Fallback accretion in the aftermath of a compact binary merger

0.35 MB·English

by  S. Rosswog

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed5February2008 (MNLATEXstylefilev2.2)
Fallback accretion in the aftermath of a compact binary merger
Stephan Rosswog,
SchoolofEngineeringandScience,InternationalUniversityBremen∗,CampusRing1,Germany;
∗JacobsUniversityBremenasofspring2007
7
0
0
2
ABSTRACT
n
Recentobservationsoflongandshortgamma-rayburstshaverevealedapuzzlingX-rayac-
a
tivitythatinsomecasescontinuesforhoursaftertheburst.Itisdifficulttoreconcilesuchtime
J
scaleswiththeviscoustimescalesthatanaccretiondiskcanplausiblyprovide.HereIdiscuss
6
theaccretionactivityexpectedfromthematerialthatislaunchedintoeccentric,butgravita-
1
tionallyboundorbitsduringacompactbinarymergercoalescence.Fromasimpleanalytical

2 model the time scales and accretion luminosities that result from fallback in the aftermath
v ofa compactbinarymergerarederived.Fortheconsideredmassrange,doubleneutronstar
0 binaries are relatively homogeneous in their fallback luminosities. Neutron star black hole
4 systems show a larger spread in their fallback behaviour. While the model is too simple to
4 make predictionsaboutthe detailed time structure of the fallback, it makes reasonable pre-
1 dictionsaboutthegrosspropertiesofthefallback.Aboutonehourafterthecoalescencethe
1 fallback accretion luminosity can still be as large as ∼ 1045 erg/s, a fraction of which will
6
be transformed into X-rays. Large-scale amplitude variations in the X-ray luminosities can
0
plausiblybe caused by gravitationalfragmentation,which for the high-eccentricityfallback
/
h shouldoccurmoreeasilythaninanaccretiondisk.
p
- 1 INTRODUCTION insteadbythedistributionofeccentricitieswithinthefallbackma-
o terialwhichisaresultofthetorquesthatactedduringthemerger
r RecentobservationsbySwiftandHeteIIhaveconfirmedthelong-
t process.Thisnaturallyleadstotimescalesthatareordersofmag-
s heldsuspicionthatshortGRBshaveadifferentcentralenginethan nitudeslargerthanusualviscousdisktimescales.
a longbursts:theyoccuratsystematicallylowerredshifts(Foxetal.
v: 2005)1,ingalaxieswith(Hjorthetal.2005)andwithoutstarfor-
i mation (Bergeretal. 2005; Barthelmyetal. 2005), the emitted
X
gamma-rayenergyissubstantiallylower comparedtolongbursts
2 ASIMPLEANALYTICALMODEL
r (Soderbergetal. 2006) and no supernova explosion seems to ac-
a
companytheburst(Bloometal.2006;Hjorthetal.2005;Foxetal. The question of fallback time scales is addressed by means of a
2005).Theseobservationsarenaturallyexplainedbythemostpop- simple analytical model whose initial conditions are taken from
ularcentralengineforshortbursts,thecoalescenceofacompactbi- a set of recent, three-dimensional smooth particle hydrodynam-
narysystem(Blinnikovetal.1984;Eichleretal.1989;Paczyn´ski ics(SPH)calculationsofthemergerofbothdoubleneutronstars
1991), but alternative models do exist (e.g. MacFadyenetal. (DNS)andneutronstarblackhole(NSBH)systems.Thedetailsof
2005; Levanetal. 2006). The observations, however, have also theinputphysicsandthenumericaltechniquesaredescribedelse-
revealed long-lasting X-ray activity after the main gamma-ray where (Rosswog&Davies 2002; Rosswog&Liebendo¨rfer 2003;
emission phase for about half of the bursts (Burrowsetal. 2005; Rosswogetal. 2003; Price&Rosswog 2006; Rosswog&Price
Nouseketal. 2006; O’Brienetal. 2006). Several possible expla- 2006; Rosswog 2006). Here we only summarise the main physi-
nations have been suggested (e.g. Kingetal. 2005; Zhangetal. cal parameters of thecalculations, seeTab. 1. A few tensof mil-
2006), among them is the fragmentation of the outer parts of an lisecondsaftersuchabinaryencountertheremnantconsistsofthe
accretiondiskduegravitationalinstabilities(Pernaetal.2006). following parts: a central object (either a neutron star-like object
Thislate-timeX-rayflaringisconsideredachallengetomostmod- orablackhole),adisk,somematerialflungintoeccentricorbits,
elsofshortGRBs.Inthisletter,IdiscusstheX-raysignaturefrom butgravitationallyboundtotheremnant(“fallbackmaterial”)and
fallback material in the aftermathof a compact binary merger. A somedynamicallyejecteddebrismaterial.Thedetailsofthegeom-
fractionofthedebrismaterialislaunchedintoeccentricorbitsand, etrydependonthetypeofsystem(DNSorNSBH)andonthemass
atsomelatertime,fallsbacktowardsthecentralremnant.Forthis ratio.Asanexample,Fig.1showsthematterdistribution22.3ms
material,thetimescalesarenotsetbyviscousdisktimescales,but afterthemergerofa1.1and1.6M⊙DNSsystem.
RestrictedbytheCourant-Friedrichs-Lewytimestepcriterion(typ-
icaltimestepsareoftheorder10−7s)theevolutionofthemerger
1 It has recently been suggested (Bergeretal. 2005) that at least 1/4 of remnant can never be calculated hydrodynamically up to the in-
theshortburstscouldlieatredshiftsz > 0.7andwouldthereforeimplie teresting time scales of hours. We therefore adopt the following
sunstantiallylargerisotropisedenergiesthanwasinferredforthefirstsetof simpleanalyticalmodel.TheSPH-particlesthatarelaunchedinto
detectedshortGRBswithafterglows. eccentric orbits, but arestill bound to the remnant, aretreated as
2 Rosswog
Table1.Runsthatareusedasaninputfortheanalyticalmodel.DNSrefers
todoubleneutronstarsystems,NSBHtoneutronstarblackholebinaries.
M1, M2 are the masses of the binary components (solar units), q is the
massratio,#part.isthetotalnumberofSPHparticles inthesimulation,
mfb,−2istheamountoffallbackmassinunitsof10−2M⊙andEfbthe
correspondingenergy.
run type M1,M2,q #part. mfb,−2 log(Efb[erg])
DA DNS 1.4,1.4,1.000 2007516 3.61 50.9
DB DNS 1.1,1.6,0.733 400904 3.21 51.0
DC DNS 1.1,1.1,1.000 211296 3.22 50.8
DD DNS 1.2,1.2,1.000 211296 2.84 50.7
DE DNS 1.3,1.3,1.000 211296 3.47 50.8
DF DNS 1.5,1.5,1.000 211296 3.36 50.9
DG DNS 1.6,1.6,1.000 211296 3.61 50.9
NA NSBH 1.4,4.0,0.350 600581 7.40 51.9
NB NSBH 1.4,6.0,0.233 208165 4.57 51.8
NC NSBH 1.4,14.0,0.100 2971627 4.05 51.7 Figure1.Contoursofthemassdensity(107−1014gcm−3)22.3msafter
ND NSBH 1.4,16.0,0.088 1005401 2.91 51.6 themergerofadoubleneutronstarsystemwith1.1and1.6M⊙.Inthis
NE NSBH 1.4,18.0,0.078 1497453 0.015 49.6 case,about0.03M⊙arelaunchedintoeccentricorbitsandwillfinallyfall
backtowardsthecentre.
testmassesinthegravitationalfieldoftheenclosedmass,M,i.e.it
isassumedthathydrodynamicpressureforcescanbeneglectedand
thedynamicsofeachparticlecanbeapproximatedasapoint-mass
two-bodyproblem.Foreachoftheseparticlestheangularmomen-
tum, J , and their energy, E , can be determined. Together with
i i
theparticlemassm M thesetwoquantitiesyieldtheorbital
i
≪
eccentricity
2E J2
e = 1+ i i . (1)
i G2m3M2
r i
Thesemi-majoraxis
GMm
a = i (2)
i − 2Ei
thenprovidesthemaximumandtheminimumdistanceofthepar-
ticleorbitfromtheorigin
rmax,i=ai(1+ei), rmin,i =ai(1 ei). (3)
−
ThetimeuntilaparticlereachesagivenradiusR canbecalcu-
dis
latedbyintegratingtheradialequation
Figure2.Asampleoffallbacktrajectoriesresultingfromadoubleneutron
dr 2 J2
dt =±smi Ei−V(r)− 2miir2 , (4) wstaarsm10e0rgkemrw. ith1.1and1.6M⊙.Thedissipationradius,Rdisinthiscase
(cid:18) (cid:19)
where V is the potential energy and the positive (negative) sign
referstothemotionawayfrom(towards)theorigin.Thetimethat where D = 4AC B2. The time that elapses from a particle’s
−
elapseswhileaparticlemovesfromaradiusr1 toaradiusr2 can presentradius,ri,untilitsenergyisdissipatedataradiusRdis is
thengivenbythecontributionswiththeappropriatesigns:
bewrittenas
τr1,r2 =±Zr1r2 √Ar2r+dBrr+C, (5) τi=(cid:26) IIrrii,,Rrmdiasx,i +Irmax,i,Rdis ffoorr ~v~vii··~r~rii><00 . (7)
Fortheradius,wherethefallbackenergyisdissipated,R ,
wheretheconstantsA = 2mEii, B = 2GM andC = −mJi22i have wetakeinthedouble neutronstarcasethediskradiusattheednisd
been introduced. The integral appearing in eq. (5) can be solved ofthenumericalsimulation.Thisisconservativeinthesensethat
analytically thediskwillshrink,seebelow,andthereforetheassumptionofa
radiusfixedat R yields shorter timescalesandlower accretion
√Ar2+Br+C B 2Ar+B r2 luminosities.Fordthiseblackholecases,IchooseR =10GM/c2,
I = + arcsin ,(6) dis
r1,r2 A 2A√ A √ D sojustoutsidetheinnermoststablecircularorbitofanon-rotating
(cid:20) − (cid:18) − (cid:19)(cid:21)r1
Fallbackaccretion 3
53 DNS 2 x 1.4
DNS 2 x 1.1
52 DNS 1.1 & 1.6
DNS 2 x 1.2
51
DNS 2 x 1.3
DNS 2 x 1.5
) 50
] DNS 2 x 1.6
s
/ NSBH 1.4 & 4
g 49
r NSBH 1.4 & 6
e
[ 48 NSBH 1.4 & 14

c NSBH 1.4 & 16
c
a47 NSBH 1.4 & 18
L
(
046
1
g
o ∝ 5/3
45 t
l
44
43
42
-3 -2 -1 0 1 2 3 4 5 6 7
log (t [s])
10
1 min 1 hour 1 day
Figure3.Fallbackaccretionluminosityforvariouscompactbinaries.Circlesrefertodoubleneutronstarsystems(DNS),trianglestoneutronstarblackhole
mergers(NSBH).Notethattherightmostpointisdeterminedbythemassresolutionofthesimulation.Forreferenceastraightlinewithslope5/3isshown.
(Schwarzschild-)blackholeatRISCO = 6GM/c2.Thedetailsof intoX-rays,LX =ǫXLacc.
thefallbacktimesandenergieschangeslightlywithR ,butnone The double neutron star cases form a rather homogeneous class
dis
of the conclusions of this paper depends on the exact numerical with respect to their fallback accretion, in all cases the fallback
valueofRdis. material is approximately 0.03 M⊙, see Tab. 1. After an initial,
Asanillustration,Ishow inFig.2asetoffallbacktrajecto- short-livedplateau, theluminosityfallsoff withtimeclosetothe
ries.Theinitialconditionsweretakenfromthedoubleneutronstar expected 5/3-power law (Rees 1988; Phinney 1989). It has to be
mergercalculationwith1.1and1.6M⊙thatisshowninFig.2.200 pointedoutthatthelastpointinthesecurvesisdeterminedbythe
randomly chosen trajectories out of the more than 5800 fallback numerical mass resolution in the hydrodynamics simulations and
particlesareplotted.Thereisabroaddistributionofeccentricities shouldthereforebeinterpretedwithsomecaution.Allotherpoints
and fallback times, while some particles return from their initial shouldbeafairrepresentationoftheoverallfallbackactivity.Typ-
position directly towards the central object, others follow highly ically, the X-ray luminosity about one hour after the coalescence
eccentricityorbitsouttodistancesofmany10000kmbeforethey isL ǫX 1044 erg/s.Fortheinvestigatedmassrange, the
X ∼ 0.1 ·
cometoahaltandfallbacktowardsthecentre. spreadintheluminositiesonehour afterthecoalescence isabout
(cid:0) (cid:1)
oneorderofmagnitude.
Theneutronstarblackholecasesshowalargerdiversity.Themass
inthefallback material of different mass ratiosvariesby about a
3 FALLBACKACCRETION factor of 500, see Tab. 1, an hour after the merger the accretion
luminositiesofthedifferentNSBHsystemsdifferbyabouttwoor-
Fig.3showstheaccretionluminosities,Lacc = dEfb/dt,derived dersofmagnitude. Alsotheinvolvedtimescaleschangestrongly
for the various DNS and NSBH systems. Here, E denotes the
differencebetweenthepotentialpluskineticenergyfabtthestartra- withthebinarymassratio.Forexample,the1.4and4M⊙ NSBH
casedoesnotproducemucheccentricfallbackmaterial.Accretion,
dius, r , and the potential energy at the dissipation radius, R .
i dis at least tothe resolvable level, isover in 0.2 s.Thisaccretion
Thecurveshavebeenobtainedbybinningtheenergiescontained ∼
periodmayproduceashortGRB,butprobablynotmuchX-rayac-
inthefallbackmaterial,E ,accordingtothecorresponding fall-
backtimes,τi,seeeq.(7).AfbfractionǫXofthisenergyischannelled tivity.The1.4and18M⊙NSBHsystemisattheotherextreme:its
4 Rosswog
peakluminosity islower bythreeordersordersof magnitudebut alpha-particles are photodissociated since this consumes about 7
extends(ataresolvablelevel)uptoaboutonehour.Themassra- MeVofthermalenergypernucleonandthusprovidesanefficient
tiosinbetweencouldpossiblyproducea(weak)GRBandextended cooling mechanism. They suggest that even neutron star-like
X-rayactivityuptoaboutonedayaftertheburst. objectsoflowmassesmayform,butconcludethattheirdisruption
would be too rapid toaccount for the observed, post-GRB X-ray
flaring.
Some the eccentric fallback material could undergo grav-
4 SUMMARY itational fragmentation into one or several bound objects.
Since there is a large spread in orbital eccentricities, it seems
A simple analytical model was presented and used to derive the
unlikely that a large fraction of the fallback material ag-
fallbacktimescalesandaccretionluminositiesintheaftermathof
glomerates into a single fragment. If fragmentation occurs,
a compact binary merger. The initial conditions are taken from a
it will do so only locally and for a small fraction of the ma-
largesetof3Dhydrodynamicalsimulations.Themainassumption
terial. A fragment of mass m will release a flare energy of
ofthismodelisthatthefallbackmaterialisballistic,i.e.thathydro- f
dynamicalpressureforcescanbeneglected.Thisisagoodapprox- E 1049erg ǫX Mc.o. mf 200km .Whensuch
f ∼ 0.1 3M⊙ 0.003M⊙ Rdis
imationforthelargestpart of thetimespent along theindividual afragmentofr(cid:0)adiu(cid:1)s(cid:16)Rf falls(cid:17)b(cid:16)acktoward(cid:17)s(cid:16)thecentr(cid:17)e,itbecomes
trajectories, therefore, the fallback time scales can be considered tidally disrupted at a distance Rtid = (Mc.o./mf)1/3 Rf, or,
valid estimates. The assumption may only break down when the if the remnant disk radius is larger than R , the fragm·ent will
tid
vicinityof remnant isapproached, butthiswillonly cause minor impactwithnearlyfree-fallvelocityonthatdisk.Inbothcases,it
correctionstothefallbacktimescaleandluminosityestimates. willtriggerviolentX-rayflaringactivity.
Fortherangeofmassratios(1.1to1.6M⊙)thatIhaveconsidered, In this picture the GRB is produced by the massive accretion
double neutron star systemsshow arather homogeneous fallback disk that can form early on in the merger. The GRB duration
behaviour.AtonehourafterthecoalescencetheirX-rayluminos- is determined by the viscous time scale on which this disk is
ityisstill 1044(ǫX/0.1)erg/s.Neutronstarblackholemergers consumed. The late-time X-ray activity, in contrast, is caused by
∼
provide a larger spread intheir fallback behaviour. Anhour after the fallback of a smaller amount, about a tenth of the disk mass,
thetidaldisruptionoftheneutronstar,theirrangeinfallbacklumi- of high-eccentricity material which is a natural outcome of the
nositiesisabouttwoordersofmagnitudes.Somemassratioscould merger. The fallback time scales are set by the distribution of
produceaGRB,butnotmuchX-rayactivity,othersmayproduce eccentricitiesinthismaterial,whichisaresultofthetorquesthat
X-rayflaringonlybutnoGRB,andyetothersystemscanproduce actedduringthemergerprocess.
both.
Since this simple approach does not account for self-gravity and
gas pressures, it tends to produce too smooth results. The shown
ACKNOWLEDGEMENTS
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