Nuclear Equation of State from Observations of Short Gamma-Ray Burst Remnants
Paul D. Lasky, Brynmor Haskell, Vikram Ravi, Eric J. Howell, David M. Coward
aa r X i v : . [ a s t r o - ph . H E ] F e b Nuclear equation of state from observations of short gamma-ray burst remnants
Paul D. Lasky, ∗ Brynmor Haskell, and Vikram Ravi † School of Physics, University of Melbourne, Parkville, VIC 3010, Australia
Eric J. Howell and David M. Coward
School of Physics, University of Western Australia, Crawley WA 6009, Australia
The favored progenitor model for short γ -ray bursts (SGRBs) is the merger of two neutron starsthat triggers an explosion with a burst of collimated γ -rays. Following the initial prompt emission,some SGRBs exhibit a plateau phase in their X-ray light curves that indicates additional energyinjection from a central engine, believed to be a rapidly rotating, highly magnetized neutron star.The collapse of this “protomagnetar” to a black hole is likely to be responsible for a steep decay inX-ray flux observed at the end of the plateau. In this paper, we show that these observations canbe used to effectively constrain the equation of state of dense matter. In particular, we show thatthe known distribution of masses in binary neutron star systems, together with fits to the X-raylight curves, provides constraints that exclude the softest and stiffest plausible equations of state.We further illustrate how a future gravitational wave observation with Advanced LIGO/Virgo canplace tight constraints on the equation of state, by adding into the picture a measurement of thechirp mass of the SGRB progenitor. PACS numbers: 26.60.-c, 97.60.Jd, 04.30.Tv
Recent observations of long and short γ -ray bursts(SGRBs) show plateau phases in the X-ray light curvesthat last hundreds of seconds [1–6] and provide evidencefor ongoing energy injection through a central engine[2, 7, 8]. The main candidate for the central enginein SGRBs is a rapidly rotating, highly magnetized neu-tron star (NS) [9–12] that forms following the coalescenceof two NSs [13–19]. Recent analytic fits to X-ray lightcurves support this “protomagnetar” interpretation of acentral engine for both long [20–23] and short GRBs [4–6]. Excitingly, some objects exhibit an abrupt cutoff inthe X-ray flux ∼
100 s after the initial trigger [5, 20, 21].This has been interpreted as the metastable protomag-netar collapsing to form a black hole.From a theoretical perspective, the coalescence of bi-nary NSs can follow a number of evolutionary paths.If the merger remnant is sufficiently massive, it imme-diately collapses to a black hole, or forms a dynam-ically unstable hypermassive NS that is supported bystrong differential rotation and thermal pressure [18, 24].Magnetic braking terminates differential rotation on theAlfv´en timescale [25, 26] implying that the object col-lapses in ∼ ∼ M ⊙ NSs [30, 31] demonstrates that the equation of state(EOS) permits massive enough NSs for supramassivestars to be created from the merger of two NSs [32]. Fi-nally, a merger remnant that is less massive than the TOV maximum mass will survive as a stable NS.In this paper, we focus on the possibility that proto-magnetars drive the plateau phases of SGRB X-ray lightcurves. The loss of rotational energy from the NS powersthe emission, and a simple spin-down model can be fit tothe light curve to obtain the initial spin period, p , andsurface dipolar magnetic field, B p , of the protomagnetar[4, 5, 11]. When an abrupt decay in X-ray luminosityis also observed, this is interpreted as the star havingspun down to the point at which centrifugal forces canno longer support its mass against gravity [20, 21]. Thetime between the initial prompt emission and the decay, t col , is hence interpreted as the collapse time of the pro-tomagnetar. Given p and B p , the time it takes the NSto collapse will depend only on its initial mass and theEOS. We thus have almost all of the ingredients neededto determine the EOS, with the exception that the initialmass of the NS is not known. In the following, we showhow one can constrain the EOS using these observationsand the observed distribution of NS masses in binary NSsystems [33–35]. We also show how the EOS constraintswill improve given a gravitational wave (GW) measure-ment of the binary inspiral (i.e., prior to coalescence)with Advanced LIGO and Virgo.We focus on the observations presented in Rowlinsonet al. [5], in which X-ray plateaus were observed followinginitial SGRB triggers using Swift . The light curves fit theprediction of a protomagnetar that is being spun downthrough dipole electromagnetic radiation [11] (as noted inRowlinson et al. [5], this is consistent with the late-timeresidual spin-down phase being driven by a relativisticmagnetar wind [36]), allowing the authors to obtain p and B p from the model.Rowlinson et al. [5] present data for a number of ob- TABLE I: The SGRB sample containing central engines usedin this article, with all data and fits from Ref. [5]. z , p , B p and t col are, respectively, the redshift, initial spin period,surface dipolar magnetic field, and collapse time. The bottomfour SGRBs do not collapse within 10 – 10 s.GRB z p B p t col [ms] [10 G] [s]060801 1 .
13 1 . +0 . − . . +1 . − . .
46 1 . +1 . − . . +1 . − . .
122 9 . +0 . − . . +10 . − . .
718 0 . +0 . − . . +0 . − . .
55 7 . +0 . − . . +0 . − . –070809 0 .
219 5 . +0 . − . . +0 . − . –090426 a . . +0 . − . . +0 . − . –090510 0 . . +0 . − . . +0 . − . – a Duration ( T jects with accurate redshift measurements. As this is re-quired to determine the rest-frame light curve, and hence p and B p , we omit any SGRBs for which the redshift isnot known. We are left with four SGRBs that collapseand four that are long-term stable , which are presentedin Table I.The values of B p and p are derived assuming elec-tromagnetic dipolar spin-down, with perfect efficiencyin the conversion between rotational energy and electro-magnetic radiation. We discuss the possibility of a lowerefficiency below. Note that a mass of 1 . M ⊙ and radiusof 10 km were also assumed, although the dependence onthese parameters is weak [4, 5]. The standard spin-downformula is [39] p ( t ) = p (cid:18) π c B p R Ip t (cid:19) / , (1)where R and I are the radius and moment of interia,respectively, of the NS. This spin-down law is implicitlyused in the fits to the X-ray light curves [5]; a devia-tion from dipole spin-down would result in a differentpower-law exponent (see also [11]). Moreover, it has re-cently been shown that randomly distributed magneticfields lead to similar spin-down luminosities than orderedmagnetic fields [40].For a given EOS, one can write the maximum gravita-tional mass, M max , as a function of the star’s rotational There were six SGRBs that are long-term stable and satisfy ourcriteria; however, two of these (GRBs 050509B and 061201) didnot show conclusive fits to the magnetar model and were there-fore labelled by Ref. [5] as “possible candidates”. We omit thesein the present analysis, although note that they are consistentwith our general conclusions. kinetic energy [39, 41], and hence p . For slow rotation M max = M TOV (cid:0) αp β (cid:1) , (2)where in Newtonian gravity β = − α is a functionof the star’s mass, radius, and moment of inertia. Weevaluate equation (2) in relativistic gravity by creatingequilibrium sequences of M max ( p ) using the general rel-ativistic hydrostatic equilibrium code RNS [42]. That is,for various values of the spin period we calculate equilib-rium sequences and find the local maximum in the M – ρ c curve (where ρ c is the central energy density) that indi-cates the maximum mass. We then calculate a functionalfit to these equilibrium sequences to get α and β for eachEOS.A supramassive protomagnetar collapses when thestar’s period becomes large enough that M p = M max ( p ),where M p is the mass of the protomagnetar. The col-lapse time, t col , is found by substituting (1) into (2) with t = t col and M max = M p . Solving for t col gives t col = 3 c I π B p R "(cid:18) M p − M TOV α M
TOV (cid:19) /β − p . (3)Equation (3) gives the time for a supramassive protomag-netar to collapse to a black hole given observed param-eters ( p , B p , M p ) and parameters related to the EOS( M TOV , R , and I ). Note that Eq. (3) does not accountfor several effects, such as how I and B p change withtime as the star spins down or how the presence of mat-ter outside the star affects the spin-down torque, a pointwe discuss below.The observations in Ref. [5] give B p , p and t col ,implying that we require M p in Eq. (3) to constrainthe EOS. We obtain M p statistically from the observedmasses of NSs in binary NS systems [33–35], where themost up-to-date measurements give M = 1 . +0 . − . M ⊙ ,with the errors being the 68% posterior predictive in-tervals [35]. Numerical simulations of binary NS merg-ers and observations of SGRBs indicate that . . M ⊙ of material is ejected during the merger [e.g., 24, 43,and references therein]. Modulo this lost mass, whichwe ignore in the following, it is the rest mass of a sys-tem that is conserved through the merger. An approxi-mate conversion between gravitational and rest masses is M rest = M + 0 . M [44], which leads to a gravitationalmass for the protomagnetar following an SGRB mergerof M p = 2 . +0 . − . M ⊙ .In Fig. 1, we plot the collapse time, t col , as a functionof the protomagnetar mass, M p , for each of the SGRBslisted in Table I. We utilize five EOSs that are consistentwith current observations and have a range of maximummasses: SLy [45] ( M TOV = 2 . M ⊙ , R = 9 .
97 km; blackcurve), APR [46] (2 . M ⊙ , 10 .
00 km; orange), GM1 [47](2 . M ⊙ , 12 .
05 km; red), AB-N [48] (2 . M ⊙ , 12 .
90 km;green) and AB-L [48] (2 . M ⊙ , 13 .
70 km; blue).
FIG. 1: Collapse time as a function of the protomagnetar mass for each of the SGRBs in Table I. The two left-hand columnsare those in which the protomagnetar collapses to form a black hole, where the collapse time is given by the horizontal dashedblack line. The two right-hand columns are those SGRBs that form stable protomagnetars. The theoretical collapse time foreach EOS is calculated from Eq. (3), where the initial spin period and magnetic field distributions are given in Table I foreach GRB. Five EOSs are shown in each panel: SLy (black), APR (orange), GM1 (red), AB-N (green) and AB-L (blue). Thedark solid curve for each EOS assumes the values of p and B p given in Table I with the 68% confidence intervals in p and B p included in the faded dashed and faded solid curves respectively. The shaded region is the protomagnetar mass distributionthat results from merging two NSs whose masses are independently drawn from the binary NS mass distribution of Ref. [35],with the 68% and 95% mass intervals represented with the vertical dashed lines. Consider GRB 060801 in Fig. 1, with B p and p givenin Table I. EOS GM1 (red curves) requires M p ≈ . M ⊙ for it to collapse 326 s following the initial burst. Onthe other hand, EOS SLy (black curves) requires M p ≈ . M ⊙ , which falls well outside the 2 σ posterior massdistribution. The quoted errors for B p and p have littleeffect on this result. Similarly, GRB 101219A requires M p ≈ . M ⊙ for AB-L and M p ≈ . M ⊙ for AB-N,which both lie at the extreme high-mass end of the dis-tribution. In this sense, all of the GRBs plotted in thetwo left-hand columns of Fig. 1 favor the intermediateEOSs. It is worth noting that the EOSs we plot are a rep-resentative sample that covers a wide range of maximummasses; many more EOSs fit into the intermediate regimethat would be satisfied by the constraints we are placingherein. For an up-to-date review of plausible EOSs seeRef. [32].It is worth paying special attention to GRB 080905A.Rowlinson et al. [5] found relatively large p , implyingslow spin-down from electromagnetic torques. In the274 s before GRB 080905A collapses, the protomagnetarhas spun down from p = 9 . p = 10 . p = 10 . ∼ − –10 − , which requires an average internal toroidal field of almost 10 G for a star with M & . M ⊙ [50]. On the other hand, the isotropic effi-ciency of turning rotational energy into electromagneticenergy is assumed to be 100%. Reducing the assumedefficiency or beam opening angle also leads to a reduc-tion of the initial spin period. Other possibilities includea chance alignment that led to a false host-galaxy iden-tification, or ongoing accretion or propellering that is af-fecting the pulsar spin-down [51]. It is clear that theseare crucial issues that have to be dealt with in a moresystematic study if our method is to be used to obtain astrong, quantitative constraint on the EOS.The two right-hand columns of Fig. 1 are thoseSGRBs that are not observed to collapse. Their relativelyhigh initial spin periods and low surface magnetic fieldstrengths imply that they do not spin down significantlyin ∼ M p ≤ M TOV , theseobjects are stable magnetars, and will never collapse fromloss of centrifugal support. On the other hand, they maystill have M p & M TOV , in which case they are unstablewith t col ≫ s. If the latter is true, these GRBs couldbe candidate “blitzars” [52] that are a proposed physicalmechanism behind fast radio bursts (FRBs) [53, 54]. Inprincipal, if the blitzar model is correct one could uti-lize the method described herein to also constrain theEOS using FRBs, although a method for determining t col would be required. A method for testing the blitzarmodel, in particular the connection between FRBs andGRBs, has recently been described in Ref. [55]. FIG. 2: Collapse time as a function of the protomagnetarmass for GRB 101219A. The five equations of state are de-scribed in the caption of Fig. 1. The three protomagnetarmass distributions represent the binary NS mass distributiononly (solid black; same as the distribution shown in Fig. 1),the posterior mass distribution from a conservative AdvancedLIGO/Virgo measurement of the progenitor chirp mass andsymmetric mass ratio (dotted black) and the posterior massdistribution from the binary NS distribution and the conserva-tive Advanced LIGO/Virgo progenitor measurement (dashedblack with shading).
Figure 1 shows what currently can be achieved giventhat the mass of the GRB remnant can only be sta-tistically inferred from binary NS observations. In thenear future, Advanced LIGO and Virgo will begin mea-suring GWs from binary NS inspirals at a rate of 0 . M =( m m ) / / ( m + m ) − / , and the symmetric mass ra-tio η = ( m m ) / ( m + m ) , where m , are the massesof the original progenitor NSs. The fractional 95% con-fidence intervals for these quantities will, at worst, be ∼
2% for M and ∼
20% for η (see Ref. [57] for details,which includes an exhaustive discussion of data-analysisalgorithms for parameter estimation from GW measure-ments). These measurements will allow m and m to beestimated. In Fig. 2 we again plot the collapse time as afunction of protomagnetar mass for GRB 101219A, butassume a hypothetical GW measurement of the mergerof two 1 . M ⊙ NSs with the aforementioned confidenceintervals for M and η .Figure 2 clearly shows that a combined measurementof the NS progenitor masses using GWs and knowl-edge of the prior NS mass distribution significantly tight-ens the constraints on the EOS. For example, assumingthe GM1 EOS (red curve in Figure 2) implies a pro-tomagnetar mass for the remnant of GRB 101219A of M p = 2 . +0 . − . M ⊙ (68% confidence interval). Whilethis is broadly consistent with the expectation of M p =2 . +0 . − . M ⊙ from current constraints on the binary NSmass distribution (see Fig. 1), this would be inconsistentwith a putative Advanced LIGO measurement of M for the merger of two 1 . M ⊙ NSs combined with binaryNS mass distribution constraints. The latter would im-ply M p = 2 . +0 . − . M ⊙ .The apparent bias in the protomagnetar mass pos-terior distribution from Advanced LIGO-only measure-ments (dotted black curve of Fig. 2) is caused by thebias in the estimation of η for some GW waveform tem-plates [57]. Individual templates can, however, be usedto estimate η with percent-level precision, which, whencombined with measurements of M , would render thebinary NS mass distributions irrelevant in constrainingprotomagnetar masses.How often does one expect a coincident GW andelectromagnetic detection of an SGRB with an X-rayplateau? Using a conservative beaming angle of 8 ◦ [19, 58, and references therein] and the Swift sampleof SGRBs corrected for dominant selection biases [59],we obtain an intrinsic rate of 820 Gpc − s − . With a bi-nary NS horizon distance for coincident Advanced LIGOand Virgo detections [56, 60] and assuming 50% of allSGRBs have X-ray plateaus [5], we get a rate of 0.2 co-incident electromagnetic and GW detections per year.The Space-based multi-band astronomical Variable Ob-ject Monitor (SVOM) has a decrease in sensitivity of afactor ∼ Swift , but the higher triggeringenergy band may be more optimal for the detection ofspectrally harder SGRBs. Assuming that these two ef-fects cancel, the increased sky coverage of SVOM over
Swift implies ∼ . ACKNOWLEDGMENTS
We are extremely grateful to Antonia Rowlinson, Lu-ciano Rezzolla and Wen-fai Fong for valuable com-ments. We also gratefully acknowledge Jordan Campwho carefully read the manuscript as part of a LIGOScientific Collaboration review (LIGO document numberP1300195). P.D.L. is supported by the Australian Re-search Council (ARC) Discovery Project (DP110103347)and an internal University of Melbourne Early CareerResearcher grant. E.J.H. acknowledges support from aUWA Research Fellowship. D.M.C. is supported by anARC Future Fellowship and B.H. by an ARC DECRAFellowship. V.R. is a recipient of a John Stocker Post-graduate Scholarship from the Science and Industry En-dowment Fund. This work was made possible through aUWA Research Collaboration Award. ∗ Electronic address: [email protected] † CSIRO Astronomy and Space Science, Australia Tele-scope National Facility, P.O. Box 76, Epping, NSW 1710,Australia[1] P. T. O’Brien et al., Astrophys. J. , 1213 (2006).[2] J. A. Nousek et al., Astrophys. 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