Magnetically driven winds from differentially rotating neutron stars and X-ray afterglows of short gamma-ray bursts

5.5 MB·

by  Daniel M. Siegel

DRAFTVERSIONAPRIL11,2014
PreprinttypesetusingLATEXstyleemulateapjv.08/22/09
MAGNETICALLYDRIVENWINDSFROMDIFFERENTIALLYROTATINGNEUTRONSTARS
ANDX-RAYAFTERGLOWSOFSHORTGAMMA-RAYBURSTS
DANIELM.SIEGEL1,RICCARDOCIOLFI1,ANDLUCIANOREZZOLLA1,2
DraftversionApril11,2014
ABSTRACT
Besidesbeingamongthemostpromisingsourcesofgravitationalwaves,mergingneutronstarbinariesalso
representaleadingscenariotoexplainthephenomenologyofshortgamma-raybursts(SGRBs). Recentobser-
vations have revealed a large subclass of SGRBs with roughly constant luminosity in their X-ray afterglows,
4
lasting 10 104s. These features are generally taken as evidence of a long-lived central engine powered by
1
−
themagneticspin-downofauniformlyrotating,magnetizedobject. Weproposeadifferentscenarioinwhich
0
2 thecentralenginepoweringtheX-rayemissionisadifferentiallyrotatinghypermassiveneutronstar(HMNS)
thatlaunchesaquasi-isotropicandbaryon-loadedwinddrivenbythemagneticfield,whichisbuilt-upthrough
r
differential rotation. Our model is supported by long-term, three-dimensional, general-relativistic, and ideal
p
A magnetohydrodynamicsimulations, showingthatthisisotropicemissionisaveryrobustfeature. Foragiven
HMNS, the presence of a collimated component depends sensitively on the initial magnetic field geometry,

0 while the stationary electromagnetic luminosity depends only on the magnetic energy initially stored in the
1 system. Weshowthatourmodeliscompatiblewiththeobservedtimescalesandluminositiesandexpressthe
latterintermsofasimplescalingrelation.

] Subjectheadings:gamma-rayburst: general–magnetohydrodynamics(MHD)–methods: numerical–stars:
E
magneticfield–stars: neutron
H
.
h
1. INTRODUCTION livedcentralengineasproposedinZhang&Me´sza´ros(2001),
p
where such emission is assumed to be powered by the mag-
- Binaryneutronstar(BNS)mergersrepresentaleadingsce-
o nario to generate the physical conditions necessary for the netic spin-down of a uniformly rotating object formed in a
r BNS merger. In general, depending on the total mass and
launchofshortgamma-raybursts(SGRBs;see,e.g.,Paczyn-
st ski1986;Eichleretal.1989;Narayanetal.1992;Barthelmy mass ratio of the binary, the binary-merger product (BMP)
a can either be a stable neutron star, a supramassive neutron
etal.2005;Foxetal.2005;Gehrelsetal.2005;Rezzollaetal.
[ star(SMNS,i.e.,astarwithmassabovethemaximummass
2011;Bergeretal.2013;Tanviretal.2013).Additionally,the
for nonrotating configurations M , but below the maxi-
3 inspiralsandmergersofBNSsarepromisingsourcesforthe TOV
mummassforuniformlyrotatingconfigurationsM ,with
v detection of gravitational waves (GWs) with advanced inter- max
4 ferometerssuchasLIGOandVirgo(Harryetal.2010;Acca- Mmax (cid:39) (1.15−1.20)MTOV, Lasota et al. 1996), a hyper-
4 diaetal.2011)thathaveapredictedrateof40yr−1 (Abadie massiveneutronstar(HMNS,i.e.,astarmoremassivethana
5 etal.2010).Coincidentelectromagnetic(EM)andGWobser- SMNS),orablackhole.3
4 vationsareneededtoconfirmtheassociationofSGRBswith InthisLetter,weproposeadifferentscenariotoexplainthe
1. BNSmergersandtounravelthephysicalprocessesinvolved. X-ray emission soon after the merger, in which the central
engineisalong-liveddifferentiallyrotatingBMPand,inpar-
0 Despite observational evidence in some cases (e.g., Burrows
ticular, anHMNS.Recentobservationsofneutronstarswith
4 etal.2006;Soderbergetal.2006;Berger2013)andarequire-
massesaslargeas 2M (Demorestetal.2010;Antoniadis
1 mentcomingfrommodelingmergerrates(Metzger&Berger (cid:39) (cid:12)
etal.2013)indicatearatherstiffequationofstate,whilethe
: 2012), itisnotknownwhetherthepromptgamma-rayemis-
v massdistributioninBNSsispeakedaround1.3 1.4M (Bel-
sionofanSGRBisalwayscollimatedinarelativisticjet. Ifit (cid:12)
i czynskietal.2008).Thiscombinedevidencesu−ggeststhatthe
X was, a large fraction of SGRB events would be missed, as
BMPisalmostcertainlyanSMNSoranHMNS.
their jets would be beamed away from us. Therefore, un-
r On the basis of these considerations and using three-
a derstanding the potential EM signatures from BNS mergers
dimensional (3D) general-relativistic ideal magnetohydrody-
and,inparticular,thenon-collimatedemission(e.g.,Yuetal.
namic (MHD) simulations, we investigate the EM emission
2013),isessential.
of an initially axisymmetric HMNS endowed with different
The Swift satellite (Gehrels et al. 2004) has recently re-
initial magnetic fields spanning the range of reasonable con-
vealedalargesubclassofSGRBsthatshowphasesofroughly
figurations. In all cases, we find a very luminous, quasi-
constant luminosity in their X-ray afterglow lightcurves,
stationaryEMemissionfromtheHMNS,whichisassociated
referred to as “extended emission” and “X-ray plateaus”
withabaryon-loadedoutflow,drivenbymagneticwindingin
(e.g., Rowlinson et al. 2013; Gompertz et al. 2014). The as-
sociatedX-rayemissioncanlastfrom10to104sandtheflu- the stellar interior. The emission, which is compatible with
the observed X-ray afterglows, is almost isotropic, though
ence can be comparable to or larger than that of the prompt
a collimated, mildly relativistic flow can be produced if the
emission. These features are taken as a signature of a long-
magnetic field is mainly oriented along the spin axis. The
1MaxPlanckInstituteforGravitationalPhysics(AlbertEinsteinInstitute), isotropicemissionispresenteveninrandom-field,initialcon-
AmMu¨hlenberg1,D-14476Potsdam-Golm,Germany
2Institutfu¨rTheoretischePhysik,Max-von-Laue-Str. 1,D-60438Frank- 3 SMNSs or HMNSs will eventually collapse to a black hole, but on
furtamMain,Germany timescalesthatcanbemuchlargerthanthedynamicalone.
2 D.M.Siegel,R.Ciolfi,L.Rezzolla
figurations, making it a robust feature of a BNS merger and (530, 1.5, 5.1) 1044erg formodelsdip-60, dip-6and
×
animportantEMcounterparttotheGWsignatures. rand,respectively.
Thetimeevolutionoftheseinitialdataisperformedusing
2. PHYSICALSYSTEMANDNUMERICALSETUP thepubliclyavailableEinsteinToolkitwiththeMcLachlan
spacetime-evolutioncode(Lo¨ffleretal.2012),combinedwith
AsatypicalHMNS,resultingfromaBNSmerger,wecon-
the fully general-relativistic ideal-MHD code WhiskyMHD
sideranaxisymmetricinitialmodelconstructedusingtheRNS
(Giacomazzo & Rezzolla 2007). The ideal-fluid equation
code (Stergioulas & Friedman 1995), assuming a polytropic
of state p = ρ(cid:15)(Γ 1) is used for the evolution, where (cid:15)
equationofstatep=KρΓ,wherepisthepressure,ρtherest- −
is the specific internal energy and Γ = 2. The star is ini-
massdensity,Γ=2,andK =2.124 105cm5g−1s−2. The
corresponding maximum gravitationa×l mass for a uniformly t6iall1y0s−u9rρrou,nwdheedrebρy a lo1w.0-den1s0i1ty5gatcmmo−sp3hiesrtehewciethntρraaltmres(cid:39)t-
rotating(nonrotating)modelis≈2.27(1.97)M(cid:12). Ourinitial m×ass denscity. We ccar(cid:39)ry out×the MHD computations within
HMNS has a mass of M = 2.43M , an equatorial radius
(cid:12) themodifiedLorenzgaugeapproach(Farrisetal.2012;Gia-
of R = 11.2km, and it is differentially rotating according
e comazzo&Perna2013),withdampingparameterξ = 2/M.
to a “j-constant” law (Komatsu et al. 1989), with the differ-
Thecomputationalgridconsistsofahierarchyofsevennested
ential rotation parameter A/R = 1.112 and central period
e boxes, extending to 105R 1180km in the x- and
Pc =0.47ms. y-directions and to ≈72R e ≈817km in the z-direction.
Tostudytheinfluenceofthemagneticfieldgeometryonthe ≈ e ≈
Thefinestrefinementlevelcorrespondstoaspatialdomainof
EM emission, we endow this model in hydrodynamic equi-
[0,18.4] [0,18.4] [0,12.8]km and covers the HMNS at
libriumwiththreedifferentinitialmagneticfieldgeometries, × ×
all times. The highest spatial resolution is h 140m, but
which are shown in the left panels of Figures 1–3. The first (cid:39)
lower-resolution simulations have been performed to check
model (hereafter dip-60) employs a dipolar configuration
convergence. Tomaketheselong-termsimulationscomputa-
very similar to the one in Shibata et al. (2011) and Kiuchi
tionallyaffordablein3Dandathighresolution,aπ/2rotation
etal.(2012),specifiedbytheazimuthalcomponentofthevec-
symmetryaroundthez-axisandareflectionsymmetryacross
torpotentialA =A (cid:36)2/(r2+(cid:36)2 /2)3/2,where(cid:36)isthe
φ 0,d 0,d thez =0planehavebeenemployed.
cylindricalradius,r2 (cid:36)2+z2,A tunestheoverallfield
0,d
strength,and(cid:36) ≡5.3R 60kmindicatestheradiallo- 3. RESULTS
0,d e
(cid:39) (cid:39)
cation of the magnetic neutral point on the equatorial plane. Figures 1–3 show snapshots of the norm of the magnetic
The second model (henceforth dip-6) is endowed with the fieldandrest-massdensitycontoursinthe(x,z)planeforthe
same field geometry, but with (cid:36) 0.53R 6km. In three models discussed above at representative times during
0,d e
(cid:39) (cid:39)
contrast to model dip-60, where the star is threaded by the evolution. Due to the differential rotation in the HMNS,
a magnetic field that is artificially uniform on lengthscales strongtoroidalfieldsaregeneratedviamagneticwinding(lin-
(cid:38) R (cf. inset in Figure 1), for model dip-6, the neutral early growing at the beginning). Within a few rotational pe-
e
pointandmaximummagneticfieldarelocatedintheinterior riods, the build-up of magnetic pressure (mostly associated
of the star, corresponding to a more realistic distribution of withsteeptoroidal-fieldradialgradients)issufficienttoover-
currents. Finally, the third model (henceforth rand) repre- comethegravitationalbindinginthevicinityofthestellarsur-
sents a “random” magnetic field, which we compute from a face, powering an outflow of highly magnetized low-density
vectorpotentialbuiltasthelinearsuperpositionof 6 104 matterwithmildlyrelativisticvelocities(cf. alsoFigure4).
∼ ×
modeswithrandomamplitudesandphases For model dip-60, since the initial magnetic field is ori-
entedalongtherotationaxisonlengthscales(cid:38) R (cf.Fig-
e
A0,r√γ (cid:88)nk ure 1, left panel), the outflow is efficiently channeled along
A = a cos(x k +2πb )
ijk (r2+(cid:36)2 )3/2 (cid:96)mn ijk· (cid:96)mn (cid:96)mn thisaxisandthussignificantlycollimated,thoughthisismore
0,r (cid:96)mn=0 the result of the ordered initial magnetic field configuration
+c sin(x k +2πd ) . (1) than of genuine self-confining processes. During the evolu-
(cid:96)mn ijk (cid:96)mn (cid:96)mn
·
tion,thenon-collimatedpartoftheoutflowatlowerlatitudes
Here, i,j,k label the points on the computational grid, tendstopushthecollimatedparttowardtheaxis, effectively
xijk denotes the associated position vectors, and (cid:96), m, n shrinking the inner opening angle of this jet-like structure
refer to a grid in wave vector k space, defined by (cf. Figure 1). This triggers irregularities in the collimated
k(cid:96)mn [0,2π/λmin,2π/(λmin +∆λ),...,2π/(λmin + outflowthatmanifestthemselvesassmallbubblesofmaterial
{(nk }1)∆≡λ)]3, wherenk = 30, λmin 3kmisthesmall- withlowdensityandmagnetizationbeingreleasedalongthe
− ≈
est wavelength employed, and ∆λ 2.2km. Furthermore, vertical axis, possibly pointing to a kink instability (Kiuchi
a(cid:96)mn,...,d(cid:96)mn are random numbe≈rs between 0 and 1. The etal.2012).
constantA0,r setsthemaximumfieldstrength,thefactor√γ, Theinitialmagneticfieldgeometryofdip-6hasamuch
withγ asthedeterminantofthespatialmetric,helpstocon- smaller curvature scale and is less effective at funneling
centrate stronger magnetic fields in regions of larger space– the outflow; the resulting collimation is thus much less
time curvature, while the denominator scales the magnetic pronounced than for dip-60. Furthermore, while model
field as r−3 for r > (cid:36)0,r 2Re. We built the random- dip-60showsacleardistinctionbetweenthecollimatedand
∼ ≈
field model to resemble the actual magnetic field configura- the non-collimated part of the outflow, this is no longer the
tion resulting from a BNS merger (cf. Rezzolla et al. 2011). case for model dip-6 (cf. Figures 1 and 2). Finally, the
In all the above cases, A0,d and A0,r are adjusted to yield model with the most realistic magnetic field configuration,
maximum initial field strengths of B = 2 1014G, with i.e.,modelrand,ischaracterizedbyanalmostisotropicout-
0
themagnetic-to-fluidpressureratiobeing ×10−5 insidethe flow (cf. Figure 3), though we cannot exclude the fact that
(cid:28)
star (cf. Figure 4). The resulting initial magnetic energy some form of collimation could emerge at later times. This
E differs with the magnetic field geometry, being E representsanimportantresultofoursimulations: theamount
M M (cid:39)
MagneticallydrivenwindsandX-rayafterglowsofSGRBs 3
250 250
60 15.0
50 200 200
14.5
40
m] 150 40 80 120 150 40 80 120
k 5 10 15 14.0
[ 30
z
100 100
20 13.5
50 50
10
0ms 30ms 60ms 13.0
0 0 0
0 10 20 30 40 50 60 0 50 100 150 200 250 0 50 100 150 200 250
x[km] x[km] x[km]
Figure1. Snapshotsofthemagneticfieldstrength(color-codedinlogarithmicscaleandGauss)andrest-massdensitycontoursinthe(x,z)planeatrepresentative
timesformodeldip-60.Magneticfieldlinesaredrawninredintheleftpanel.TheleftmostinsetshowsamagnificationoftheHMNS,theotheronesshowa
horizontalcutatz=120km.
250 250
60 14.0
50 200 200
13.5
40
m] 150 40 80 120 150 40 80 120
k 5 10 15 13.0
[ 30
z
100 100
20 12.5
50 50
10
0ms 40ms 60ms 12.0
0 0 0
0 10 20 30 40 50 60 0 50 100 150 200 250 0 50 100 150 200 250
x[km] x[km] x[km]
Figure2. SameasFigure1,butformodeldip-6.
250 250
60 14.0
50 200 200
13.5
40
m] 150 40 80 120 150 40 80 120
k 5 10 15 13.0
[ 30
z
100 100
20 12.5
50 50
10
0ms 45ms 60ms 12.0
0 0 0
0 10 20 30 40 50 60 0 50 100 150 200 250 0 50 100 150 200 250
x[km] x[km] x[km]
Figure3. SameasFigure1,butformodelrand.
ofcollimationinthemagnetic-drivenwindfromtheBMPwill Definingtheisotropicluminosityas
dependsensitivelyonthemagneticfieldgeometryandcould
(cid:73)
beabsentifthefieldisrandomlydistributed. L dΩ√ g(TEM)r , (2)
In all of the configurations considered, the magnetized EM ≡− − t
baryon-loaded outflow has rest-mass densities 108 r=Rd
109gcm−3andisejectedfromthestarwithveloci∼tiesv/c−(cid:46) where dΩ is the solid-angle element, g is the determinant of
0.1, in the isotropic part, and v/c (cid:46) 0.3, in the collimated thespacetimemetric,andTEM istheEMpartoftheenergy-
part. µν
momentum tensor, all of the different initial magnetic field
geometriesyieldhigh,stationaryEMluminositiesassociated
4 D.M.Siegel,R.Ciolfi,L.Rezzolla
20 20 20 0
0ms 0ms 0ms
2
15 15 15 −
4
m] 4 8 12 4 8 12 4 8 12 −
k 10 10 10
[
z 6
−
5 5 5
8
−
60ms 60ms 60ms
0 0 0
0 5 10 15 20 0 5 10 15 20 0 5 10 15 20
x[km] x[km] x[km]
Figure4. Logarithmicmagnetic-to-fluidpressureratioandrest-massdensitycontoursformodeldip-60(left),dip-6(middle),andrand(right).
withtheoutflowofL 1048 1050ergs−1 (cf.Figure5, need for a fudge factor, χ, that is 1 for the more realistic
EM ∼ − ∼
upperleftpanel).Modeldip-60hasaluminositythatistwo magnetic field configurations rand and dip-6, and 100
∼
ordersofmagnitudelargerthanthoseofmodelsdip-6and for dip-60, showing that initial models with the same B
0
rand, which have surprisingly similar luminosities. These but different field geometries lead to luminosities that differ
differences can be understood in terms of the different mag- byordersofmagnitude(cf.upperleftpanelofFigure5). The
neticenergiesassociatedwiththethreemodels(seebelow). role of the equatorial radius, R , is to define the volume in
e
We have also verified that when computed at sufficiently whichtheconversionofrotationalenergyintoEMenergyoc-
large distances, the luminosity depends only weakly on the curs, while the period, P, sets the timescale of such conver-
radius R , chosen to compute the integral in Equation (2) sion.
d
(seeFigure5,lowerleftpanel). Theremainingdiscrepancies Equation (3) still holds if expressed in terms of quan-
among different R choices are due to the bulk of the non- tities referring to the stage when the outflow has become
d
collimatedpartoftheoutflowhavingnotyetcrossedtheouter roughly stationary, by substituting χ(B /1014G)2 with
0
detection spheres at t = 60 ms. The effective bulk speed of (B¯/1015G)2,whereB¯ isthemagneticfieldstrengthreached
theoutflowatlargerdistances(asinferredfromthetimedelay in the outer layers of the star (R and P are essentially un-
e
itnionthsepoenesdeotfo(cid:46)fL0E.M1ca,tdduieffteoreanntoRndz)eriossamtmaollseprhtehraenrtehset-mejaescs- cahsathnigsendefwromexptr=ess0i)o.nTdhoeefsundogtedfaecpteonrdisonnothloenmgeargnneectiecssfiaerlyd,
densityfloor. Thiseffectisillustratedinthelowerrightpanel geometry.
of Figure 5, where a lower floor yields a higher bulk speed Two remarks should be made. First, scalings similar to
(cf. black and red lines). Note, however, that the stationary Equation (3) have already been reported in the literature
luminosityisnotaffectedbytheatmospherelevelevenwhen (e.g., Meier 1999; Shibata et al. 2011), where, however, no
thelatterischangedbyoneorderofmagnitude. Furthermore, discussion was made on the dependence of the luminosity
the lower right panel of Figure 5 reports the luminosity for on the initial magnetic field geometry and thus on the im-
threedifferentresolutions, showingthattheoutflowspeedis portance of the factor χ. Ignoring this dependence can eas-
affectedbutthestationarylevelsareinreasonablygoodagree- ily lead to over/underestimates of the luminosity by orders
ment. of magnitude. Second, the scaling with radius and period in
Anotherimportantconclusiontobedrawnfromoursimu- Equation(3)areverydifferentfromthoseresultingfromthe
lationsisthat,keepingthebackgroundfluidmodelfixed,the magneticdipolespin-downemissionconsideredinthemodel
EMluminosityessentiallydependsontheinitialmagneticen- of Zhang & Me´sza´ros (2001), where the luminosity is given
ergy ofthe system. Inthis way, the difference oftwo orders byL B2R6P−4. Thesedifferencesinscalingmaybe
ofmagnitudebetweentheluminositiesofmodeldip-60and usedEtoMd∝iscrimineatebetweenthescenarioproposedhereand
those of models dip-6 and rand (cf. upper left panel of thatbyZhang&Me´sza´ros(2001).
Figure 5) is simply due to the corresponding difference in
E .ThisisdemonstratedintheupperrightpanelofFigure5, 4. MAGNETIC-DRIVENWINDSANDX-RAYEMISSION
M
where we show the luminosity of a model dip-60 with an Defining the conversion efficiency of EM luminosity into
initialmagneticfieldstrengththatisrescaledtomatchtheini- observed X-ray luminosity as η Lobs/L , we can use
tialmagneticenergyofmodeldip-6. Oncetheinitialmag- ≡ EM EM
theresultsofoursimulationsforcharacteristicradiiandperi-
netic energies coincide, the resulting stationary luminosities ods(R 106cmandP 10−4s)todeducethatmagnetic
arethesameandthereforeindependentofthemagneticfield field stere∼ngths in the range∼B¯ η−1/2 1014 1016G are
geometryanddegreeofcollimation. needed to produce luminosities∼Lobs ×1046 1−051ergs−1,
OurresultsleadtoasimpleexpressionfortheEMluminos-
EM ∼ −
whichcharacterizetheextendedemissionandX-rayplateaus
ityfromtheBMP,
ofSGRBs(e.g.,Rowlinsonetal.2013;Gompertzetal.2014).
(cid:18) B (cid:19)2(cid:18) R (cid:19)3(cid:18) P (cid:19)−1 Assuming an efficiency of, e.g., η 0.01 0.1 yields mag-
L 1048χ 0 e ergs−1, (3) netic field strengths B¯ 1014 ∼1017G.−These strengths
EM(cid:39) 1014G 106cm 10−4s ∼ −
are not reached in the progenitor neutron stars, but they can
whereP denotesthe(central)spinperiodandB denotesthe be built from much weaker initial magnetic fields in a num-
0
initial maximum magnetic field. Our simulations reveal the ber of ways: the compression of stellar cores (Giacomazzo
MagneticallydrivenwindsandX-rayafterglowsofSGRBs 5
49
50
s] s]48
/ /
erg48 erg47
[ [
) )
M M
E46 E46
L L
g( dip-60 g(
lo44 dip-6 lo45 dip-6
rand dip-60rescaled
44
51 50
s]50 s]48
/ /
g g
er49 er
)[ 80km )[46
M M
E48 100km E
Log(47 112500kkmm Log(44 ddiipp--66,1.2h
l 200km l dip-6,1.5h
46 250km 42 dip-6,atm5
0 10 20 30 40 50 60 0 5 10 15 20 25 30 35 40
time[ms] time[ms]
Figure5. Topleft:EMluminositiesforthedifferentsimulations. Topright:luminositycomparisonbetweenmodeldip-6andmodeldip-60withrescaled
initialmagneticfieldtomatchtheinitialmagneticenergyofmodeldip-6. Bottomleft: luminositiesextractedatdifferentradiiformodeldip-60. Bottom
right:impactofresolutionandafactorof10loweratmospherethresholdonsimulationsofmodeldip-6.
etal.2011),theKelvin–Helmholtzinstabilitydevelopingdur- power the EM emission (Zhang & Me´sza´ros 2001; Gao &
ing the merger (Price & Rosswog 2006; Baiotti et al. 2008; Fan 2006). Note that a similar evolution also applies if the
Andersonetal.2008;Giacomazzoetal.2011;Zrake&Mac- differentiallyrotatingBMPisanSMNSatbirth.
Fadyen 2013), magnetic winding and the magnetorotational Considering a reference luminosity of L
instability (MRI; Siegel et al. 2013). In particular, the MRI 1048ergs−1, the timescale needed to exhaust thEeM rese∼r-
might enhance the EM emission, though its saturation level voir of rotational energy T 5 1052erg of our BMP is
is unknown and its role in determining luminosity levels re- readilygivenbyτ (cid:46) T/L∼ ×5 104s. Throughfitting
mainsunclear.Unfortunately,resolvingtheMRIinlong-term the data reportedEiMn Table 3EoMf∼Row×linson et al. (2013) we
global simulations is out of reach. In our simulations only findapower-lawcorrelationbetweentheobservedplateaulu-
magneticwindingisatplay,yetmagneticfieldsareamplified minositiesanddurations: Lobs[ergs−1] 1052(τ [s])−a,
byatleastoneorderofmagnitude. where a = 1.36 0.E1M1. Given∼a luminEMosity of
Thetimescaleforthepersistenceofdifferentialrotationand L = Lobs/η 1±048ergs−1, the fit gives a duration
the survival time of the BMP is still very much uncertain EM EM ∼
of the plateau emission τ (104/η)1/as 5 103s
(Lasky et al. 2014) and it is hard to estimate the timescale EM ∼ ∼ ×
for η 0.1. It is reassuring that even these crude estimates
τ over which these luminosities can be sustained. Sim- ∼
EM provide emission timescales that are well below the upper
ulations show that the timescale for the extraction of angu-
lar momentum via GWs from a bar-deformed BMP is (cid:46) 1s limitofT/LEM ∼ 5×104s,thusshowingthecompatibility
withtheobservations.
(Baiotti et al. 2008; Bernuzzi et al. 2013; but see also Fan
Wealsonotethattheestimatesmadeabovearebasedonthe
etal.2013). Hence,iftheBMPsurvivesformorethan 1s,
∼ assumptionthattheobservedemissionisessentiallyisotropic.
itmusthavebeendriventowardanalmostaxisymmetricequi-
IfthereiscollimationwithinasolidangleΩ ,theduration
libriumbyGWemission. Abackoftheenvelopeestimateof coll
τ estimated from the fit would be reduced by a factor of
magnetic braking leads to a timescale of 1 10 s for the EM
magneticfieldsconsideredhere(Shapiro2∼000)−. Thisiscon- (Ωcoll/4π)1/a.
∼
firmed by the evolution of the angular velocity in our simu-
lations, which shows that Ω/Ω˙ 10 s in the stellar interior. 5. CONCLUSIONS
∼
Bothofthesetimescalesshouldbemeantaslowerlimitsand
UsingidealMHDsimulationswehaveinvestigatedtheEM
differentialrotationwillberemovedontimescalesthatcould
emission of an initially axisymmetric HMNS endowed with
be 10 times larger. However, even 100 s are not sufficient
different initial magnetic field configurations, spanning the
to explain X-ray luminosities lasting 103 104 s. A viable
range of geometries expected from BNS mergers. Despite
−
scenario is one in which the BMP is an HMNS at birth, but
thedifferentinitialconfigurations, wehavefoundthat, inall
evolves into an SMNS while differential rotation is removed
cases, differential rotation in the HMNS generates a strong
via magnetic braking and rest mass is lost via the wind (our
toroidal magnetic field and a consequent baryon-loaded out-
simulationsshowmass-lossratesof 10−3 10−2M(cid:12)s−1). flow with bulk velocities (cid:46) 0.1c. This emission is almost
∼ −
Inthiscase,theBMPcansurviveonmuchlongertimescales
isotropic, thoughanadditionalcollimated, mildlyrelativistic
as it needs less angular momentum to prevent gravitational
flow is produced if the initial magnetic field has a dominant
collapse. Once differential rotation has been removed (or is
dipolecomponentalongthespinaxis.
very small), the spin-down via dipolar emission through the
Sincetheemissionweobserveemergesasarobustfeature
global magnetic field produced over these timescales would
ofaBNSmergerandgiventheconsistencyoftheluminosity
6 D.M.Siegel,R.Ciolfi,L.Rezzolla
levels and duration with the observations, we conclude that Farris,B.D.,Gold,R.,Paschalidis,V.,Etienne,Z.B.,&Shapiro,S.L.
the proposed physical mechanism represents a viable expla- 2012,PhRvL,109,221102
nationfortheX-rayafterglowsofSGRBs. Fox,D.B.,Frail,D.A.,Price,P.A.,etal.2005,Natur,437,845
Gao,W.-H.,&Fan,Y.-Z.2006,ChJAA,6,513
Gehrels,N.,Chincarini,G.,Giommi,P.,etal.2004,ApJ,611,1005
We thank F. Pannarale and F. Galeazzi for discussions, Gehrels,N.,Sarazin,C.L.,O’Brien,P.T.,etal.2005,Natur,437,851
W. Kastaun for help with the visualizations, and the ref- Giacomazzo,B.,&Perna,R.2013,ApJL,771,L26
eree for valuable comments. R.C. is supported by the Hum- Giacomazzo,B.,&Rezzolla,L.2007,CQGra,24,S235
Giacomazzo,B.,Rezzolla,L.,&Baiotti,L.2011,PhRvD,83,044014
boldt Foundation. Additional support comes from the DFG
Gompertz,B.P.,O’Brien,P.T.,&Wynn,G.A.2014,MNRAS,438,240
Grant SFB/Transregio 7, from “NewCompStar”, COST Ac- Harry,G.M.2010,CQGra,27,084006
tionMP1304,andfromHICforFAIR.Thesimulationshave Kiuchi,K.,Kyutoku,K.,&Shibata,M.2012,PhRvD,86,064008
been performed on SuperMUC at LRZ Garching and on Komatsu,H.,Eriguchi,Y.,&Hachisu,I.1989,MNRAS,237,355
DaturaattheAEI. Lasky,P.D.,Haskell,B.,Ravi,V.,Howell,E.J.,&Coward,D.M.2014,
PhRvD,89,084006
Lasota,J.-P.,Haensel,P.,&Abramowicz,M.A.1996,ApJ,456,300
REFERENCES Lo¨ffler,F.,Faber,J.,Bentivegna,E.,etal.2012,CQGra,29,115001
Meier,D.L.1999,ApJ,522,753
Metzger,B.D.,&Berger,E.2012,ApJ,746,48
Abadie,J.,Abbott,B.P.,Abbott,R.,etal.2010,CQGra,27,173001
Narayan,R.,Paczynski,B.,&Piran,T.1992,ApJL,395,L83
Accadia,T.,Acernese,F.,Antonucci,F.,etal.2011,CQGra,28,114002
Paczynski,B.1986,ApJL,308,L43
Anderson,M.,Hirschmann,E.W.,Lehner,L.,etal.2008,PhRvL,100,
Price,R.H.,&Rosswog,S.2006,Sci,312,719
191101
Rezzolla,L.,Giacomazzo,B.,Baiotti,L.,etal.2011,ApJL,732,L6
Antoniadis,J.,Freire,P.C.C.,Wex,N.,etal.2013,Sci,340,1233232
Rowlinson,A.,O’Brien,P.T.,Metzger,B.D.,Tanvir,N.R.,&Levan,A.J.
Baiotti,L.,Giacomazzo,B.,&Rezzolla,L.2008,PhRvD,78,084033
2013,MNRAS,430,1061
Barthelmy,S.D.,Chincarini,G.,Burrows,D.N.,etal.2005,Natur,438,
Shapiro,S.L.2000,ApJ,544,397
994
Shibata,M.,Suwa,Y.,Kiuchi,K.,&Ioka,K.2011,ApJL,734,L36
Belczynski,K.,O’Shaughnessy,R.,Kalogera,V.,etal.2008,ApJL,680,
Siegel,D.M.,Ciolfi,R.,Harte,A.I.,&Rezzolla,L.2013,PhRvD,87,
L129
121302
Berger,E.2013,arXiv:1311.2603
Soderberg,A.M.,Berger,E.,Kasliwal,M.,etal.2006,ApJ,650,261
Berger,E.,Fong,W.,&Chornock,R.2013,ApJL,774,L23
Stergioulas,N.,&Friedman,J.L.1995,ApJ,444,306
Bernuzzi,S.,Dietrich,T.,Tichy,W.,&Bruegmann,B.2013,
Tanvir,N.R.,Levan,A.J.,Fruchter,A.S.,etal.2013,Natur,500,547
arXiv:1311.4443
Yu,Y.-W.,Zhang,B.,&Gao,H.2013,ApJL,776,L40
Burrows,D.N.,Grupe,D.,Capalbi,M.,etal.2006,ApJ,653,468
Zhang,B.,&Me´sza´ros,P.2001,ApJL,552,L35
Demorest,P.B.,Pennucci,T.,Ransom,S.M.,Roberts,M.S.E.,&Hessels,
Zrake,J.,&MacFadyen,A.I.2013,ApJL,769,L29
J.W.T.2010,Natur,467,1081
Eichler,D.,Livio,M.,Piran,T.,&Schramm,D.N.1989,Natur,340,126
Fan,Y.-Z.,Yu,Y.-W.,Xu,D.,etal.2013,ApJL,779,L25