Gas squeezing during the merger of a supermassive black hole binary

0.43 MB·

by  Alice Cerioli

MNRAS000,1–11(2016) Preprint18January2016 CompiledusingMNRASLATEXstylefilev3.0
Gas squeezing during the merger of a supermassive black
hole binary
Alice Cerioli1⋆, Giuseppe Lodato1† and Daniel J. Price2
1Dipartimento di Fisica, Universita`degli Studi di Milano, Via Celoria 16, 20133, Milano, Italy
2Monash Centre for Astrophysics and School of Physics& Astronomy, Monash University,Vic 3800, Australia
6
1
0 Accepted2016January4.Received2015December21;inoriginalform2015July28
2

n
ABSTRACT
a
J We study accretion rates during the gravitational wave-driven merger of a binary
supermassive black hole embedded in an accretion disc, formed by gas driven to the
4
1 centre of the galaxy. We use 3D simulations performed with phantom, a Smoothed
ParticleHydrodynamicscode.Contrarytopreviousinvestigations,weshowthatthere

] is evidence of a“squeezingphenomenon”,caused by the compressionof the inner disc
E
gaswhenthesecondaryblackholespiralstowardstheprimary.Thiscausesanincrease
H in the accretion rates that always exceed the Eddington rate. We have studied the
. main features of the phenomenon for a mass ratio q = 10−3 between the black holes,
h
including the effects of numerical resolution, the secondary accretion radius and the
p
disc thickness. With our disc model with a low aspect ratio, we show that the mass
-
o expelled from the orbit of the secondary is negligible (< 5% of the initial disc mass),
r differentto the findings of previous2Dsimulations with thickerdiscs. The increasein
t
s the accretion rates in the last stages of the merger leads to an increase in luminosity,
a
making it possible to detect an electromagnetic precursor of the gravitational wave
[
signal emitted by the coalescence.

1
Key words: accretion,accretiondiscs–blackholephysics–hydrodynamics–meth-
v
6 ods: numerical – gravitationalwaves
7
7
3
0
1 INTRODUCTION orbital evolution of the binary (Armitage & Natarajan
.
1 2002; Escala et al. 2005; Dottiet al. 2007; Cuadra et al.
0 There is clear evidence that most galaxies harbour 2009; Lodato et al. 2009) and since the two black holes
6 a supermassive black hole (Magorrian et al. 1998;
are expected to interact strongly with the surrounding
1 Ferrarese & Merritt 2000). The current hierarchical
material, we can investigate the possibility of identifying
: formation models state that our Universe evolves through
v electromagnetic signatures of such interaction.
the merger of smaller galaxies to form larger ones.
i
X During a galaxy-galaxy merger, the two supermassive The theoretical modelling of electromagnetic sig-
nals from merging supermassive black hole binaries has
r black holes approach each other through dynamical fric-
a tion and form a bound binary (Begelman et al. 1980; developed in the last few years. Numerous mechanisms
for eliciting an electromagnetic signature from coalesc-
Milosavljevi´c & Merritt 2001). Supermassive black hole
ing binaries and recoiling remnants have been proposed
pairs evolve by losing their angular momentum and energy
(Armitage & Natarajan 2002; Milosavljevi´c & Phinney
byacombinationofvariousprocesses,includinginteraction
2005; Shapiro 2010); see Schnittman (2013) for a review.
with a gas disc, until they reach a separation small enough
This rapid theoretical development is justified by the hope
that the main mechanism of shrinking can take over as
ofdetectinggravitationalwavessignalsbytheimprovement
gravitational radiation (Peters 1964). Mergers of gas rich
ofgroundbasedobservatoriesandtheprospectsofspacede-
galaxies, through gravitational torques, can drive a large
tectors,such aseLISA.Groundbased detectionsof galactic
amount of gas in thecore of theremnant galaxy, where the
events will provide a test of the difficult task of extracting
supermassive black hole pair resides. Gas rapidly settles
physicalquantitiesfrom waveforms. However, thedetection
into a thin accretion disc surrounding the two bound
of gravitational waves from merging supermassive binaries
black holes. The gas rich environment may accelerate the
will dependon thelaunch of space-based interferometers.
Independent of the detection of gravitational waves,
⋆ E-mail:[email protected] mergingsupermassiveblackholesmightbealreadydetected
† E-mail:[email protected] intheelectromagneticdomainifsomepeculiarfeatureindi-
(cid:13)c 2016TheAuthors
2 A. Cerioli , G. Lodato and D. J. Price
cating the merger can be identified. In this paper we study where G is the gravitational constant and c is the speed of
theaccretionratesoftheblackholesimmediatelybeforethe light.Thisdifferentialequationcanbesolvedanalyticallyto
final coalescence, to understand if the gas present between give
the two is squeezed and finally accreted very rapidly or if
t 1/4
it manages to escape immediate accretion. The possible in- a(t)=a 1− , (2)
creaseoftheaccretion ratesinthelast stages ofthemerger 0 τ
(cid:18) (cid:19)
can lead to an increase in luminosity, making it possible where τ is the time-scale to reach merger from an initial
to detect an electromagnetic precursor of the gravitational separation a=a , definedby
0
wavessignalemittedbythecoalescence. Thecoincidentde-
tection of both the gravitational wave and electromagnetic τ ≡ 5 c5a40 . (3)
signatures allows for a possible high precision measurement 256G3Mp3q(1+q)
of black hole masses, spins and the properties of the host
Since the sign of (1) is negative, the emission of gravi-
environment.
tational waves causes the binary separation to decrease.
Electromagnetic identification ofthesourcewould con-
The time-scale in eq(3) depends strongly on the separa-
firm themerger and accretion physics. This idea was firstly tion a . For q = 10−3, τ becomes comparable to the Hub-
0
investigated by Armitage & Natarajan (2002), using a one ble time only at a ≈ 10−2pc. At larger separations, other
0
dimensionalmodel.Theyfoundevidenceofsuper-Eddington
mechanisms need to be identified to allow the orbital de-
flares in the luminosity of the binary, caused by the rapid
cay of the binary. Dynamical friction against the stellar
accretion of theinnerdisc.Two dimensional simulationsby
background is effective only for large separations a ≥ 1pc
Baruteau et al. (2012) suggested that the gas in the inner
(Begelman et al. 1980; Milosavljevi´c & Merritt 2001). The
disc could actually flow across the secondary’s gap through
evolution from a ≈ 1pc is probably due to the interac-
horseshoe orbits back to the outer disc, excluding the pos-
tion with a gas disc (Escala et al. 2005; Dotti et al. 2007;
sibility of enhanced accretion luminosity. Here we broaden
Lodato et al. 2009; Tazzari & Lodato 2015). Angular mo-
theanalysisprovidinganewpictureofthisphenomenonex-
mentumexchangewithagasdiscdominatesthebinaryevo-
aminingthecaseofthinnerdiscs.Wemodeltheevolutionof
lutiondowntotheso-called“decouplingradius”,wheretidal
the inner accretion disc using three dimensional Smoothed
torquesequalthetorqueduetogravitationalwaveemission
Particle Hydrodynamics(SPH)simulations.
(Milosavljevi´c & Phinney 2005).
Thepaperisorganizedasfollows:inSection2weintro-
Beyond decoupling the gas outside the secondary’s or-
duce the physics behind the gravitational decay of a super-
bit is unable to evolve on the rapid time-scale of gravita-
massive black hole binary due to emission of gravitational
tional decay so it remains frozen behind the secondary. A
radiation.InSection3wegiveanoverviewon initialcondi-
simple estimate of the decoupling radius can be derived
tionsanddescribethesystemimplementedinournumerical
by equating the the gravitational decay migration rate to
simulations. We show our results in Section 4, where we
the viscous radial drift velocity, assumed to be the ve-
illustrate the effects of increasing the resolution, changing
locity of the secondary black hole in the disc dominated
the disc thickness and lowering the accretion radius of the
phase (Armitage & Natarajan 2002), or by equating the
smaller black hole. Wediscuss ourresults anddraw conclu-
gravitational decay time τ to the viscous time-scale t .
ν
sions in Sections5 and 6.
Milosavljevi´c & Phinney(2005)self-consistentlysolveasta-
ble standard α-model for the binary-disc evolution, and
equatingthegravitationaldecaytime-scalewiththeshrink-
2 PHYSICAL PROBLEM
ingtime of gas they estimate thedecouplingradius as
When a binary is formed by unequal mass black holes, i.e.
when the mass ratio q = Ms/Mp between the secondary adec ≈117 α −0.34 Mp 0.08 4q 0.42Q,
and primary mass is small, the secondary black hole exerts GMp/c2 (cid:16)0.1(cid:17) (cid:18)106M⊙(cid:19) (cid:18)(1+q)2(cid:19)
a tidal torque on the disc, pushing away gas from its orbit (4)
and creating a gap in the accretion disc around the pri-
where Q is a factor of order unity. After the decoupling,
mary.The disc is then split intoan inner disc and an outer
τ < t , so the inner edge of the outer disc cannot fol-
disc(Lin & Papaloizou 1979).This particular configuration ν
low the secondary as it rapidly spirals in. Therefore, if be-
istypicalofprotostellardiscs,wherethegapopeningprocess
tweentheblackholesthereisnogas, thefinalmergerisex-
is associated with planetary migration (Lin & Papaloizou
pectedtooccurinvacuum(Milosavljevi´c & Phinney2005).
1986).Ifthetwoblack holesreach separations smaller than
On the other hand, if gas is still present at least around
approximately 10−3 pc,theystart to lose energy andangu-
the primary black hole, the tidal torques of the secondary
lar momentum because of the emission of gravitational ra-
force the gas to be squeezed and rapidly accreted by the
diation (Peters & Mathews 1963). Peters (1964) found the
primary, causing a sudden enhancement of the accretion
equationsfor theaveragetime variationsof theeccentricity
rate(Armitage& Natarajan 2002;Chang et al. 2010).This
e, the separation a, the energy and theangular momentum
would increase the disc’s bolometric luminosity, constitut-
ofthebinaryinthegenericcasee6=0.Ifthebinaryreaches
ing an electromagnetic precursor to thegravitational waves
coalescence through the interaction with a disc, we expect
signal emitted by the final merger. This is the possibil-
anyeccentricitytobedampedbydissipativeeffects.Wethus
ity considered by Armitage & Natarajan (2002), who per-
consider thecase e=0, where
formedonedimensionalsimulationstomodelthesecondary
64G3M3q(1+q) migration in a gaseous disc. However, twodimensional sim-
a˙(t)=− p , (1)
5 c5a3(t) ulations performed by Baruteau et al. (2012) suggest that
MNRAS000,1–11(2016)
Gas squeezing during the merger of a SMBHB 3
duringthegravitationaldecaytheinnergasisprogressively fixedduringthecomputation;thismeansthatthetwoblack
funnelled to the outer disc following horseshoe trajectories holes donot changetheir mass following accretion.
passingthroughthesecondaryorbit.Thiswouldimplythat The accretion radius of the primary black hole R
acc,p
there should be no significant increase in the binary lumi- is taken to be equal to the inner radius of the accretion
nosity prior to the merger with the choice of their initial disc R , so that all mass approaching the black hole will
in
conditions. We discuss this issue again in Section 4, in the be eventually accreted. In our simulations we chose R =
in
light of theresults of our simulations. 2GM /c2. Then, we can assume that this is the position
p
of the last stable orbit for a non-maximally rotating black
hole. However, no effects related to a Kerr black hole were
3 INITIAL CONDITIONS taken into account in the implementation. The secondary
accretionradiusR ischosenforcomputationalefficiency,
acc,s
Theevolutionoftheaccretiondiscandtheblackholeismod-
and we will discuss the effect of varying it in Section 4.4.
elledusingSmoothedParticleHydrodynamics(SPH),seefor In phantom, when particles approach a black hole by a
exampleMonaghan(1992,2005);Price(2012)andRosswog
distance less than its accretion radius, they are considered
(2009). SPH is a numerical method to solve the hydrody-
accreted. The mass and the momentum of the particles are
namics equations in Lagrangian form, interpolating quanti-
not added to the black holes because we are treating the
ties over a finite set of N point particles. Thus, resolution
binary as an external potential. However, this is negligible
dependson thenumberof particles, and it is automatically
anyway since the black holes are much more massive than
higher in denser regions. The simulations presented in this
thegas disc we considered.
paperhavebeencarriedoutwiththethreedimensionalcode
The gas follows a locally isothermal equation of state,
phantom, written by Daniel Price (Lodato & Price 2010;
where the sound speed c is described by the power law
s
Price & Federrath2010). c = c R−γ with γ = 0.75 and where R is the radial
s s,0
The system consists of an unequal mass black hole bi-
distance from the centre of mass of the binary in cylin-
nary on a circular orbit (with masses M and M for the
p s drical coordinates. The disc surface density is modelled as
primary and secondary black hole, respectively) and of an Σ=Σ R−δ, with δ=1.5.
0
initially axisymmetric circumprimary disc that lies in the
WemodelaShakura& Sunyaev(1973)viscositybyus-
same plane of the two black holes. The relative position r
ing the SPH artificial viscosity formalism. The value of the
ofthesecondaryblackholewithrespect totheprimaryhas
viscosity parameter is α =0.01, set bytherelation
SS
components r = a(t)cosϑ(t) and r = a(t)sinϑ(t). The
x y
evolution of theangular position ϑ(t)is given by
1 hhi
α = αAV , (7)
SS 10 H
G(M +M )
ϑ˙(t)= p s , (5)
s a(t)3 where αAV is the coefficient of artificial viscosity
which can beintegrated to give (Lodato & Price 2010). The other two quantities in equa-
tion(7)arehhi,theazimuthallyaveragedsmoothinglength
8τ G(M +M ) t 5/8
ϑ(t)=− p s 1− . (6) of SPH particles and H = cs/Ω, the disc thickness defined
5 s a30 (cid:18) τ(cid:19) by the sound speed and the Keplerian velocity Ω. The cho-
In the centre of mass system, the positions of the pri- sen disc model with δ = 1.5 and γ = 0.75 ensures that the
mary and the secondary black hole are, respectively, r′ = discis uniformly resolved since we have:
p
−M /M r and r′ =M /M r.
sOurtostimulatiosns staprt wtiotth a circumprimary disc only. h∝ρ−1/3 ∝ Σ −1/3 ∝R3/4. (8)
We decided to neglect the outer circumbinary disc because H
(cid:18) (cid:19)
we always choose an initial separation that is smaller than
the decoupling radius of the system. The outer disc would The smoothing length h is thus proportional to the disc
befrozenfarbehindthesecondary,soitwouldnotaffectthe thickness H ∝ R3/4, so that the ratio hhi/H is constant
dynamics inside the secondary’s orbit. However, during the with radius. The ratio hhi/H is a good estimator of how
simulations, in some cases we can see that a small amount well our simulation resolves verticaldisc scale height.
ofgas(theexactfractionwillbediscussedlater)isfunnelled For our chosen value of α and q, the decoupling ra-
outsidetheorbitofthesecondary,formingalowmassouter dius would be adec ≈25GMp/c2, with little dependenceon
disc. the mass scale. We have experimented with several choices
The mass ratio q=Ms/Mp has been fixed to q=10−3 of the initial separation a0. We have noticed that during
for all of the simulations. This choice is mostly dictated the early evolution of the system there is no or little evo-
by the available computational resources. A higher q cor- lution in the accretion rate onto the primary until the bi-
responds to a larger decoupling radius, but this is roughly nary separation decreases to ≈ 5GMp/c2. In order to save
balancedbyacorrespondingdecreaseinthenumberoforbits computational time, we decided to start our simulations at
required to merge from a given initial separation. However, a0 = 4.75GMp/c2. We have also performed a number of
ahigherq alsoleadstoamuchstrongerperturbationtothe simulations with a fixed, non decaying binary, in order to
circumprimarydisc,sinceahighermasssecondaryproduces assess the radius at which the circumprimary disc is trun-
awidergapinthedisc.Thismeansthathighermasssecon- catedbythesecondary’stidalforce.Foraninitialseparation
dariesmustbeplacedfurtherawayinordertoavoidastrong a0 =4.75GMp/c2, this turns out to be Rout =0.86a0, that
perturbationtotheinitialconditions,leadingtoaconsider- we takeas outer discradius in all oursimulations.
ablylongercomputationaltimetomerger.Themassratiois Inthefollowing,weusecodeunitsformass,lengthand
MNRAS000,1–11(2016)
4 A. Cerioli , G. Lodato and D. J. Price
SIMULATION Npart a0 αSS H/R(R=Rin) Rin=Racc,p Racc,s Rout αAV hhi/H
ref 5·105 4.75 0.01 0.01 2 0.2 4.1 0.1114 0.8975
resol 1 1·106 4.75 0.01 0.01 2 0.2 4.1 0.1404 0.7124
resol 2 2·106 4.75 0.01 0.01 2 0.2 4.1 0.1768 0.5655
asp 1 5·105 4.75 0.01 0.015 2 0.2 4.1 0.1460 0.6850
asp 2 5·105 4.75 0.01 0.02 2 0.2 4.1 0.1768 0.5655
asp 3 5·105 4.75 0.01 0.05 2 0.2 4.1 0.3259 0.3068
asp 4 5·105 4.75 0.01 0.0095 2 0.2 4.1 0.1077 0.9287
sec 1 5·105 4.75 0.01 0.01 2 0.05 4.1 0.1114 0.8975
sec 2 5·105 4.75 0.01 0.01 2 0.01 4.1 0.1114 0.8975
Table 1. Summary of the simulations performed, showing the name of the simulation, the resolution given by the number of SPH
particlesNpart,theinitialseparationa0 betweentheblackholes,thedesiredShakura-SunyaevviscosityparameterαSS,thevalueofthe
disc aspect ratio H/R at the radius R = Rin, the inner disc edge equal to the primary accretion radius Rin = Racc,p, the secondary
accretionradiusRacc,s,theouterdiscedgeRout,thevalueoftheartificialviscosityparameterαAVandtheratiohhi/H,theapproximated
meansmoothinglengthoverthediscscaleheight.
time given by: t[days]
0 10 20 30 40 50
10−2
M =M ,
0 p M˙p/Mdisc
R0 = GcM2p, (9) M˙s/Mdisc
10−3
GM
T0 = c3p. M)]p
G
Duetofinitesizeof blackholeaccretion radii,thefinal 3c/( 10−4
merger will occur at a time M[disc
τ =τ(a )−τ(a¯) (10) /
m 0 ˙M
10−5
where a¯ = R +R . We refer to this as the merger
acc,p acc,s
time in the following discussion. For example, considering
an initial separation a =4.75, R =2 and R =0.2,
0 acc,p acc,s 10−6
the decay time is τm = 9475.7, corresponding to ≈ 54 days 0 2000 4000 6000 8000
and to 145.7 Torb(a0), where Torb(a0) is the orbital period t[GMp/c3]
of thesecondary black hole at theinitial separation.
Figure 1. Primary and secondary accretion rates of simulation
Table 1 summarizes the initial conditions, giving the
resol 2.Timetisexpressedincodeunitsandtheaccretionrates
nameofthesimulation,theresolution(givenbythenumber
are normalised with disc mass Mdisc. The sharp spikes mark
of SPH particles Npart), the initial separation a0 between episodes of strong and sudden accretion. The upper axis shows
the black holes, the desired Shakura-Sunyaev viscosity pa- timeexpressedindaysforaprimarymassMp=108M⊙.
rameter α ,thevalueof thediscaspect ratio H/R at R ,
SS in
the inner edge of the disc (equal to the primary accretion
radius R =R ), the secondary accretion radius R ,
in acc,p acc,s 4.1.1 The squeezing phenomenon
the disc outer edge R , the value of the artificial viscos-
out
ityparameterαAV setbythecodeandtheratiohhi/H,the Inourreferencesimulationsthesecondaryblackholerapidly
average smoothing length overthedisc scale height. shrinks towards the primary, squeezing the inner disc gas.
Figure 1 shows the primary and secondary accretion rates
M˙ and M˙ , normalised by the initial disc mass M . Af-
p s disc
terashortinitialtransientwheretheinitiallyaxisymmetric
4 RESULTS disc readjusts to the imposed tidal torques, the accretion
rates approach a quasi steady configuration where M˙ is
4.1 Reference run p
almost one order of magnitude higher than M˙ . As the bi-
s
We first present the results of our reference calculation, nary shrinks, the secondary accretion rate starts to grow
where we set H/R(R = R ) = 0.01 (Table 1). Then, we approaching the primary’s value. As the merger eventually
in
study the effects of increasing the resolution (Section 4.2), occurs, thereis a strong spikein theprimary rate M˙ , that
p
changingthediscaspectratio(Section4.3)andloweringthe reaches a maximum value almost two orders of magnitude
secondary accretion radius(Section 4.4). above the quiescent value. The inner disc shows a spiral
The reference simulation ref will always be consid- structure due to the tidal interaction with the secondary.
ered as the starting point for the other simulations. How- Thisimpliesthatthesubsequentaccretion ofregionsofdif-
ever, since simulation ref has been run with low resolution ferentdensitymodulatestheaccretionratesandthesurface
(N =5·105),wealsopresenttheresultsofthesimulation densitycurves.Thisexplainsthepresenceoftwomaximain
part
resol 2 with N =2·106. The qualitative results are the the plot of the accretion rate of the primary black hole M˙
part p
same in both simulations. (Figure 1).
MNRAS000,1–11(2016)
Gas squeezing during the merger of a SMBHB 5
t=7200 t=8000 t=8600
t=8800 t=9000 t=9250
GM / c2
p
-15 -14 -13 -12 -11 -10
log column density
Figure 2. Snapshots of column density in simulation resol 2. The two black holes are indicated by circles, whose size is proportional
to the accretion radius (Racc,p = 2GMp/c2 and Racc,s = 0.2GMp/c2). In the snapshots time is expressed in code units. The binary
separationdecreases froma=3.44GMp/c2 att=7200GMp/c3 toa=2.43GMp/c2 att=9250GMp/c3
During the decay, the secondary black hole sweeps up
1.6 thegastowardstheprimary,causingtheformationofspikes
t=5000,a=3.99 in the disc surface density. Figure 3 shows the evolution of
1.4
t=6250,a=3.71 the azimuthally averaged surface density Σ(R) at different
units] 1.2 tt==77050205,,aa==33..5303 tbiimnaersy. Tsehpeasruatrifoanceddeecnresiatsyesinfcrroemaseasbbouyta4fatcoto2r.6o.fF3igausrteh2e
ry 1.0 t=8025,a=3.14 shows snapshots of column density of the simulation in the
a
arbitr 0.8 tt==88567255,,aa==22..8896 lpolwanmexayss.Touhteecrodloiuscrstchaaltefhoarsmbeedenouchtsoisdeensiencoornddearrtyo’sseoerbthite.
11−[ 0.6 t=8725,a=2.81 Thesnapshotsattimest=8600andt=9000coincidewith
10 t=9075,a=2.57 thetwo spikesin theaccretion rate M˙ .
× p
R) 0.4
Σ( The discsurface densityspikes arecaused by therapid
0.2
shrinking of the secondary. The inner disc is unable to vis-
cously respond to the increasingly rapid infall of the black
0.0
2.0 2.5 3.0 3.5 4.0 hole,sothegasissqueezedandpushedinwards.Themecha-
R[GMp/c2] nismisthesameasforplanetsopeninggaps.Theinteraction
of the secondary with the inner disc results in an exchange
Figure 3.SnapshotsofthesurfacedensityΣ(R)forthesimula-
tion resol 2. For each snapshot, the time t and the separation a ofangularmomentumbetweenthesatelliteandthegas.Gas
(thatroughlycoincideswiththepositionRsofthesecondary)are particlesloseangularmomentumandshiftontoinnerorbits.
indicated. Gas is squeezed by the rapid decay of the secondary
blackholetowardstheprimary. Theforcedaccretionofthegasontotheprimaryblack
hole, marked by peaks in the accretion rate, causes an in-
MNRAS000,1–11(2016)
6 A. Cerioli , G. Lodato and D. J. Price
y
nsit
1.0 -10 n de
m
u
2 ol
0.8 g c
o
l
0.6 Min/MREF
∆Mout/MREF
∆Macc,p/MREF 1 -12
0.4 ∆Macc,s/MREF
y
0.2
0.0 0 -14
4000 5000 6000 7000 8000 9000
t[GMp/c3]
Figure 4. Simulation resol 2: Inner disc mass evolution in the
last part of the simulation. Min represents the inner disc mass.
∆Macc,p and∆Macc,s arethemassaccretedbytheprimaryand
by the secondary in the time interval considered. ∆Mout is the -3 -2 -1
increaseintheouterdiscmassstartingfromtimet=4000.MREF x
istheinnerdiscmassattimet=4000.Fromtheplotwecaninfer
that only the 0.8% of the disc mass is expelled outside (green Figure 5. Horseshoe orbits, simulationresol 2. Arrowsindicate
curve)andthattheinnerdiscmassisaccretedalmostcompletely particlesvelocityinthereferencesystemco-rotatingwiththesec-
bythetwoblackholes. ondaryblackhole.Aheadandbehindthesecondaryblackholewe
canseetheinversionpoints,wherethedirectionofparticlesmo-
tionchanges.Particleslocatedinsidetheorbitofthesecondary’s
crease of the disc accretion luminosity just prior to the orbitaremovingquicker than theblackhole, whiletheparticles
merger. outside are slower. Particles located in the corotation region ex-
Figure 4 shows the mass accreted by the primary, erthorseshoeorbits andthey canend upinadifferent regionof
the disc after the execution of a turn if the horseshoe region is
∆M /M (blue curve), where M is the inner disc
acc,p REF REF
asymmetric.
mass at time t = 4000 (no significant evolution of the ac-
cretion rates is observed before this time, see Fig. 1). This
showsthebehaviourinferredfrom M˙ ,growingsuddenlyin
two steps, corresponding to the twopspikes in M˙ . At the velocity map, the arrows show a counterclockwise motion.
p
By contrast, the slower particles in the outer disc are re-
endofthecalculation,79.9%oftheinnerdiscmassM is
REF
ceding from the secondary. The transfer of gas is possible
accreted bytheprimaryblack hole, while19.3% isaccreted
because the horseshoe orbits are not closed as predicted by
by the secondary. Only 0.8% of the disc mass is expelled
theory,butinsteadtheorbitsareasymmetricduetothein-
outside the binary orbit. This is a remarkable result since
ward drift of the secondary, meaning that the width of the
this means that the merger of two black hole happens in
horseshoe region is larger behind the black hole, similar to
a relatively gaseous environment, in contrast to what was
what occurs for planets in discs (Lubowet al. 1999). The
found by Baruteau et al. (2012) in their thickerdisc case.
streamlines near the inversion point behind the black hole
do not bring back gas in the horseshoe region, but chan-
4.1.2 Horseshoe orbits nel it outside. The inverse process, i.e. material transferred
from the outer disc to the inner is also possible. However,
Figure 5 shows a close-up of the region near the secondary
only gas located near the disc edge can embark on horse-
black hole in Cartesian coordinates at time t = 8725, with
shoe orbits to be funnelled through the gap, so the process
velocities shown in the frame rotating with the secondary
is unidirectional due to the very low mass of the outer disc
black hole. Hence arrows indicate the direction of particle
and to the presence of the gravitational decay. The outer
motionwithrespecttothesecondary.Massisfunnelledout-
discwouldalsobelowmassinarealisticscenario, sincethe
sidetheorbitofthesecondaryblackholethroughhorseshoe
outer circumbinary disc would be frozen far away from the
orbits (Murray & Dermott 1999), as seen in the reference
black holes in the post-decoupling phase. The mass located
frame of the rotating smaller black hole. Horseshoe orbits
intheouterdiscisthusonlythemasstransferred outwards
are equilibrium orbits that lie around the secondary orbit
from theinnerdisc.
and encircle the Lagrangian points L , L and L in the
3 4 5
equipotentialsurfacesoftheeffectivepotentialofarotating
binary system.
4.2 Effect of resolution
In the inertial frame the secondary and the disc rotate
counterclockwise around the primary, located near thecen- Figure 6 shows the accretion rates of the primary mass for
tre of mass, since we choose a small mass ratio. Particles three different resolutions (see Table 1). The two spikes in
located in the inner disc, in Keplerian rotation, are mov- M˙ are present at each resolution, indicating this is not a
p
ing more rapidly with respect to the secondary, so, in the numericalartefact.
MNRAS000,1–11(2016)
Gas squeezing during the merger of a SMBHB 7
t[days] t[days]
0 10 20 30 40 50 0 10 20 30 40 50
10−2
ref:Npart=5·105 asp3:H/R=0.05
resol1:Npart=106 asp2:H/R=0.02
10−3 resol2:Npart=2·106 10−3 asp1:H/R=0.015
M)]p M)]p raesfp:4H:/HR/=R0=.010.0095
G G
3c/( 10−4 3c/(
M[disc M[disc 10−4
/ /
˙Mp ˙Mp
10−5
10−5
10−6
0 2000 4000 6000 8000 0 2000 4000 6000 8000
t[GMp/c3] t[GMp/c3]
Figure 6. Primaryaccretion rates at differentresolutions. The Figure7. AccretionratesM˙p/MdiscfordifferentvaluesofH/R.
simulation resol 2 presents the higher accretion rate. The blue In the simulations asp 2 and asp 3 all the inner disc mass is
curvereachaspikesthatisafactor72higherthanthemeanvalue accreted or expelled before the establishment of the squeezing
fortimeinferiortot=5000. The spikeofthe primaryaccretion mechanism, that on the contrary is visible in the other cases.
rateofresol 2 is4.5timesthespikeoftheresolutionref,theone UpperaxisshowstimeindaysforaprimarymassMp=108M⊙.
with lower resolution. The upper axis shows time expressed in
daysforaprimarymassMp=108M⊙.M˙p isnormalisedbythe
initialdiscmassMdisc. may affect the quantity of gas expelled outside the orbit of
thesecondary.
We first try to increase the disc thickness (disc aspect
SIMULATION asp 4 ref asp 1 asp 2 asp 3 ratio)insimulationsasp 1,asp 2,asp 3 andtodecreaseitin
simulation asp 4.Theirinitialconditionsarelisted inTable
H/R(R=Rin) 0.0095 0.01 0.015 0.02 0.05 1; theydiffer only for thechosen valueof H/R.
Macc(TOT)/Mdisc 0.980 0.978 0.966 0.954 0.949 Figure7showstheaccretion ratesoftheprimaryblack
Mout/Mdisc 0.020 0.022 0.034 0.046 0.051 hole for different values of the disc aspect ratio H/R, nor-
t˜ 9250 9250 9250 8875 7500 malised by initial disc mass Mdisc. In simulations asp 2
and asp 3, corresponding to the values of H/R = 0.02 and
H/R = 0.05, respectively, we do not see any effects of the
Table 2. Total accreted mass M and mass funnelled
acc(TOT) forced compression because all mass is swallowed before it
outside Mout for different values of H/R. Masses are expressed
could be squeezed by the secondary, due to the increased
as fraction of the total initial disc mass. For each simulation we
definethetimet˜atwhichtheinnerdiscmassMin/Mdiscdecreases viscosity.
below10−4. The accretion rates M˙p and M˙s increase in magnitude
with increasing disc thickness in the first part of the simu-
lations. Forthe thinnerdiscs of ref and asp 4, M˙ presents
p
Thesimulationresol 2 (Npart =2·106)hasthemostsig- the two spike feature encountered before, while the gas in
nificant enhancement in the primary accretion rate, shown asp 1 is too little to producea notable peak.
inFigure6inunitsoftheinitialdiscmassMdisc1.M˙p/Mdisc In Table 2 we show the total accreted mass Macc(TOT)
growsfrom3·10−5 to1.1·10−3 (byafactorof37)inthefirst and the mass funnelled outside M for different values of
out
spike and reaches 2.15·10−3 (a factor of 72 higher) in the H/R. Masses are expressed as fraction of the total initial
secondspike.Similarly,thesecondaryaccretionratespikeis discmassM .Foreachsimulationwedefinethetimet˜at
disc
50 times higher than theinitial value(Figure 1). which theinner disc mass M /M decreases below 10−4.
in disc
The huge enhancements in the accretion rates can be FromTable2wecandeducethefollowingtrend:whenH/R
consideredthesignatureofablackholemergerinagaseous decreases, M increases and consequently, M de-
acc(TOT) out
environmentandlaterwecommentontheconsequencesthis creases. In other words, if the disc is thick, gas can more
couldhaveontheaccretionluminosityofthesourceinareal- easily flow outwards through thegap.
isticscenario(Section5.2).Asalreadyobserved,thesqueez-
ingmechanismcanleadtoapossibleobservableelectromag-
netic signal preceding the gravitational waves that will be 4.4 Effect of the accretion radius
intercepted bynext generation detectors.
Here we present the effects of reducing the secondary ac-
cretion radius. We expect a smaller accretion radius of the
secondaryblackholetoreduceM˙ andincreasethefraction
4.3 Effect of disc thickness s
of mass expelled. We set the accretion radius R = 0.2
acc,s
The availability of a three dimensional code such as phan- in most of the simulations. According to Ayliffe& Bate
tom allows us to study the dependence of the squeezing (2009), we could estimate the size of the secondary’s accre-
phenomenononthediscthickness.Thethicknessofthedisc tionradiusasits centrifugalradiusandfindR ≈R /3,
acc,s H
MNRAS000,1–11(2016)
8 A. Cerioli , G. Lodato and D. J. Price
innerdiscmassisverylow(M /M =7.88·10−5 inref)
in REF
0.20 because at time t = 9250 the binary is close to the final
ref:Racc,s=0.2 merger. Most of the disc mass MREF has been accreted by
sec1:Racc,s=0.05 theblackholesandonlyasmallfractionhasbeenchannelled
0.15 sec2:Racc,s=0.01 outside (third row). Not surprisingly, the mass funnelled to
theouterdisc∆M increasesbydecreasingR ,asseen
out acc,s
EF in Figure 8, where ∆Mout is expressed in unit of MREF.
MR ∆Mout/MREF =2.30·10−2 for ref andgrows to9.97·10−2
M/out 0.10 for seRced2u.cingR haslittleeffectonM˙ (∆M /M
∆ acc,s p acc,p REF
does not change notably) and if anything increases the pri-
0.05 maryrate.Alarger outerdiscmass,byconstrast,decreases
the secondary accretion rate, since the additional expelled
mass is the mass that is not accreted by the smaller black
0.00 hole.
4000 5000 6000 7000 8000 9000
t[GMp/c3]
Figure8.Outerdiscmassevolutioninthelastpartofthesimu-
lation.Thefractionofmassfunnelledtotheouter discincreases 5 DISCUSSION
for decreasing Racc,s. ∆Mout is expressed in unit of MREF, the
innerdiscmassattimet=4000. 5.1 Comparison with previous work
Armitage & Natarajan(2002)showtheevolutionofasuper-
SIMULATION ref sec 1 sec 2 massiveblackholebinarywithMp =5·108M⊙andq=0.02,
consideringacombinationofdiscdrivenmigrationandgrav-
Racc,s 0.2 0.05 0.01 itational decay in a one dimensional code. In this idealized
Min/MREF 7.88·10−5 6.34·10−5 8.42·10−5 situation, the final stages of the merger drive rapid accre-
∆Mout/MREF 2.30·10−2 5.24·10−2 9.97·10−2 tion of the inner disc, reaching an enormous accretion rate
∆Macc,p/MREF 0.698 0.704 0.710 of 10T6Mhe⊙w/oyrrk. was revisited by Baruteau et al. (2012), who
∆Macc,s/MREF 0.279 0.243 0.190 improved the set up by using a two dimensional hydrody-
namicalcode.Theyconsideredabinarywiththesamemass
Table 3. Disc masses at time t = 9250, for different values of ratio of Armitage & Natarajan (2002) to facilitate compar-
Racc,s. MREF is the inner disc mass at time t = 4000 for every ison. Baruteau et al. (2012) show that the increase in the
simulation. Min is the inner disc mass. ∆Mout is the increase outer disc mass ∆M is about 80% of the mass in the in-
out
in the outer disc mass starting from time t = 4000. ∆Macc,p nerdiscandthatonly20%isaccretedbytheprimary.This
and ∆Macc,s are the mass accreted by the primary and by the small quantity does not cause any increase in the accretion
secondaryinthetimeintervalconsidered.
luminosity and it does not excite theformation of spikes in
thesurface density of thedisc.
where R is the Hill radius of the black hole. The Hill ra- By contrast, all our thin disc simulations confirm that
H
dius is given approximately by a(q/3)1/3. R is then di- if gas is present between the black holes it is squeezed
acc,s
rectly proportional to the separation a, so it decreases as and rapidly accreted by the primary during the decay.
the black holes come closer. For our choice of initial con- We attribute the difference with respect to Baruteau et al.
ditions, at a = 4.75GM /c2 the Hill radius should be (2012) mostly to an effect related to the disc thickness.
0 p
R ≈0.33GM /c2. Assuming that the secondary accretion Baruteau et al. (2012) adopt H/R=0.08 (evaluated at the
H p
radiusisoforderthecentrifugalradius,welowerR from initial separation a ). Our reference case has H/R = 0.008
acc,s 0
0.2 to 0.05 and to 0.01 in order to resolve it during the (evaluatedata ),anorderofmagnitudesmaller.Indeedwe
0
secondary’s inspiral. Considering ref with R = 0.2 as findthat,asH/Risincreased,moremasstendstoflowpast
acc,s
starting point and then we decrease the secondary’s accre- the secondary orbit into the outer disc. This is expected,
tion radiusto R =0.05 in sec 1 and to R =0.01 in since for larger H pressure forces are stronger and tend to
acc,s acc,s
sec 2.DetailsaregiveninTable1.Allthreesimulationshave overcome the gravitational torque of the secondary. At a
a resolution of 5·105 particles. We point out that it is not givenmassratio, forsmalleraspectratio,thewakesexcited
possibletolowertheaccretionradiustoomuchbecauseitis by the secondary are more tightly wound; they therefore
oneofthecausesfortheslow-downofthecode.Bylowering deposit their energy and angular momentum in a region of
R there would be many particles in the vicinity of the the disc (“shocked region”) that is closer to the secondary
acc,s
secondary black hole that evolve on very short dynamical andthathassmallerradialextent(Rafikov2002).Uponap-
times.Thecodewill consequentlybecomeslower becauseit proachingthesecondary,thegasintheinnerdiscclosetothe
has to evolve on smaller time steps to follow the evolution horseshoe separatrix then undergoes a larger deflection in-
of such particles. wardswhichlimitstheamountofgasthatcanundergoout-
Table3liststhefractionsofthemassinsideandoutside ward U-turnsrelative to the secondary. We cannot directly
thesecondary’s orbit and theaccreted mass by single black compareourresultstoBaruteau et al.(2012)becauseforthe
holes at time t=9250. Masses are expressed in unit of the large H/R used by them the whole disc would be accreted
inner disc mass M at time t = 4000. The fraction of before the merger (see our simulations asp 3 and asp 2).
REF
MNRAS000,1–11(2016)
Gas squeezing during the merger of a SMBHB 9
However,privatecommunicationwiththeauthorconfirmed the primary mass and the second one is the ratio between
usthat2Dhydrodynamicalsimulationswith thesamecode theunit of mass and time introduced in Section 3.
asinBaruteau et al.(2012)andtheinitialconditionsofour WechooseMp =108M⊙ andMs =qMpwithq=10−3.
ref run (thin disc and small mass ratio) give very similar Tazzari & Lodato (2015) estimated, using simple 1D diffu-
results. This strengthens the hypothesis that the squeezing sionmodels,thatfora108M⊙ primary(butdifferentq)the
phenomenonmostlydependsontheaspectratioratherthan innerdiscmassatdecouplingisoforder1M⊙ (thisestimate
3D vs 2D or thesecondary accretion radius. ismuchlargerthanpreviousestimatesofthesamequantity
In contrast to Baruteau et al. (2012) and by Chang et al. (2010), see Tazzari & Lodato (2015) for a
Armitage & Natarajan (2002), we allow accretion onto discussion oftheorigin ofthediscrepancy).Wethuschoose
the secondary, meaning that we consider particles falling 1M⊙ as theinitial mass of theinner disc.
insidethesecondaryaccretion radiustobeaccreted,evenif Table 4 shows the primary accretion rates at the ref-
wedonotaddtheirmassandmomentumtotheblackhole. erence time t = 4000 and at the times of the two spikes
This means that some of themass that would flow through forsimulation resol 2. Theaccretion rates relativetospikes
the secondary’s orbit will be captured by the black hole. are higher than the Eddington value; the highest being
Exploringtheeffectsofchangingtheaccretionpropertiesof M˙ /M =30.53.
p Edd
the secondary, we find (Section 4.4) that it does not affect Tazzari & Lodato(2015),neglectingmassleakagefrom
howmuchmass flows totheouterdisc, butonly whetherit theinnerdisc,foundpeakluminositiesoforder1−20times
is accreted or not by the secondary. The primary accretion the Eddington luminosities. In our 3D simulations we find
rate remains essentially unchanged. comparable results, even with different masses and param-
Ourchoiceofthediscaspectratiointheinnerdiscwas eters. This confirms the hypothesis of a strong luminosity
dictated by the previous work of Lodato et al. (2009) and outburstbefore themerger.
Tazzari & Lodato (2015). Figure 4 of Lodato et al. (2009) For a merger of supermassive black holes in the Virgo
clearly show that in the circumprimary disc it is of the or- cluster, located at a distance ≈ 17 Mpc from Earth, the
der of 10−3, while in the outer disc it is an order of magni- absolute luminosity associated to themain spike before the
tude larger. We tried to extrapolate a trend for H/R from coalescencewouldbe4.8·1047 erg/s,assumingasolarmass
theworkofTazzari & Lodato(2015)andwefoundthatthe disc, q = 10−3 and a primary mass Mp = 108M⊙. Its ap-
disc aspect ratio is nearly constant around 10−4−10−3 in parent bolometric magnitude would be ≈ 0.2. This highly
theparameterspace explored in thepaper. Thuswe expect energetic and transient state has an evolution time-scale of
merging binaries to have thinner discs, as in our case, and theorderoffewdays(seeFig.1).Theeventshouldbesim-
to give a strong electromagnetic signal from the squeezing ilarinluminositytotidaldisruptioneventsandsupernovae;
of theinner disc. afocused studywould givesomeelementstodistinguish its
In the model by Lodato et al. (2009), no accretion is peculiar features. We expect the emitted signal to cover a
allowed from thecircumbinarydiscontothecircumprimary wide range of frequencies in the electromagnetic spectrum,
disc,whichleadstomassaccretionratesthroughthecircum- in particular in the optical and X-rays. Dottiet al. (2006)
primary disc of ∼ 0.001M˙ . If the circumprimary disc is summarized the possible electromagnetic signature of the
Edd
replenished significantly above this value, this would result coalescence for mass ratios q >0.01 that can be detectable
in a larger valueof H/R. infutureX-raymissions.Hereweshowthat,ifthereissome
residual gas, a bright precursor can be also observed for an
extrememass ratio.
5.2 Physical units
In our discussion so far we have used dimensionless code
units or normalised quantities; here we switch to physical 6 SUMMARY AND CONCLUSIONS
unitstointerprettheresultsinaphysicalscenario.Wecom-
We have studied the evolution of gas accretion during the
pareourresultsfortheaccretionratestothecriticalEdding-
tonvalueM˙ .WeassumethattheluminosityL,generated gravitationaldecayofasupermassiveblackholebinary,with
bytheaccreEtidodnofgasbythebinary,isgivenbytheformula amassratioofq=10−3,inordertoinvestigatethepossibil-
ity of a strong enhancement of luminosity due to the rapid
L=ηM˙c2, (11) accretion of fossil gas between the black holes in the last
phaseof coalescence.
where η is the efficiency coefficient. The critical Eddington
Foranunequalmassbinary,thesecondarymasssweeps
luminosity and associated Eddington rate are given by:
up gas from its orbit, creating a gap in the disc and acting
L =1.3·1038 Mp erg, (12) as a dam. Mass transfer through the gap is not completely
Edd M⊙ s suppressed (D’Orazio et al. 2013; Duffellet al. 2014), since
(cid:18) (cid:19)
somegascancrossthelowdensityregionthroughhorseshoe
M˙Edd =4.5·10−8 MM⊙p 0η.1 Myr⊙. (13) orbits, as seen in thereference frame of the secondary.
(cid:18) (cid:19)(cid:18) (cid:19) Wefocusedourattentiononthecaseofasupermassive
If we choose Mp = 108M⊙ and η = 0.1 we have LEdd = blackholebinarywith massratioq=10−3,choosinginitial
1.3·1046erg/s and M˙Edd = 4.5M⊙/yr. In our simulations separations a0 smaller than thedecouplingradius.
accretionratesareexpressedinunitsoftheinitialdiscmass. Wefoundthatthesecondaryblackholerapidlyinspirals
Toswitchtophysicalunits,wemultiplythesedimensionless towards the primary sweeping up the gas in the inner disc.
quantities by two factors: M /M and c3/G, where the Thisdeterminestheformation ofa narrowspikein thedisc
disc p
firstonerepresentsthechosenvalueofdiscmassinunitsof surface density that is pushed inwards, while only a small
MNRAS000,1–11(2016)
10 A. Cerioli , G. Lodato and D. J. Price
SIMULATION timet M˙p/Mdisc M˙p/(M⊙/yr) M˙p/M˙Edd
resol 2 4000 2.9·10−5 1.86 0.41
8608(firstspike) 1.1·10−3 70.5 15.67
9007(secondspike) 2.14·10−3 137.39 30.53
Table 4. Primary accretion rates at the reference time t = 4000 and at the times of the two spikes for simulation resol 2. The third
column shows the dimensionless M˙p/Mdisc, the fourthcolumn shows the accretion rate inunits of M⊙/yr, the fifth columnshows M˙p
inunitsoftheEddingtonrateM˙Edd=4.5M⊙/yr.TheaccretionratesatthespikesarelargerthanEddington.
ThedependenceonH/R ofthesqueezingphenomenon
7.0% is themain reason for the discrepancy with Baruteau et al.
Mout/Mdisc (2012), who considered a thicker disc. The dependence of
6.0% the enhancement on disc thickness may provide a method
forobservationallyinferringH/Rfrommeasurementsofthe
5.0%
accretion rate.
4.0% The fraction of mass funnelled outside depends also,
butmoreweakly,onthesecondaryaccretionradius.Wefind
3.0% that the primary accretion rate remains nearly unchanged
when decreasing the secondary accretion radius, while the
2.0% secondary accretion rate is reduced. Consequently there is
more mass available to be dynamically expelled and this is
1.0%
exactlywhatwefoundinourresults.However,theincrease
in theexpelled mass is always restricted to a few percent.
0.0%
0.01 0.02 0.03 0.04 0.05
Theaccuracyofourresultsdependsonresolution.Due
H/R
to the limitation of computational resources, we performed
Figure9. PercentageoftheexpelledmassMout overtheinitial mostofoursimulationsbyemploying5·105SPHparticles.In
discmass Mdisc versus discthickness. H/R is computed at R= twocasesweraisedtheresolutionto106 and2·106particles.
Rin.PointsrepresentthefivesimulationslistedinTable2. Wefindthatlowresolutionsimulationstendtounder-resolve
the accretion rate, hence our simulations are a lower limit
onthetrueenhancement.Inthehighresolutioncasewefind
amount of mass is funnelled outside. The inner disc mass
thatinthelaststagesofthemergertheaccretionluminosity
isquicklyaccretedjustbeforethemerger,causingasudden
ofthebinaryisincreasedbyafactorofabout70withrespect
increaseintheaccretionratethatalwaysexceedsEddington
to the initial constant one. This corresponds to a super-
limit.
Eddington luminosity. The primary accretion rate reaches
AsnoticedbyBaruteau et al.(2012),wefoundthatgas the valueof M˙ =30.53M , for a primary mass of M =
p Edd p
particlesarechannelledtotheouterdiscthroughhorseshoe 108M⊙ and an accretion efficiency η=0.1.
orbits as seen in the reference frame corotating with the
secondary. However, we found that the expelled mass is a Welimited oursimulationstoarestrictedregion ofpa-
negligible fraction of thetotal initial discmass. This means rameter space, in particular we considered only the case of
that the efficiency of the funnelling mechanism depends on mass ratio q = 10−3. A wider investigation would require
thediscthickness.BydecreasingthediscaspectratioH/R, more computational resources.
thetotalaccreted masswith respect totheinitialtotaldisc
It would be useful to compare our results with general
mass increases, while the mechanism of mass funnelling is
relativisticsimulationsthataccountforthedistortedgeom-
suppressed.ByloweringH/RatR=R from 0.05to0.01,
in etry near the binary.We could then understand if it is safe
themassfunnelledoutside(M )decreasesfrom 5% ofthe
out to neglect relativistic correction to the potential of the bi-
totalinitialdiscmassto2%(seeFig.9).Theexpulsiontends
naryaswe havedoneinourwork. Thepossibility of sucha
to flatten for larger H/R because inner disc mass is swal-
comparison already exists (Gold et al. 2014).
lowedbeforethesqueezingtakesplace.Alargeraspectratio,
assuming the validity of the alpha viscous model, implies a In conclusion, with our three dimensional simulations
larger kinematic viscosity that causes a faster viscous drift wepredictthatelectromagnetic signal from acoalescing bi-
oftheinnerdisc.Inourmodel,thealphaviscosityisfixedby nary should be detectable. Even if the study was carried
initial conditionsthrough the artificial viscosity mechanism outwithlowresolutionandweinvestigatedonlythecaseof
implementedinphantom,butthelargeraspectratiobroad- q = 10−3, the results are interesting because we find that
enstheradialextentoftheshockedregionandSPHparticles only a small amount of inner disc mass can be funnelled
are more easily accreted by the central black hole. On the outside, contrary to previous results. Of course the study
contrary,whenH/Rissmaller,theshockedregionisthinner could be improved by increasing the resolution and by in-
and it is located closer to the secondary black hole because vestigating the parameter space widely. This will allow to
wakesaremoretightlywound(Rafikov2002).Thereforepar- understandthephenomenoninthecaseofvariouscombina-
ticles close to the inner separatrix of the horseshoe region tions of mass ratio and disc thickness, in view of a possible
are more easily deflected inwards ratherthan outwards. detection in future surveys.
MNRAS000,1–11(2016)