Black hole-neutron star mergers: effects of the orientation of the black hole spin

0.48 MB·English

by  Francois Foucart

Blackhole-neutron starmergers: effects ofthe orientationofthe blackholespin
Francois Foucart,1 Matthew D. Duez,1,2 Lawrence E. Kidder,1 and Saul A. Teukolsky1,3
1CenterforRadiophysicsand SpaceResearch, CornellUniversity, Ithaca, NewYork, 14853, USA
2DepartmentofPhysics&Astronomy, Washington StateUniversity, Pullman, Washington99164, USA
3Theoretical Astrophysics350-17, CaliforniaInstituteofTechnology, Pasadena, California91125, USA
Thespinofblackholesinblackhole-neutronstar(BHNS)binariescanhaveastronginfluenceonthemerger
dynamicsandthepostmergerstate;awidevarietyofspinmagnitudesandorientationsareexpectedtooccurin
nature. Inthispaper,wereportthefirstsimulationsinfullgeneralrelativityofBHNSmergerswithmisaligned
blackholespin. Wevarythespinmagnitudefroma /M =0toa /M = 0.9foralignedcases,and
BH BH BH BH
wevarythemisalignmentanglefrom0to80◦ fora /M = 0.5. Werestrictourstudyto3:1massratio
BH BH
1 systemsanduseasimpleΓ-lawequationofstate.Wefindthatthemisalignmentanglehasastrongeffectonthe
1 massofthepostmergeraccretiondisk,butonlyforanglesgreaterthan≈ 40◦. Althoughthediskmassvaries
0 significantly with spin magnitude and misalignment angle, we find that all disks have very similar lifetimes
2 ≈100ms. Theirthermalandrotationalprofilesarealsoverysimilar.Foramisalignedmerger,thediskistilted
withrespecttothefinalblackhole’sspinaxis. Thiswillcausethedisktoprecess, butonatimescalelonger
n
thantheaccretiontime. Inallcases,wefindpromisingsetupsforgamma-rayburstproduction: thedisksare
a
J hot,thick,andhyperaccreting,andabaryon-clearregionexistsabovetheblackhole.

6
PACSnumbers:04.25.dg,04.40.Dg,04.30.-w,47.75.+f,95.30.Sf

]
E
H I. INTRODUCTION radius, and the BH spin. Current estimates from population
synthesis models suggest that most systems are likely to be
.
h Black hole-neutronstar (BHNS) binary mergers presenta formedwithablackholeof 10M⊙.Relativisticsimulations
p to date have considered ca∼ses of relatively low mass black
remarkable opportunity to study strongly-curved spacetime
o- andsupernuclear-densitymatterinthemostextreme,dynam- holes ( 2 7M⊙ ) [6–13], for which the NS is expected
∼ −
r icalconditions. BHNSbinariesincompactorbitsemitstrong to disruptoutside the innermoststable circular orbit(ISCO),
t making disk formation more likely. These simulations have
s gravitational waves, and they are expected to be one of the
a mainsourcesforAdvancedLIGOandVIRGO[1,2]. Current foundcasesofmassivediskformation,withMBH 3 4M⊙
[ ∼ −
resultinginthelargestdisks[12].TheNSradiusistheparam-
estimates for the eventrates of binary mergerscoming from
2 populationsynthesismodelspredictthatAdvancedLIGOwill eterrelatedto theequationofstate thathasthelargesteffect
v onthewaveformandpost-mergerdisk[13],withlargerradii
see about10 BHNS/yr, althoughuncertainties in the models
3 resultinginlargerdisks[8,13].
allow for a large range of potentialeventrates, 0.2 – 300
0 ∼
BHNS/yr[3]. Thesegravitationalwavescontain,inprinciple, The spin of the black hole can have a strong influence on
2
4 awealthofinformationontheirsource,suchasthemassand themerger. TheISCOissmallerforprogradeorbitsarounda
. spin of the black hole (BH) and the mass and radius of the spinningBH than for orbitsarounda nonspinninghole. Be-
7
neutronstar(NS).InferredpropertiesoftheNScouldbeused causediskformationisexpectedtobemorelikelyifNStidal
0
toconstraintheNSequationofstate.Informationinthewaves disruption occurs outside the ISCO than if it occurs inside,
0
1 canonlybeextracted,however,bycomparisonwithaccurate BH spin can facilitate disk formation. With high BH spin,
: numerically-generated predictions that provide the expected it is even plausible that BHNS binaries with the most likely
v
waveformforeachpossibleBHNSsystem. mass ratios ( 7:1) give rise to substantial disks [14]. The
i ∼
X Mergers of BHNS binaries have also been proposed magnitude of the BH spin is largely unconstrained by pop-
r as potential progenitors of short-hard gamma-ray bursts ulation synthesis models, as it comes mostly from the spin
a (SGRB)[4]. TheoriginofSGRBsisnotyetknown,although acquiredduringformationoftheblackholeinacore-collapse
itiscertainthattheenginesarecompactandlocatedatcosmo- event[15]. Theeffectofalignedandantialignedspinswasin-
logicaldistances,andthereisevidence(suchastheirpresence vestigatedinfullgeneralrelativityforthe3:1mass-ratiocase
in nonstar forming regions) to support a mechanism differ- by Etienne et al. [12]. They found that a large aligned spin
entfromthatassociatedwithlong-softGRBs, namelystellar (andcorrespondinglysmallISCO)leadstoamuchmoremas-
corecollapse. ForBHNSmergers,thegenerationofaSGRB sive post-merger disk. For example, for aBH/MBH = 0.75
ispossibleonlyif theremnantblackholeissurroundedbya and MBH 4M⊙, a disk of Mdisk 0.2M⊙ can be ob-
∼ ∼
massive, hot, thick accretion disk. Also, to obtain relativis- tained.ForBHNSbinarieswithmassiveblackholes(MBH
∼
tic jets anda beamedoutflow, a regionmostlydevoidof any 10M⊙),formingadiskmayinfactonlybepossibleifthehole
matterisnecessary(seee.g.[5]andreferencestherein). Only isspinning.
numerical simulations in full general relativity with realistic There is no reason to expect the black hole spin to be
microphysics can determine if these conditions are likely to alignedwiththeorbitalangularmomentum. Populationsyn-
beobtained. thesis models predict a relatively wide distribution of orien-
Whetheradiskformsornotwilldependonthepremerger tations, with about half of the binaries having a misalign-
characteristicsofthebinary,especiallytheBH mass, theNS ment between the BH spin and the orbital angular momen-
2
tumoflessthan45◦whentheinitialBHspinisa /M = malism (XCTS) [17, 18]. If we write the spacetime metric
BH BH
0.5 [15]. Misalignment can reduce or reverse the BH spin as
effectsdescribedabove. Thiscanbeunderstoodbyconsider-
ingprogradeorbitsoftestparticleswithsmallradialvelocity, ds2 = gµνdxµdxν
whichbecomeunstablefartherawayfromtheBHforinclined = α2dt2+ψ4γ˜ (dxi+βidt)(dxj +βjdt), (1)
ij
−
orbitsthanforequatorialorbitsofthe same angularmomen-
tum. Misalignment will also produce qualitatively new ef- theinitialdatatobedeterminedincludethelapseα,theshift
fects,includingtheprecessionofthepremergerorbitalplane vectorβi,theconformalfactorψ,theconformal3-metricγ˜ij
and BH spin. The influence of misalignmenthas been stud- and the extrinsic curvature Kµν = −21Lngµν (where Ln is
iedfor10:1mass-ratiobinariesintheapproximationthatthe the Lie derivative along the normal n to the t = 0 slice).
spacetimeisassumedtobeKerr. Rantsiouetal.[16]showed The constraintscan be expressed as a set of 5 coupledellip-
that,inthisapproximation,diskscanbeformedonlyatrela- tic equations for the lapse, shift, and conformal factor [18].
tivelylowinclinationsandonlyfornearextremalblackholes. Thephysicalpropertiesofthesystemarethendeterminedby
Ultimately, though, simulations in full general relativity are the choiceof the remainingfree parameters: the trace of the
neededtoaccuratelymodelsuchsystems. extrinsic curvatureK = gµνKµν, the conformalmetric γ˜ij,
Inthispaper,wereportonfullyrelativisticstudiesofmis- their time derivatives ∂tK and ∂tγ˜ij, and the matter stress-
alignedBHNSbinaries.Welimitourselvestosmallmasssys- energytensorTµmνatter.
tems (MBH 4.2M⊙) and a simplified equation of state, The system of elliptic equations is solved using the spec-
but we consid∼er a significant range of black hole spin mag- tral elliptic solver SPELLS developedby the Cornell-Caltech
nitudesandorientations. WeconfirmtheresultsofEtienneet collaboration[19], and initially used to constructinitial data
al[12]regardingtheeffectsofanalignedBHspin. Formis- for binary black hole systems by Pfeiffer et al. [18, 20]. A
aligned spins, we find that the misalignment angle can have detailedpresentationofthemethodsusedfortheconstruction
astrongeffectonthepost-mergerdiskmass,butonlyforan- of BHNS initial data was givenin Foucartet al. [21]. Here,
gles greaterthanaround40◦. Althoughthe disk mass varies we limit ourselves to a brief summary plus a description of
greatlywithBHspin,mostotherdiskpropertiesareverysimi- thechangesmadetoaccommodatethepossibilityofarbitrary
lar,includingtheaccretiontimescale,thelocationofthemax- spinorientation.
imum density, the average temperature, and the entropy and Asthesystem isexpectedto beinitiallyin a quasiequilib-
angularmomentumprofiles.Thedisksareallthick,eachwith riumstate,withthebinaryinalow-eccentricitycircularorbit
a heightH to radius r ratio H/r 0.2, nearly independent of slowly decreasing radius, we work in a frame comoving
ofr. Crucially,theyallhaveabary≈on-clearregionaboveand withthebinaryandsetthetimederivativestozero: ∂tK = 0
belowtheBH.ThedisksaremisalignedwiththefinalBHspin and∂tγ˜ij = 0. Asforγ˜ij andK,wemakeachoiceinspired
by 15◦. TheydoprecessabouttheBHspinaxis,butwith- by the results of Lovelace et al. [22] for binary black hole
out≤reachingafixedprecessionrate. Indeed,thesteady-state systems. ClosetotheBH,themetricmatchesitsKerr-Schild
precession time scale is expected to be significantly longer valuesforaBHofthedesiredmassandspin,whileawayfrom
thantheaccretiontimescale. theBH, the conformalmetricisflatandK = 0. Thetransi-
Thispaperisorganizedasfollows.InSec.II,wediscussthe tionbetweenthesetworegionsisdonebyusingthefollowing
methodused toconstructthe verygeneralBHNS initialdata prescription:
we use. We also discuss in detail the improvements to our
γ˜ = δ +[γKS(a ,v ) δ ]e−λ(rBH/w)4, (2)
evolution code that have increased the accuracy by an order ij ij ij BH BH − ij
of magnitude over the results presented in Duez et al. [11]. K = KKS(a ,v )e−λ(rBH/w)4, (3)
BH BH
WethenpresentourrundiagnosticsinSec.III. Thedifferent
r r
casestobeevolvedaredescribedinSec.IV. Wethenpresent λ = BH− AH , (4)
r /q r
the resultsof the simulationsin Secs. V andVI. Finally, we NS − BH
drawconclusionsinSec.VII. vBH = Ωrot cBH, (5)
×
where the KS subscript refers to the Kerr-Schild values,
r (r ) is the coordinatedistance to the center of the BH
BH NS
II. NUMERICALMETHODS
(NS), r the coordinate radius of the BH apparent hori-
AH
zon, a /M is the dimensionless spin of the BH, q
BH BH
A. Initialdata M /M aconstantoftheorderofthemassratio,c th∼e
NS BH BH
coordinatelocationoftheBHcenterwithrespecttothecenter
For numerical evolutions of Einstein’s equations, we de- ofmassofthesystem,andwissomefreelyspecifiablewidth,
composethespacetimeunderstudyintoafoliationofspace- chosen so that the metric is nearly flat at the location of the
like hypersurfaces parametrized by the coordinate t. Ein- NS.Theparameterλ,whichisdesignedtoimposeaflatback-
stein’sequationscanbewrittenintheformofhyperbolicevo- groundatthelocationoftheNS,issetto forr <qr .
NS BH
∞
lutionequationsplusasetofconstraintsthathavetobesatis- Boundaryconditionsareimposedatinfinityandontheap-
fiedoneacht=constantslice. Ourinitialdataatt=0must parenthorizonoftheBH(sincetheinsideoftheBHisexcised
bechosensuchthatitsatisfiestheseconstraints.Weconstruct fromourcomputationaldomain).Theboundaryconditionsat
initialdatausingtheExtendedConformalThinSandwichfor- infinity are chosen so that the metric is asymptotically flat,
3
while the innerboundaryconditionsfollowthe prescriptions
of Cook and Pfeiffer [23], which make the inner boundary s.5i20
0.3 s.5i40
anapparenthorizoninquasiequilibrium. Thereissomefree-
dom in these boundary conditions: on the apparenthorizon,
the conformallapse is not fixed (we set it to the value of an
isolatedKerrBH),andtheshiftisdeterminedonlyuptoaro- 0.2
M
tationtermβ′i = βi+ǫijkΩBjHxk. ThevalueofΩBH deter- COz
minesthespinoftheBH,buttheexactrelationbetweenΩBH
andthespinisunknownapriori;togetthedesiredBHspin, 0.1
wehavetosolveiterativelyforΩBH. Ontheouterboundary,
theshiftcanbewrittenas
β =Ωrot×r+a˙0r+vboost (6) 00 2 4 6
T(ms)
whereΩrot allowsforaglobalrotationofthecoordinates,a˙
0
for a radial infall with velocity v = a˙ r, and v for a
0 boost FIG. 1: Evolution of the position of the center of mass along the
boost. As an initial guessfor the orbitof the binary,we can z-axisduringinspiral(solidline),comparedtothemotionexpected
settheradialvelocityatt=0to0(a˙0 =0).Thisassumption, fromtheboostv∞z givenintheinitialdata(dottedline).
aswellasthequasiequilibirumformalism,clearlyneglectsthe
evolutionoftheorbitovertimethroughtheradialinfallofthe
binary and the precession of the orbital plane. Both effects ΩBH isalignedwiththeorbitalangularmomentum. Instead,
are,however,actingoverrelativelylongtimescales: overits all3componentsofΩBHaresolvedfor. Wealsoabandonthe
firstorbit,eventhebinarywiththemostinclinedspinconsid-
assumption of equatorial symmetry, and controlthe position
eredhere(s.5i80inthelatersections)goesthroughlessthan
ofthe BH alongthe z-axisofthe orbitalangularmomentum
10%ofafullprecessionperiodoftheBHspinwhilethecoor-
by requiringthat the NS is initially movingin the xy-plane,
dinateseparationbetweenthecompactobjectsisreducedby
withitscenterinthez = 0plane(thez coordinateofthelo-
about20%. One known effectof the quasi-circularapproxi-
cation of the BH is chosen so that the conditioneˆ . h = 0
mationisthatthebinarywillhaveanonzeroeccentricity.The z ∇
issatisfiedatthecenteroftheNS).Finally,toguaranteethat
eccentricitycan bedecreasedbymodifyingthe initialvalues
Pz =0weimpartaboosttothewholesystemthroughthe
of a˙ and Ωrot [24] (see also [21] for an application of that ADM
0 boundarycondition at infinity: vz = vz . The center of
method to BHNS binaries) as long as the initial eccentricity boost ∞
massisthenexpectedtohaveaglobalmotionduringinspiral
andorbitalphasecanbeaccuratelymeasured. Here,weonly
correspondingto thatboost, and we check in Fig. 1 thatthis
applythistechniquewhenthespinoftheBHisalignedwith
isindeedthecase. Byaddingtheseconditionstotheiterative
the orbitalangular momentum of the binary. For precessing
procedureusedtogenerateBHNSinitialdata,weareableto
binaries,significantlyreducingtheeccentricitywouldrequire
obtain high-precision initial configurations for arbitrary val-
alargerinitialseparationforwhichtheeffectsofeccentricity,
uesoftheorientationoftheBHspin.
precessionandradialinfallcanbeproperlydisantangled.
In the presence of matter, additionalchoices are required.
We assume that the fluid is in hydrostaticequilibriumin the
comovingframe,andrequirethatits3-velocityisirrotational. B. Evolution
Thefirstconditiongivesanalgebraicrelationbetweentheen-
thalpy h of the fluid, its 3-velocity v , and the metric g ,
i µν The simulations presented here use the SpEC code devel-
while the secondleadsto anotherelliptic equationdetermin-
oped by the Cornell-Caltech-CITA Collaboration [25]. To
ingthevelocityfield. Theseequationsarecoupledtothecon-
evolveBHNSsystems,thetwo-gridmethoddescribedinDuez
straints: the whole system can only be solved throughan it-
etal.[11]isused.Einstein’sequationsareevolvedonapseu-
erative method. For a BH with a spin aligned with the total
dospectralgrid,usingthefirst-ordergeneralizedharmonicfor-
angularmomentumofthesystem,thatmethodisdescribedin
mulation[26],whilethehydrodynamicalequationsaresolved
Foucartetal.[21]:wesolveforthemetricusingSPELLS,then
onaseparatefinitedifferencegridcalledthe“fluidgrid”.The
determinethenewvalueoftheenthalpyh, aswellastheor-
bitalangularvelocityΩrot(chosensothatthebinaryisinqua- hydrodynamicequationsarewritteninconservativeform
sicircularorbit),thepositionoftheBHintheequatorialplane
(so thatthe totallinearmomentumPADM vanishes), andthe ∂tU+∇F(U)=S(U). (7)
freeparameterΩz (todrivethespinoftheBHtoitsdesired
BH
value). Finally,wesolveforthevelocityfieldthroughtheel- To compute the flux F on the faces of each finite difference
lipticequationimposinganirrotationalconfiguration,andgo cell,weusethethird-ordershockcapturingPPMreconstruc-
backtothefirststep. tionmethod[27].Moredetailsonthenumericalmethodsused
InordertoconstructinitialdataforBHswithaspinthatis canbefoundinDuezetal.[11]. However,sincethepublica-
notalignedwiththeorbitalangularmomentumofthebinary, tionof[11],severalimportantimprovementshavebeenmade
a few changes are necessary. First, we do not assume that tothecode,whicharedescribedinthefollowingsubsections.
4
1. Dynamicregridding regionfarawayfromit. Theexactformofthemapis
r, r <r
To accurately evolve a BHNS binary while determining A
its gravitational wave emission, simulations have to resolve r′ =f(r) f(rA)+rA, rA <r <rB
−
events occurring at very different scales. When the neutron λ(r rB)+f(rB) f(rA)+rA, r >rB
star is disrupted and a disk forms, we expect shocks in the − − (8)
disk,andsteepdensityandtemperaturegradientsclosetothe 
BH.Butthedisruptionofthestaralsoleadstothecreationof f(r)=r(ar3+br2+cr+d), (9)
alongtidaltailwhichcaninitiallycontainupto5 10%of
−
theinitialmassofthestarandexpandhundredsofkilometers wherethecoefficients(a,b,c,d)arechosensothatthe
awayfromthecenteroftheBH[13]. Clearly,boththesharp, mapisC2atr andr andλischosensothatthegrid
A B
small-scalefeaturesaroundtheblackholeandthelarge-scale isofthedesiredsize. Theradiir andr arefixedfor
A B
tidaltailshouldbeproperlyresolvedifwewanttofollowthe thewholeevolutionandwilldeterminerespectivelythe
formationofanaccretiondisk. Furthermore,toextractgrav- minimum resolution in the neighborhoodof the black
itational waves accurately, the evolution of the gravitational holeandthecharacteristiclengthscaleofthetransition
fields should extend to the wave zone, in regions where no betweenthehighandlowresolutionregions.
matteratallispresent.
Becauseweusedifferentgridstoevolvethemetricandthe
fluid variables, the spectral grid on which we solve the gen- 2. Excision
eralized harmonic equations can be extended into the wave
zonewhilethefluidgridusedfortherelativistichydrodynam- We find that our hydrodynamicscode is more stable near
icalequationsonlycoverstheregionwherematterispresent. the excision zone if we switch from PPM to the more dif-
Ourearliersimulations[11]werelimitedtononspinningblack fusiveMC reconstruction[29] in the vicinity of the excision
holes. In that case, most of the matter was rapidly accreted zone. Therefore, we replace the face values determined by
ontotheholeandthetidaltailsandaccretiondisksweresmall PPMreconstruction,u PPM withaweightedaverage:
R,L
enoughthat manually expandingthe fluid grid at a few cho-
sentimestepsallowedusto resolvetheevolutionofthefluid u =fu MC +(1 f)u PPM , (10)
R,L R,L R,L
−
atareasonablecomputationalcost. Forspinningblackholes,
this is no longer the case: cost-efficient evolutions require a wheref = 1 forr < r 2r andf = e−[(r−r1)/r1]2 for
1 ex
gridwithpointsconcentratedinthehigh-densityregionsclose r >r . ∼
1
to the BH, and coarser resolution in the tail. Furthermore, The MC face-value computation must be altered when its
astheevolutionofthefluidishighlydynamical,interrupting regularstencilwouldextendintothe excisedregion,anddo-
thesimulationwheneverthefinitedifferencegridisnolonger ingthisproperlyturnsouttobeimportantforstability. Con-
adaptedto the fluid configurationbecomesimpractical. One sideraone-dimensionalproblemwithgridpointsx =n∆x.
n
solutionwouldbetouseanadaptivemeshrefinementscheme, Then the face-value reconstruction of the function u from
i
similar to the codes used by Yamamoto et al. [28] and Eti- theleftu = u andfromtherightu =
L,i−1/2 i−1/2−ǫ R,i−1/2
enneetal.[12]. Inourcode,wechooseinsteadtouseamap u mustbeadjustedasfollows.
i−1/2+ǫ
between the fluid grid and the pseudospectral grid that con- (i)Ifxi isintheexcisionzone,butxi−1 isoutside,
centratesgridpointsintheregionclosetotheblackholeand setuL,i+1/2 =ui−1anduR,i+1/2 =ui−1
automaticallyfollowstheevolutionofthefluid. (ii)Ifx isintheexcisionzone,butx isoutside,
i i+1
Todothis,wemeasuretheoutflowofmatteracrosssurfaces setu =u andu =u
L,i−1/2 i+1 R,i−1/2 i+1
closetobutinsideofthefluidgridboundaries.Assoonasthe (iii)Ifxi isoutsidetheexcisionzone,butxi−1isinside,
outflowacrossoneofthesesurfacescrossesagiventhreshold setu =u
L,i−1/2 R,i−1/2
(chosensothattheamountofmatterleavingthegridoverthe (iv)Ifx isoutsidetheexcisionzone,butx isinside,
i i+1
wholesimulationisnegligiblecomparedto thefinalmassof setu =u
R,i+1/2 L,i+1/2
theaccretiondisk),thegridexpands.Theoppositeisdoneon Wealsoobservedthatthestabilityofourcodeclosetothe
fixedsurfacesfartherawayfromthegridboundariestoforce excisionsurfacewasstronglyaffectedbythedetailsofthein-
thegridtocontractwheneverthefluidismovingawayfroma terpolationmethodchosenforthecommunicationfromfinite
boundary.Themapitselfisthecombinationof: differencetospectralgridinthatregion.Previously,theinter-
polationstencilwasshiftedawayfromtheexcisionboundary
(i) Atranslationofthecenterofthegrid,tofollowthegen- untiltheentirestencilwasoutoftheexcisionzone.Thiscould
eralmotionofthefluid leadtounstableevolutionsorlargeinterpolationerrorsifthe
excisionregionhappenedtobelocatedclosetotheboundary
(ii) Alinearscalingofeachcoordinateaxis,toadapttoits betweentwosubdomainsofthefinitedifferencegrid,andac-
expansionsandcontractions ceptable stencils could only be found far from the point we
wereinterpolatingto—orcouldnotbefoundatall(tolimit
(iii) Aradialmapsmoothlytransitioningfromahighresolu- MPIcommunications,thestencilhastobeentirelycontained
tionregionclosetotheblackholetoalowerresolution in one subdomain). Currently, we limit the displacement of
5
thestenciltoamaximumof3gridpointseparations. Ifthere 2
is no good stencil within that distance, we decrease the or-
deroftheinterpolation,andkeepdoingsountilanacceptable
stencilisfound. 1.5
Another interpolation method would be to forbid any dis-
placementofthestencil,andimmediatelydroptolowerorder 2m)
c
aBsotshooanlgoasritphamrtsoafretheequsatellnycriolbliuesst,wbuitthwinhethneteesxtceidsioonnaznoance-. 1810 g/ 1
tualBHNSmergerthefirstappearedtoperformbetteratmain- Σ ( t0
t+0.5ms (full)
tainingasmoothsolutionandlowconstraintviolationsonthe 0.5 t0+0.5ms (hydro)
0
excisionsurface. Accordingly,wechoseitasourstandardin- t + 1.1ms (full)
0
terpolationmethodanduseditforallsimulationspresentedin t + 1.1ms (hydro)
0
thispaper. 0
25 50 75 100
R (km)
3. Coordinateevolution FIG.2: Averagesurfacedensityofthefluidatagivendistancefrom
thecenterofmassofthesystem,forthesimulations.5i20described
In the generalized harmonic formulation, the evolution of in Sec. IV. The surface density is plotted at time t0 = tmerger +
the inertial coordinates x is given by the inhomogeneous 10.3ms,whenwebegintoevolvethesystemusingthefixed-metric
a
approximation, as well as 0.5ms and 1.1ms later. We see that the
waveequation
profileisvery similarfor both evolution methods, even though the
b x =H , (11) diskitselfisnotinastationaryconfiguration.
b a a
∇ ∇
where isthecovariantderivativealongx . Theevolution
b b
∇
of the function Ha(xb) can be freely specified, but its value a stronginfluenceon thebehaviorof thesystem. Innumeri-
ontheinitialslicet = 0isdeterminedbytheinitialdata(the calsimulations,wecanthusextractsomeinformationonthe
lapseandshiftatt = 0fixtheinitialevolutionofthegauge). long-termbehaviorofthefinalblackhole-accretiondisksys-
Whilethebinaryspiralsin,wechoose∂tHa(t,x˜i)=0inthe tembyneglectingtheevolutionofthemetricandonlyevolv-
coordinate frame x˜i comoving with the system. In our pre- ing the fluid (c.f. [30]). Using this approximation, our code
viouspaper[11], wechangedthegaugeevolutionduringthe runsabout4timesfaster.
mergerphasebydampingH exponentiallyinthecomoving
a To test the limitations of this method, we evolve the cou-
frame:
pled system 1ms past the time at which we begin the ap-
∼
H (t,x˜ )=e−(t−td)/τH (t ,x˜ ), (12) proximate,fluid-onlyevolution.Welookfordifferencesinthe
a i a d i
accretion rate or the characteristics of the disk (temperature,
wheret is the disruptiontime—the time atwhich we begin density,inclination)betweenthetwomethodsused. Aslong
d
damping—andτ isadampingtimescale oforder10M (M aswewaitforthepropertiesoftheblackholetosettle down
beingthetotalmassofthesystem). Furtherexperimentation before switching to the approximate evolution scheme, the
hasshownthatitisbetternottochangeH neartheexcision two methods show extremely good agreement — except for
a
zone.Inourcurrentsimulations,weset thehighestspinconfiguration,whichleadstoamassivedisk
that cannot be evolved accurately by fixing the background
H (t,x˜ )
a i = Q(r˜)+[1 Q(r˜)]e−(t−td)/τ (13) metric. InFig.2weshowtheevolutionofthedensityprofile
H (t ,x˜ ) −
a d i ofadiskusingboththefullGRevolutionandthefixed-metric
n o
Q(r˜) = e−(r˜/r˜ex)2+1 (14) approximation. Theevolutionoftheaccretiondiskismostly
unaffectedbythechangeofevolutionscheme.
during the mergerphase, where r˜is the distance to the cen-
ter of the black hole in the comoving frame, and r˜ is the
ex
excisionradius.
III. DIAGNOSTICS
4. Evolutionswithfixedmetric Anindispensabletestofnumericalaccuracyisconvergence
withgridresolution. We haveevolvedmostofthecasesdis-
During the merger of a BHNS binary, both the spacetime cussedbelowatthreeresolutions. We callthese Res1, Res2,
metricandthefluidconfigurationarehighlydynamical. Ein- andRes3;theycorrespondto1003,1203,and1403gridpoints,
stein’sequationshavetobesolvedtogetherwiththeconserva- respectively,onthefluidgridandto693,823,and953 collo-
tivehydrodynamicsequations,andtheevolutionofthatcou- cationpointsonthepseudospectralgrid.
pledsystemiscomputationallyintensive.However,afewmil- The black hole is described by its irreducible mass
lisecondsafter merger,the BH remnantsettles into a quasis- Mirr, its spin SBH, and its Christodoulou mass MBH =
tationarystate asit accretesslowly fromthesurroundingac- M2 +S 2/(4M2 ). The spin S is computed using
irr BH irr BH
cretiondisk. Then,theevolutionofthemetricdoesnothave theapproximateKillingvectormethod[22].
p
6
To monitor the nuclear matter, we first measure the bary- angularmomentumJ aregivenby
disk
onicmassM onthegridasafunctionoftime. Initially,this
b
willbe the baryonicmassofthe neutronstar. Afterthe tidal Lµν = xµTν0 xνTµ0 d3x (17)
destructionofthestar,itwillbethesumoftheaccretiondisk −
Z
andthetidaltailmasses. Atlatetimes(morethan10msafter (cid:0) (cid:1)
Sµ = Tµ0d3x (18)
merger),itwillbethebaryonicmassofthe disk. Theaccre-
tiontimescaleτdisk isMb/(dMb/dt). Zǫ LµνSσ
J = µνσi (19)
disk,i
We also analyze the heating in the disk. This is done 2√ SαSα
−
through both an entropy and a temperature variable. When J .J
β(r) = arccos BH disk(r) (20)
evolvingwith microphysicalequationsof state, the tempera- J J
tureandentropyareprovideddirectly. Forthisstudy,weuse (cid:20)| BH|| disk(r)|(cid:21)
J J
a Γ-law equation of state, so we need a way to estimate the γ(r) = arccos BH× disk(r) .eˆ . (21)
J J y
entropyandtemperature.Theentropymeasureswedefineas (cid:20)| BH× disk(r)| (cid:21)
s = log(κ/κ ), where κ is the polytropic constant obtained
i These parameters determine the inclination and the preces-
fromtherelationbetweenthepressureandthebaryondensity
sion ofthe disk with respectto the spinof the blackhole: if
(P = κρΓ), and κ is its initial, cold value. To estimate the
0 i J =J eˆ ,thentheorbitalangularmomentumofthedisk
physical temperature, T, we assume that the thermal contri- BH BH z
atradiusriswrittenas
butionto the specific internalenergyǫ = ǫ ǫ(T = 0)is
th
−
givenbyasumofidealgasandblackbodycomponents: J (r)=J (r)(sinβcosγeˆ +sinβsinγeˆ +cosβeˆ ).
disk disk x y z
(22)
Another usefulpropertyof the disk is its scale height, H.
3kT aT4
ǫ = +f , (15) Foradiskwithexponentiallydecreasingdensity,H isdefined
th 2mn ρ by ρ = ρce−z/H. Here, however, the vertical profile of the
diskissignificantlymorecomplex,andvariousdefinitionsof
H couldbeconsidered. Weusethespreadofthedensitydis-
where mn is the nucleon mass, and the factor f reflects the tributionρ(θ,φ)onasphereofconstantradiusrwithitspolar
numberof relativistic particles, and is itself a functionof T. axisalongJ (r)anddefine
disk
(See [8, 12], who also make this assumption.) For the most
part, we will reportdensity-averagedvaluesofs andT. For ρ(r)[θ(r)]2dS
example,thedensity-averagedentropyis H(r)=rtan−1 µ . (23)
s R ρ(r)dS !
R
The parameter µ is somewhat arbitrary. For an exponential
ρ(r)s(r)dV
hsi= R ρ(r)dV . (16) pfororfiθle<µH∼/r0a.5n,dwρh=ile0footrhaercwoinsset)anwtedheanvseityµpro3fi.leT(hρe=disρk0s
∼
observed in our simulations are somewhat in between these
R
twoextremes. Accordingly,wemaketheapproximatechoice
µ=1.
To launcha GRB, a baryon-cleanregionabovethedisk is
Tomeasuretheaccuracyofoursimulations,wemonitorthe
probablyneeded.Thisdoesnotmeanthatawidercleanregion
ADM Hamiltonianand momentumconstraints, and the gen-
isalwaysbetter,sinceathickdiskcanhelpcollimatetheout-
eralized harmonic constraints [26]. At our middle res-
flowingjet—butwewanttodeterminewhethersucharegion kCk
olution, peaks below 1% for all cases and is less than
existsornot. Toestimatethebaryon-pooropeningaboveour kCk
0.1%duringmostoftheinspiral. WealsomonitortheADM
disks,wedefinetheopeningangleθ . Thisanglespecifies
clean mass M and angular momentum J . An important
thewidestconeorientedalongtheBHspininwhichthecon- ADM ADM
check of our simulations is that the changesin these quanti-
dition ρ ρ is everywheresatisfied: if θ (r,φ) is the
≤ cut clean ties match the flux of energy and angular momentum in the
openinganglewithinwhichwehaveρ ρ atradiusrand
≤ cut outgoing gravitational radiation, which we reconstruct from
azimuthalcoordinateφ,thenθ =min [θ (r,φ)].In
clean r,φ clean theNewman-Penrosescalarψ asinRef.[32].
thesesimulations,thenumericalmethodrequiresatmospheric 4
corrections to be applied starting at ρ = 6 108g/cm3, so
×
thatitisimpossibletoreliablypredictthebehaviorofmatter
IV. CASES
belowthatthreshold.Therefore,wesetρ =3 109g/cm3.
cut
×
Forprecessingbinarieswealsocomputethetiltβandtwist Inordertoassesstheinfluenceoftheblackholespinonthe
γ ofthedisk,asdefinedbyFragile&Anninos[31]. Ifxµ are disruptionandmergerofBHNSbinaries,westudyconfigura-
the inertial coordinates, Tµν the stress-energy tensor, ǫ tionsforwhichallotherphysicalparametersareheldconstant.
µνσi
the Levi-Cevitatensor(with itslastindexlimitedto nonzero The mass of the black hole is M = 3M , where M
BH NS NS
values),J theangularmomentumoftheBHandeˆ anar- is the ADM mass of an isolated neutron star with the same
BH y
bitraryunitvectororthogonaltoJ , thenβ, γ andthedisk baryon mass as the star under consideration, and the initial
BH
7
coordinateseparationisd=7.5M,withM =MBH+MNS. Case aBH/MBH φBH ΩinitM Eb/MADM JADM/MADM2 tmerger
Forthenuclearequationofstate,weusethepolytrope
s0 0 - 4.16e-2 9.5e-3 0.66 7.5ms
s.5i0 0.5 0 4.11e-2 1.01e-2 0.91 11.4ms
P =(Γ 1)ρǫ=κρΓ+Tρ (24)
− s.9i0 0.9 0 4.13e-2 9.6e-3 1.13 15.0ms
s.5i20 0.5 20 4.09e-2 1.01e-2 0.90 10.5ms
where T is a fluid variable related to, but not equal to, the s.5i40 0.5 40 4.10e-2 1.01e-2 0.87 9.9ms
physical temperature. We set Γ = 2 and choose κ so that s.5i60 0.5 60 4.11e-2 9.9e-3 0.82 9.0ms
the compactionof the star is C = MNS/RNS = 0.144. For s.5i80 0.5 80 4.13e-2 9.6e-3 0.76 7.7ms
polytropicequationsofstate,thetotalmassofthesystemdoes
nothavetobefixed: resultscaneasilyberescaledbyM (see TABLEI: Descriptionofthecasesevolved. a /M istheini-
BH BH
e.g. Sec. II-FofFoucartetal.[21]). However,wheneverwe tialdimensionlessspinoftheBH,φ isitsinclinationwithrespect
BH
choosetointerpretourresultsinphysicalunits(ms,km,M⊙), totheinitialorbitalangularmomentumandEbistheinitialbinding
wewillassumethatMNS =1.4M⊙ (M =5.6M⊙). Forthat energy.tmergerisdefinedasthetimebywhichhalfofthematterhas
choice,theneutronstarhasaradiusR = 14.6km,andthe beenaccretedbytheBH.Differencesintheinitialangularvelocity
NS
andbindingenergyarewithinthemarginoferroroftheinitialdata:
initialseparationisd=63km.
atthisseparationtheeccentricityreductionmethodcanrequirevari-
The different initial configurations and black hole spins
ations of Ω of order 1%, and modifies the binding energy by a
studied are summarized in Table I. We consider 3 dif- init
fewpercent.
ferent magnitudes of the dimensionless spin a /M =
BH BH
(0,0.5,0.9), all aligned with the orbital angularmomentum.
Then, we vary the inclination angle φ between the spin
BH Nearly identical systems have been studied both by Shi-
of the black hole and the initial angular velocity of the sys-
bata et al [9] and by Etienne et al [12]. The former found
tem, Ωrot. Considering that most BHNS binary systems
an insignificant disk after merger, while the latter found 4%
are expected to have φ 90◦ (Belczynski et al. [15]),
with about half of the BbiHnar≤ies at φ 40◦, we choose of the NS mass still outside the hole 300M (∼ 8ms) after
φ = (20◦,40◦,60◦,80◦). The oBrHien≤tation of the com- merger. BothgroupsfoundafinalBHspinofs = 0.56. We
BH findadiskmassof3.7%at300M aftermerger,smallerthan
ponent of the BH spin lying in the orbital plane could also
in [12], butcloser to this resultthan to thatin [9]. Our final
have measurable consequences. For example, Campanelliet
BHspinis0.56,inagreementwithbothpreviousstudies.
al.[33] showedthatthesuperkickconfigurationfoundinbi-
InFig.3,weshowtheevolutionofM and s fortheen-
naryblackholesystemsissensitivetothedirectionofthemis- b h i
tiremergerphaseforthethreeresolutions. Reassuringly,the
alignedcomponentoftheBHspin.ForBHNSbinaries,kicks
different resolutions give very similar results, with the two
arerelativelysmall,andwearemoreinterestedinthecharac-
higherresolutionsbeingclosesttogether. Thebaryonicmass
teristicsofthefinalblackhole-disksystem. Afterlookingat
is initially constant before accretion starts. Then, as the NS
differentorientationsfor φ = 80◦, we find that the influ-
BH is disrupted and the core of the star is swallowed, M drops
enceoftheorientationofthemisalignedcomponentoftheBH b
rapidly. Itnextlevelsoffwhiletheremainingmatterisinan
spinisnegligiblecomparedtotheinfluenceofφ . Forthis
BH accretiondiskandanexpandingtidaltail. Whenthetidaltail
firststudyofmisalignedspins,wewillthuslimitourselvesto
falls back onto the disk, there is a second phase of rapid ac-
configurationsforwhichtheinitialspinliesintheplanegen-
cretion, after which the accretion rate settles down to a low
erated by the initial orbital angular momentum and the line
value. At the end of the simulation, the accretiontime scale
connecting the two compact objects. As the different initial
is τ 55ms, implying that the total lifetime of the disk
configurationsdo notuse thesame backgroundmetric, there disk ∼
would be around 75ms. At late times, the deviation in M
isnoguaranteethattwobinarieswiththesameinitialcoordi- b
betweenresolutionsbecomessomewhatlarger,indicatingthat
nate separationcanbe directlycompared. A better compari-
ourerrorshaveaccumulatedtoabout0.1%oftheinitialmass.
sonbetweeninitialconfigurationsistheorbitalangularveloc-
Forthepurposesofthispaper,thisisadequate,sincethedisk
ityofthesystem. InTableI, weshowthatallconfigurations
onthesetimescalesisaffectedbymagneticandradiationpro-
have initial angular velocity within 1% of each other. This
cessesnotincludedinthesimulations. However,futurelong-
istheleveloferrorthatweexpectfromthequasiequilibrium
termdisksimulationswillrequirehigheraccuracy.
methodforbinariesatthisseparation.
As for the entropy, at the beginning of the merger it only
deviates from zero because of numerical heating during the
inspiral. As expected, this numericalheating is significantly
V. THENONSPINNINGCASE:ATESTOFOUR lowerathigherresolutions.Thepost-mergerheatingisnotnu-
ACCURACY merical,butaphysicalconsequenceofshocksinthediskand
thedisk-tailinterface.Aconfirmationthattheheatingisphys-
Asanexample,weconsiderthecases0,inwhichtheBHis icalisthatitisnearlythesameforallresolutions,anditsmag-
initiallynonspinning.Weevolvethiscaseateachofourthree nitude is much larger than the numericalheating. When the
resolutions.Afterashort(twoorbits)inspiral,theneutronstar disk settles, thereis no furthershockheating, so the entropy
is disrupted,andmostofthe matteris quicklyswallowedby levels off. This indicates that the heating due to numerical
the black hole. The remainder expands into a tidal tail and viscosityissmallcomparedtoshockheating. Unfortunately,
thenfallsbacktoformanaccretiondisk. this is notthe same as saying that the numericalviscosity is
8
1
ResR 1es 1 M,i 1
M/Mbb,0 0.1 RReessRR 23eess 23 /MDMAD0.995 RRReeesss 123
MA GW
0.99
0.01
10 1
M,i
1 D 0.95
> A
<s /JM 0.9
0.1 D
A
J 0.85
0.01 0.8
0 5 10 15 20 -5 0 5
(t-t )(ms) (t-t )(ms)
merger merger
FIG.3: BaryonicmassMbnormalizedbyitsinitialvalueMb,0and FIG.4: MADMandJADM(normalizedtotheirinitialvalues)com-
averageentropyhsiforthreeresolutions. pared tothe changes expected from the gravitational radiation flux
forthreeresolutions.
irrelevantaltogether. However,theclosenessofτ ateach
disk
resolutionindicatesthatthisviscosityisnotthemaindriving Case M /M hTi β(Σ ) r(Σ ) θ H(Σ )
disk NS disk max max clean r max
forceoftheaccretion. Theaveragetemperature T behaves
in a way similar to the entropy. Starting from lohwivalues, it s0 5.2% 3.0MeV 0◦ 50km 50◦ 0.20
s.5i0 15.5% 3.5MeV 0◦ 50km 35◦ 0.25
increases after the merger and stabilizes around 3MeV. All
s.9i0 38.9% 5.6MeV 0◦ 20km 8◦ 0.18
resolutionsshowthesame T growth,andallleveloffatthe
h i s.5i20 14.5% 3.6MeV 2◦ 50km 40◦ 0.20
samevalue. Afterlevelingoff,though,hTidisplays0.1MeV s.5i40 11.5% 3.8MeV 4◦ 50km 30◦ 0.22
oscillationsthatdonotconvergewell,anotherindicationthat s.5i60 8.0% 3.7MeV 7◦ 50km 40◦ 0.20
ouraccuracyissufficientforsomebutnotallpurposes. s.5i80 6.1% 3.6MeV 8◦ 50km 50◦ 0.25
In Fig. 4, we plot the ADM energyand orbital-axisangu-
lar momentum measured on a surface 75M from the center TABLEII: Propertiesof the accretion tori at late time. The mass
ofmass ofthe system. Also plottedis theevolutionof these ofthediskM (baryonmassoutsidetheexcisionregion),which
disk
quantities expected from the gravitational radiation through decreases continuously due to accretion onto the BH, is measured
this surface. Overall, the agreement is quite good, although at tmerger + 5ms. Even at late times, all quantities still show
there is some deviation in M a while after the merger. oscillationsof∼10%.
ADM
This seems to be associated with an increase in constraint
violations at the merger time. The relative constraint viola-
tions,asmeasuredby ,peakslightlybelow1%atthemid-
kCk
dle resolution. The correspondingvalues for the ADM con- duration of the inspiral varies with the spin of the BH, with
straints are 1 –2 %, before both constraints fall back to low the componentof the BH spin along the orbitalangularmo-
values. ThedeviationsinMADMhappenaroundthetimethis mentumdelayingthemerger.Themergertimetmerger,which
constraint-violatingpulsereachesther =75M surface. we define as the time at which 50% of the matter has been
We have checked several other quantities, including the accretedontotheBH,islistedforallcasesinTableI.
blackholemassandspinandthegravitationalwaveform.All
The resulting accretion disk is highly asymmetric, and
oftheseshowverygoodconvergence.
evolves in time. When the disk first forms, around 5ms af-
ter merger, it creates a torus of matter with its peak surface
density(baryondensityintegratedovertheheightofthedisk)
VI. RESULTS atr(Σ ) 30kmandatemperatureT 1 2MeV. Then,
max
∼ ∼ −
asmatteraccretesfromthetidaltailandshocksheatthefluid,
ThegeneralbehaviorofoursimulationsistypicalofBHNS thediskexpandsquickly. About10 20msaftermerger,the
−
binariesfor which the NS is disruptedoutside the innermost disk starts to settle into a stable, slowly accreting state. To
stable circular orbit of the BH. From an initial separation of comparethedifferentconfigurationsstudiedhere,welookat
60km, the compactobjects go through 2-3 orbits of inspiral the properties of this late-time stable configuration, listed in
drivenbytheemissionofgravitationalwaves. Whenthedis- Table II. In Table III, we givethe characteristicsofthe final
tance between them has been reduced to about 30 40km, blackhole,aswellasthekickvelocity,theenergycontentof
−
tidalforcescausetheneutronstartodisrupt.Mostofthemat- the emitted gravitationalwaves, and the peakamplitudeof a
terisrapidlyaccretedintotheblackhole,whiletherestisdi- dominant(2,2)modeofthe waves. [The(2,2)and(2,-2)are
videdbetweenalongtidaltail,expandingabout200kmaway the strongest modes, with nearly equal amplitude. See Sec-
from the black hole, and a developing accretion disk. The tionVIBonthehighermodes.]
9
Case M /M a /M v (km/s) E /M rMΨ2,2
BH BH BH kick GW 4 s=0.0
1
s0 0.97 0.56 53 0.98% 0.020 s=0.5
s=0.9
s.5i0 0.94 0.77 60 0.92% 0.012
s.9i0 0.89 0.93 52 0.95% 0.009
s.5i20 0.95 0.76 60 0.89% 0.012
s.5i40 0.96 0.74 61 0.91% 0.013 Mb,0
s.5i60 0.96 0.71 54 0.95% 0.014 M/b 0.1
s.5i80 0.97 0.66 67 0.95% 0.017
TABLE III: Properties of the post-merger black hole and gravita-
tionalwaves.
0.01
-5 0 5 10 15
(t-t ) (ms)
A. Effectsofspinmagnitude merger
FIG.5: Evolutionofthetotalbaryonicdensityoutsidetheholefor
To testtheeffectsofBH spinmagnitude,we compareour
threedifferentalignedBHspins.
results for s0, s.5i0, and s.9i0. A comparison of this type
hasalreadybeenperformedbyEtienneetal[12]. Ourcases
aredifferent,though: unlikethem,wedonotconsideranan-
spin is increased, as shown in Fig. 5. This is in qualita-
tialignedcase,butwedopushtheBHspintoaslightlyhigher
tive agreementwith Etienne etal. About400M(11ms)after
levelinourruns.9i0.
mergertheyfindM /M 20%forana /M = 0.75
b b,i BH BH
Runs.9i0presentedspecialnumericalchallenges.InSPEC, system, while for our a ∼/M = 0.9 system, we find
BH BH
thesingularityandinnerhorizonoftheBHhavetobeexcised
M /M 35%atasimilartime.
b b,i
fromthe numericalgrid while the apparenthorizonmustre- ∼
main outside the excision surface. But for nearly extremal
blackholestheregionbetweentheinnerhorizonandtheap-
parenthorizonbecomesverynarrow. Toperformexcisionin B. Effectsofspinorientation
suchcases,theexcisionboundarymustnearlyconformtothe
apparenthorizon.Wedothisbyintroducingacoordinatemap Most BHNS binaries are expectedto have at least a mod-
intheinitialdatasothatthehorizonisinitiallysphericalonthe erate misalignmentbetweenthe spin of the BH andthe total
pseudospectralgrid. We thenuseourdualframecoordinate- angular momentum [15], and this should affect all stages of
controlmethod[34]tofixthelocationofthehorizonthrough- thebinaryevolution.
out the whole simulation. For lower spins this is not neces- Duringinspiral,theorbitalangularmomentumandtheBH
sary,andweonlybegintocontrolthehorizonlocationatthe spin precess around the total angular momentum of the sys-
time of neutron star disruption. That modification excepted, tem. The evolutionof the coordinatecomponentsof the BH
cases.9i0wassimulatedinexactlythesamewayastheother spin for the s.5i80 case is shown in Fig. 6. Over the two
cases. The deviationbetween the results at resolutionsRes2 orbits of inspiral, the spin goes through about a quarter of
and Res3 is somewhatlarger than in the other cases (though a precession period. The qualitative evolution of the spin is
still quitesmall). To be moreexact, the relative deviationin welldescribedbypost-Newtoniancorrections(seee.g.[37]),
the disk massbetweenLev2and Lev3is about9%( 3%of even though our simulation uses a different gauge choice.
∼
theNSmass) fors.9i0whileitwasabout5%fors0, andthe As for aligned spin, the infallvelocity variesbetween cases:
deviationinmergertimeisabout4%fors.9i0butonly0.3% the larger the componentof the spin aligned with the angu-
fors0. Thedifferenceindicatesthathighresolutionisneeded lar momentum,the slower the inspiral. Nottoo surprisingly,
whenstudyingsuchextremecases. we find a monotonic decrease of the merger time with in-
Wefindthatsystemswithhighera /M spiralinmore creasingmisalignmentangleφ ,witht (φ ,a )
BH BH BH merger BH BH
slowly: from the same initial separation, s0, s.5i0, and s.9i0 t (0,0)forφ 90◦. →
merger BH
→
takeroughly2,3,and3.7orbits,respectively,beforeNSdis- Thedisruptionofthestarandformationofadisk,shownin
ruptionbegins. Thiseffectexistsinthepost-Newtoniantreat- Fig. 7, proceed somewhat differently from what is observed
ment [35] and it has already been seen both in binary black for nonprecessing binaries. As before, the disruption of the
hole (e.g. [36]) and BHNS [12] simulations. Because of the starisaccompaniedbytheformationofalongtidaltail. But
prolongedinspiralforthehigh-spincases, moreangularmo- because of the inclination of the BH spin, the orbital plane
mentumisradiated: 0.11M2fors0vs0.14M2fors.9i0. Ad- ofthefluidcontinuestoprecessafterdisruption. Becausethe
ditionally,astheBHbecomesnearlyextremal,increasingthe precession rate varies with the distance to the hole, the tail
spinbecomesmoreandmoredifficult. Thisisreflectedinthe andthediskdonotremaininthesameplane. Whilefornon-
finalspinoftheBH:whileaBH/MBHincreasesfrom0to0.56 precessingbinariesmatter fromthe tail falls back within the
fors0,itonlyincreasesfrom0.9to0.93fors.9i0. orbital plane of the disk, here matter is added to the disk at
We also confirm that the post-merger accretion disk mass an angle varyingin time. This significantlymodifiesthe na-
increases significantly as the magnitude of the aligned BH ture of the tail-disk interactions. At small inclination angles
10
0.5 tained for aligned spins; here too, a large component of the
spinalongtheorbitalangularmomentumworksagainstlarge
0.4 Ψ amplitudes.Additionally,thecontributionofsubdominant
4
modescanbecomesignificantatlargeinclinations. InFig.9,
PN1
0.3 PSN (2NR) we show the ratio of the amplitude of the scalar Ψ4(l,m) to
x
Si Sy (NR) theamplitudeofthe(2,2)modeforvarious(l,m)modesand
S (NR) for φ = 20◦,60◦. The modes most strongly affected by
0.2 z BH
the precession of the binary are the (2,1) and (3,2) modes,
the first reaching half the amplitude of the dominant mode
0.1
around merger for the s.5i60 simulation. Analytical predic-
tions for the effect of a precessing trajectory on the modal
0
0 2 4 6 decompositionofthegravitationalwavesignalhavebeende-
t (ms)
rivedbyArunetal.[39]. Wefindqualitativeagreementwith
theirresultsifweassumethatthecompactobjectsfollowthe
FIG.6: Comparison oftheevolutionof theCartesiancomponents
of the spin for an initial inclination of 80◦ in our simulation (NR) trajectoriesobtainedfromournumericalsimulations. Inpar-
andforthefirst-andsecond-orderPost-Newtonianexpansions(resp. ticular,wenotethatforprecessingbinaries,thefrequencyof
1PNand2PN).ThePNvaluesareobtainedbyintegratingtheevo- the(2,1)modeiscloserto2Ωrot thantoΩrot. The(2,1)and
lutionequationsforthespingiven in[37], usingthetrajectoryand (2,2)modeshavesimilarfrequencies,sothattheratiooftheir
currentspinofthenumericalsimulation. amplitude computed using the scalar Ψ is close to the re-
4
sult one would obtain by using the gravitational strain h in-
stead. This will not be true for the (3,3) mode, which has a
(φ 20 40◦), we have a direct collision between the de- frequency Ω(3,3) 3Ωrot: since Ψ4 = ∂2h/∂t2, we have
velo∼ping d−isk and the tidal tail, while for larger inclinations h(3,3)/h(2,2) (4∼/9)Ψ4(3,3)/Ψ4(2,2).
∼
thedisk isinitially formedof layersofhigh-densitymaterial In Fig. 10, we plot the gravitational strain h as observed
atdifferentangleswithrespecttotheblackholespin. from a distance of 100Mpc for the simulations s.5i0 and
s.5i80.Thethreewaveformscorrespondtoobservationpoints
Themassofthedisk,plottedinFig.8,decreasesasthein-
whoselinesofsightareinclinedby0◦,30◦ and60◦ withre-
clinationofthebinaryincreases. Thetransitionbetweenlow
spect to the initial orbital angular momentumof the system.
and high mass disks is continuous, but more rapid at large
inclinations: for φ < 40◦, 10 15% of the initial mass Overtheshortinspiralconsideredhere,theeffectsofpreces-
BH
− sionarerelativelysmall. Themaindifferencevisibleinthese
of the star ( 0.15 0.2M⊙) remains either in the tail or
∼ − waveforms is the slower inspiral experienced by the binary
in the disk 5ms after merger. This is roughly similar to the
with aligned spin. We can also note that in the misaligned
diskformedforφ = 0. Athigherinclinations,thesizeof
BH
configuration the star does not disrupt as strongly as in the
thediskdropssharply,toabout5%oftheinitialmassofthe
aligned case, causing the cutoff of the wave emission to oc-
star. Weexpectthediskmasstobeevenlowerforantialigned
spins (φ > 90◦), though such configurations appear less curatalatertime. Asaconsequence,thecutofffrequencyof
BH
thewaveislargerformisalignedspinsthanforalignedspins.
likelytobefoundinastrophysicalsystems. Thesechangesin
Thisexplainswhythe amplitudeofthegravitationalstrainh
thediskmasswiththeorientationoftheBHspinshowsome
is comparable for both configurations, while the misaligned
similaritieswiththeresultsofRantsiouetal.[16],obtainedin
case showed a significantly larger amplitude when the wave
thesmallmass-ratiolimit(q = 1/10)byusingastaticback-
was measured using the scalar Ψ (a similar effect occursif
groundmetric. They foundthat for a disk to be formed, the 4
conditionφ < 60◦ hastobesatisfied. Asoursimulations both spins are aligned but of differentmagnitudes). Finally,
BH
theprecessionoftheorbitcausesthegravitationalwaveemis-
useamassratioq = 1/3,whichismorefavorabletothefor-
sionofthemisalignedconfigurationtopeakatanon-zeroin-
mationofadisk,itisnottoosurprisingthatevenhighinclina-
clinationwithrespecttotheinitialorbitalangularmomentum.
tions leave us with a significantdisk; however,the influence
ofinclinationremainsimportantforφ > 40◦. Thesefac- Here,atthetimeofmergerthewavemeasuredataninclina-
BH tionof30◦isslightlylargerthanat0◦. Fortheseeffectstobe
torsare particularlyusefulwhenconsideringthepotentialof
morevisible, andin particularfora fullprecessionperiodto
the final remnant to be a short gamma-ray burst progenitor.
beobservable,longersimulationsarerequired( 10orbits).
Since the influence of a misaligned BH spin is only felt for
∼
φ > 40◦, the majority of BHNS binaries can form disks
BH
about as massive as is predicted by simulations that do not
takeintoaccounttheinclinationoftheBHspin. C. Post-mergeraccretiondisks
Thegravitationalwave signalis also significantlyaffected
bythevalueofφ . WeexpandtheNewman-Penrosescalar The evolution of the accretion disk over time scales com-
BH
Ψ extracted at R = 75M using the spin-weighted spheri- parabletoitsexpectedlifetimeislikelytobesignificantlyin-
4
cal harmonics −2Ylm and choosing the polar axis along the fluencedbyphysicaleffectsthatarenottakenintoaccountin
initialorbitalangularmomentumofthebinary.Thepeakam- oursimulations,mostlythemagneticeffectsandtheimpactof
plitude of the dominant (2,2) mode of Ψ will increase for neutrinocooling. Wedonotexpecttheresultsofoursimula-
4
large inclinations, as could be expected from the results ob- tionstoaccuratelyrepresentthedetailsofthelate-timeevolu-