# Retrograde Accretion and Merging Supermassive Black Holes

###### Abstract

We investigate whether a circumbinary gas disc can coalesce a supermassive black hole binary system in the centre of a galaxy. This is known to be problematic for a prograde disc. We show that in contrast, interaction with a retrograde circumbinary disc is considerably more effective in shrinking the binary because there are no orbital resonances. The binary directly absorbs negative angular momentum from the circumbinary disc by capturing gas into a disc around the secondary black hole, or discs around both holes if the binary mass ratio is close to unity. In many cases the binary orbit becomes eccentric, shortening the pericentre distance as the eccentricity grows. In all cases the binary coalesces once it has absorbed the angular momentum of a gas mass comparable to that of the secondary black hole. Importantly, this conclusion is unaffected even if the gas inflow rate through the disc is formally super–Eddington for either hole. The coalescence timescale is therefore always , where is the secondary black hole mass and the inflow rate through the circumbinary disc.

###### keywords:

accretion, accretion discs – black hole physics – galaxies: formation – galaxies: active – cosmology: theory## 1 Introduction

Astronomers now generally agree that the centre of every reasonably large galaxy contains a supermassive black hole (SMBH). Moreover the mass of this hole correlates (at least at low redshift) with properties of the host galaxy (Ferrarese & Merritt 2000; Gebhardt et al. 2000; Häring & Rix 2004). In the hierarchical picture of structure growth, small galaxies merge to produce large ones, promoting accretion on to their central SMBHs, and apparently causing these holes to coalesce.

The favoured mechanism for driving the holes closer is dynamical friction. However it is unclear that this can bring them close enough for gravitational wave losses to complete the coalescence, since the frictional process itself scatters away the stars causing it, and refilling of the loss cone is apparently too slow. This is often called the ‘final parsec problem’, as dynamical friction typically stalls at such separations between the holes (Milosavljevic & Merritt, 2003).

A possible way of overcoming this problem is interaction with gas orbiting in a disc just outside the SMBH binary (Armitage & Natarajan 2005; MacFadyen & Milosavljević 2008; Lodato et al. 2009; Cuadra et al. 2009). There has also been much discussion of cases where an SMBH binary is embedded in a disc (Escala et al. 2005; Dotti et al. 2007,2009). It is implicitly assumed that dissipative torques make the disc coplanar with the binary. Studies of the circumbinary disc problem have so far considered prograde discs, i.e. those rotating in the same sense as the binary. Then tidal interaction with the binary turns the disc into a decretion disc, which transports angular momentum outward, but with little inward mass transport (Lodato et al., 2009). If the disc mass is large enough to carry away the binary angular momentum, a decretion disc is vulnerable to the self–gravitational instability (Lodato et al., 2009). This can rob the disc of the gas it needs to drive further angular momentum loss, halting the binary shrinkage.

This makes it doubtful that a prograde disc can ever in practice shrink the binary separation from pc to the point ( pc) where gravitational wave losses can drive it to coalescence. However the separation of the SMBH binary is much smaller than the interacting galaxies themselves so it is highly unlikely that the central gas flows are always prograde. These flows also receive randomly–directed injections of energy and momentum from star formation and supernovae, suggesting that retrograde flows are as likely as prograde. This kind of chaotic accretion gives a plausible picture of the mass and spin evolution of a central accreting black hole (King & Pringle 2006;King et al. 2008;Hobbs et al. 2010) (but see Berti & Volonteri 2008; Mayer et al. 2007 for an alternative picture). It also implies that if nothing else manages to drive an SMBH binary to coalescence, it is highly likely that at some point there will be a retrograde coplanar disc surrounding the binary. This situation would arise if for example an earlier episode of prograde gas accretion failed to coalesce the binary, and was followed by a retrograde accretion event. As we will see, retrograde discs behave quite differently from prograde ones, and may offer a solution to the final parsec problem.

## 2 Prograde versus Retrograde

We start by contrasting the main qualitative features of the prograde and retrograde cases. These stem from the physics of the interaction between the binary and the disc, where dissipative torques try to share the angular momenta. If the disc is prograde, this interaction shrinks the binary but moves the inner edge of the disc outwards, reducing the torque shrinking the binary. If instead the disc is retrograde, the effect is to shrink the binary, but also to move the inner edge of the disc inwards (see Appendix A).

A prograde disc becomes a decretion disc, transporting angular momentum outwards with little net mass transport (Lodato et al., 2009). A retrograde disc instead remains an accretion disc, transporting angular momentum outwards and mass inwards. As we shall see, the long–term evolution of the disc–binary system is radically different in the two cases.

In particular, the disc–binary torque is quite different. For a prograde disc the tidal interaction occurs mainly through resonances. These occur when

(1) |

where is the binary orbital frequency, is the disc angular velocity and is the wave mode number (for example see the analysis in Papaloizou & Pringle 1977). So there are resonances at radii where

(2) |

where and have the same sign. Resonances outside the binary orbit (i.e. with ) correspond to the positive sign in the denominator of equation 2 and so appear at radii where

(3) |

The dominant interaction then involves the 2:1 (more strictly 1:2) resonance.

By contrast, in a retrograde disc and have opposite signs, and equation 1 requires , so there are no resonances in a circumbinary retrograde disc. The disc–binary interaction is direct, as the inner edge of the disc starts to impinge on the secondary black hole. A retrograde circumbinary disc remains an accretion disc whose material is gravitationally captured by the binary, directly reducing its angular momentum. This is inherently more promising for shrinking the binary towards coalescence than the prograde case, where the binary dams up the disc.

It is important to understand that ‘capture’ simply means that the gas orbits a particular hole, and so has added its (negative) angular momentum to the binary orbit. It does not imply that the relevant hole must actually accrete this gas (although it may). Once captured the gas is bound to the hole and thus may be treated as a single body. Some or all of this captured gas can be expelled, for example by radiation pressure. Provided that this process is isotropic in the frame of the hole this does not change its orbital angular momentum, and so has very little effect on the orbital dynamics and eventual coalescence (cf equation 22 below).

## 3 Where does the mass go?

The effect of a retrograde circumbinary disc differs in detail depending on how the captured mass is distributed between the two black holes. Accordingly we look at the reaction of test particles to the binary. To make things simple we first consider a circular binary with a low mass ratio, i.e. . Then to first order we can treat the primary as fixed and the secondary as following a circular orbit of radius around it with velocity . The circumbinary disc gas is on circular orbits with velocity at each radius . We assume that the disc creeps slowly inwards from a large radius by the usual viscous evolution. As it is counter–rotating we can ignore all resonant effects and assume everything is ballistic until orbits begin to cross and fluid effects appear. We define the effective radii of the two holes as the radii within which gas particles are captured by each hole, e.g. by forming a disc around one or other of them, so that this captured gas has the same net specific angular momentum as the relevant hole. Evidently cannot in practice be larger than the individual Roche lobes for each hole.

For the disc particles orbiting closest to the interaction with is initially hyperbolic and so we can use the impulse approximation. Here the relative velocity is approximately . If the disc edge is at a radius , where , the impulse approximation shows that the disc particle acquires an inward radial velocity

(4) |

The inner edge of the circumbinary disc is not significantly perturbed if its distance from the secondary’s orbit is large enough that , i.e. , or equivalently . From this we conclude that the secondary cleanly pulls the gas from the unperturbed inner edge of the disc at provided that , i.e.

(5) |

Comparing this with the Roche lobe constraint

(6) |

where is a dimensionless factor, we see that the secondary captures almost all the gas for mass ratios

(7) |

Equivalently we can define a Safronov number (Safronov, 1972) as used in discussing accretion of planetesimals from a protoplanetary disc:

(8) |

where are the escape velocity from the secondary’s effective radius, and its orbital velocity respectively. This measures how much the gas is gravitationally perturbed before being captured. Small implies little perturbation. We see that the criterion for secondary capture is just , that is, the gas is cleanly captured without significant perturbation.

For larger mass ratios the flow becomes more complex and it is likely that some gas falls towards the binary centre of mass, producing some form of primary capture. In numerical simulations (Section 6) we will find that even in this case most of the gas is captured by the secondary. So in general the secondary captures most of the mass, and only in a major merger with does the primary capture a non–negligible amount from a retrograde circumbinary disc.

## 4 Eccentricity growth

A retrograde circumbinary disc can decrease both the energy and angular momentum of the SMBH binary, and so change its eccentricity. A simple argument shows how this happens. We consider a slightly eccentric binary orbit (the orbit is never exactly circular if it is shrinking) and again for simplicity assume that the mass ratio is sufficiently extreme that we can regard the primary black hole as effectively fixed, and only the secondary as interacting with the disc. None of these restrictions will affect our conclusions (see Appendix A for a fuller discussion).

Momentum conservation shows that capture of disc gas always reduces the secondary’s orbital velocity (see equation 18). This holds both for a direct collision, or (more commonly) if the secondary captures gas into a bound disc around itself. The secondary’s mass cannot decrease in these interactions. Its specific orbital energy therefore always drops, so that the binary semimajor axis decreases. But for mass capture into a symmetrical disc around the secondary near apocentre, the new orbit must retain the same apocentre as the old one (it must pass through this point, and the radial velocity remains zero there). Given the decrease of the semimajor axis, this means that capture near apocentre tends to increase the orbital eccentricity (since remains constant). The eccentricity is evidently

(9) |

where is the amount of mass the secondary has captured. Exactly the same reasoning shows that for mass capture near pericentre, the quantity has to stay constant despite a further decrease in – in other words, the eccentricity must decrease here by about the same amount () it increased at apocentre.

The secondary obviously captures at all points in between apo– and pericentre, but the effects are opposed for significant times. If the eccentricity is initially small these times are nearly equal, and stays small as the orbit shrinks, provided that is below a certain value we will derive shortly.

If on the other hand the orbit is initially quite eccentric, or the mass grows significantly in one orbit, the pericentre may be too small to allow disc interaction. Then grows with constant, and the pericentre distance goes as

(10) |

The binary therefore coalesces once the original semimajor axis has halved. We show below (equation 22) that this occurs once the secondary has absorbed the (negative) angular momentum of disc gas with mass comparable to its own. If gas flows inwards through the circumbinary disc at the rate the timescale for coalescence is

(11) |

We stress again that this does not require either hole to accrete this gas but only to capture the gas into a bound orbit around the hole. In particular, can formally exceed the Eddington accretion rate for either hole without affecting the orbital shrinkage. If the accretion rates on to either hole were super–Eddington then the captured gas would be blown away by radiation pressure from the disc(s) around the secondary (and possibly primary) black hole, without significantly changing the binary orbital evolution (cf equation 22).

The critical eccentricity separating cases where the binary remains almost circular from those of growing eccentricity depends on the surface density distribution of the circumbinary disc. In a real three–dimensional disc with scale height and aspect ratio the surface density tails off over a length–scale . This suggests that the critical eccentricity dividing these two cases is just

(12) |

Thus any binary starting with , or achieving it by capturing a large mass (comparable to the secondary’s) in one orbit, must become very eccentric. We note that even a preceding episode of prograde accretion can leave the binary with an eccentricity exceeding (e.g. Cuadra et al. 2009), so that growth to high eccentricity is very likely if accretion is chaotic.

Our conclusions about eccentricity growth agree with those of Dotti et al. (2009), who considered a related but different problem. A secondary black hole was injected with significant eccentricity into a pre–existing dense circumnuclear disc surrounding a primary black hole. The secondary’s orbit was initially retrograde with respect to this interior disc. However the secondary was able to interact with enough gas in less than one orbit that it cancelled all of its angular momentum. The secondary then briefly had approximately zero angular momentum before capturing more gas and so changing its angular momentum to prograde. In this paper we restrict ourselves to accretion events on much smaller scales than in Dotti et al. (2009). This difference in lengthscales is important. At the smaller scales we consider, a disc with mass would probably be self–gravitating, and this change of angular momentum sign is unlikely to occur. Our simulations agree with this conclusion: once interaction with the disc has cancelled the orbital angular momentum (and thus much of the orbital energy) of the secondary, the binary coalesces.

## 5 Orbital evolution with a retrograde circumbinary disc

We can now make analytic estimates of the orbital evolution as the binary interacts with an exterior disc. For simplicity we again assume that , so that the secondary has specific orbital energy and angular momentum where

(13) |

and

(14) |

We have seen above that in many cases the binary eccentricity grows quite strongly. In the limit the secondary interacts with the disc only very near apocentre . Here its velocity is purely azimuthal, with

(15) |

which by equation 14 gives

(16) |

Near apocentre the secondary interacts with disc material moving with azimuthal velocity , with

(17) |

We assume that a mass of disc matter is captured into orbit about the secondary near apocentre, as we discussed above, so that all of its orbital angular momentum is transferred to the secondary. To allow for mass loss from the subsequent accretion process on to this hole (if for example this is super–Eddington, or mass interacts gravitationally with the secondary but is all accreted by the primary) we assume that the effective mass of the hole plus disc becomes , with . Then conservation of linear momentum gives

(18) |

where is the new apocentre velocity of the secondary plus its captured gas disc. The changes in orbital specific energy and semi–major axis are given by

(19) |

Combining equations 16,17,18 and 19 gives

(20) |

to lowest order in . We noted above that if the interaction is confined to the immediate vicinity of apocentre then the apocentre distance stays constant in the subsequent evolution. Then and , so that the LHS of equation 20 is simply and therefore equation 20 becomes

(21) |

Using , where was the mass of the secondary hole when was zero, and is the total mass since transferred from the disc, this integrates to give

(22) |

Hence in this approximation the binary coalesces (i.e. ) once the disc has transferred a mass equal to the secondary black hole, i.e. after a time (cf equation 11), independently of the fraction . This means that mass loss has no effect in slowing the inspiral. We note that equation 20 implies that the energy dissipated in the shrinkage of an eccentric binary is

(23) |

per binary orbit, which is less than that produced by viscous dissipation in the disc (this has to pull in a mass larger than on each orbit).

All other cases give similar timescales for coalescence. For example if the orbit stays circular we can set in equation 20 and find

(24) |

so that . If the secondary gains the transferred mass we have , while if the transferred mass (but not the angular momentum) ends up on the primary we have . Shrinking the binary from pc to pc, where gravitational radiation rapidly coalesces it, requires the transfer of between 2 and 9 times the original mass of the secondary in the two cases. This is larger than in the eccentric case because the torque on the binary now decreases with (indeed the required mass would be formally infinite without gravitational radiation).

## 6 Simulations

We now use Smoothed Particle Hydrodynamics (SPH) simulations to compare with the analytical arguments above.

### 6.1 Code setup

We use a fully 3D conservative Lagrangian implementation of the SPH algorithms, for example Springel & Hernquist (2002); Price (2005); Rosswog (2009). We neglect gas self-gravity and so are able to use linked–lists of particles rather than the usual tree for neighbour finding (Deegan, 2009). We assume an isothermal disc, i.e. , where the temperature and hence sound speed is constant for all particles at all times. We make this choice for simplicity, and anticipate that the interaction of gas with the binary is not greatly affected by it. In particular we noted in Section 5 above (cf equation 23) that binary shrinkage does not greatly increase the heating of the disc. We integrate a ‘live’ binary which feels the back reaction of the gas and generates self–consistent orbits for the binary and the gas.

Our aim here is to understand the dynamical interaction of the disc with the binary, rather than long–term behaviour such as the timescales for coalescence, which we know depends on the total mass transferred through the disc (see Section 5 above). Accordingly we use only the standard SPH artificial viscosity (Rosswog, 2009) and do not include a physical viscosity. Our simulations run only for a few tens of binary orbits, so that the viscosity scheme employed should not affect our results.

Our code units are dimensionless and take the initial separation and mass of the binary as unity. The period of the binary in its initial configuration is thus . The general disc setup is gas, of mass , spread from to with a surface density following a power law in radius, i.e. with typically . The initial vertical structure in the disc is a Gaussian. We set the sound speed in the gas to a value ensuring that our neglect of self–gravity is justified. Thus we arrange that the Toomre parameter (Toomre, 1964) exceeds a minimum value () throughout the disc. Typically we arrange for a disc of mass . The particles are all on initially circular orbits.

The simulations detailed below were repeated with half a million,
one million and two million particles and were deemed to have
converged. All of the simulations detailed below initially contain
one million particles.^{1}^{1}1E-mail: chris.nixon@astro.le.ac.uk

^{1}

^{1}footnotetext: Movies of these simulations are available and can be found at:

www.astro.le.ac.uk/users/cjn12/RetroBinaryMovies.html

### 6.2 Where does the mass go?

Two main parameters govern the gas flow for a retrograde disc–binary system: the binary mass ratio and the ‘capture’ radius of each binary component. As we discussed in Section 3, with mass ratios , both the primary and secondary can interact with the inner edge of the disc so we should expect gas capture by both objects. However if , an inward–moving disc reaches the secondary before the primary. Then the secondary’s effective radius may be large enough for it to capture all of the gas. If instead this radius is small the gas may be perturbed towards the primary rather than captured. In Section 3 we estimated the critical value dividing these cases (equation 5). We study the effects of changing parameters here.

It is of course currently impossible to follow gas accretion down to the innermost stable circular orbits of the two black holes while simultaneously following the gas flow from the circumbinary disc. The analytic work above suggests that the secondary hole is always surrounded by a gas disc denser than the circumbinary disc and any gas flow from it, which therefore captures any gas impacting it. If initially there had been no gas around the secondary, gas entering would be captured (but not necessarily accreted). This gas would spread to form a disc around the secondary, on a timescale short with respect to the mass transfer timescale in the circumbinary disc. The secondary’s disc then forms an obstacle for further inflow from the circumbinary disc. As further mass is captured in this way, this disc can become at most as large as some fraction of the Roche lobe (cf equation 6). In the language of planetary dynamics this condition is formally equivalent to taking the secondary’s Safronov number, defined as (planetary escape velocity/orbital velocity)/2, (e.g. Safronov 1972; Hansen & Barman 2007) to be of order .

From this analysis we expect the capture radius of the holes to be of order or smaller than the Roche lobe. For completeness we also consider the possibility of still smaller capture radii. Once a particle moves within the capture radius it is assumed to have impacted upon the disc assumed to be present inside the capture radius. This gas could be accreted by the black hole or expelled by radiation pressure from the innermost parts of the disc. In either case it is clear that once it has been ‘captured’ it plays no significant further role in the binary dynamics. Accordingly we add its mass and momentum to the relevant hole and remove it from the simulation.

#### 6.2.1 Secondary capture

The first simulation has a mass ratio of and effective radii 0.15, 0.2 for the secondary and primary. Equation 5 implies that the secondary should capture the inner edge of the circumbinary disc without significantly perturbing the remaining disc particles. From equation 8 the secondary’s Safronov number is 2/3.

We start with the retrograde disc extending from 1 to 2 in radius, and a circular binary.

At first the disc inner edge is slightly perturbed, but not enough to provide any noticeable accretion onto the primary. After one binary orbit the system settles and the secondary smoothly captures the inner edge of the disc. Figure 1 shows the state of the simulation at . At this time, of the particles accreted, the secondary has captured (149759 particles) and the primary has captured (3 particles).

#### 6.2.2 Primary capture

The second simulation also has a mass ratio of , but this time we use an effective radius of 0.05 for the secondary with again 0.2 for the primary capture radius, so that its Safronov number is . Equation (5) implies that the secondary should perturb the disc particles significantly without directly accreting all of them.

The initial disc is exactly the same as for the simulation in Section 6.2.1. The binary is again initially circular.

At first, the disc inner edge is significantly perturbed, with some particles on orbits passing close to the primary. In the first 3 – 4 orbits the gas flow is very chaotic. After this the system settles into a quasi–steady state. Both holes capture from the retrograde circumprimary disc (as shown in figure 2). The secondary both disturbs and captures particles from the outer edge of the circumprimary disc, and perturbs more particles into it. The circumprimary disc is warped, eccentric and precessing. This is probably a result of particle noise destroying the symmetry about the orbital plane, as might indeed happen in a realistic situation.

At time the secondary has captured (147642) particles and the primary has captured (8193 particles). So even in this case the secondary still takes most of gas captured from the circumbinary disc. The binary probably coalesces once the disc has transferred a mass , so if the disc is more massive than this, the coalesced hole eventually accretes the remainder. Clearly if the secondary’s mass or its capture radius is made arbitrarily small, it would accrete very little, and most of the mass would be captured by the primary. However it is clear that this requires extreme choices of these parameters.

#### 6.2.3 Dual Capture

Here we look at the interaction of an near equal–mass binary with the disc. We use and the same disc as in Sections 6.2.1 and 6.2.2. Again the binary is initially circular. Here we use capture radii of 0.01 for both holes to allow formation of circumprimary and circumsecondary discs.

During the initial 3 – 4 binary orbits the flow is very chaotic, with mass captured by both holes. As more gas is captured (but this time not ‘accreted’) by both holes, circumprimary and circumsecondary discs are formed which persist throughout the simulation. The discs are supplied by streams from the circumbinary disc. We show the simulation at time in figure refboth. At this time the primary has captured (57969) and the secondary (62189) of the accreted particles. Together with the previous simulation, this shows that the primary can gain significant mass only if the mass ratio is close to unity. We note again that the discs are not planar, and show a significant tilt with respect to the binary plane. Again this is probably a result of particle noise removing the symmetry about the orbital axis. Globally angular momentum is conserved but locally the streams that supply these discs need not be planar.

### 6.3 Eccentricity growth

The analytic arguments of Section 4 suggest that capture from a retrograde circumbinary disc at apocentre and pericentre of the binary orbit increase and decrease its eccentricity respectively. Here we show three SPH experiments exploring this.

The first simulation has an initially circular binary, with , where the secondary is embedded just inside the inner edge of a retrograde disc. In code units the disc is spread radially from 0.8 to 1.5. The disc has an initial mass .

The second simulation also starts with the secondary embedded in the disc. However this time the inner edge of the disc extends much further in, to 0.1.

The third simulation starts with a retrograde disc interior to the binary, i.e. a circumprimary disc extending from 0.1 to 1.0. In this case we start with initial eccentricity 0.5 and the binary at apocentre – so the secondary begins outside the circumprimary disc but plunges into it before it reaches pericentre.

All of these simulations have capture radii 0.1, 0.01 for the primary and secondary respectively.

We show in figure 4 how the eccentricity of the binary evolves in the three cases. In the first simulation the binary captures enough gas at apocentre on the first orbit to plunge inside the inner edge of the disc. This means that the eccentricity growth at apocentre cannot be moderated by decay at pericentre. The eccentricity therefore grows as shown in figure 4. In the second simulation the secondary is still capturing when it reaches pericentre and so the eccentricity decays. It is notable that the eccentricity never returns precisely back to zero in this case. This happens because the secondary captures unequal amounts of gas at apo– and pericentre. This is reasonable, as its velocity is higher at pericentre and so it has less time to interact with the gas, In addition the mass and angular momentum at each radius is not constant.

In the third simulation the secondary can initially only capture from the circumprimary disc at pericentre and so the eccentricity decays. After approximately orbits the secondary has damped out most of its initial eccentricity to an almost circular orbit. After a short interaction with the inner parts of the disc it reaches its inner edge. Here there is a brief interval of eccentricity growth as it captures from this inner edge of a (now) circumbinary disc.

All of the results of these simulations agree with the analytic arguments of Section 4.

## 7 Discussion

We have seen that accretion from a retrograde circumbinary disc can be considerably more effective in shrinking a supermassive black hole binary system than accretion from a corresponding prograde disc. Coalescence occurs if the secondary black hole captures angular momentum from a gas mass comparable to its own.

The reason for this effectiveness is the absence of orbital resonances in a retrograde disc. This cannot react to tidal torques from the binary and become a decretion disc, which is what tends to slow the evolution in the prograde case (Lodato et al. 2009; Cuadra et al. 2009; Dotti et al. 2007; MacFadyen & Milosavljević 2008). Instead, the disc directly feeds negative orbital angular momentum to the secondary black hole. An important aspect here is that there is no restriction on the rate at which this can occur. In particular gas can flow inwards at rates higher than Eddington for even the primary black hole without any significant effect on the amount of gas required to coalesce the binary.

Capture to the primary hole is negligible unless the mass ratio is very close to unity, and is never dominant (even if ‘capture’ actually leads to ‘accretion’, which is not required). This may be important in interpreting attempts to observe merger events. In many cases the binary may develop very high eccentricity before gravitational wave emission coalesces it. The latter rapidly damps the eccentricity during the final inspiral, but leaves a residual value which is likely to be significant in any merger event detectable by gravitational wave observatories such as LISA. If the secondary hole actually accretes the captured mass it will acquire a large spin (Kerr parameter ) antiparallel to its binary orbit, which would be potentially detectable in the LISA waveform. Moreover if the mass ratio is very close to unity it is conceivable that the primary hole might do the same. This could in principle favour high black hole spin as a result of major mergers (e.g. in giant ellipticals), as sometimes proposed. However we should recall that the holes may well not gain much mass, particularly if this is captured at super–Eddington rates, and the primary definitely does not gain mass (and thus spin up) unless the mass ratio is very close to unity. For these reasons it seems unlikely that coalescences induced by retrograde gas flows produce rapidly spinning merged holes except in rare cases.

We caution finally that so far we have not considered two important effects. First, we assumed that the retrograde disc was coplanar with the binary, whereas in reality counter–alignment must occur over a viscous dissipation timescale. Second, self–gravity can deplete the circumbinary disc of gas and reduce its ability to shrink the binary (cf. Lodato et al. 2009 in the prograde case). Since coalescence by a retrograde disc requires , and self–gravity effects appear unless , where is the disc aspect ratio, it appears that the last parsec problem is so far alleviated only for mergers with . This paper does suggest how coalescence might work for larger mass ratios. The secondary hole has to absorb negative angular momentum from a gas mass comparable to its own, preferably in an eccentric orbit. Although gas self–gravity is inevitably important in such an event, a retrograde flow has the advantage that there is no limit on the mass inflow rate. We shall investigate these two effects in future work.

## Acknowledgments

We thank Walter Dehnen and Frazer Pearce for useful discussions on SPH. We thank the referee for encouraging us to investigate the conditions for orbital eccentricity growth. We acknowledge the use of splash (Price, 2007) for the rendering of the SPH plots. CJN and PJC acknowledge STFC studentships. Research in theoretical astrophysics at Leicester is supported by an STFC Rolling Grant.

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## Appendix A Binary–Disc interaction

To understand the interaction between the gas and the binary we consider the case where the disc particles can be perturbed before impacting upon the secondary. The relative velocity of a disc particle before interacting is . After the interaction it gains a radial velocity (equation 4). In the frame of the energy of the particle is conserved and so

(25) |

so that

(26) |

Back in the inertial frame, this implies that after the interaction the radial velocity of the disc particle is and the azimuthal velocity is , i.e.

(27) |

The particle was on an orbit with eccentricity , specific energy and specific angular momentum . After the interaction the particle’s specific kinetic energy is where

(28) |

Thus after the interaction the specific energy of the particle orbit is increased to

(29) |

where

(30) |

and the particle’s specific angular momentum is decreased to

(31) |

where

(32) |

We note that the angular momentum of the particle has the opposite sign to that of the secondary, as it must.

The perturbed particle now has semi-major axis larger than , by an amount . Thus

(33) |

and also for comparison

(34) |

The eccentricity of the particle’s new orbit is given by

(35) |

So we find

(36) |

From this we conclude that the interaction with the binary increases the particle’s orbital energy and decreases its orbital angular momentum, leading to an eccentric orbit. Now there are two possibilities for the disc. Either the disc now becomes eccentric itself. Or the orbits of the perturbed particles begin to cross and therefore the particles collide, lose energy and share angular momentum.

As the disc can radiate away the excess energy associated with the eccentric disc orbits (cf equation 23) it is clear that the overall effect on the disc of the interaction with the binary is to shrink the disc. Similarly for the binary we conclude that the binary loses both orbital energy and angular momentum and hence shrinks.