Disc formation from tidal disruptions of stars on eccentric orbits by Schwarzschild black holes

1.1 MB·English

by  Clément Bonnerot

Mon.Not.R.Astron.Soc.000,1–14(2014) Printed19October2015 (MNLATEXstylefilev2.2)
Disc formation from tidal disruptions of stars on eccentric
orbits by Schwarzschild black holes
5 Cl´ement Bonnerot1⋆, Elena M. Rossi1, Giuseppe Lodato2 and Daniel J. Price3
1
1Leiden Observatory,Leiden University,PO Box 9513, 2300 RA, Leiden, the Netherlands
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2Dipartimento di Fisica, Universit`a Degli Studi di Milano, Via Celoria, 16, Milano, 20133, Italy
2
3Monash Centre for Astrophysics and School of Mathematical Sciences, Monash University,Clayton, Vic 3800, Australia

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Accepted ?.Received?;inoriginalform?

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1
ABSTRACT

] The potential of tidal disruption of stars to probe otherwise quiescent supermassive
E
black holes cannot be exploited, if their dynamics is not fully understood. So far,
H
the observational appearance of these events has been derived from analytical ex-
. trapolations of the debris dynamical properties just after disruption. By means of
h
p hydrodynamical simulations, we investigate the subsequent fallback of the stream of
- debris towards the black hole for stars already bound to the black hole on eccentric
o orbits.Wedemonstratethatthedebriscircularizeduetorelativisticapsidalprecession
r
which causes the stream to self-cross. The circularization timescale varies between 1
t
s and 10 times the period of the star, being shorter for more eccentric and/or deeper
a
encounters. This self-crossing leads to the formation of shocks that increase the ther-
[
malenergyofthe debris.Ifthis thermalenergyis efficiently radiatedaway,the debris

2 settle in a narrow ring at the circularization radius with shock-induced luminosities
v of ∼ 10−103LEdd. If instead cooling is impeded, the debris form an extended torus
5
located between the circularization radius and the semi-major axis of the star with
3 heating rates ∼ 1−102LEdd. Extrapolating our results to parabolic orbits, we infer
6 that circularization would occur via the same mechanism in ∼ 1 period of the most
4
bound debris for deeply penetrating encounters to ∼ 10 for grazing ones. We also
0
. anticipate the same effect of the cooling efficiency on the structure of the disc with
1 associated luminosities of ∼ 1−10LEdd and heating rates of ∼ 0.1−1LEdd. In the
0
latter case of inefficient cooling, we deduce a viscous timescale generally shorter than
5
the circularizationtimescale. This suggests an accretionrate through the disc tracing
1
the fallback rate, if viscosity starts acting promptly.
:
v
i Key words: blackhole physics – accretiondiscs – hydrodynamics– galaxies:nuclei.
X
r
a
1 INTRODUCTION of the disc (Giannios & Metzger 2011). A handful of candi-
date TDEs have been detected so far in these bands (e.g.
Most galaxies have been found to contain a supermassive
Esquej et al. 2008; Komossa et al. 2004; Gezari et al. 2009;
black hole (SMBH) at their centre orbited by stars. If one
van Velzen et al. 2011; Zaudereret al. 2011).
of these stars wanders within the tidal radius of the black
hole,thetidalforceoftheblackholeexceedsthestar’sself-
gravity and the star is torn apart. Such an event is called TDEs are powerful tools to detect SMBHs in other-
a tidal disruption event (TDE). After the disruption, the wise quiescent galaxies. Furthermore, they can in principle
stellardebrisevolvetoformanelongatedstreamofgasthat be used to estimate the properties of both the black hole
fallsbacktowardstheblackhole.Inthestandardpictureof and the disrupted star and to probe the physics of accre-
TDEs,thesedebristhencircularizetoformanaccretiondisc tion and relativistic jets. In practice, deriving such con-
wheretheyaccreteviscouslyemittingathermalflaremainly straints from observations is challenging as it requires a
in the UV to soft X-ray band (Lodato & Rossi 2011). This precise understanding of the dynamics of TDEs. The lat-
flare could also be accompanied by a radio signal associ- terhasbeenthefocus ofmanyanalytical andnumericalin-
ated to a relativistic jet originating from the inner region vestigations undertaken since the 80s. A distinctive feature
of the pioneering works by Lacy et al. (1982), Rees (1988),
Evans& Kochanek (1989) and Phinney (1989) is a t−5/3
⋆ E-mail:[email protected] decrease of the rate at which the stellar debris fall back
(cid:13)c 2014RAS
2 Cl´ement Bonnerot, Elena M. Rossi, Giuseppe Lodato and Daniel J. Price
towards the black hole.1 The same slope is generally fit- dard α parametrization (Shakura& Sunyaev 1973). Other
ted to the observed TDE light curves (Esquejet al. 2008; investigations considered the possibility of a geometri-
Gezari et al. 2008, 2009; Cappelluti et al. 2009) although it callythickdisc(Loeb & Ulmer1997;Coughlin & Begelman
is only expected in the X-ray band (Lodato & Rossi 2011). 2014).
It assumes that the accretion rate, and therefore the lumi- Numerical simulations of tidal disruptions
nosity, traces the fallback rate of the debris. In turn, this (Evans& Kochanek 1989; Lodato et al. 2009) have of-
requires that the debris circularize to form a disc and that tenusedsmoothedparticlehydrodynamics(SPH)primarily
this disc is accreted faster than it is fed from the fallback because of its ability to deal with large regions of space
stream.However,themechanismdrivingthecircularization devoid of gas. This techniqueis also well suited tosimulate
is still unknown. the subsequent fallback of the debris towards the black
Various effects are likely to be involved in this pro- hole. The computational cost of such a simulation scales
cess, whose common feature is to dissipate the kinetic en- with the total number of SPH particles used to model the
ergy of the debris, injecting a large amount of thermal en- stream. To accurately follow their evolution, each part of
ergy into the newly formed disc. However, the efficiency of thestream mustcontain enoughparticles. Thiscondition is
this energy transfer is certainly dependent on the mech- hard to fulfill for a long stream because the SPH particles
anism considered. The main candidates are the following are spread on a large volume. The length of the stream
(Evans& Kochanek 1989; Kochanek 1994): increases with the stellar to black hole mass ratio q and
with the eccentricity e of the star. The typical values of
(i) Pancake shock: As the star is disrupted, part of its these parameters are q = 106 and e = 1. This extreme
material is accelerated out of the initial orbital plane. As
value of e comes from the fact that most disrupted stars
a result, the debris inside the stream have a range of incli-
are scattered from the sphere of influence of the SMBH,
nations. Their orbits are therefore vertically focussed and
which is much larger than the tidal radius (Frank & Rees
intersecttheorbitalplaneclosetopericentre.Atthispoint,
1976). For the Milky Way, the ratio of these two distances
thestream isstronglycompressed,leading totheformation is ∼105. However, as noted by Ayalet al. (2000), following
of a pancake shock.
thefallback of the debrisfor these typicalvalues of q and e
(ii) Self-crossing: Whentheyreach pericentre,thedebris
numerically is acomputational burden.Intheir simulation,
experience changes of their apsidal angles driven by rela-
the leading part of the stream is composed of very few
tivistic apsidal precession or hydrodynamical effects. This
SPH particles that come back almost one by one towards
can lead to the self-crossing of the stream: as the leading
the black hole. As a result, the evolution of these particles
parts move away from the black hole after pericentre pas-
cannot be followed accurately. Even if they use a relatively
sage, they collide with the part that is still falling back,
lownumberofparticles(N =4295),thisissueissoextreme
generating shocks.
that it is likely to persist for larger particle numbers. The
(iii) Shearing: As the stream comes back to pericentre,
high computational cost of such a simulation is not limited
the orbits of the debris are radially focussed due to their
to SPH butgeneralizes to other computational techniques.
large range of apocentres but small range of pericentres.
Inthefewotherinvestigationsofthisproblem,eitherq
This effect is similar to a passage into a nozzle. The debris
or e has therefore been lowered. The first option has been
experience shearing at this point as they have a range of
chosen by several authors (Ramirez-Ruiz & Rosswog 2009;
apsidal angles and eccentricities induced by the disruption
Guillochon et al.2014)whoinvestigatedq=103 ande=1.
of the star. This effect is enhanced by relativistic apsidal
The physical motivation in this case is the tidal disruption
precession:astheyhavearangeofpericentredistances,the
of astar byan intermediate mass black hole. Both pancake
orbits of the debris precess by different angles, leading to
shock and shearing were shown to be inefficient at circular-
furthershearing.
izingthedebrisinthiscase.Instead,themaineffectseenin
Ifan accretion discforms from thedebris, itsstructure these simulations is the expansion of the stream caused by
and evolution are two additional uncertainties in the mod- its pericentre passage. Circularization is more likely driven
elsofTDEs.Theydependontherelativeefficiencyofthree byself-crossing eitherinducedbythisexpansion orbyrela-
processes(Evans & Kochanek1989):circularization,viscous tivistic apsidal precession. Hayasaki et al. (2013) made the
accretion and radiative cooling. Denoting the timescales of secondchoice,asinthepresentpaper,whichcorrespondsto
theseprocessesbytcirc,tviscandtcoolrespectively,threelim- tidal disruptions of bound stars. They considered q = 106
iting regimes are to be expected. In the case tvisc < tcirc, and e = 0.8 and found that self-crossing driven by rela-
viscosity may be important during the circularization pro- tivisticprecessionefficientlydrivesthecircularizationofthe
cess.Ifinsteadtvisc>tcirc,accretionbeginsonlyonceadisc debrisin this case.
is formed. In this case, if tcool <tcirc, the disc cools during Several mechanisms are able to put a star on a bound
its formation and is therefore geometrically thin. If instead orbitenteringthetidalradiusofaSMBH.Thefirstinvolves
tcirc < tcool, the disc puffs up while it forms due to its ex- aunequalmassSMBHbinaryinteractingwithasurrounding
cess of thermal energy. Many authors have assumed that stellar cusp. In this situation, bound stars can be scattered
the disc is geometrically thin or slim (Strubbe& Quataert inside the tidal radius of the primary black hole through
2009; Cannizzo & Gehrels 2009; Cannizzo et al. 2011; a combination of Kozai interactions and close encounters
Lodato & Rossi2011;Shen & Matzner2014)usingthestan- withthesecondaryblackhole(Chen et al.2009,2011).The
second mechanism takes place during the coalescence of a
SMBH binary. Due to the anisotropic emission of gravita-
1 Thisratewaslatershowntobedependentonthestellarstruc- tionalwavesinthisphase,theremnantblackholeundergoes
ture(Lodatoetal.2009;Guillochon&Ramirez-Ruiz2013). a kick. It can lead to close encounters between the black
(cid:13)c 2014RAS,MNRAS000,1–14
Disc formation from TDEs 3
SPH simulation
hole and the surrounding stars leading to their tidal dis-
ruptions (Stone& Loeb 2011). Furthermore, mean-motion
resonances can pull surrounding stars inwards during the Relativistic potential
final phase of the SMBH merger (Seto& Muto 2010) en-
hancing the rate of tidal disruptions of bound stars by this Schwarzschild metric
mechanism.Thethirdmechanismtakesplaceafterthetidal
separation of a stellar binary by a black hole, which places
one of the components on an eccentric orbit. Due to both
encounterswiththeotherstarsorbitingtheSMBHandgrav-
itationalwaveemission,itsorbitshrinksandcircularize.Un-
der certain conditions, it can then be tidally disrupted by
theSMBH on a boundorbit (Amaro-Seoane et al. 2012).
Inthispaper,weuseSPHsimulationstoinvestigatethe
circularization process leading to the formation of the disc.
Wealsocharacterizethestructureandevolutionofthisdisc.
Relativistic apsidal precession, involved in the circulariza-
tion process, is treated exactly while most previous stud-
ies (Ramirez-Ruiz& Rosswog 2009; Guillochon et al. 2014;
Hayasaki et al.2013)usedanapproximatetreatmentofthis
effect. The parameters that are likely to play a significant
roleinthecircularizatonprocesshavealsobeenvaried.Two
extremecoolingefficiencies,encodedintheequationofstate
(EOS)ofthegas,havebeenconsidered.Theeffectoftheor-
bit of the star has also been examined, by varying both its
penetration factor and its eccentricity. Our results demon-
strate that circularization is a fast process, happening in a 47 R
g
few orbits of the debris around the black hole, and driven
mostly by relativistic apsidal precession. They also confirm
the expected effect of the cooling efficiency on the struc- RRRRRRRRRRRRRRRRRRRRRRRRRR
tttttttttttttttttttttttttt
ture of the resulting disc. In addition, we found different
channelsofcircularization fordifferentorbitsofthestar.In Figure1.Trajectoryofthecentreofmassofthegasinthesimu-
particular, byincreasing itseccentricity, wegot insightinto lationcorrespondingtomodelRI5e.8(blacksolidline)compared
thecircularizationprocessatworkinthestandardparabolic to that of a test particle on the same orbit in the relativistic
case. potential(reddashedline)andintheSchwarzschildmetric(blue
This paper isorganized as follows. InSection 2,we de- dottedline).Allthetrajectoriesstartattheinitialpositionofthe
scribe the SPH simulations performed. The results of these starindicatedbythebrownpointontherightofthefigure.The
blackarrowspecifiesthedirectionofmotion.Thetrajectoryofthe
simulations are presented in Section 3 varying the follow-
testparticleintherelativisticpotentialandintheSchwarzschild
ing parameters: the gravitational potential, the EOS, the
metricis followedfor 4orbits. That of the centre of mass of the
depthoftheencounterandtheeccentricityofthestar.The
gasinthesimulationisfollowedduringthedisruptionofthestar
effect of the resolution on these results is also examined.
andthe fallback ofthe debris until t/P⋆ =8. The transitionbe-
These results are discussed in Section 4 and compared to tween the disruption and the fallback phase, treated with two
other studies by Shiokawa et al. (2015) and Hayasaki et al. different codes, is indicated by the green dash perpendicular to
(2015) that appeared at thesame time as this paper. thistrajectory.Afterafewpericentrepassages,themotionofthe
gasisaffectedbyhydrodynamicaleffectsandthetrajectoryofits
centreofmassdiffersfromthatofatestparticle.
2 SPH SIMULATIONS
We investigate the tidal disruption of a star of mass M =
⋆
1M⊙ and radius R⋆ = 1R⊙ by a non-rotating black hole
of mass Mh = 106M⊙. In this configuration, the tidal ra-
dius is Rt =R⋆(Mh/M⋆)1/3 =100R⊙. Several initial ellip- tween 100R⊙ and 500R⊙, corresponding to orbital periods
tical orbits of the star are considered. They have different P⋆ =2π GMh/a3⋆ −1/2 between 2.8h and 31h.
pericentre distances Rp defined via the penetration factor Boththedisruptionofthestarandthesubsequentfall-
(cid:0) (cid:1)
β = Rt/Rp by setting β = 1 and β = 5. It corresponds to backofthedebristowards thedisruptionsitearesimulated
pericentre distances Rp = 100R⊙ and Rp = 20R⊙ respec- using SPH. To increase efficiency, the simulations of these
tively.For β=1, only an eccentricity e=0.8 is considered. two phases are performed separately and make use of two
For β = 5, two different values e = 0.8 and e = 0.95 are different codes, each adapted to a specific phase. For the
investigated. For these orbits, all the debris produced by disruption phase, we use a code that takes into account
the disruption stay bound to the black hole. This is gener- self-gravity(Bate et al.1995).Forthefallbackphase,wedo
ally the case if e < ecrit = 1−(2/β)(Mh/M⋆)−1/3, where notconsiderself-gravityandmakeuseofthehighlyefficient
ecrit =0.996 and0.98forβ =5andβ =1respectively.The codePHANTOM(Price & Federrath2010;Lodato & Price
semi-major axis a⋆ =Rp/(1−e) of these orbits ranges be- 2010) that is optimized for studying non-self-gravitating
(cid:13)c 2014RAS,MNRAS000,1–14
4 Cl´ement Bonnerot, Elena M. Rossi, Giuseppe Lodato and Daniel J. Price
problems2.Thischoiceislegitimatebecausetheself-gravity the precession angle for a test particle on a given orbit is
forceisonlyneededduringthedisruptionphasewhereitop- the same in this potential and in the Schwarzschild met-
posesthetidalforceoftheblackhole.Inthefallbackphase, ric. This is an improvement compared to most previous in-
the gravitational interactions between the debris is negligi- vestigations that used an approximate treatment of this ef-
ble compared to their hydrodynamical interactions and the fect(Ramirez-Ruiz& Rosswog2009;Guillochon et al.2014;
tidalforceoftheblackhole.Thedisruptionphaseisfollowed Hayasaki et al. 2013).
untilthemost bounddebriscome backtopericentre. Their For the disruption phase, the star is initially placed at
properties arerecorded at this point andconstitute theini- theapocentreofitsorbit,adistanceRa =(1+e)Rp/(1−e)
tial conditions for the simulation of the fallback phase. As from the black hole. Its initial velocity va is determined by
this paper’s primary focus is on the fallback of the debris, conservation of specific orbital energy and angular momen-
the disruption phase is only simulated to get these initial tum between Ra and Rp. In the Keplerian and relativistic
conditions and not discussed in detail. potentials, thespecific orbital energy is given by
Thecodeusedforthedisruptionphasehasalreadybeen
1 GM
adoptedbyLodato et al.(2009)tosimulatetidaldisruptions εK= 2 vr2+vt2 − Rh, (3)
of stars on parabolic orbits. In thepresent paper,we follow
thesame procedure and numerical setupbut consider ellip- 1(cid:0) R(cid:1) 2 R GM
ttihcealsotarbritassinastpeoaldy.trAoplsiocusspihnegrethweistahmγe m=et5h/o3d,cwonetaminoidnegl εR = 2"(cid:18)R−2Rg(cid:19) vr2+(cid:18)R−2Rg(cid:19)vt2#− Rh, (4)
100Kparticles.However,asexplainedinsubsection3.4,the and thespecific angular momentum by
fallback phase is simulated at a higher resolution with the
stream of debriscontaining 500K particles. lK =Rvt, (5)
fallbaInckorpdhearset,oPrHesAolNveTOthMe,shinocclkusd,etshethceodsteanudseadrdfoarrttihfie- lR = Rvt , (6)
R−2Rg
cial viscosity prescription that depends on the parameters
αAV and βAV. In addition, theMorris & Monaghan (1997) respectively. The initial velocity of the star at apocentre is
switch is implemented to reduce artificial dissipation away therefore
ftrwoomvashluoecsksα.ATVoatnhdisαeAnVd, αacAcVordisinagllotwoeadstoourvcaeryanbdetdweeceany vK = GMh 1/2 2Rp 1/2 (7)
equation. Inmthinis paperm,awxe use αAmVin = 0.01, αAmVax =1 and a (cid:16) Ra (cid:17) (cid:18)Ra+Rp(cid:19)
βAV =2. = GMh 1/2(1−e)1/2, (8)
The orbits considered for the star have pericentre dis- Ra
RtGapMnc=he/sc92c.4ooRmfgpthafeorarbblβlaec=kwhi1tohlaen.tdhMeβorge=rapv5rie.tcaTitsiheolenyr,aelfRoprrea,=diru4esl7aRtRigvgiastn=idc vaR = GRM(cid:16)ah 1/2(cid:17)Ra((Ra+R21p/)2(RRpp(R−a2−Rg2)R+g)2RgRp2)1/2,(9)
(cid:16) (cid:17)
effects must significantly affect the motion of the gas when dependingon the potential.
itpassesatpericentre.Oneoftheseeffectsisrelativisticap- In order to demonstrate the correct implementation of
sidalprecession,amechanisminvolvedinthecircularization the relativistic potential into the SPH codes, we anticipate
processcausingtheorbitofatestparticletoprecessateach the next section and analyse the motion of the gas in the
pericentre passage by a given precession angle. For the or- simulation associated to model RI5e.8. In this model, the
bitsconsidered,theprecessionanglevariesbetween13.5and relativistic potential is used and the orbit of the star has a
89.7 degrees for β = 1 and β = 5 respectively. In order to penetration factor β = 5 and an eccentricity e = 0.8. Fig.
investigatetheinfluenceofrelativisticeffects,theblackhole 1 shows the trajectory of the centre of mass of the gas in
ismodelledbyanexternalpotentialthatiseitherKeplerian the simulation corresponding to model RI5e.8 compared to
orrelativisticbothinthedisruptionandthefallbackphase. that of a test particle on the same orbit in the relativistic
In the following, the variables are labelled by the letter K potential and in the Schwarzschild metric. The trajectory
if they are computed in the Keplerian potential and by the of the test particle in the relativistic potential and in the
letter R if they are computed in the relativistic potential. Schwarzschild metric is followed for 4 orbits. That of the
These potentials are respectively given by centreofmassofthegasisfollowedduringboththedisrup-
ΦK =−GMh, (1) Ttiohneotfratjheectsotrayraonfdthtehetefastllbpaacrktioclfethisetdheebrsiasmuentiinltt/hPe⋆r=ela8-.
R
tivisticpotentialandintheSchwarzchildmetric,confirming
ΦR =−GMh − 2Rg R−Rg v2+ vt2 , (2) thatapsidalprecessionistreatedaccuratelyintherelativis-
R R−2Rg R−2Rg r 2 tic potential as found by Tejeda & Rosswog (2013). During
(cid:18) (cid:19)(cid:20)(cid:18) (cid:19) (cid:21)
the disruption and the beginning of the fallback phase, the
where vr and vt are the radial and tangential velocity of
centreofmassofthegasandthetestparticlehavethesame
a test particle respectively. We adopt this relativistic po-
trajectory, validating the implementation of the potential
tential, designed by Tejeda & Rosswog (2013), because it
into the SPH codes. After a few pericentre passages, the
contains an exact treatment of apsidal precession around
twotrajectoriesstarttodifferasthehydrodynamicaleffects
a non-rotating black hole. In other words, the value of
on the debris become dominant. This evolution will be in-
vestigated in detail in the nextsection.
2 Self-gravityisnowimplementedintoPHANTOM.However,it The accretion onto the black hole is modelled by an
wasnotavailableatthebeginningofthiswork. accretion radius fixed at theinnermost stable circular orbit
(cid:13)c 2014RAS,MNRAS000,1–14
Disc formation from TDEs 5
t / P = 0 t / P = 1.2 t / P = 3 t / P = 8
⋆ ⋆ ⋆ ⋆
R
t
t / P = 0 t / P = 1.2 t / P = 3 t / P = 8
⋆ ⋆ ⋆ ⋆
R
t
6 7 8
log Σ (g/cm2)
Figure 2. Snapshots of the fallbackof the debris at different times t/P⋆ =0,1.2,3 and 8 for models RI5e.8 (upper panel) and KI5e.8
(lower panel). For these models, the period of the star P⋆ is 2.8h. The colours correspond to the column density Σ of the gas whose
valueisindicatedonthecolourbar.Thewhitepointrepresents theblackhole.Thedashedwhitecircleonthelastsnapshot represents
thecircularizationradiusgivenbyequations (12)and(13)fortheKeplerianandrelativisticpotentials respectively.
of 6Rg for a non-rotating black hole. Particles entering this 3 RESULTS
radius are removed from thesimulations.
Inthis section, we present theresults of thesimulations for
thefallback phase3. The timeis scaled bytheperiod of the
star P with the starting point (t/P = 0) being when the
⋆ ⋆
most bound debris of the stream come back to pericentre.
The circularization process is investigated through the evo-
lution of both the position and the specific orbital energy
The simulation of the fallback phase is performed for
of the debris. They will be respectively compared to the so
two different equations of state (EOS) for the gas: locally
called circularization radius and specific circularization en-
isothermal and adiabatic. For the locally isothermal EOS,
ergy.
each SPH particle keeps their initial specific thermal ener-
These two quantities are defined as follows. Before the
gies. Physically, these two EOS represent the two extreme
disruption,thedistributionofangularmomentumofthegas
cases for the rate at which an excess of thermal energy is
insidethestarissharplypeakedaroundthatofthestar.As
radiated away from thegas. For the locally isothermal one,
this distribution is not significantly affected by the disrup-
everyincreaseofthermalenergyisinstantaneouslyradiated
tion,thedebrisstillhavesimilar angular momenta.Assum-
away while for the adiabatic one, none of this energy is ra-
ing that they then move from their initial eccentric orbits
diated.
tocircularorbitseachofthemconservingtheirangularmo-
mentum,thesecircularorbitswillformatsimilarradii.Ital-
lowstodefineacharacteristiccircularizationdistance,called
thecircularization radius, obtained byequatingthespecific
angular momentum of the star, given by equations (5) and
Four parameters are therefore considered to simulate
the fallback of the debris: the potential, the EOS, the pen-
etration factor β andtheeccentricity e.Thevaluesofthese 3 Movies of the simulations pre-
parameters for the eight models investigated in this paper sented in this paper are available at
are shown in table 1. http://home.strw.leidenuniv.nl/˜bonnerot/research.html.
(cid:13)c 2014RAS,MNRAS000,1–14
6 Cl´ement Bonnerot, Elena M. Rossi, Giuseppe Lodato and Daniel J. Price
1
Table 1.Nameandparametersofthedifferentmodels.
t / P⋆ = 0
Model Potential EOS β e ε|)circ 10-1 t / P⋆ = 2
RI5e.8 Relativistic Locallyisothermal 5 0.8 / |⋆ t / P⋆ = 4
KI5e.8 Keplerian Locallyisothermal 5 0.8 M t / P⋆ = 6
RKAA55ee..88 KReelpalteirviiasntic AAddiiaabbaattiicc 55 00..88 εM / d ( 10-2 t / P⋆ = 8
RI1e.8 Relativistic Locallyisothermal 1 0.8 d
RA1e.8 Relativistic Adiabatic 1 0.8
RI5e.95 Relativistic Locallyisothermal 5 0.95 10-3
RA5e.95 Relativistic Adiabatic 5 0.95 0 0.2 0.4 0.6
(ε - ε ) / |ε |
circ circ
1 1
Model RI5e.8
εεε<> - ) / ||circcirc 0.5 Model KI5e.8 εεM / d (M / ||)⋆circ 1100--21
( d
0 10-3
0 2 4 6 8 0.5 0.6 0.7 0.8
t / P⋆ (ε - εcirc) / |εcirc|
Figure 3.Evolutionoftheaveragespecificorbitalenergyofthe Figure 4. Specific orbital energy distributions of the debris at
debrisformodelsRI5e.8andKI5e.8.Forthesemodels,theperiod different times t/P⋆ = 0,2,4,6 and 8 for model RI5e.8 (upper
ofthestarP⋆is2.8h.Theaveragespecificorbitalenergyisshown panel)andKI5e.8(lower panel).Forthese models,theperiodof
relative to the specific circularization energy given by equations the starP⋆ is2.8h. Thespecific orbitalenergy isshown relative
(14)and(15)fortheKeplerianandrelativisticpotentialsrespec- tothespecificcircularizationenergygivenbyequations(14)and
tively. (15)fortheKeplerianandrelativisticpotentials respectively.
(6)evaluatedatapocentre,andthespecificcircularangular
is the strongest for the orbit with β =5 and e=0.8 where
momentum given by
the precession angle reaches 89.7 degrees. This is because
lcK =(GMhR)1/2, (10) thisorbithasthelowestpericentredistanceRp =9.4Rgand,
to a smaller extent, the lowest eccentricity. We investigate
lR = (GMh)1/2R . (11) the role of relativistic precession for a star on this orbit
c (R−3Rg)1/2 by comparing models RI5e.8 and KI5e.8. In model RI5e.8,
the relativistic potential is used and apsidal precession is
This yields a circularization radius of
taken into account. In model KI5e.8, the debris are forced
RcKirc= GRMa2vah2 =(1+e)Rp, (12) tbootmhomveodonelsc,loaseldocoarlblyitsisbotyhuersminaglaEKOeSpliesriaadnoppotetdenftoiarl.thIne
RcRirc= Ra4va2+(Ra4va2(−12G2GMMhh(R(Raa−−22RRg)g2)2Rg+Ra4va2))1/2,(13)gbariss.WcanedbiescsuesesnminodFeilg.R2I5(eu.8ppfierrstp.aTnhele),ewvohliucthiosnhoowfsthsenadpe--
whereva isobtainedfromequations(7)or(9)dependingon shots of their fallback towards the black hole at different
the potential. The associated specific orbital energy, called timest/P =0,1.2,3and8.Thestream remainsunaffected
⋆
specificcircularizationenergy,isequaltothespecificcircular untilt/P ≃1.2,correspondingtoitssecondpericentrepas-
⋆
orbitalenergyevaluatedatthecircularizationradius.Inthe sage. At this time, the stream self-crosses due to apsidal
two potentials, it is given by precession leading to the formation of shocks that convert
kinetic energy into thermal energy. However, as a locally
GM GM
εKcirc=−2RcKirhc =−2(1+eh)Rp, (14) isstoatnhtearnmeoaulsElyOrSemisovuesdedfr,otmhistheexgcaesss.Tthheermneatleeffneecrtgoyfitsheinse-
εR =−GMh RcRirc−4Rg . (15) shocks is therefore to reduce kinetic energy. This results in
circ 2RcRirc (cid:18)RcRirc−3Rg(cid:19) a decrease of the average specific orbital energy of the de-
bris, whose evolution is shown in Fig. 3 (solid black line).
As self-crossings occur at each pericentre passage, this de-
crease continues for t/P & 1.2 through periodic phases.
3.1 Impact of relativistic precession ⋆
These phases of decrease also become progressively steeper
Asmentionedintheprevioussection,relativisticprecession as the self-crossings involve a larger fraction of the stream.
modifiesthetrajectoryofthedebrisatpericentre.Thiseffect Accordingly, the specific orbital energy distribution, shown
(cid:13)c 2014RAS,MNRAS000,1–14
Disc formation from TDEs 7
t / P = 0 t / P = 1.2 t / P = 3 t / P = 8
⋆ ⋆ ⋆ ⋆
R
t
6 7 8
log Σ (g/cm2)
Figure 5.Snapshots ofthefallbackofthedebrisatdifferenttimest/P⋆=0,1.2,3and8formodelRA5e.8.Forthismodel,theperiod
ofthestarP⋆ is2.8h.ThecolourscorrespondtothecolumndensityΣofthegaswhosevalueisindicatedonthecolourbar.Thewhite
pointrepresentstheblackhole.Thedashedwhitecircleonthelastsnapshotrepresentsthecircularizationradiusgivenbyequation(13)
fortherelativisticpotential.Thedotted whitecirclerepresentsthesemi-majoraxisofthestar.
1 1
t / P⋆ = 0
εεdM / d (M / ||)⋆circ 1100--21 tttt //// PPPP⋆⋆⋆⋆ ==== 2468 εεε(<> - ) / ||orbcirccirc -01 ANloln-accreted
Accreted
10-3
0.4 0.6 0.8 1 0 2 4 6 8
(ε - εcirc) / |εcirc| t / P⋆
1 Figure 7. Evolution of the average specific orbital energy of all
thedebris,thenon-accretedandaccretedonesformodelRA5e.8.
Forthismodel,theperiodofthestarP⋆is2.8h.Theaveragespe-
/ l)⋆⋆ 10-1 ceinfiecrgoyrbgiitvaelnenbeyrgeyquisatsihoonw(n15re)lafotrivtehteortehlaetsivpiesctiificcpcoitrecnutliaarli.zation
M
dl (
M / 10-2 value similar to the specific circularization energy (Fig. 3,
d
solid black line). By thistime, they havesettled intoa thin
andnarrow ringof radiuscomparable to thecircularization
10-3 radius (Fig. 2, upperpanel).
1 1.2 1.4 1.6
As can be seen from Figs. 2 (upperpanel) and 3 (solid
l / l⋆
black line), the final specific orbital energy of the debris
Figure 6.Specificorbitalenergydistribution(upperpanel)and is somewhat larger than the specific circularization energy,
specific angular momentum distribution(lower panel) of the de- whichresultsin aringforming slightly outsidethecircular-
bris at different times t/P⋆ = 0,2,4,6 and 8 for model RA5e.8. ization radius. These small discrepancies are due to redis-
Forthismodel,theperiodofthestarP⋆ is2.8h.Thespecificor- tribution of angular momentum between the debris during
bitalenergyisshownrelativetothespecificcircularizationenergy
theshockswhereafraction ofthem(3%at t/P =8) loses
given byequation (15) forthe relativisticpotential. The specific ⋆
enough angular momentum to be accreted onto the black
angularmomentumisshownrelativetothatofthestargivenby
hole. This causes an excess angular momentum shared by
equation(11)evaluated atapocentre.
the remaining debris, that therefore settle into circular or-
bits at radii larger than thecircularization radius. Notably,
in Fig. 4 (upper panel), shifts towards lower energies. Ow- these discrepancies are reduced when the resolution of the
ing to this orbital energy decrease, the debris progressively simulations is increased, as will be shown in subsection 3.4.
move from their initial eccentric orbits to circular orbits. WenowdiscussmodelKI5e.8forwhichtheevolutionof
At t/P ≃8, their average specific orbital energy reaches a thedebris,showninFig.2(lowerpanel),isverydifferent.As
⋆
(cid:13)c 2014RAS,MNRAS000,1–14
8 Cl´ement Bonnerot, Elena M. Rossi, Giuseppe Lodato and Daniel J. Price
1.5 1
t / P = 8
⋆ |z / R| < 0.5
t
0.5 < |z / R| < 1
1 t
z / Rt (F / F)pgr 0.5 11 .<5 <|z |/z R/ tR| <t| <1 .25
0.5
0 0
0 1 2
0 1 2
R / R
t
R / R
t
1
-8 -7.5 -7
log ρ (g/cm3) F / F)cgr 0.5
(
Figure8.Cross-sectionintheR−zplaneofthetorusformedat
t/P⋆ =8formodelRA5e.8.z represents theheightwithrespect
to the midplane and R the cylindrical radius. The distances are
normalized by the tidal radius. The black hole is at the origin. 0
Thecolourscorrespondtothedensityρofthegaswhosevalueis 0 1 2
indicatedonthecolourbar. R / R
t
Figure 9. Azimuthally-averaged ratios of the pressure (upper
thereisnoapsidalprecession,thestreamdoesnotself-cross. panel) and centrifugal (lower panel) forces to the gravitational
Therefore,nokineticenergyisremovedfromthedebrisand forceprojectedinthesphericalradialdirectionasafunctionofthe
theirspecificorbitalenergyremainsconstant(Fig.3,dashed cylindricalradiusR att/P⋆ =8formodel RA5e.8. Theseratios
redline).Theirspecificorbitalenergydistributionalsostays are shown for different heights z with respect to the midplane:
peakedarounditsinitialvaluealthoughitspreadssomewhat |z/Rt| < 0.5, 0.5 < |z/Rt| < 1, 1 < |z/Rt| < 1.5 and 1.5 <
due to tidal effects (Fig. 4, lower panel). Consequently, the |z/Rt|<2.Thedistancesarenormalizedbythetidalradius.
debris do not move to circular orbits and settle instead at
t/P ≃ 8 into an elliptical strip centred around the initial
⋆
orbit of the star (Fig. 2, lower panel). a spread in the distributions of specific orbital energy (up-
ThecomparisonofmodelsRI5e.8andKI5e.8showsthe perpanel)andangularmomentum(lowerpanel).Thelatter
fundamentalroleofapsidalprecession inthecircularization also presents a tail at large angular momenta that reaches,
process. Our more accurate treatment of apsidal precession at t/P ≃8, the specific circular angular momentum at the
⋆
confirms theresults of Hayasakiet al. (2013). semi-major axis of the star, indicated in the lower panel of
Fig. 6 by a vertical black dotted line. Accordingly, the de-
brissettle intoa thick and extendedtoruswith most of the
3.2 Influence of the cooling efficiency
gaslocatedbetweenthecircularization radiusandthesemi-
Themodelsdiscussedintheprevioussubsection(RI5e.8and major axis of the star (Fig. 5). At this time, the majority
KI5e.8) use a locally isothermal EOS for the gas. We now of the debris (91%) are still bound to the black hole as in-
discuss models RA5e.8 and KA5e.8, that use an adiabatic dicated by their negative specific orbital energies (Fig. 6,
EOS for the gas instead. The potential used is relativistic upperpanel).
formodelRA5e.8andKeplerianformodelKA5e.8.Thestar Remarkably,asignificantfractionofthedebrisarebal-
is on thesame orbit as before with β =5 and e=0.8. listicallyaccretedontotheblackholeduringtheformationof
WediscussmodelRA5e.8first.Theevolutionofthede- thetorus.Thefractionofaccretedgasgrowsroughlylinearly
brisastheyfallbacktowardstheblackholeisshowninFig. toreach26%att/P ≃8.Fig.7showstheevolutionofthe
⋆
5 at different times t/P =0,1.2,3 and 8. The stream self- average specific orbital energy of all the debris (solid black
⋆
crosses at t/P ≃1.2 and experiences shocks which convert line),thenon-accreted (dashedred line) and accreted (long
⋆
kinetic energy into thermal energy. This early evolution is dashed blue line) ones. When all the debris are considered,
similartomodelRI5e.8.However,thesubsequentbehaviour their orbital energy decreases due to the shocks where it is
differs. As the EOS is adiabatic, the thermal energy pro- converted into thermal energy. However, it does not reach
duced by the shocks is not removed but kept in the debris. the circularization energy as part of the thermal energy is
Asmoreself-crossingsoccur,morethermalenergyisinjected transferred back into kinetic energy during the expansion.
into the stream which expands under the influence of ther- Theaccreteddebrisarethosewhichexperiencedthelargest
mal pressure. During the expansion, the ordered motion of decrease of kinetic energy during the shocks as can be seen
the stream is suppressed and the debris depart from their from their lower orbital energy. They also have the largest
initial eccentric orbits. This is achieved via a redistribution thermal energy which is therefore advected onto the black
of their orbital parameters which can be seen in Fig. 6 by hole.Thethermalenergyofthetorusisthusreduceddueto
(cid:13)c 2014RAS,MNRAS000,1–14
Disc formation from TDEs 9
both its expansion and the accretion of the hottest debris. 3.3 Dependence on the orbit of the star
Insteadofdecreasing,theorbitalenergyofthenon-accreted
So far, the fallback of the debris has been investigated for
debris stays constant. This is dueto theaccretion of debris
a fixed orbit of the star with β = 5 and e = 0.8. We now
with low orbital energies which results in an excess orbital
considertwoneworbitsobtainedbydecreasingthepenetra-
energy shared among the non-accreted debris. As the ac-
tionfactor(subsection3.3.1)andincreasingtheeccentricity
creted debris also have the lowest angular momenta, their
(subsection 3.3.2). Throughout this section, the potential
accretion leads to an increase of the angular momentum of
is fixed to relativistic. Models RI5e.8 and RAe.8, discussed
theremainingdebriswhichresultsinaslightshiftofthespe-
abovefortheinitialorbitandthispotential,willbeusedas
cific angular momentum distribution to larger values (Fig.
reference.
6, lower panel).
Amorepreciseanalysisofthetorusformedatt/P =8
⋆
can be done by examining its internal configuration. Fig. 8
3.3.1 Decreasing the penetration factor
shows a cross-section in theR−z planewhere z represents
theheightwithrespecttothemidplaneandRthecylindrical
Westartbydecreasingthepenetrationfactortoβ =1keep-
radius. It exhibits funnels around the rotation axis. It also
ingtheeccentricitytoitsinitialvaluee=0.8.Thisdecrease
contains a dense inner region at z/Rt .0.5 and R/Rt ≃ 1 of the penetration factor corresponds to an increase of the
that corresponds to the semi-major axis of the star. This
pericentre distance from Rp = 9.4Rg to Rp = 47Rg. We
dense region is surrounded by a more diffuse one that ex-
investigate the fallback of the debris for a star on this new
tendsfromclosetotheblackholetoR/Rt ≃2.5andz/Rt ≃ orbit by discussing models RI1e.8 and RA1e.8. A locally
1.5. To assess the internal equilibrium of the torus, we plot
isothermalEOSisusedformodelRI1e.8whileanadiabatic
in Fig. 9 azimuthally-averaged ratios of the pressure force
EOS is used for model RA1e.8. The relativistic potential is
(upperpanel)andcentrifugalforce(lowerpanel)tothegrav-
used in both models.
itational force projected in the spherical radial direction as
We discuss model RI1e.8 first. Snapshots of the evolu-
afunctionofRandfordifferentintervalsofz:|z/Rt|<0.5, tion of the debris are shown in Fig. 10 at different times
0.5 < |z/Rt| < 1, 1 < |z/Rt| < 1.5 and 1.5 < |z/Rt| < 2. t/P = 0,2.2,4,6,8 and 16. The first self-crossing of the
⋆
The ratio of the pressure force to the gravitational force is
stream occurs at t/P = 2.2 that is at its third pericentre
⋆
roughly (Fp/Fg)r ≃ 0.2 in the entire torus implying that passage. It causes the formation of shocks that reduce the
it is not hydrostatically supported. However, the ratio of
kinetic energy of the debris as a locally isothermal EOS is
thecentrifugal force to the gravitational force is larger. For
used. However, as the stream passes further from the black
|z/Rt| < 0.5, it presents a maximum of (Fc/Fg)r ≃ 1 at hole,relativistic precession is weakerthanfor model RI5e.8
R/Rt ≃0.3 and decreases at larger radii. The same depen- withaprecessionangledecreasingfrom89.7to13.5degrees.
dence exists for regions further from the midplane but for
Consequently,theshockshappenfurtherfromtheblackhole
lowermaximaof(Fc/Fg)r ≃0.5−0.8. Therefore, thistorus andinvolvepartsofthestreamwithlowerrelativevelocities.
is mostly centrifugally supported against gravity with this
For this reason, they are less efficient at removing kinetic
supportbeingstrongerinitsinnerregion.Astheyarebound
energy from the debris. This results in a slower and more
totheblackhole,theregionsthatarenotsupportedagainst
gradual decrease of their average specific orbital energy, as
gravitywillstopexpandingandcollapseatalatertime.This
seen from Fig. 11 that shows itsevolution for model RI5e.8
collapse is likely to cause the formation shocks in theouter
(solidblackline)andRI1e.8(dashedredline).Att/P ≃16,
⋆
part of the torus which would increase hydrostatic support
theaveragespecificorbitalenergyofthedebrisstabilizesat
in thisregion.
avaluesimilartothecircularizationenergy(Fig.11,dashed
FormodelKA5e.8,thestreamdoesnotexperienceapsi- red line) and the debris settle into a thin and narrow ring
dalprecession. However,it isheatedwhen itpasses at peri- ofradiuscomparable tothecircularization radius(Fig.10).
centre.Thisisduetotheformationofthepancakeshockde- As for model RI5e.8, the origin of the small discrepancies
scribed in the Introduction.As aresult, it expandsroughly with the specific circularization energy and the circulariza-
byafactorof2ateachpericentrepassagebothintheradial tionradiusareduetotheaccretion ontotheblackholeofa
and vertical directions. It causes the stream to self-cross at fraction of the debris (3 % at t/P = 16) with low angular
⋆
t/P⋆ ≃ 4 that is after its fifth passage at pericentre. The momenta.
subsequent evolution is similar to model RA5e.8. The de- For model RA1e.8, the evolution of the debris is very
bris settle into a thick and extended torus at t/P⋆ ≃ 10. similar to model RA5.8. The shocks produced by the self-
This channel of disc formation is similar to that found by crossings of the stream leads to its expansion. The debris
Ramirez-Ruiz& Rosswog (2009). Therefore, relativistic ap- settle into a torus with most of them located between the
sidal precession is not the only factor that can lead to cir- circularization radius and the semi-major axis of the star.
cular orbits of the debris, but it is the most efficient and Theonlyeffectofmodifyingtheorbitistochangethevalue
operates regardless of the EOS used for thegas. of these two limiting radii.
AscanbeseenbycomparingmodelsRI5e.8andRA5.8,
the cooling efficiency determines the structure of the disc
formed duringthecircularization process. Whileathinand
3.3.2 Increasing the eccentricity
narrow ring forms at the location of the circularization ra-
dius for an efficient cooling, an inefficient cooling leads in- Wenowkeepthepenetrationfactortoitsinitialvalueβ =5
steadtotheformationofathickandextendedtoruslocated while increasing the eccentricity to e = 0.95. The fallback
between the circularization radius and the semi-major axis ofthedebrisisinvestigated for thisneworbit bydiscussing
of the star. modelsRI5e.95andRA5e.95whichuserespectivelyalocally
(cid:13)c 2014RAS,MNRAS000,1–14
10 Cl´ement Bonnerot, Elena M. Rossi, Giuseppe Lodato and Daniel J. Price
t / P = 0 t / P = 2.2 t / P = 4
⋆ ⋆ ⋆
R
t
t / P = 6 t / P = 8 t / P = 16
⋆ ⋆ ⋆
5 6 7
log Σ (g/cm2)
Figure 10.Snapshots ofthefallbackofthedebrisatdifferenttimes t/P⋆ =0,2.2,4,6,8and16formodelsRI1e.8.Forthismodel,the
periodofthestarP⋆ is31h.ThecolourscorrespondtothecolumndensityΣofthegaswhosevalueisindicatedonthecolourbar.The
whitepointrepresentstheblackhole.Thedashedwhitecircleonthelastsnapshotrepresentsthecircularizationradiusgivenbyequation
(13)fortherelativisticpotential. Apsidalprecessionisweaker forthismodelthan formodelRI5e.8owingtoalargerpericentre, which
causesthestreamtoself-crossfurtheroutfromtheblackhole.Atthislocation,relativevelocitiesarelower,weakeningtheshocks.The
debristhereforemoveslowertocircularorbits.
isothermal andanadiabatic EOS.Therelativistic potential 1
is adopted for both of them. Model RI5e.8
s0h,0o.t1sW5o,ef0.td2hi5es,c0fua.s3lsl3b,fia0cr.ks5toamfntdohde1e.ldeTRbhIr5eise.sa9tt5re.daiFmffiegrfi.ern1st2ttcsirmhooeswssests/sPint⋆saep=l-f εε) / ||circcirc 0.5 MMooddeell RRII15ee..895
at t/P⋆ ≃ 0.15 just after the first passage of its leading > -
ε
partatpericentre.Whenthisoccurs,most ofthedebrisare <
(
still falling back towards the black hole away from the self-
crossingpoint.ThisisdifferentfrommodelRI5e.8,forwhich
0
thisself-crossing happenslaterandwhenmostofthedebris 0 5 10 15
are already beyond the self-crossing point (see Fig. 2). The t / P⋆
reason for these differences is that the stream is longer for
model RI5e.95 than for model RI5e.8 owing to the larger Figure11.Evolutionoftheaveragespecificorbitalenergyofthe
debris for models RI5e.8, RI1e.8 and RI5e.95. For these models,
eccentricity. After the first self-crossing, the debris located
beyondtheself-crossingpointareexpelledleavingtherestof the periods of the star P⋆ are respectively 2.8h, 31h and 22h.
Theaveragespecificorbitalenergyisshownrelativetothespecific
thestreamfreetomovearoundtheblackhole.Thisinduces
circularization energy given by equation (15) for the relativistic
a second self-crossing at t/P ≃ 0.33. These self-crossings
⋆ potential.
createshocksthat removekineticenergyfrom thedebrisas
(cid:13)c 2014RAS,MNRAS000,1–14