X-Ray Afterglow Plateaus of Long Gamma-Ray Bursts: Further Evidence for Millisecond Magnetars S. X.Yi1,2,Z. G. Dai1,2,X. F. Wu3,4,5, andF. Y. Wang1,2 4 1Schoolof AstronomyandSpaceScience, NanjingUniversity,Nanjing,China;[email protected] 1 0 2KeylaboratoryofModernAstronomyand Astrophysics(Nanjing University),Ministryof 2 Education,Nanjing210093,China n a J 3PurpleMountainObservatory,ChineseAcademyof Sciences, Nanjing210008,China 8 4ChineseCenter forAntarcticAstronomy,ChineseAcademyof Sciences, Nanjing,210008,China ] E H 5JointCenter forParticleNuclear Physicsand CosmologyofPurpleMountain . Observatory-NanjingUniversity,ChineseAcademyofSciences, Nanjing210008,China h p - o ABSTRACT r t s a [ Many long-duration gamma-ray bursts (GRBs) were observed by Swift/XRT to 1 haveplateausintheirX-rayafterglowlightcurves. Thisplateauphasehasbeenargued v tobeevidenceforlong-lastingactivityofmagnetar(ultra-stronglymagnetizedneutron 1 0 stars) central engines. However, the emission efficiency of such magnetars in X-rays 6 is still unknown. Here we collect 24 long GRB X-ray afterglows showing plateaus 1 1. followed by steep decays. We extend the well-known relationship between the X-ray 0 luminosity L and spin-down luminosity L of pulsars to magnetar central engines, 4 X sd 1 and find that the initial rotation period P ranges from 1 ms to 10 ms and that the 0 : v dipolemagneticfieldBiscenteredaround1015 G.Theseconstraintsnotonlyfavorthe i X suggestion that the central engines of some long GRBs are very likely to be rapidly r rotating magnetars but also indicate that the magnetar plateau emission efficiency in a X-rays iscloseto100%. Subjectheadings: gamma-rayburst: general—radiationmechanisms: nonthermal— stars: pulsars 1. Introduction Many gamma-ray bursts (GRBs) observed by Swift/XRT present plateaus prior to the sub- sequent power-law decay phase in their early X-ray afterglows (Zhanget al. 2006; Nouseket al. – 2 – 2006). The plateaus generally appear in 100-1000s since the GRB trigger with a typical slope α ∼ 0.5 (Lianget al. 2007), where the flux of the plateau evolves as F ∝t- α1. The distribution 1 of the observed temporal decay slope α (defined in F ∝t- α2) after the plateau, ranging from less 2 than 1 to much greater than 1 (even to 10), is quite diverse. According to the standard model for GRB afterglows (for a recent complete reference see Gao et al. (2013b)), it is hard to under- stand some observed α with large values. Unless the plateau happens to be followed with the 2 jet-like phase, which seems to be unlikely, the external shock models can not explain α > 1.75 2 (for details see next section). Therefore, it is now known that there are generally two types of plateaus. The first one is “external plateau”, characterized by a normal decay (α ≤ 1.75) after 2 theplateau, which is currently understoodas beingdue toenergy injectioninto theexternal shock (Dai &Lu 1998a,b; Zhang &Mészáros 2001). A tight correlation for X-ray plateaus between the breaktimeT andthecorrespondingX-rayluminosityL wasrecentlydiscoveredbyDainottiet al. b X (2010) and Xu& Huang (2012), who selected the sample with the slope of the follow-up decay phase generally less than 1.5. The second type is called “internal plateau”, characterized by a steep decay (α > 1.75) after the plateau, which might be originated from internal dissipation of 2 magnetic energy continuously blew out from the central engine (Trojaet al. 2007). One possi- ble candidate of the central engine responsible for external energy injection as well as internally dissipativemagneticenergy is an ultra highlymagnetized and rapidly rotating neutron star, which is also called magnetar (Thompson& Duncan 1995). The rotation energy of the magnetar can be tapped through magnetic dipole radiation (MDR) and/or relativistic leptonic wind (Dai 2004). This speculation can be realized if the initial rotation period P and dipole magneticfield B of the 0 central neutron stars are found to be consistent with our expectation (Fan & Xu 2006; Dai & Liu 2012). Sincethemagnetarmodelisalmosttheonlysuccessfulmodelforinternalplateaus1,assuming dissipativemagnetic energy is from MDR, one can derive the initial period P and magnetic field 0 strengthBifthespin-downluminosityL andspin-downtimescaleT ofthemagnetarareknown. sd sd Rowlinsonet al. (2013) applied this method by assuming the emission efficiency η ≡ L /L = rad sd 100% to fit the observed X-ray plateaus, where L is the total bolometric luminosity in the 1- rad 104 keV in the cosmologically rest frame extrapolated from the observed X-ray luminosity L X measured by Swift/XRT. Zhang (2009) considered five remarkable plateaus with sharp drops as a sampleto discussmagnetars as thecentral engineofGRBs, and foundthat theluminosityemitted inX-raybandisafractionofthetotalspin-downluminosity. Inthispaper,wecollectallSwiftlong GRBs with a steep decay after the plateau and infer the stellar parameters based on the magnetar 1Matter-dominatedenergyinjectionis also possible for externalplateaus, which is impulsivelyejected and does not need long-lived central engine activity. It only requires a large variation in Lorentz factor. For large sample applicationsseeNouseketal.(2006)andZhangetal.(2006). – 3 – model. We assumethat the end timeofthe plateau phasecorresponds to the spin-downtimescale T and thatη .100%isan adjustableparameter. sd To more understand the physics behind η, we draw lessons from persistent X-ray emission of normal pulsars. The dissipation of the rotation energy of a normal pulsar to its persistent X- ray radiation could be similar to or the same as a millisecond magnetar in a GRB, in which both spindownthroughmagneticdipoleradiation. UnlikeGRBmagnetars,theemissionefficiencyofa pulsarinX-raycanbedirectlycalculatedastheobservedX-rayluminosityL dividedbythespin- X downluminosityL . In ordertounderstandthemechanismbywhichthestellarrotationenergy is sd converted into X-ray emission, a tight correlation between L and L of pulsars has been widely X sd studied(Seward & Wang1988;Becker &Truemper1997;Possentiet al.2002;Cheng et al.2004). Because distinctcomponentsofX-ray emissionhavedifferent originsfornormal pulsars,we here focusonthenonpulsedcomponentofX-rayemissionfrompulsarwindnebulae(PWNe). Asstated above,sincebothmillisecondmagnetarsinGRBsandnormalpulsarsspindownthroughmagnetic dipoleradiation, weassumethat theyhavethesamecorrelation between L and L . Evidencefor X sd this assumption is as follows. (1) Gavriilet al. (2008) found that the the dipolar magnetic field of the young pulsar PSR J1846.0258 is about 4.9×1013 G, which is higher than those of normal pulsars,butlowerthanthoseofmagnetars. Moreover,thedetectionofmagnetar-likeemissionfrom thispulsarsuggeststhatthereisacontinuumofmagnetar-likeactivitythroughoutallneutronstars. (2) Vink&Bamba (2009)found that the L -L correlation ofthe magnetarcandidateanomalous X sd X-ray pulsar1E1547.0-5408is similarto that of PWN pulsars. In this paper, therefore, we extend the L -L correlation from normal pulsars to magnetars, and obtain the spin-down luminosity of X sd magnetarsby usingtheobservedluminosityofaplateau. The structure ofthe paper is as follows. In the next section, we introducethe selection of the pulsar and plateau samples, and carry out empirical fittings to the observed plateau light curves. The correlation in and between pulsars and GRBs are calculated and discussed in section 3. Our conclusionsand discussionare presentedin thelast section. 2. SampleSelection and LightCurve Fitting ThenonthermalnonpulsedX-rayemissionfromrotation-poweredpulsarshasbeenstudiedin the context of emission from PWNe. Here we mainly investigate the nonpulsed X-ray emission fromPWNe. WeexcludeX-raypulsarspoweredbyaccretionfrombinarycompanions,andcollect X-ray observational data of 101 pulsars with PWN from the published literature (Possentiet al. 2002; Cheng et al. 2004; Li et al. 2008; Kargaltsev&Pavlov 2010). We find a correlation of L - X L withthe101PWN sample(seenextsection). Thiscorrelationalsoindicatesthefraction ofthe sd rotationalenergy lossofapulsargoingintoX-ray emission. – 4 – X-ray plateaus are a common phenomenon in the afterglow observations. Much work has beendonefortheoreticalexplanationsandstatisticanalysisforshallowdecays(Dai &Lu1998a,b; Zhang& Mészáros 2001; Dai 2004; Lianget al. 2007; Dai &Liu 2012; Yu& Dai 2007; Yuet al. 2010; Dainottiet al. 2010; Xu& Huang 2012). In the external shock models, usually the decay slope after the plateau is α =(3p- 2)/4 if the environment is an interstellar medium (ISM) with 2 aconstantdensity,orsometimes(almostunlikelyinX-rayaftertheplateauphase)α =(3p- 1)/4 2 if the environment is a stellar wind, where p is the power-law index of the energy distribution of shock-accelerated electrons. The typical value of p is about 2.3, however, it can range from 2.0 to 3.0 or even more smaller and larger. Therefore, the typical value of α is ∼ 1.2 and the 2 maximal allowable value by the model is 1.75. The possibilityof the coincidence that the plateau happenstobefollowedbythejet-likephaseisextremelysmall. Eveninthiscase,α =α +0.75∼ 2 1 1.3 for an ISM environment and α = α +0.5 ∼ 1.0 for a wind environment, as long as the jet 2 1 sideways expansion can be neglected. If the jet sideways expansion play the role, the value of α = p is typically 2.0- 3.0. As can be seen, the above values of α can not explain the large 2 2 decay slope after the plateau observed in some GRBs. Internal plateaus with large α are thought 2 to be due to magnetic energy dissipation at small radii, so that when the central engine ceases the decay timescale (equivalent to decay slope) is very short. In this paper, we focus on internal plateaus and thecriterion to be internal plateaus is α >1.75. We havecollected 24 long duration 2 GRB (T ≥ 2 s) X-ray plateaus with this judgement. Some of the GRBs in this sample have no 90 redshiftmeasurements,andweadoptpseudo-redshiftestimatedbytheL - T correlationforthem X b (Dainottietal. 2010). We suppose that the ending of an X-ray plateau corresponds to the spin- downtimeofthemagnetar. Thecentrifugalforcereducesasthemagnetarspinsdownsignificantly, the magnetar collapses into a black hole due to the imbalance of the gravity and outward forces2. It is likely that theending ofthe plateau, thespin-downand collapseof themagnetarcoincidently happen atthesametime. we have collected 24 remarkable X-ray afterglow light curves in our sample. We apply a smooth broken power-law and an extra power-law to fit the light curves. The fitting results are summarized in Table 1. Generally speaking, the break (ending) time (104- 105 s) of internal plateausislongerthanthat(103- 104 s)ofnormal/externalplateaus. Thebreaktimeoftheinternal plateau (T )is assumedtobethespin-downtimeofamagnetar(T ), i.e. b sd 3c3I 3c3IP2 T = = 0 , (1) sd B2R6Ω2 4π2B2R6 0 where Ω = 2π/P is the initial angular frequency, I is moment of inertia, R is the stellar radius, 0 0 2Recently, Zhang (2014) applied this scenario to interpretfast radio bursts (FRBs), a new type of cosmological transients,althoughthephysicalnatureisstillunknown. – 5 – and cisthespeed oflight. TheisotropicX-ray luminosityat thebreak timeT iscalculated by b 4πD2F L = L X , (2) X (1+z)1- β wherez istheredshift,D istheluminositydistance,F istheobservedX-ray flux at theend time L X of the plateau phase, and β is the spectral index of the X-ray afterglow which can be found from theSwift/XRTwebsite(Evanset al.2009). Thespin-downluminosityofapulsar/magnetarcan be expressedas IΩ2 L = 0, (3) sd 2T sd whent ≪T . Withequations(1)and(3),weobtaintheinitialperiodandthedipolemagneticfield sd ofthepulsar/magnetaras 3c3I2 1/2 B= (4) (cid:18)2R6L T2(cid:19) sd sd and 2π2I 1/2 P = . (5) 0 (cid:18)L T (cid:19) sd sd With the fitting results (see Table 1) and assuming I = 2×1045 g cm2, R = 1×106 cm, we can constraintheinitialperiod (P )and thedipolemagneticfield strength(B)ofthepulsar/magnetar. 0 3. The L - L Correlation inPulsarsand GRBs X sd It has been found that L and L in pulsars have a power-law relationship, but different au- X sd thorshaveobtaineddifferentpower-lawindices(Seward &Wang1988;Becker &Truemper1997; Possentiet al. 2002; Cheng et al. 2004; Liet al. 2008). By analyzing observed X-ray data of 101 PWN pulsars from the published literature, we find that (see Figure 1) L and L have a tight X sd correlation L 1.28±0.05 L =10- 13.56±1.90× sd ergs- 1. (6) X (cid:18)ergs- 1(cid:19) Thus, the corresponding conversion efficiency of the rotational energy of a pulsar into nonpulsed X-ray emissionis L L 0.28±0.05 η = X =10- 13.56±1.90 sd , (7) L (cid:18)ergs- 1(cid:19) sd showingthattheefficiency η is dependenton thespin-downluminosity. TheconversionefficiencyoftherotationalenergyofamagnetarintoX-rayemission,inorder to interpret X-ray plateaus, is unknown. Some papers, such as Rowlinsonet al. (2013), gener- ally adopted the efficiency of the rotational energy into the 1- 104 keV emissionas 100% in their – 6 – calculations. However, their extrapolation from X-ray to 1- 104 keV is based on the X-ray spec- tral index, which may not be applicable beyond the XRT band. In this paper we consider X-ray plateaus followed by steep decays as central engine afterglows from magnetars, and assume that such magnetars and rotation-powered pulsars have the same mechanism that X-ray emission are from internal dissipation of Poynting flux. Evidence for this assumption comes from the possi- ble fact that pulsars and magnetars may have the same L -L correlation (Gavriilet al. 2008; X sd Vink& Bamba 2009),thatis,theL andL ofthemagnetarcandidate,theanomalousX-raypul- X sd sar1E1547.0-5408,satisfythecorrelationinPWNpulsars. Therefore,weextendthecorrelationof L and L from rotation-powered pulsars to magnetars. The corresponding conversion efficiency X sd oftherotationalenergy ofa magnetarintoX-ray emissionis alsogivenbyequation(7). Thespin-downluminosityduringX-rayplateauscan becalculatedbyequation(6), whereL X is the luminosity at the end of the plateau (see Table 1). There are some GRBs without redshift measurement. In these cases, their redshifts can be estimated by the correlation between L and X T from Dainottiet al. (2010). With the derived spin-down luminosity, we can further calculate b the initial period P and the dipole magneticfield strength B of a magnetar with equations (4) and 0 (5). Table 1 shows that the derived initial spin period of the magnetars ranges from 1 to 10 ms, whichiswellconsistentwiththevaluesexpectedinthemagnetarformationhypothesis. Thedipole magnetic field B of Table 1 is in the range of 1014- 1015 G, which is consistent with the magnetic field ofsoftgamma-rayrepeaters and anomalousX-ray pulsars. Figure 2 shows the magnetic field and spin period for both long and short GRB candidates. The magnetars could be roughlydividedby B=5×1015 G into two different samples, short GRB candidates above the line and long GRB candidates below the line. One caveat is that there are some GRBs with extended emission included in the sample plotting Figure 2. Because their dis- tributionissimilartothatoftheshortones(Rowlinsonet al.2013;Gompertzetal.2013),wecon- sider them as one subset of the short GRB candidates. Compared with the long GRBs, the short GRB candidates tend to have higher magnetic fields. From our statistics, we find the initial spin period mainly in the range 1- 10 ms and thedipolemagneticfield in therange 5×1014- 5×1015 G for the long GRB magnetars. These values are all reasonable, implying that internal plateaus couldbepowered byacentral spinning-downmagnetar. 4. Conclusions and Discussion TheX-rayplateauscanbeexplainedasbeingduetocontinuousenergyinjectionfromcentral engines after the prompt bursts and rapidly rotating, ultra-strongly magnetized pulsars are good candidates of such GRB central engines. In this paper, we have collected 24 remarkable long- GRB X-ray plateaus followed by sharp drops. We assumed that the X-ray plateaus are powered – 7 – by internal magnetic energy dissipation of Poynting flux from a magnetar and the sudden drop is caused by the spin-down and collapse of the magnetar. On the other hand, we gathered the X-ray observationaldataonL andL of101PWNpulsarsfromthepublishedliterature,andfittedthem X sd with a power law function (Figure 1), L =10(- 13.56±1.90)(L /ergs- 1)(1.28±0.05)ergs- 1. We assumed X sd thatmagnetarsandrotation-poweredpulsarsmayexperienceacommoninternaldissipationmech- anism. Thus, we extended the correlation of L and L from rotation-powered pulsars to X-ray X sd plateaus. We find that for the magnetar candidates in the 24 long GRBs, the initial period P is 0 about 1 to 10 ms,whilethedipolemagneticfield strengthB is in therangeof 1014 to 1015 G. This resultimpliesthatthecentral enginesofsomelongGRBs are millisecondmagnetars. Millisecond magnetars are not only proposed as the central engines of some long GRBs, but also they may survive from some binary neutron star mergers that power short GRBs. The long-lastingactivityofthecentralmagnetars havebeen suggestedtointerpret theX-ray flares and plateaus following some short GRBs (Dai et al. 2006; Fan& Xu 2006; Rowlinsonet al. 2010; Rowlinsonet al.2013)andthestatisticalpropertiesofX-rayflaresfrombothlongandshortGRBs (Wang &Dai 2013). Recently, such a survived massive millisecond magnetar scenario has been studiedtopredictabrightmulti-wavelengthafterglow(Gao etal. 2013a)andinvokedtoexplainan unusualenergetictransientPTF11agg(Wang & Dai 2013;Wuet al. 2014). 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