestimated to be detectable up to several hundreds Mpc(e.g. Flanagan & Hughes 1998; Kobayashi & M´esz´aros2003). GWs eventually detected after a chirp and dur-ing an electromagnetic plateau of a short GRB, wouldadd a significant piece of information, probing whethera magnetar is formed in the coalescence, rather than aBH.It is finally worth adding some few considerationsmore. First, in the scenario we are proposing here, somecorrelations do exist between the electromagnetic plateauand the GW signal, that could be explored in future anal-yses so as to test up to which level those may help theGW signal search. For example, a measurement of theinitial frequency of the GW signal, for a given NS massand radius, would allow one to derive an estimate β GW for the actual value of β (see also Fig. 5 in Lai & Shapiro1995). In the ellipsoidal approximation, β GW would pre-dict a specific evolution of the bar, as e.g. the expectedvalue of Ω eff ( β GW ) = Ω( β GW ) − Λ( β GW ) during thenearly constant phase. At the same time, the luminos-ity of the afterglow plateau, for a given NS mass, ra-dius and magnetic field strength, would also allow oneto estimate the value of Ω eff during the constant phase,which could thus be checked for consistency with thevalue Ω eff ( β GW ) inferred from the GW measurements.Next, some considerations are required on the fate ofthe bar after the final Dedekind state is reached. In theabsence of dipole losses, the evolution of the NS along thesequence would have maintained Ω nearly constant up toa time T GW , of the order of few secular growth times, τ GW ≃ × − s [ M/ (1 . M ⊙ )] − [ R / (10km)] ( β − β sec ) − (Lai & Shapiro 1995). Here β is referred to theinitial Maclaurin configuration, and it’s determined bythe choice of ¯ C . In our case, ¯ C = − .
41 and β = 0 . τ GW ≃
335 s. As evident from the black-dash-dotted line in the lower panel of Fig. 3, for such value ofthe circulation, when only GW losses are considered, onehas T GW ≃ (1 − × s ≃ (3 − τ . The addition ofmagnetic dipole losses speeds-up the process, so that the http://exist.gsfc.nasa.gov/ star reaches the stationary football configuration some-what earlier (Fig. 3, lower panel, black-solid line). Afterthe end of the secular evolution, we do not know whatthe fate of the bar is. As Lai & Shapiro (1995) have un-derlined, while the star approaches a Dedekind ellipsoidthe gravitational evolution timescale increases, eventu-ally becoming comparable to the viscous dissipation one.When this happens, ¯ C is not conserved anymore and thestar is expected to be driven along a nearly-Dedekindsequence to become a Maclaurin spheroid, since this isthe only final state that does not radiate GWs or dissi-pate energy viscously. The addition of magnetic dipolelosses would speed up such evolution, and further spin-down the final Maclaurin state. We thus expect to haveΩ eff decreasing at some point after the constant − ¯ C evo-lution, with the dipole luminosity L dip also decreasingaccordingly. Correspondingly, the energy injected intothe fireball will start decreasing (eventually entering inthe q < − − ¯ C evolution, are in agreement with those typi-cally observed in GRBs. CONCLUSION
We have discussed a possible scenario where a newlyformed magnetar is left over after a GRB explosion, andexplored the hypothesis of its being subject to a secu-lar bar-mode instability, including in the spin down thecontributions of both radiative losses by magnetic dipoleemission and by GWs. Following the analytical treat-ment of Lai & Shapiro (1995), we have shown that forreasonable values of the physical parameters, the typicalproperties of GRB afterglow X-ray plateaus may be re-produced. A consequence of this is that, on the relativelylong timescale 10 − s of the electromagnetic plateau,the advanced LIGO/Virgo interferometers may detect acorresponding GW signal up to d ∼
100 Mpc, by carry-ing out matched searches. Such a signal would be asso-ciated with an afterglow light curve plateau from a longsub-luminous GRB, or from a short GRB, with isotropicenergy . erg, which is typical of most nearby GRBsdetected. For the more energetic GRBs, a bar-mode GWsignal may be detected without a visible plateau in theafterglow.In conclusion, although there are considerable uncer-tainties about the evolutionary path of newborn mag-netars, our analysis indicates that the scenario pro-posed here is a plausible and interesting possibility, lead-ing to an efficient GW emission process which is ac-companied by a distinctive electromagnetic signature.Thus, in view of the impending commissioning of theadvanced LIGO and Virgo, we consider that it wouldbe highly worthwhile to test this possibility throughmatched electromagnetic-GW data searches.We are grateful to Benjamin Owen for important com-ments and valuable suggestions on this scenario, and forhelping improve the manuscript. AC thanks Fulvio RicciRB afterglow plateaus and Gravitational Waves 9for crucial support during this project, Giovanni Mon-tani for important discussions, Cristiano Palomba forvery helpful suggestions, and Christian D. Ott for use-ful comments. This work was supported by the “Fon-dazione Angelo della Riccia” - bando A.A. 2007-2008 & 2008-2009 (AC), and by NSF PHY-0757155 & NASANNX08AL40G (PM). AC gratefully acknowledges thesupport of the Penn State Institute for Gravitation andthe Cosmos (IGC).erg, which is typical of most nearby GRBsdetected. For the more energetic GRBs, a bar-mode GWsignal may be detected without a visible plateau in theafterglow.In conclusion, although there are considerable uncer-tainties about the evolutionary path of newborn mag-netars, our analysis indicates that the scenario pro-posed here is a plausible and interesting possibility, lead-ing to an efficient GW emission process which is ac-companied by a distinctive electromagnetic signature.Thus, in view of the impending commissioning of theadvanced LIGO and Virgo, we consider that it wouldbe highly worthwhile to test this possibility throughmatched electromagnetic-GW data searches.We are grateful to Benjamin Owen for important com-ments and valuable suggestions on this scenario, and forhelping improve the manuscript. AC thanks Fulvio RicciRB afterglow plateaus and Gravitational Waves 9for crucial support during this project, Giovanni Mon-tani for important discussions, Cristiano Palomba forvery helpful suggestions, and Christian D. Ott for use-ful comments. This work was supported by the “Fon-dazione Angelo della Riccia” - bando A.A. 2007-2008 & 2008-2009 (AC), and by NSF PHY-0757155 & NASANNX08AL40G (PM). AC gratefully acknowledges thesupport of the Penn State Institute for Gravitation andthe Cosmos (IGC).