Tidal Disruptions of White Dwarfs from Ultra-Close Encounters with Intermediate Mass Spinning Black Holes

1.7 MB·English

by  Roland Haas

DRAFTVERSIONJANUARY24,2012
PreprinttypesetusingLATEXstyleemulateapjv.8/13/10
TIDALDISRUPTIONSOFWHITEDWARFSFROMULTRA-CLOSEENCOUNTERS
WITHINTERMEDIATEMASSSPINNINGBLACKHOLES
ROLANDHAAS1,2,ROMANV.SHCHERBAKOV3,TANJABODE1,PABLOLAGUNA1
DraftversionJanuary24,2012
ABSTRACT
Wepresentnumericalrelativityresultsoftidaldisruptionsofwhitedwarfsfromultra-closeencounterswitha
2 spinning,intermediatemassblackhole.Theseencountersrequireafullgeneralrelativistictreatmentofgravity.
1
Weshowthatthedisruptionprocessandpromptaccretionofthedebrisstronglydependonthemagnitudeand
0
orientation of the black hole spin. However, the late-time accretion onto the black hole follows the same
2 decay,M˙ t- 5/3,estimatedfromNewtoniangravitydisruptionstudies. Wecomputethespectrumofthedisk

n formedfro∝m the fallback material using a slim disk model. The disk spectrum peaksin the soft X-rays and
a sustainsEddingtonluminosityfor1- 3yrsafterthedisruption. Forarbitraryblackholespinorientations,the
J
disrupted material is scattered away from the orbital plane by relativistic frame dragging, which often leads

0 to obscurationof the inner fallbackdisk by the outflowingdebris. The disruptioneventsalso yield bursts of
2 gravitationalradiationwithcharacteristicfrequenciesof 3.2Hzandstrainamplitudesof 10- 18forgalactic
∼ ∼
intermediatemassblackholes. Theoptimisticrateofconsideredultra-closedisruptionsisconsistentwithno
]
E sourcesfoundinROSATall-skysurvey. ThefuturemissionslikeWide-FieldX-rayTelescope(WFXT)could
observedozensofevents.
H
Subjectheadings:accretion–blackholephysics–gravitationalwaves–hydrodynamics–relativity–radiation
h. mechanisms:general–X-rays:bursts
p
-
o 1. INTRODUCTION arisesfromthetidalcompressionperpendiculartotheorbital
r plane. The compression leads to the formation of a strong
t Tidal disruptions of stars by black holes (BHs) are fasci-
s natingandviolentcosmicevents,releasingcopiousamounts shockthattriggersapowerfulshortoutburst(Kobayashietal.
a 2004;Guillochonetal.2009;Rosswogetal.2009). Insome
of energy in electromagnetic radiation and, in some cases,
[ instances, e.g. ultra-close disruptions, the tidal compression
alsoaccompaniedbypotentiallydetectablegravitationalwave
could be strong enough to producea thermonuclearignition
1 emission. The scattered debrisfrom a stellar disruptionwill
v yield different radiation patterns depending on the orienta- of the star. In such cases, the detonation of the star would
9 tion of the orbit. This variety of radiation patterns will beobservedasanunder-luminoussupernova(Rosswogetal.
8 provide clues about both the BH and the internal struc- 2009).
3 ture of the disrupted star. As the debris from a tidal The variety of proposed radiative signatures often dis-
4 agree with each other, calling for precise dynamical model-
disruption showers back into the BH to form an accre-
. ing of tidal disruptions. Earlyhydrodynamicsimulationsby
1 tion disk, for instance, it will radiate close to or above
Evans&Kochanek(1989)confirmedaflatdistributionofde-
0 the Eddington luminosity for some duration, with the spec-
2 trumpeakingatUV/X-rayfrequenciesdependingontheBH bris mass over its energy range, yielding the predicted t- 5/3
1 mass (Rees 1988; Evans&Kochanek 1989; Cannizzoetal. law of fallback accretion rate (Rees 1988). Other numeri-
: 1990; Bogdanovic´etal. 2004; Strubbe&Quataert 2009). calinvestigationsaddressedthetidalcompressionandrelated
v
In addition, the unbound debris will quickly scatter over shockformation;someofthesestudiesusedsmoothedparti-
i
X a large volume, and may be illuminated by the accre- clehydrodynamics(SPH)techniques(Kobayashietal.2004;
r tion disk, leading to an optical irradiation spectrum domi- Rosswogetal. 2009) and others hydrodynamic grid-based
a nated by lines (Bogdanovic´etal. 2004; Sesanaetal. 2008; codes(Guillochonetal.2009). ThestudiesbyRosswogetal.
Strubbe&Quataert2009;Clausen&Eracleous2011). (2009) included nuclear reactions with simplified networks.
When a main sequence star is disrupted, the super- On the other hand, Bogdanovic´etal. (2004) looked at the
Eddington outflow is by itself hot enough to be a sig- formation of spherical envelopes Loeb&Ulmer (1997) us-
nificant source of optical emission resembling supernova ing SPH simulations, but the study did not find any con-
radiation (Kasen&Ramirez-Ruiz 2009; Strubbe&Quataert clusive evidence of a spherical structure. At present tidal
2009). In these circumstances, instead of outflowing de- disruption simulations that capture all the relevant physics
bris, a steady optically thick envelopemay reprocessthe in- are quite challenging. The simulations require handling,
ner disk radiation (Loeb&Ulmer 1997; Ulmeretal. 1998). in addition to hydrodynamics, effects from general relativ-
As a consequence, neither the inner disk nor the irradiated ity, nuclear reactions, and magnetic fields. Thus, many im-
debris emission is visible. Another tidal disruption effect portant questions are still left unanswered. In particular,
there are very few simulations that accountfor generalrela-
1CenterforRelativisticAstrophysics,SchoolofPhysics,GeorgiaInsti- tivisticeffects(Lagunaetal.1993a,b;Kobayashietal. 2004;
tuteofTechnology,Atlanta,GA30332,USA Bogdanovic´etal. 2004). The inclusion of magnetic fields
2Theoretical AstroPhysicsIncludingRelativity, California Instituteof mayleadtojetformation.
Technology,Pasadena,CA91125,USA Verylikely,theBHsinvolvedinatidaldisruptionwillhave
3Hubble Fellow, Department ofAstronomy, University ofMaryland, spins misaligned with the orbital angular momentum of the
CollegePark,MD20742-2421,USA
2 Haas,Shcherbakov,Bode,&Laguna
incoming star. Thus the disk formed from the tidal debris (GWs)andtheelectromagneticspectrum(Gould2011)from
willhaveangularmomentumthatisalsomisalignedwiththe a disruption event could potentially guide observations that
BH spin direction(Rees1988). For thindisks, the Bardeen- provide evidence for the existence of IMBH as well as in-
Petterson effectwill align the BH spin with the angularmo- sightsintothestructureoftheWDinvolved.
mentumoftheinnerregionofthedisk(Bardeen&Petterson Our study uses the full machinery of numerical relativity
1975),butforthethickdisksexpectedfromtidaldisruptions, to solve the Einstein equations of general relativity and hy-
the alignment may not happen (Papaloizou&Pringle 1983; drodynamics. We do not include nuclear reactions. For the
Fragileetal. 2007; Stone&Loeb 2011). This misalignment ultra-close encounters of our interest, with R few R , a
p g
∼
couldhaveimportantconsequencesontheemissionchannels general relativistic description of gravity is needed. How-
from the disruption, e.g. jet formation. The spin of the BH ever,giventhe massratioM /M 10- 3 ofthe bodiesin-
wd bh
will also producedistincteffectsin an ultra-closeencounter, volvedandencountersnotdrivenby≃GWemission,onedoes
when the pericentric radius Rp is comparableto the gravita- notneedtoaccountfordynamicalgeneralrelativisticgravity.
tional radius R =GM /c2 of the BH. In this strong grav- The present study could have been carried out by doing hy-
g bh
ityregime,the detailsofthedisruptionwilldependnotonly drodynamicsonthefixedspace-timebackgroundprovidedby
onthestellar radius, stellar equationof state, andorbitalen- the IMBH. However, this would have implied developing a
ergy, but also on the magnitude and orientation of the spin generalrelativistichydrocodeonafixedbackground,acode
of the BH. In particular, the properties of the accretion will similartotheSPHcodedevelopedbyoneofustoinvestigate
depend on the BH spin since its magnitude determines the the tidal disruption of main sequence stars by supermassive
location of the innermoststable circular orbit (ISCO) radius BHs(Lagunaetal.1993a,b). We decidedinsteadtotakead-
r , rangingfromr =9R for a counter-rotatingmaxi- vantageofournumericalrelativitycodeMAYA. Thecodehas
ISCO ISCO g
mallyspinningBHtor =1R foraco-rotatingmaximally demonstratedexcellentperformanceinhandlingfluidflowsin
ISCO g
spinning BH. The value of spin determines whether the star thevicinityofBHsinourstudiesofbinaryBHmergersinas-
followingthesame orbitgetsdisrupted(Kesden2011). Fur- trophysicalenvironments(Bodeetal.2008;Bodeetal.2009;
thermore,a misalignedrotatingBH will drag aroundnearby Bodeetal. 2011). The other advantage of using the MAYA
debris(Bardeenetal.1972;Shapiro&Teukolsky1986),and codeis the ability to obtainthe GW signaldirectlyfromthe
will effectivelypush thismaterialaway fromthe orbitaldis- simulation,withouthavingtotakerecoursetothepossiblyin-
ruptionplane. Theneteffectwillbetheformationofashell- accuratequadrupoleformulaortheapplicationofpostNew-
likedisruptiondebrisengulfingtheBH,asopposedtothetra- tonian(PN)approximationstospace-timescontainingaspin-
ditionalS-shapedebrisobservedinNewtoniangravitysimu- ningBH.
lations(Evans&Kochanek1989). Thepaperisorganizedasfollows. In§2,wedescribehow
Thepresentworkaimsatexploringtheexcitingregimeof self-consistent general relativistic initial data is constructed.
ultra-close encounters (i.e. R few R ) of a white dwarf Results from the dynamics of the disruption events are dis-
p g
(WD) with an intermediate ma∼ss black hole (IMBH). Our cussedin§3. Weestimateelectromagnetictransientradiation
main goal is to investigate observational signatures due to in§4usingaslimdiskmodeltocomputethespectrumduring
strong gravity effects that may shed light on the presence thefallbackphase. WefindthatthesourcesshineatEdding-
of IMBHs. We consider a carbon-oxygen WD with mass tonluminosity 1041ergs- 1forabout1- 3yrs,andthenfade
Mwd=1M⊙ andradius(Nauenberg1972) approximately∼as t- 5/3. We also find that there is a 50/50
∝
chancethattheinnerdiskwillbeobscuredbyoutflowingde-
M - 1/3 M 4/3 3/4 brisfora fullymisalignedBH spin. In§ 5, we calculatethe
Rwd=1.56R⊕ Mwd 1- Mwd gravitationalwavesignalproducedduringtheencounters. In
(cid:18) ch(cid:19) " (cid:18) ch(cid:19) # §6weestimatetheeventratesanddiscusstheobservational
=0.86R (1) prospects. Wediscusstheprogressandthelimitationsofour
⊕
studyin§7.
withM =1.44M theChandrasekharmassandR =6,960
ch ⊙ ⊕
km the Earth mean radius. We fix the mass of the IMBH to
Mbh =103M⊙. With these choices, Rwd 4Rg 6,000km. 2. INITIALDATAANDCODETESTS
≃ ≃
Therefore, the WD and IMBH can be numerically modeled
When using a fully general relativistic code, such as our
with comparablegridresolutions. Theother importantscale
MAYAcode,thatisbasedona3+1formulationoftheEinstein
intheproblemisthetidaldisruptionradiusR. Foroursetup
t equations, the initial data for the dynamical space-time are
(Rees1988),
comprisedofthespatialmetricγ andtheextrinsiccurvature
ij
R R M - 1/3 M - 2/3 Kij of the initialspace-time hypersurface. All tensor indices
t 40 wd wd bh , (2) are spatial indicesand, in this section only, we use units for
Rg ≃ (cid:18)0.86R⊕(cid:19)(cid:18)1M⊙(cid:19) (cid:18)103M⊙(cid:19) whichG=c=1.Themetricγijcharacterizesthegravitational
potentialsoftheIMBHandWD,andthetensorK provides
gaveusampleroomtocarryoutdeeppenetrationencounters. ij
the “velocity” of the metric, or the embeddingof the space-
Currently,thereisalargebodyofevidencefortheexistence likehypersurface,inthespace-time.4
ofbothsolarmassaswellassupermassiveBHswithmasses
Althoughγ andK aredominatedbytheIMBH,itiscru-
intherangeof105–108M . IMBHsare,however,themiss- ij ij
⊙ cialtoaccountforthecontributionsfromtheWDinorderto
inglinkbetweenstellarmassandsupermassiveBHs. Today,
correctlyincludetheself-gravityoftheWD.Thecomponents
onlytentativeevidenceexists(Irwinetal.2010;Farrelletal.
of γ and K cannot all be freely specifiable. They need to
ij ij
2009;Davisetal.2011). Ontheotherhand,WDsarethought
to be abundant in spiral galaxies (Evansetal. 1987; Reid
2005) and globular clusters (Gerssenetal. 2002). Thus the 4 Forareview onnumerical relativity, werecommendthetextbooks by
identification of distinct signals in both gravitational waves Alcubierre(2008)andBaumgarte&Shapiro(2010)
Tidaldisruptionofwhitedwarfsbyintermediatemassblackholes 3
satisfythefollowingellipticequations Thus, equation (3) becomes an equation for u. In (12), the
8∆φ+φ- 5A¯ijA¯ij=- 16πφ- 3ρ¯∗, (3) fisorlsutttieornm;iasntdhethfleatthsiprdacteersmoluutiionncltuod(e3s)b;oththestehceognrdavisittahteioBnHal
∂ A¯ij=8πJ¯i, (4) potentialoftheWDaswellastheinteractioneffectswiththe
j
BH.
where To test our initial data method, and in particular to assess
γij=φ4δij (5) tihneitiiamlpdoarttaanfocreoafWthDeWinDiti-aBllHyiantterreasct;titohnaetfifse,cJt¯si,=w0ecaonmdpthuutes
Kij=A¯ijφ- 10 (6) A¯wd=0. Inaddition,weassumeanon-spinningBH,i.e.A¯bh=
ij ij
ρ =ρ¯ φ- 8 (7) 0.TheHamiltonianconstraint(3),withthehelpof(12),takes
∗ ∗
Ji=J¯iφ- 10. (8) theform
∆¯u=- 16πφ- 3ρ¯ . (13)
Above,φisaconformalfactor,δ =diag(1,1,1)theflatspa- ∗
ij
tial metricand ∆its associated Laplace operator. Inwriting Thesolutionto(13)isshowninFigure1. Forcolorversions
Eqs. (3) and (4), we have made the customary assumption of the figures in this paper see the electronic edition of the
thattheinitialphysicalmetric γ is conformallyflatandK Journal. AlsoinFigure1isthesolutionu toequation(13)
ij ij wd
istrace-free. Eqs.(3)and(4)are,respectively,theconformal withouttheBH;thatis, withM =0in(12). Thedifference
bh
versions of the so-called Hamiltonian and momentum con- betweenuandu isentirelyduetothegravitationalinterac-
wd
straintsofgeneralrelativity(York1979). Thesourcesρ and tionoftheWDwiththeBH.
∗
Jiare,respectively,thetotalenergyandmomentumdensities.
OurapproachtosolvetheHamiltonianandmomentumcon- 4
u
straintsis to specify the “free”data in Eqs. (3) and (4) from
3.5 uwd
thesolutionstoanisolated,spinningBH,andaboostedWD
(Löffleretal.2006).
3
Let us consider first the solution for a boosted WD. The
sources ρ¯∗ and J¯i in Eqs. (3) and (4) are obtained from 410 2.5
×
the stress-energy tensor for a perfect fluid (see chapter 5 in
u
Baumgarte&Shapiro2010). Theyread 2
P
ρ¯ =ρ 1+ε+ W2- P, (9) 1.5
∗
ρ
(cid:18) (cid:19)
J¯i=ρ 1+ε+P W2vi, (10) 1 -1 -0.5 0 0.5 1
ρ
(cid:18) (cid:19) r/Rwd
where W = (1- γijvivj)- 1/2 is the Lorentz factor and vi FIG.1.—Solutionutoequation(13)foraWDatrestadistance15Rwd
awayfromanon-spinningIMBH.Thesolutionuprovidesnotonlythegrav-
the boost velocity of the WD. In (9) and (10), ρ is the
itational potential of the WD but also includes the full non-linear interac-
rest-mass density, P the pressure, and ε the specific in- tionswiththeIMBH.Forcomparison,weplotalsouwd,thesolutiontoequa-
ternal energy density. The values for those quantities are tion(13)foraWDinisolation.
obtained from Tolman-Oppenheimer-Volkoff (TOV) solu-
tions (Oppenheimer&Volkoff 1939; Tolman 1939) with a We also tested the ability of ourmethodto producestable
polytropicequationofstateP=KρΓ. WeuseΓ=5/3appro- isolated WD models and to conserve rest mass. We found
priateforanon-degenerategas,whichissufficienttocapture that,oversixdynamicaltimes,thecentraldensityoftheWD
thedynamicsduringtheinspiralanddisruptionphases. Dur- did not change more than 1.2% of its initial value and rest
ingtheevolution,weswitchtoagamma-lawequationofstate masswasconservedatthe0.1%level. Thisisinspiteofthe
P=ρε(Γ- 1). WedenotethesolutionstoEqs.(3)and(4)in adaptivemeshrefinementinfrastructureweuse(CARPET)not
¯
theabsenceoftheBHbyφ andAwd. beingdesignedtoaccuratelypreserverestmassconservation
wd ij
Nowweconsiderthesolutiontotheconstraintsforaspin- wheninterpolatingbetweendifferentrefinementlevels.
ning BH. In the absence of the WD, ρ¯∗ =0 and J¯i =0. In 3. SIMULATIONRESULTS
thiscase,themomentumconstraint(4)canbesolvedanalyti-
3.1. TidalDisruptionSimulationParameters
cally(Bowen&York1980). Thesolutionreads
Our study consists of a series of six simulations. As
A¯bijh= r33 ǫkilSllklj+ǫkjlSllkli (11) amnednttihoeneWd Dbefhoarse,athmeaIsMs BMH h=as1aMmaasnsdMrbahdi=us10R3M⊙,
wd ⊙ wd
≃
withli=xi/randǫ the(cid:0)three-dimensionalL(cid:1)evi-Civitasym- 6,000km(Hamada&Salpeter1961). TheWD massandra-
bol.Above,SiisthiejkspinvectoroftheBH. diuscorrespondtoacentraldensityρinitial=1.33 107gcm- 3,
With the solutionsA¯wd andA¯bh athand, it is clear thatthe Γ=5/3,andpolytropicconstantK=2.41 10- 9×(cm3g- 1)Γ- 1.
linearsuperpositionA¯ ij=A¯bh+A¯ijwd isthesolutiontothemo- AllthesimulationsstartwiththeWDin×aparabolicorbit,in
ij ij ij theNewtoniangravitysense, a distance1.5R 60R along
t g
mentum constraint (4) in the presence of both a WD and thenegativexˆ-axisawayfromtheIMBH.Theo≃rbitalangular
aA¯iBj=HA.¯biNjh+exA¯t,wijdwaendsoρ¯l∗vegivtheenbHya(m9i)l.toWneiaunsecothnestpraoipnutla(4r)anwsaitthz minoFmigeunrteum2.oftheWDisalignedwiththeˆz-axis,asdepicted
Table 1 lists the configurations used in the simulations
M
φ=1+ bh +u. (12) where β = R/R is the penetration factor for an orbit with
2r t p
4 Haas,Shcherbakov,Bode,&Laguna
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) lessthan4%overthe courseofthe simulations. RunsB6Su
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)S(cid:0)(cid:1)(cid:0)(cid:1) and B6S0 display the worst mass conservation. As we shall
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)L (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) see, these are the cases in which most of the material of the
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) star (&95%) escapes the BH, forming an expanding cloud
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) that reaches the coarsest mesh refinements. Notice also that
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)z (cid:0)(cid:1)q (cid:0)(cid:1)y(cid:0)(cid:1)(cid:0)(cid:1) mostofthe errorsin massconservationaccumulateafterthe
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) disruption. Run B6Sd is the only run that loses mass. This
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1) R (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)f(cid:0)(cid:1)(cid:0)(cid:1) isthecaseinwhichtheIMBHaccretesalmostallofthestar
(cid:0)(cid:1)(cid:0)(cid:1) t (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)x duringtheinitialpassageoftheWD.
(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1) R
p
0.07
0.06 B6S0 B6Sd B6Sa
FIG.2.—Configurationsetupofoursimulations.TheWDislocatedadis- B6Su B6Si B8Sa
tance1.5Rt 60Rgalongthenegativexˆ-axisawayfromtheBH.Theorbital 0.05
angularmom≃entum~LoftheWDisalignedwiththeˆz-axis.Theorientationof 0.04
theBHspinS~isdeterminedbythestandardθandφanglesinarighthanded d 0.03
w
sphericalcoordinatesystem. M 0.02
pericentricdistance R , anda⋆= S~/M2 is the magnitudeof M/
p | | 0.01
thedimensionlessspinparameteroftheIMBH.Theanglesθ d
0
and φ determine the direction of the BH spin. They are the
standardanglesinaright-handedsphericalcoordinatesystem -0.01
as depictedin Figure2. We use the followingconventionto -0.02
label the cases we studied. If a case is labeled "BxSy", the -0.03
WD hadpenetrationfactorβ =x. Inaddition,thevalueofy -2 -1 0 1 2 3 4 5 6 7
denotesanon-spinningBH(y=0)ortheBH’sspinorientation: timeafterdisruption(s)
(up,down,in-plane,arbitrary)fory=(u,d,i,a),respectively. FIG.3.— Relative gain/loss of rest mass in the computational domain,
δM/Mwd=(M- Mwd)/Mwd,whereMisthetotalrelativistic restmassand
Run β a⋆ θ φ Mwd theinitialWDrestmass. Thetotalrelativistic restmassMtakesinto
B6S0 6 0.0 0◦ 0◦ accountfluxesintotheBH.Theonlysimulationrundisplaying anetmass
lossisthecaseB6Sdinwhichalmostallthematerialaccretespromptly. In
B6Su 6 0.6 0◦ 0◦ theothercases,mostofthemassgainhappensintheshorthighcompression
B6Sd 6 0.6 180◦ 0◦ phasewhentheWDisatitsperiapsispointclosetotheBH.
B6Si 6 0.6 90◦ 0◦
B6Sa 6 0.6 63◦ 90◦
B8Sa 8 0.6 63◦ 90◦
3.2. Tidalcompression
TABLE1 The first stage in a tidal disruption encounter consists of
SIMULATIONPARAMETERS:β=Rt/RpISTHEPENETRATIONFACTOR, the deformations that the incoming star experiences due to
Ba⋆LA=C|S~K|/HMOL2EIS,θTHAENDDIφMAENRESITOHNELEASNSGSLPEISNBPAETRWAMEEENTE~SRAONFDTTHHEECOERNBTIRTAALL cthoemtpideatilnfgortcideas.l Idneftohremoartbioitnasl.plOanneeoofftthheemstairs,tthidearel satrreettcwho-
ANGULARMOMENTUM(ˆz-AXIS)ANDTHERADIALDIRECTIONJOINING
THEIMBHANDTHEWD(xˆ-AXIS),RESPECTIVELY. ingalongtheradialdirectionjoiningtheIMBHwiththeWD.
Theotheristidalcompressionperpendiculartothisradialdi-
rection. In the direction perpendicular to the orbital plane,
In all the simulations, we employeight levels of mesh re- thereisalsotidalcompression. Thiscompression,orsqueez-
finementwithradii1.24Rg 2ℓ(0 ℓ 7)centeredattheBH ing, is stronger than that in the orbital plane because in the
× ≤ ≤
location. Inadditionto theserefinementlevels, we surround latter the radialdirection changesrapidly as the star reaches
the WD during the pre-disruptionphase with five additional periapsis, thus modifying the “contact point” in the star of
nested boxes of radii 4.96R 2ℓ (0 ℓ 4). The resolu- the in-plane compression. We concentrate here on the com-
g
× ≤ ≤
tion on the finest refinement is R /19.35, the resolution on pression in the direction perpendicular to the orbital plane.
g
thelevelcoveringtheWDisR /9.675.Thismeshrefinement This compression could, in principle, generate enough heat
g
setup becomes insufficient when the WD begins to be dis- to detonatethe star (Luminet1985;Luminet&Marck1985;
rupted. Atthispoint,weturnofftheboxestrackingtheWD Luminet&Pichon1989) by ignitingnuclearreactionsatthe
andconstructalargermeshwithresolutionR /9.675thatin- coreoftheWD(Rosswogetal.2009). Toinvestigatewhether
g
cludesboththeIMBHandthehighlydistortedWD.Oncethe anuclearignitionislikelytohappenfromtidalcompression,
expandingdebrishasclearedtheinnerregionandthedensity weplotinFigure4 theinstantaneousmaximumtemperature
hasdropped,weturnoffthislevelagaintospeedupthesim- of the star as a function of the density at that location. To
ulation. a good approximation, this density is also the instantaneous
The mesh refinements affect our ability to conserve rest maximumdensity. Theinstantaneousmaximumtemperature
mass throughout the computational domain, in particular if is computedas the mass-density-weightedtemperatureaver-
the meshes are moving, created, or removed. Figure 3 dis- ageoftheportionofthestarwhosenumericalcellsthathave
playstherelativegain/lossofrestmassin thecomputational thelargestinternalenergyandmakeupfor10%ofitsmass.
domain,δM/M =(M- M )/M ,whereMisthetotalrela- We calculatetemperaturesT fromthethermalspecificen-
wd wd wd
tivisticrestmassandM theinitialWDrestmass. Thecom- ergyas(seeLeeetal.(2005))
wd
putedvalueofMtakesintoaccountthemasslossthroughthe
3
BHhorizonandtheouterboundaryofthecomputationaldo- m n ε = k T(N +N )n +a T4, (14)
p nuc th B ion e nuc BB
main.Wefindtolerablemassconservation,withviolationsof 2
Tidaldisruptionofwhitedwarfsbyintermediatemassblackholes 5
10 β∗=Rt/R∗p. Recallthattheorbitalparameters,andinpartic-
ularR ,usedtosettheWDinaparabolicorbitwereobtained
p
9.5 using Newtonian gravity. Because of relativistic effects, the
K) 9 tign<tdyn valueof R∗p will differ fromthe Newtonianestimate Rp, and
()x tign>tdyn thusβ6=β∗. Inoursimulations,wedefineR∗p asthedistance
Tma 8.5 B6S0 of closest approach of the point within the WD where Tmax,
( B6Su or equivalentlyρ , is found. Notice that forthe B6Su and
g max
lo 8 BB66SSdi B6Sd cases, Tmax almost doubles and ρmax triples when go-
ing from a spin aligned with the orbital angular momentum
B6Sa
7.5 B8Sa toonethatisanti-aligned. Thisisbecauseintheanti-aligned
case (B6Sd), the WD penetrates closer, β∗ =11, than in the
7
6 6.5 7 7.5 8 alignedcase(B6Sd),β∗=9.4. Figure5showsthemaximum
log(r max)(gcm-3) toefmtpheeraatcutruealTpmeaxneatnradtiomnaxfaimctuomr βd∗enfsroitmy ρTmaabxleas2.a fBuenccatuiosne
FIG.4.—Instantaneousmaximumtemperatureasafunctionofthedensity of the small range of values covered by β∗, it is not possi-
atthesamelocation. Wecomputethedensity-weightedtemperatureaverage
ofthehottestcellsinthegridwhichmakeup10%ofthetotalrestmassof ble to verifythe scaling Tmax β2 and ρmax β3 suggested
∝ ∝
theWD.Thedottedblacklineseparates regions inparameterspacewhere byCarter&Luminet(1982)
theignition timeasestimated fromFigure2ofDursi&Timmes(2006)is
larger/smaller than the dynamical timescale of the WD,i.e. the region of
10
successful explosion from the region where the explosion fails. Densities 9 B6S0 B6Si
icsgmansaietlsiloeinrnvttheimastnieg1ai0nte7dtgh,citsmher-e3iggianorinetionanroettipbmraeessesecdnatolenintisgFinmigopuflretehle2inWaenaDdretihsxetmraeupsctoihmlastahitooensrt.eforIrnthtaahlnel r/xinitial 5678 BB66SSud BB68SSaa
thedynamicaltimescaletdyn=1/√Gρ&0.38softheWD. rma 34
with 12
K)11
εth=ε- KΓin-iti1alρΓ- 1, (15) 9(10x1089
ma 7
wherethesecondterminequation(15)isthe“cold”specific T 6
internalenergyintheabsenceofshocks. Ne isthenumberof 7.5 8 8.5 9 9.5 10 10.5 11
electronspernucleonmakingup the gas, Nion is the number b ⋆
ofionspernucleon,n isthenumberdensityofnucleonsin
thegas,mp istheprotnounc(nucleon)mass,kBistheBoltzmann ofFβIG∗.f5o.r—theMvaaxluimesumindTeanbsleity2.ρmax(top)andtemperatureTmaxasafunction
constant,anda hereistheradiationconstant.Forsimplicity
BB
we assume N =1/14, for an equalmixture of oxygenand
ion
carbonatoms. ForN , whichdependsontheionizationstate 3.3. Outflowingdebrisandspin
e
oftheplasma,weuseNe=0forT .5 106KandNe=1/2 Allcasesshowedthatthemassandorientationofthedebris
×
for5 106K.T .1 1010K. Thatis,weignoreanypartial after disruption is strongly affected by the BH spin. This is
× ×
ionizationstatesoftheconstituentatomsatlowtemperatures. becausethespindeterminesthelocationofISCOradius,and
In all cases investigated, the maximum temperature and it is also responsible for frame-draggingeffects. In particu-
densitydisplayedinthetracksofFigure4reachhighenough lar,frame-draggingpullsthematerialoutoftheorbitalplane,
values to trigger nuclear burning in the star (cf. Figure 2 therebychangingtheorientationoftheresultingdisk.
of Dursi&Timmes (2006)). Interestingly, the temperature- We first discuss three disruptions: disruption by a non-
density tracks in Figure 4 are qualitatively similar to those spinningIMBH(caseB6S0), byanIMBHwith spinaligned
byRosswogetal.(2009),whoincludednuclearnetworks. (case B6Su), and by an IMBH with spin anti-aligned (case
B6Sd) relative to the orbital angular momentum (see Ta-
Run β∗ Tmax(109K) ρmax/ρinitial facc funb ble 2). For the spinning BH cases, the magnitude of the di-
B6S0 8.9 8.6 7.5 68% 19% mensionless spin parameter a∗ was kept at 0.6. With this
B6Su 9.44 6.6 9.3 <1% 60% spin magnitude, rISCO = 5,670km for the case B6Su, and
B6Sd 11 12 3.2 >99% <0.5% r = 11,600km for the case B6Sd. In the non-spinning
ISCO
B6Si 7.6 8.6 3.3 65% 22% case, r = 9,000km (McClintocketal. 2011). All three
ISCO
B6Sa 9.1 6.0 8.1 2% 67% cases had penetration factors β = R/R = 6. From equa-
t p
B8Sa 10 9.1 4.0 43% 34% tion(2), R =40R =60,000km, thusa penetrationfactorof
t g
6 translates to a pericentric distance R 10,000km. We
TABLE2 p ≃
ACTUALPENETRATIONFACTORβ∗MEASUREDFROMTHE then expect that the WD in the retrograde case B6Sd will
SIMULATIONS.MAXIMUMTEMPERATURETmaxANDCOMPRESSION passwithintheISCO;intheprogradecaseB6Suthestarwill
STρA/ρRINAICTICALREWTIETDHDρUINRITIINALG=TH1.E33FI×RS1T072sgcAmFT- 3E.RFDRIASCRUTIPOTNIOfNAC,CUNOBFOTUHNED mWoDstwlyilsltgayrazoeutbsyidteheofIStCheOI.SBCeOca;uasnedrin th>eRcasienBth6eSB06tShde
FRACTIONfUNBATTHEENDOFTHESIMULATIONAT∼6s. case,almostallofthestarisaccretedbyIStChOeBHpsoonafterthe
disruption. ForthecasesB6S0andB6Su,ontheotherhand,
In Table 2, we list the maximum temperature T and most of the stellar debris escapes direct capture, expanding
max
compression ρ /ρ . Here T and ρ are the max- awayfromthehole. Thedebriscloudhasashaperesembling
max initial max max
imum temperature and density attained over the course of a thick circular arch. This arch-like cloud eventually closes
the evolution. We also list the “actual” penetration factor upandformsadiskaroundtheBH.
6 Haas,Shcherbakov,Bode,&Laguna
˙
In all cases, the leading edge of the tidally-disrupted star onto the BH with a characteristic rate of accretion M
wraps around the BH and crashes into the trailing material t- 5/3 (Rees 1988; Phinney 1989). This rate is maintaine∝d
creating a hot region. For the B6Su, B6Sd and B6S0 cases, as long as dM/dε , with ε being the specific kinetic en-
kin kin
thishotregionismostlycontainedwithintheequatorialplane ergy,isapproximatelyconstant(Lodatoetal.2009). Wefol-
andmovesoutwardsfromthecentralregionintheformofa lowRosswogetal.(2009)andcomputethefallbacktimefor
spiralingdisturbanceinthedebrisdisk,eventuallyappearing each fluid element at a time long after disruption, based on
on the debris surface. Figure 6 shows snapshots in the or- dataavailableattheendofthesimulation.Atthistime,hydro-
bitalplane ofthe density (toprow) andtemperature(bottom dynamicalinteractionsbetweenthefluid elementsaresmall,
row)forrunsB6Si andB6Su. Thesnapshotsin the leftcol- and each fluid element moves on an almost geodesic orbit.
umnareforthecaseB6Si 0.5secondsafterdisruption.The Figure 8 shows the results of these calculations. Clearly we
middleandrightcolumnsc∼orrespondtocaseB6Su 0.5sec- recover the expected t- 5/3 behavior for return times longer
∼
ondsand 5.9secondsafterdisruption,respectively. Inthe thant&200s. NoticethatearliesttimesinFigure8arecom-
∼
alignedcaseB6Su,thehotregionformsaspiralwhichmoves parabletothelatetimeaccretionratesfoundduringthesimu-
outwardsfromthepointwheretheleadingedgeoftheWDin- lationinFigure7.
tersectedwiththetailmaterial. Atlatetimesthisfeaturehas
beenwashedout. InrunB6Sithehotregionismuchsmaller; 4. ELECTROMAGNETICSIGNATURES
framedraggingbytheBHhaspulledmaterialoutoftheplane
Ultra-close(i.e. R fewR ) tidaldisruptionsareviolent
intoashellsurroundingtheBH. p∼ g
events. The conversion of even a small fraction of gravita-
For the cases in which the spin of the BH is not aligned
tionalenergyintolightwouldresultinapowerfulelectromag-
with the orbitalangularmomentumofthe binary(i.e. B6Si,
neticsignature.Inthissectionwewillusethehydrodynamics
B6Sa and B8Sa cases), frame draggingis effectivelyable to
resultsfromoursimulationstomakeorderofmagnitudeesti-
wrapthe debrismaterialaroundtheBH. Thiseffect–which
matesofthetidaldisruptionflare’sluminosity,itscharacteris-
is notcapturedby Newtoniancalculations– greatly changes
ticphotonenergy,andthefallbackmaterial’selectromagnetic
thegeometryofthedebris,blanketingthecentralregionwith
signatureasiteventuallyformsaslimaccretiondisknearthe
a shell-like cloud of gas. We will address the observational
BH.
consequencesofthematerialsurroundingtheBHinthenext
Whenthe WD descendsinto the potentialwell ofthe BH,
section. Theeffectofthe misalignmentis evidentfromFig-
with the typical dynamical timescale of 1 s, it acquires
ure6 in the remarkabledifferencesofthe spiralpatternsob- energy in excess of 1053erg. Even if∼a tiny fraction of
servedbetweentheleftandmiddlesnapshotsat 0.5seconds ∼
∼ this energy gets converted into radiation on this dynamical
afterdisruption.TheleftpanelsarefortheB6Sicaseinwhich
timescale, it would lead to a huge luminosity which would
the BH spin direction is perpendicularto the orbital angular
be noticeable at cosmologicaldistances. Furthermore, since
momentum. Ontheotherhand,themiddlepanelsareforthe
theWDundergoesasubstantialaccelerationinthedeepgrav-
B6SucaseinwhichtheBHspinisalignedwithorbitalangu-
itational potential of the BH, the streams of matter from
lar momentum. The observeddifferencesare a consequence
the disruption will approach speeds V (GM /R )1/2
ofhowthedebrisrespondstotheorientationoftheBHspin. p h p
≃ ≃
c(βR /R)1/2 0.4β c, comparable to the speed of light.
g t 6
≃
Thestreamsoftidaldebrislatercollideandtransformapor-
3.4. Promptandlateaccretion
tionoftheirkineticenergyintothermalenergy. Beingheated
We observe an initial promptphase of accretion when the by this process in excess of 109 K, the dense hot plasma
star swings by the BH. This prompt accretion phase is fol- quicklyproducesphotons(Krolik&Piran2011). Thesepho-
lowedbyanintermediatephaseinwhichmaterialinitiallyde- tons, however,are effectivelytrappedby the debris, capping
flectedbytheBH slowsdownandbeginstoaccrete. Atlate the luminosity around the Eddington luminosity (Eddington
times,longaftertheendofoursimulations,thecharacteristic 1926)
t- 5/3fallbackbehaviortakesover.
4πcGM m M
Figure 7 shows the flow of matter through the IMBH’s L = h p =1.4 1041 h ergs- 1, (16)
horizonduringthefirstsixsecondsafterthedisruptiontakes Edd σ × 103M
T (cid:18) ⊙(cid:19)
place. The instant labeled t is the time at which the star
tidal where m is the proton mass and σ is the Thomson cross-
enters the tidal radius. The horizontal dashed line labeled p T
section. Notice that the equilibrium between the gravita-
“noise” denotesthe levelof accretiondue to the atmosphere
tional and spherical radiation fields is reached at a luminos-
usedtomodelthevacuumregions. Weobservethatthemag-
nitude of the prompt accretion rate decreases the more pro- ity L=2LEdd=2.8 1041ergs- 1 due to composition effects.
×
ThesystemcanbyfarexceedL inthepresenceofacolli-
gradeanorbitis. BothB6S0andB6Sudisruptatcomparable Edd
matedoutflow,especiallyinthefirstseveralminutesafterthe
distancesfromthecentralBH,butthepeakaccretionratedif-
disruption.Weelaborateonthistidaldisruptionphenomenon
fersbyalmosttwoordersofmagnitude.
inasubsequentpaper(Shcherbakovetal. 2012,inprep). In
AsalreadypointedoutinSection3.3,theBHspinstrongly
thepresentmanuscriptwelimitourselvestoelectromagnetic
affectswhatfractionofthestarisaccretedontotheBHsoon
signaturesatlatetimesofmonthstoyears.
( 2 s) after the star disrupts. Table 2 shows this accreted
∼ When the accretion rate falls below the Eddington rate
fractionforeachcase.IntheB6Sdcase,thestariscompletely
˙
accreted.Ontheotherhand,thedebrisintheB6SuandB6Sa MEdd, the accretion becomes radiatively efficient, radiating
cases escapes almost in its entirety during the flyby. Notice mostly in X-rays with luminosity LEdd. Assuming an ef-
∼
alsofromTable2thatthepromptaccretionisalsocorrelated ficiencyofǫ=0.1,thisluminositytranslatesintoanaccretion
withthemaximumdensityρ theWDisabletoreachasa rate
max
consequenceofthetidalcompression.
The late-time behavior consists of material raining back M˙ = LEdd =2.3 10- 5 Mh M yr- 1. (17)
Edd 0.1c2 × 103M ⊙
(cid:18) ⊙(cid:19)
Tidaldisruptionofwhitedwarfsbyintermediatemassblackholes 7
FIG.6.—Snapshotsintheorbitalplaneofthedensity(toprow,ingcm- 3)andtemperature(bottomrow,inK)forrunsB6SiandB6Su. Thesnapshotsinthe
leftcolumnareforthecaseB6Si 0.5secondsafterdisruption.ThemiddleandrightcolumnscorrespondtocaseB6Su 0.5secondsand 5.9secondsafter
disruption,respectively. ∼ ∼ ∼
1010 ttidal 106
B6S0 B6Sa 104
108 B6Su B8Sa
B6Sd noise 102
1−) 106 B6Si 1−) 100 t−5/3
yr⊙ 104 yr⊙ 10−2
M M
˙M( 102 ˙M( 10−4 BB66SS0u
10−6 B6Sd
100 B6Si
10−8 B6Sa
10−2 10−10 B8Sa
-2 -1 0 1 2 3 4 5 6 7 100 102 104 106
timeafterdisruption(s)
timeafterdisruption(s)
FIG.7.—Accretionrateduringthesimulation. ttidalindicatestheapproxi- FIG.8.—Massaccretionratescomputedfromtheestimatedfallbacktimes
matetimewhentheWDenterswithinthetidalradiusRt.Thehorizontaldot- ofthefluidelements. Thesolidlinerepresentsthet- 5/3 falloffpredictedin
tedblacklineindicatesthemassaccretionwhichweobserveinitiallyaround
Rees(1988);Phinney(1989).
theblackhole. Itisapurelynumericalartifactfromtheatmosphereusedto
modelthevacuumregions.
4.1. SlimDiskModel
For our study we employ a slim disk model, developed
˙
ThetimeelapseduntiltheaccretionratedropstoMEddcanbe andsummarizedinAbramowiczetal.(1988);Lasota(1994);
estimatedbasedonthefallbacktimesreportedin§3.Wehave Kawaguchi (2003); Sadowski (2009), of a simple one-zone
demonstratedthatthe famoust- 5/3 fallbacklaw (Rees1988) vertically-integrateddynamicalmodel.Wecomputemodified
is also presentin ourultra-closeencounters. From Figure8, black-bodybremsstrahlung emission from the disk. For our
we observe that the accretion rate in all but the B6Sd case radiationestimates,weassumethattheangularmomentumof
dropsbelowM˙ attimesoft 1- 2yrs. Thusthetidal themarginallybounddebrisaccretingatlate times, andthus
Edd Edd
∼
eventswe areconsideringcan, inprinciple,shineatEdding- theangularmomentumofthedisk,arealignedwiththeorbital
tonluminositiesformanyyears. Theactualluminositycould angularmomentumoftheincomingWD.Thediskmodelwill
be substantially sub-Eddington,though, if outflowing debris then,inprinciple,bemisalignedwiththespinoftheBH.We
produced while the BH swallows matter at super-Eddington presumethat,despitethemisalignment,theslimdiskextends
ratesobscurestheaccretionflow. Inthissectionwewillfirst down to ISCO radiuscomputedfor the aligned case. In our
estimatethediskemissionwithinaslimdiskmodelandthen situationitischallengingtofindamorereasonablesetofas-
consider the added effects of obscuration by the outflowing sumptionsfor a geometrically thick slim disk. The ultimate
debris. answers require performing full numericalrelativity simula-
8 Haas,Shcherbakov,Bode,&Laguna
tions of the fallback regime, a task requiring modeling dy- avoidtheinfiniteincreaseofthedensityneartheISCOinher-
namicaltimescales 106 times longer than those addressedin enttoearlyslimdiskmodels(Abramowiczetal.1988).
thepresentstudy. Theaccretiondiskemitspredominantlybremsstrahlungra-
Wetakeourheavywhitedwarftoconsistofapproximately diation. Since the scattering cross-section in the disk is
50% carbon and 50% oxygen as suggested by stellar evo- much higher than the absorption cross-section, the emis-
lution computations by Mazzitelli&Dantona (1986). The sion spectrum is described by a modified black body
same heavy white dwarf composition was assumed by both spectrum (Rybicki&Lightman 1979; Czerny&Elvis 1987;
Rosswogetal.(2009)andKrolik&Piran(2011). Thusthere Kawaguchi 2003). The photon production rate via
is 1/14nuclei and 1/2 electrons per one nucleonwith mass bremsstrahlung is (Rybicki&Lightman 1979; Katzetal.
m . Intermsofnucleondensity,theionandelectronden- 2010)
p
≃
sitiesare
8 m c2
1 1 Q= α Z2n n σ c e Λ g cm- 3s- 1, (23)
nion= 14nnuc and ne= 2nnuc, (18) rπ f ion e T skBTe eff eff
respectively.Thecorrespondingnon-relativisticheatcapacity whereΛeff geff 1,αf 0.00729isthefinestructurecon-
∼ ∼ ≈
ofneutralandfully-ionizedgaspersinglenucleonoftemper- stantandσT istheThomsoncross-section.Withthehighden-
atureT are sities, the productionrate Q is high so the radiation quickly
dominatesinternalenergyandpressure.
3 6
c = k T and c = k T, (19) Let us estimate the equilibrium temperature of this
neutral B ion B
28 7 radiation-dominatedflow. Thetypicalnucleondensityinthe
1re3s.p6eecVtiv=e6ly8.0TeVhefoerntehregyasosfumfuelldicoonmizpaotisointioisn,EwChNeOre=Z<Z27>is simimuumlaotifonnmvian,rniuecsfro8mn1n0uc20∼cm10- 235.cTmh-e3tnyepaicrathleviBriHal,tteomapmeirna--
thechargeofthenucleus.ACNOWDisthusfullyioniz≈edat ture is T 101∼1 K.×As photons are produced, the equilib-
ion
T >T 5 106K. Themolecularmassofthefully-ionized riumimplie∼s
ion
plasmais∼µ=×1.75,anditsdensityisρ=mpnnuc. 6n k T =a T4 , (24)
A fully self-consistent model of a slim disk requires an 7 nuc B ion BB ph
estimate of the disk scale-height H. For a disk in the
leadingtophotontemperaturesT 3 108K.Thetimescale
super-Eddington regime, H/r =1 (Abramowiczetal. 1988; ph∼ ×
forreachingequilibriumcanbeestimatedas
Kawaguchi 2003; Strubbe&Quataert 2009). In our simu-
3llao0tciRaotgnesdf,rbocemirtwctuheleeanrBizthHaet.ioIWSnCeoOfthrdeaerdebifruoissrerhIcSaCopOnpseaindnsedrwthaiethtrmiunanxaciamdteiudsmtadnircsake- teq∼ akBBBTTpph4hQ(1Tph) =9×1010Tni9no7u/n/c88 ∼10- 8s, (25)
dius r = 30R . The viscosity parameter is taken to be
max g which is significantly shorter than the dynamical timescale
α=0.1. We solve the equations for the central temperature
t 1s.
as detailed below, taking into account the effects of viscous orb∼
We work in the diffusion approximation of high optical
heating,emission,andthefiniteheatcapacityofmatter. For
depthtofindthesurfacetemperature.Thecorrespondingscat-
simplicity, we followShakura&Sunyaev(1973) and set the
teringopacity,τ 1,ismuchlargerthanthebremsstrahlung
viscousstressattheISCOtozero.Theenergyreleaseperunit sc≫
absorptionopacityτ . Theelectronscatteringopacityis
diskareaisthen abs
σ n
F+= 3rgM˙c2 1- rISCO 1/2 . (20) κsc= mpTnneuc ≈0.2cm2g- 1, (26)
8πr3 r
(cid:20) (cid:16) (cid:17) (cid:21) whereastheabsorptionopacityis
Theaccretionrateisexpressedas
(1- e- x)e6h2Zρ 2π
M˙ =4πrHρcvr, (21) κabs=3ckB3mem2pTs7/2xc3rkBme
whereρc istheone-zonedensityandvr >0istheinflowve- 4.4 1025(1- e- x)ρ
locity. = × c (27)
We adopt a beta disk model where the viscous stress T7/2x3
s
is given by t = αp with p being the gas pres-
rφ g g where x=hν/(k T) (Shapiro&Teukolsky 1986). The total
sure (Sakimoto&Coroniti 1981). A beta disk model is one B
emittedpowerthroughonesideofthediskwithsurfacetem-
of three options. In the standardShakura&Sunyaev(1973)
model,t p =p +p ,whereasinthe“meanfield”mod- peratureTs isthen
φ tot g rad
∝
eTlhsetrφe∝is√nopgdpetofitn(isteiveeea.gn.swDeornyee&t aDbaovuitsw20h0ic8hfmoroaderelvbieetwte)r. F- = aB4BcTs4 κ¯κ¯abs, (28)
describesstressinaradiation-dominatedaccretiondisk. The r sc
betadiskmodelt =αp assumptionleadstoaradialvelocity where, in a one-zone model, the central density is taken
rφ g
ofplasma as a proxy for the surface density. Here κ¯ denotes
mean opacities. We use here the Rosseland mean opacity
k T
vr≈αcs=αsµBmpc, (22) (ShTahpeireone&rgTyeflukuoxldskeyliv1e9r8e6d)t,oκ¯tahbse=su3r.f2ac×e1is022ZρcTs- 7/2.
wherecsistheisothermalsoundvelocity.Theexpression(22) a cT4
for radial velocity, though not fully self-consistent, helps to Fsurf=F- = B3B τc . (29)
sc
Tidaldisruptionofwhitedwarfsbyintermediatemassblackholes 9
Tosustaintheradiationflowthroughthesurface,thisenergy emitted.Astheaccretionratedropswithtime,theluminosity
flux is equal to the radiated flux. Thus from (28) and (29) (theareaunderthecurve)alsodrops.
we obtain that T T16/9. In a slim disk, cooling does not
s c
∝
balanceheating,F+=F- . Theresidualenergyisstoredwithin
6
theaccretingmaterial.Theenergybalancecanthenbewritten
as
Q=F+- F- , (30)
wheretheresidualenergyperunitsurfaceareaisgivenas
4 1 dT 1 dρ
Q=∆U+A= Ha T4 3 c - c , (31)
3 BB c T dt ρ dt
(cid:18) c c (cid:19)
wheredt =- dr/v . Wesolvetheequations(21)and(22)for
r
thecentraldensityρ . ThecentraltemperatureT isobtained
c c
fromequation30withthehelpof(20),(29)and(31). Given
thecentraldensityandtemperature,we calculatethesurface
temperatureT from(28)and(29)aswellasthecorrespond-
s FIG.10.—Spectrumfortheslimdiskformedfollowingthetidaldisrup-
ingspectrum. tionatvarioustimes. ThespectrashowncorrespondstothecaseB6Suasin
Figure 9 shows the dependence of surface temperature T Figure9. Fromtoptobottom: att=2months,t=8months,t=3years,
s
t=10years,andt=30years,respectively. Theinnerboundaryischosento
onradiusRfortheaccretionratesinferredfromtheoutputof
theB6Susimulation. Latertimescorrespondtoloweraccre- bdeecartearIsSeCsO. =3.83Rg.Thespectrumgetsharderwithtime,buttheluminosity
tionrates. ThetemperatureT is,ingeneral,higherathigher
s
accretionrates, i.e. earliertimes. Thisisbecausetheenergy TheintegratedluminosityisshowninFigure11forthecase
releaseperunittimegoesupwithaccretionratewhilethera- B6Su.Atlatetimes,i.e.t&2years,theluminosityispropor-
˙
diatingsurfaceareastaysconstant. Atverylow M, ourslim tionaltotheaccretionrateanddecreasesast- 5/3.Specifically,
diskaccretionmodelresemblesthestandardthindiskmodel we estimate a luminosity L 0.05M˙ c2 for our disk model.
(Shakura&Sunyaev1973)soTs 0attheISCO.Incontrast, Notice that the estimated lum≃inosity in a fully general rela-
→
Ts at the ISCO saturates at the highest accretion rates. With tivisticmodelisL 0.091M˙ c2 forathindiskwithoutradia-
little energy release, F+ F- , the energy density is propor- ≃
≫ ˙ tivetransfereffects(Bardeenetal.1972). Atearlytimest .
tionaltotheaccretionrateQ=F+ M. Inturn,thescattering 2yearstheluminositysaturatesatabouttheEddingtonvalue
∝ ˙
opacityisalsoproportionaltotheaccretionrateτsc ne M. L =1.4 1041ergs- 1.Theluminosityattimest<2months
TheenergydensityofradiationisproportionaltoT∝c4. I∝nad- isEnddot relia×ble, since the slim disk model may not be an ad-
dition there is a factor of M˙ in both the numerator and the equate representation of accretion at rates M˙ > 4000M˙ .
Edd
denominatoroftheexpression(29)forthesurfaceflux.Inthe Forinstance,whentheoutflowfromtheinnerdiskispresent,
˙
end,Fsurf Mintheradiativelyinefficientregime. theluminositymayfurtherincrease(Owocki&Gayley1997;
∝
Shaviv2001;Begelman2001;Dotan&Shaviv2011).Sucha
computationis,however,beyondthescopeofthepaper.
FIG.9.—SurfacetemperatureoftheslimdiskforthecaseB6Suatvarious
timesafterthedisruption.Fromtoptobottom:att=8months,t=2months,
t=3years,t=10years,andt=30years,respectively. Astimeprogresses
˙ ˙ ˙ ˙
theaccretionratedecreasesfromM 4000MEddtoM 4MEdd.Theinner FIG.11.—Lightcurve(solidline)oftheslimdiskfollowingthetidaldis-
diskboundaryischosentobeatrISC∼O=3.83Rg. ∼ ruption in the case B6Su. The inner boundary is chosen to be at rISCO=
3.83Rg.Theluminositydecreaseswithtime,firstslowly,thensharplybelow
Eddington(horizontaldashedline).Thebehaviorasymptotestoapower-law
The corresponding disk spectra are shown in Figure 10. L=0.05M˙c2 t- 5/3atlowluminosity(inclineddashedline),whenallther-
From top to bottom, νLν as a function of energy are shown malenergyis∝radiatedonthewaytoISCO.Themodelcannotbetrustedif
at2, 8months,then3, 10and30years,respectively.LikeT, luminositysignificantlyexceedsEddingtonvalue.
s
the hardtailof the spectrumalso saturatesat thehighestac-
cretionrates. Ontheotherhand,thelow-energytaildoesnot We consider now the radiative properties of disks from
˙ ˙
saturateatM 4000M . Thisisbecausetheradiationtime theothercaseswithdifferentpenetrationparametersβ,spins
Edd
isstill shorter∼thenor comparabletothe advectiontimescale magnitudes a⋆, and spin orientations. Figure 12 shows the
at large distances from the BH, where the low-energytail is disk spectra at t =1 yr after the disruption for all the cases
10 Haas,Shcherbakov,Bode,&Laguna
studied. In all cases, we assumed a slim disk with inner timet 6safterthedisruption. Theunboundmattertravels
≈
boundaryatr andouterboundaryat30R . essentially along straight lines at t 6 s so Figure 13 ade-
ISCO g
≈
The spectra in Figure 12 are differentbecause of two fac- quately represents the angular distribution of matter at late
tors. The first contributing factor is the accretion rate M˙, times. White color in Figure 13 indicates there is no matter
whichweparameterizebythefallbacktime.Moredebrisfalls traveling in that direction whereas all darker colors indicate
backontotheBHintheB6Sacasethanintheothercases(see thepresenceofgas.
Figure 8). Thereforethe spectrum for the B6Sa case is sys- Theopticaldepthisveryhighatdirectionsthatarenotde-
tematicallybrighter.Similarly,theloweraccretionrateforthe notedbywhite.Theinnerdiskwouldbecompletelyobscured
B6Sicasemakesitsspectrumsystematicallyweakerthanthe alongthosedirections. TheleftpanelinFigure13showsthe
others. Wefoundthatonecanachievecoincidenceofspectra distribution of the scattered debris for the B6Si case. No-
among all a⋆ =0.6 cases with a time shift. That is, at times ticethatalthoughthedebrisisscatteredalloverthesky,there
when the accretion rates M˙ are identical for different simu- aresomepreferentialdirectionsmanifestasasinusoidinthe
lations, the spectra are also identical. The spectrum for the (cosθ,φ) plane. On the other hand, in the case B6Su shown
B6Sacaseataboutt=16monthsisidenticaltothespectrum ontherightpanel,thedebrisisscatteredpreferentiallyinthe
fortheB6Sucaseatt=12monthsandtothespectrumfrom disruptionplane. ThustheorientationofBHspindetermines
the B6Si case at about t =9 months. Also, more matter is thedirectionofdebrisoutflowforultra-closeencounters.
swallowed by the BH initially in the B8Sa case when com- We consider now the situation in which we view a WD-
paredwith the B6Sa case, butat later times, less matter that BHsystemwithanunknownorientation.Recallthatwehave
will be eventually accreted is scattered. Thus the fall back assumed that the disk at late times is fully aligned with the
time islower fortheB8Sa case, resultingin a less luminous initial disruption plane. As discussed before, the disk can,
spectrum. The spectra also vary because of changes in the inprinciple,beviewedbothedge-onorface-on,althoughthe
ISCO radius. Forinstance, the ISCO is furtherfromthe BH edge-ondiskisself-obscuredandhasapeculiarspectrum(see
intheB6S0casewithspina⋆=0. Sincelittleenergyrelease Figure 12). The probabilitythe disk is obscured by the out-
istakingplaceinsider =6R ,itsspectrumissignificantly flowingdebrisisproportionaltothesolidanglesubtendedby
ISCO g
softer. thedebris.Thissolidanglecorrespondstothenormalizedsur-
face area on the (cosθ,φ) plane (see Figure 13). There may
4.2. Obscuration in principle be four different types of obscuration: i) edge-
onself-obscureddiskalsoobscuredbyoutflowingdebris,ii)
Theedge-onviewofaslimdiskisself-obscured.Thespec-
edge-onself-obscureddiskunobscuredbyoutflowingdebris,
trumofaself-obscureddiskisnoteasytodetermine.Wecan
iii)face-ondiskobscuredbyoutflowingdebris,andiv)face-
approximate the spectrum by assuming that the thick outer
oncompletelyunobscureddisk.
edgeatr=30R emitsmodifiedblackbodyradiationtheway
g Whentheheightofthephotosphereequalsthediskheight
the top of the slim disk does. The corresponding edge-on
H=r(inclinationθ=45◦),theprobabilitytoencounteraself-
spectrumisincludedinFigure12astheblackdottedline. It
obscurededge-ondiskis p =71%. Theobscurationprob-
issofter,sinceonlythematerialfarfromtheBHcontributes. edge
abilitybyoutflowingdebrisisdepictedinFigure14forboth
Thesurface areaof the thickouteredgeis large, though,re-
edge-onandface-ondisks.Thepanelsshowtheprobabilityas
sultinginaraisedlow-energytail.
afunctionofdebrislocationatt=6safterthedisruption. At
thattime, the boundarybetweenthe boundandthe unbound
matter is located at 150R . Thus, this obscurationprobabil-
g
ity is also the obscuration probability p for days to years
obs
later. Theobscurationprobabilityp ofanedge-ondisk
obs|edge
canbeinferredfromtheleftpanel. Sincemostdebrisisscat-
teredinthedisruptionplane,thisprobabilityisrelativelyhigh.
Thevaluesreach p 50%, practicallyindependentof
obs|edge
≈
spinvalue,spinorientationorpenetrationfactor. Theobscu-
ration probability of a face-on disk p can be inferred
obs|face
from the right panel. Only the fully misaligned case B6Si
scatterssignificantamountofdebrisperpendiculartothedis-
ruptionplane, which leads to relatively high p 40%
obs|face
≈
forthissimulation. Partiallyandfullyalignedsimulationsall
scattermatterwithin45◦ fromthedisruptionplane,andthus
FIG.12.— Spectra ofourslim diskmodels att =1yr: B6Sucase (top cannot obscure the face-on disk, so that pobs|face =0%. The
solid/black),B6Sicase(bottomsolid/green),B6Sacase(short-dashed/blue), totalobscurationprobabilityis
B8Sa(long-dashed/blue),B6S0case(dot-dashed/red).Differencesaredueto
changingspinandfallbacktime.Dottedlinecorrespondstoanedge-onview p = p p +p (1- p ). (32)
obs obs|face face obs|edge face
ofaslimdiskforb6ap6simulation. Theedge-ondiskhasasofterspectrum
andacomparableluminosity. Thevaluesare pobs 50%fortheB6Sicaseand pobs 30%
≈ ≈
for the B6Su, B6Sa, B8Sa and B6S0 cases. The misaligned
Besides exploringthe properties of a slim disk, we inves- simulation with β =8 yields a marginally higher p =
obs|edge
tigatedwhethertheemissionfromthediskisvisiblethrough 52% comparedto a similar simulation with β =6 for which
theoutflowingdebrissurroundingthedisk. Figure13shows p = 44%. As expected, deeper penetration results in
obs|edge
the(cosθ,φ)directionsinwhichthedebrisisscatteredinthe wider scattering of debris. However, surprisingly, partially
“sky” from the BH point of view. We considered only the alignedandfullyalignedsimulationswithβ=6anda⋆=0.6
material outside a sphere of radius 150R , correspondingto yieldalmostthesameobscurationprobabilities.
g
the boundary between the bound and the unboundmatter at We have previously stated that if there is debris along the