The evolution of a supermassive retrograde binary embedded in an accretion disk
P. B. Ivanov, J. C. B. Papaloizou, S.-J. Paardekooper, A. G. Polnarev
aa r X i v : . [ a s t r o - ph . H E ] A p r Baltic Astronomy
The evolution of a supermassive retrograde binary embedded in anaccretion disk
P. B. Ivanov , J. C. B. Papaloizou S.-J. Paardekooper A. G. Polnarev Astro Space Centre, P. N. Lebedev Physical Institute,84/32 Profsoyuznaya st., Moscow, 117997, Russia; [email protected] [email protected] DAMTP, University of Cambridge,Wilberforce Road, Cambridge CB3 0WA, UK Astronomy Unit, Queen Mary University of London,Mile end Road, London, E1 4NS, UK
Received:
Abstract.
In this note we briefly discuss the main results of a recent study ofmassive binaries with unequal mass ratio q for which he orbit is circular. Theorbit is embedded in an accretion disk with its orbital rotation being in theopposite sense to that of the disk gas. A more complete presentation of theseresults is published elsewhere (Ivanov et al. 2014).It is shown that when the mass ratio is sufficiently large the binary opens agap in the disk. However, the mechanism of gap formation has a very differentcharacter to that applicable to the prograde case. The binary is found to migrateinwards due to interaction with the disk with a characteristic timescale, t ev ,of the order of M p / ˙ M , where M p is the mass of less massive component ofthe binary, henceforth referred to as the perturber, and ˙ M is the accretionrate through the disk. When q ≪ ∼ ˙ M , while the accretion rate to the perturber is smaller, being ofthe order of q / ˙ M . However, we remark that the accretion rate to the perturbercan be significantly amplified during the late stages of the orbital evolution ofa supermassive binary black hole which are determined by gravitational waveemission. Additionally we estimate a typical time duration for which, bothelectromagnetic phenomena associated with accretion onto the perturber, andgravitational waves emitted by the binary could be detected by a future spaceborne interferometric gravitational wave antenna with realistic parameters.The study should be extended to consider orbits with significant eccentric-ity, for which the formation of a gap through the action of torques associatedwith waves launched at Lindblad resonances becomes possible. Also, when theaccretion disk has a non zero inclination with respect to the orbital plane of aretrograde binary at large distances, this inclination may increase on a timescalethat can be similar to, or smaller than t ev . This is also an aspect for future study.
Key words:
Accretion disks: -binaries, Hydrodynamics, Galaxies: quasars:supermassive black holes, Planet-disk interactions78
P. B. Ivanov et al
Supermassive black hole binaries (SBBH) may form as a consequence of galaxymergers, see e.g. see e.g. Komberg 1968, Begelman, Blanford & Rees 1980. Sincethe direction of the angular momenta associated with the motion of the binaryand the gas in the accretion disk is potentially uncorrelated, the binary may beon either a prograde or retrograde orbit with respect to the orbital motion in thedisk when it becomes gravitationally bound and starts to interact with it.The prograde case has been considered by many authors beginning with Ivanov,Papaloizou & Polnarev (1999), hereafter IPP, and Gould & Rix (2000). Theretrograde case has received much less attention, with relatively few numericalsimulations available to date, see e.g. Nixon, King & Pringle (2011) and Nixonet al. (2011). However, the retrograde case may be as generic as the progradecase when the interaction of SBBH with an accretion disk is considered. Notethat although the disk is likely to be inclined with respect to the binary orbitalplane initially, alignment on a length scale corresponding to the so-called alignmentradius is attained relatively rapidly, the direction of rotation of the disk gas beingeither retrograde or prograde with respect to orbital motion, depending on theinitial inclination, see e.g. IPP.Here we review recent results to be published in detail elsewhere (Ivanov, Pa-paloizou, Paardekooper and Polnarev 2014, hereafter IPPP) on the the evolutionof retrograde SBBH. A variety of analytical and numerical techniques were em-ployed. For simplicity, a binary in a circular orbit that was coplanar with the diskwas assumed for the most part. However, the case of an eccentric binary was alsobriefly discussed. The main emphasis is on the case of a small mass ratio q. How-ever, this is taken to be sufficiently large that the disk is significantly perturbedin the neighbourhood of the binary orbit.We describe our numerical approach to the problem of the interaction of SBBHwith an accretion disk in Section 2 and a simple analytical approach for calculat-ing the orbital evolution of SBBH in Section 3. Various associated effects andphenomena are discussed in Section 4. Finally in Section 5 we summarise ourresults.
In this section we consider numerical simulations for which the perturber is massiveenough to significantly perturb the accretion disk and open a surface densitydepression called hereafter ’a gap’ in the vicinity of its orbit. For that we requiremass ratio, q, of the perturber with mass M p to the dominant mass M , to belarger than ∼ . H/r p ) , where r p is the radius of perturber’s orbit and H isthe disk semi-thickness. We consider values of q of 0 .
01 and 0 .
02 below. In some See IPPP for the opposite case of a low mass perturber, which is insufficiently massive toopen a gap. retrograde binary embedded in accretion disk ∝ r − / and scaled so that the total mass interior to the initialorbital radius of the perturber was 10 − in units of the dominant central mass. . . Figure 1: log Σ contours for q = 0 .
02 with softening length 0 . H after 50 orbits(left panel) and after 100 orbits (right panel). In these simulations the companion,its position in each case being at the centre of the small red circle located withinthe gap region, was allowed to accrete. The width of the gaps slowly increaseswhile the accretion rates, on average, slowly decrease with time. Short wavelengthdensity waves in the outer disks are just visible. Note that values of log Σ belowthe minimum indicated on the colour bar are plotted as that minimum valueThe perturber was initiated on a retrograde circular orbit of radius r whichis taken to be the simulation unit of length. For simulation unit of time we takethe orbital period of a circular orbit with this radius. We use two different valuesof the softening length b s . For the “standard case” b s = 0 . H was adopted andthe for the case of “small” softening b s = 0 . H was adopted. For other details seeIPPP.The structure of the disk gaps for q = 0 .
02 and q = 0 .
01 is illustrated inthe surface density contour plots presented in Figs. 1 and 2 at various times.The runs respectively correspond to the strongest and weakest gap forming casesconsidered in this section. Note that the gap is indeed significantly wider anddeeper for q = 0 .
02 as expected and in addition the gap edges define significantlynon circular boundaries. Material crossing the gap in the form of streamers is alsopresent. Note that an animation of the process of gap formation can be found onthe website http://astro.qmul.ac.uk/people/sijme-jan-paardekooper/publications.The semi-major axis is shown as a function of time for q = 0 .
02 and q = 0 .
01 forsmall softening and for q = 0 .
01 with standard softening in Fig. 3. The behaviourdepends only very weakly on whether the perturber is allowed to accrete fromthe disk or not. At early times the cases with q = 0 .
01 have the migration rates80
P. B. Ivanov et al . .
Figure 2: As in Fig. 1 but for q = 0 .
01 with softening length 0 . H after 100orbits (left panel ) and 800 orbits (right panel). As the mass ratio is lower in thiscase compared to that of Fig. 1 the gap in the disk is narrower. The companion,indicated by a small red circle is found in general to orbit closer to the inner diskedge at earlier times. In the left hand panel the companion grazes the inner edgeslightly above the x axis for x < . This enhances the accretion rate at that stage. S e m i - m a j o r a x i s time(orbits) Figure 3: Semi-major axis, in units of the initial orbital radius, as a function oftime for q = 0 .
02 and q = 0 .
01 for small softening and for q = 0 .
01 with standardsoftening. Two curves without imposed crosses, which are very close together, areshown for each of these three cases. The uppermost pair of curves corresponds to q = 0 .
01 with standard softening and the lowermost pair for q = 0 .
01 with smallsoftening. The central pair corresponds to q = 0 .
02 with small softening. Thelower of the pair of curves for the cases with small softening correspond to runswith accretion from the disk included. For the case with standard softening thissituation is reversed. The straight lines which have imposed crosses are obtainedadopting the initial Type I migration rate. The line with the more widely sep-arated crosses corresponds to q = 0 .
01 with small softening while the other linecorresponds to q = 0 .
01 with standard softening. retrograde binary embedded in accretion disk q = 0 . , the initial migration rate is a factor of twosmaller than the expected type I migration rate with the effects of gap formationbeing noticeable immediately. Note that at longer times the migration rates for q = 0 .
01 with different softening lengths slow to become approximately equal aswould be expected if the migration was governed by the viscous evolution of thedisk. On the other hand, the larger open inner boundary radius adopted for thesimulations with smaller softening, on account of necessary numerical convenience,results in a relatively larger angular momentum loss from the system as materialpasses through and this may also affect the orbital evolution (see below). In allcases the characteristic time scale becomes comparable to or greater than that forthe viscous evolution of the disk.
A very simple approach to the problem of calculation of orbital the evolution ispossible when the pertuber mass is larger than a typical disk mass in a regionof size r p (see IPPP). In this case the orbital evolution timescale t ev exceeds thelocal timescale for viscous evolution of the disk, t ν . After the perturber has beenpresent in the disk for a time that is larger than t ν , but smaller than t ev , thedisk structure at radii r ∼ r p should be close to a quasi-stationary one. In thissituation, the mass flux ˙ M and the specific angular momentum at the inner diskmay be assumed to be functions of time only with a characteristic time scale forchange being much larger than t ν . In addition, in the limit q ≪ , the annulus in the vicinity of perturber, whereimpulsive interaction with the disk gas operates, is very small, with a typicaldimension ≪ r p . Therefore, in the simplest treatment of the problem, we describethe influence of the perturber on the disk as providing a jump condition on thesurface density, to be applied at the perturber’s orbital location, in a disk otherwiseevolving only under the influence of internal viscosity.As indicated above, the mass flux through the gap is approximately constantin this limit. Furthermore, it can be easily shown (e.g. IPPP) that when the massflux is fixed, stationary solutions depend only on one constant of integration, h ∗ ,which is proportional to the flux of angular momentum through the disk throughthe relation ˙ L = ˙ M Ω r h ∗ . The region of the disk for which r slightly exceeds r p should attain Σ( r p + ) ∼ r p to lose angular momentum and be transferred to the inner regionthrough the gap. This means that the flux of angular momentum through thedisk at radii r > ∼ r p , ˙ L + , should be ∼ ˙ M p GM r p and we must accordingly set h ∗ = p r p /r . On the other hand, the flux of angular momentum through the inner disk, at82
P. B. Ivanov et al r < ∼ r p , ˙ L − i, should be equal to the angular momentum accreted per unit timeby the component with the dominant mass, M . Assuming that r p is much largerthan the size of the last stable circular orbit around that component, we can set˙ L − ≈ . Since the total angular momentum of the system is conserved and that, forsmall enough inner boundary radius, there is no angular momentum flux throughthe inner disk, the outward angular momentum flux through the outer disk, T ,must be equal and opposite to the torque acting on the perturber due to the disk,the latter being − T. Thus we have T ≈ − ˙ M ( t ) p GM r p (1)where ˙ M ( t ) >
0, and, accordingly,
T < M ( t ) ≈ const being equal to the mass flux at infinity. In this case, using (1) and the law ofangular momentum conservation we get r p = r exp( t/t ev ) , t ev = M p M . (2)When the disk has a finite extent as in our numerical simulations, a simple ap-proach to the calculation of the dependence of ˙ M on t is possible for a disk witha constant kinematic viscosity. A comparison of the results based on analytic andnumerical methods is shown in Fig. 16 of IPPP, which demonstrates excellentagreement between the methods. So far we have assumed that the eccentricity of the binary is zero. In this case itcan be easily shown that there are no outer Lindblad resonances and the standardmechanism of gap opening by a torque carried by waves launched at resonancesis absent. The situation is different, however, in case of an eccentric retrogradebinary, which can be formed both when SBBH and planetary systems are con-sidered, see e.g. Polnarev & Rees (1994), Papaloizou & Terquem (2001) for thecase of SBBH and planetary systems, respectively. In this case the Lindblad res-onances are present although the amplitude of the torque is suppressed comparedwith the prograde case. Provided the gap (or cavity) is formed through the actionof the resonances, its structure is quite different from that discussed above andcan resemble the prograde case discussed in IPP. Namely, the action of resonancessupplies positive angular momentum to the disk gas, thus leading to accumula-tion of the gas at distances exceeding r p , and accordingly, formation of gap orcircumbinary cavity. In order to estimate the importance of this effect we use thetheory of Goldreich & Tremaine (1979) and the gap opening criterion discussed inLin & Papaloizou (1979) and Artymowicz & Lubow (1994). The condition of gap retrograde binary embedded in accretion disk e l,mcrit , where l and m correspond to a Fourier harmonics withtemporal and azimuthal mode numbers m and l , respectively. We have e , − crit ≈ . α / ∗ q − / ∗ δ / ∗ (3)and e , − crit = 0 . α / ∗ q − / ∗ δ / ∗ , (4)for m = 1, l = − m = 2, l = −
1, respectively, where α ∗ = α/ − , q ∗ = q/ − and δ ∗ = ( H/r ) / − . Since the critical eccentricities are of the orderof 0 . − . δ ∼ .
05, and, accordingly, δ ∗ ∼
50. In this case wehave the critical eccentricities formally exceeding unity for α ∗ = 1, and, therefore,this effect is unlikely to operate unless α is very small. The mass flux to the perturber is estimated in IPPP as˙ m ∼ q / ˙ M . (5)It was also shown by IPPP that this estimate agrees with numerical simulationsprovided the results obtained by the numerical approach are averaged over severalorbital periods. On the other hand the numerical approach shows that the massflux can change by order of magnitude or more on the orbital timescale. Notethat this variability may lead to some important consequences since it can leadto luminosity variability on the same time scale provided accretion efficiency issufficiently large. Also note that equation (5) shows that the mass flux onto theperturber is smaller than the mass flux to the primary component provided q ≪ In the case of SBBH there is an additional important mechanism for driving orbitalevolution through emission of gravitational waves. For a circular orbit and q ≪ t gw can be easily obtained from expressions givenby e.g. Landau & Lifshitz (1975) and from equation (2). We first remark that t ev can be written as t ev ≈ · (cid:18) q − M ˙ M − (cid:19) yr, (6)where q − = q/ − , M = M/ M ⊙ , and ˙ M − = ˙ M / (10 − M ⊙ yr − ). Fromthe condition t gw < t ev we find that gravitational waves determine the orbital84 P. B. Ivanov et al evolution when r p < r gw ( I ) = r g (cid:18) cqt ev r g (cid:19) / ≈ . q − M ( ˙ M − ) / pc. (7)Note that the orbital period at r p ∼ r gw ( I ) being given by P orb ≈ r / − M − / yr ,where r − = r p / (10 − pc ) is expected to be of the order of a few years. From thedefinition of r gw ( I ) and (6) it also follows that t gw = (cid:18) r p r gw ( I ) (cid:19) t ev . (8)Another important length scale, r gw ( ν ) , is determined by the condition that thetime scale for orbital evolution due to gravitational radiation be less than the timescale for viscous evolution of the disk, or t gw ( r p < r gw ( ν ) ) < t ν . For this lengthscale we obtain r gw ( ν ) = r g " √ q αδ / ≈ · − M ( q − ) / α − / ∗ δ − / ∗ pc. (9)When r < r gw ( ν ) , from the point of view of the perturber, the disk gas is trans-ferred from the inner the disk to the outer disk which is the opposite direction tothat considered above. However, arguments leading to the expression (5) remainessentially the same if instead of the accretion rate through the disk, ˙ M , the rateof transfer of the disk gas through perturber’s orbit, ˙ M tr , is adopted. Note that˙ M tr is defined in the frame, where perturber is at rest. We can estimate it as˙ M tr ∼ M d ( r < r p ) /t gw , where the disk mass inside the perturber’s orbit. Asdiscussed above the disk inside the perturber’s orbit may be approximated as astationary accretion disk, characterised by the accretion rate ˙ M , and therefore, itsmass can be estimated as M d ( r < r p ) ∼ ˙ M t ν . Taking these considerations intoaccount we obtain˙ m ∼ q / M d ( r < r p ) t gw ∼ q / ˙ M t ν t gw ∼ q / ˙ M r gw ( ν ) r . (10)This indicates that the accretion rate onto the secondary can exceed that onto theprimary, ∼ ˙ M , provided that r < r crit = q / r gw ( ν ) . (11)Since the power of q in (11) is small, we have that typically r crit ∼ r gw ( ν ) . Let us assume that the future space-borne gravitational wave antenna will havesensitivity h = 10 − ˜ h − in the frequency range ω min = 10 − ˜ ω − Hz < ω gw < ω max = 10 − ˜ ω − Hz (12) retrograde binary embedded in accretion disk h − = h / − , ˜ ω − = ω min / − Hz , ˜ ω − = ω max / − Hz are dimen-sionless constants, and we expect the antenna to be sensitive to gravitational waveswith a typical amplitude 10 − and typical frequencies 10 − − − Hz . On theother hand, when SBBH orbit is approximately circular we have ω gw ≈ ω orbit = 2( GM ) / r − / and hence r = r g ( c √ /r g ω gw ) / . (13)From (13) and the conditions on ω gw given by (12) one obtains the followingconstraints on the orbital radius during this final stage: β min < r/r g < β max , where β min = (cid:16) √ r g ω max /c (cid:17) − / and β max = (cid:16) √ r g ω min /c (cid:17) − / . (14)Another constraint is obtained from a comparison of the amplitude of the emittedgravitational waves, | h αβ | , with h . Using the quadrupole formula (Landau &Lifshitz, 1975) to make an order of magnitude estimate, one obtains h ∼ (2 G/ c L ) ¨ D αβ ∼ (2 G/ c L )(3 / qM r ω gw = ( G/c L ) qM r (4 GM ) /r = qr g /rL > h , (15)where L = L ×
100 Mpc is the distance to the binary and ¨ D αβ is the second timederivative of the quadrupole tensor. Noting that r > r st = 3 r g , where r st is theradius of the last stable circular orbit for the Schwarzschild metric, the conditionsfor the gravitational radiation from the binary to be detectable can be written inthe form max [3 , β min ] < rr g < min [ β ∗ , β max ] , where β ∗ = qr g h L . (16)These constraints are compatible if β ∗ > , β ∗ > β min and β max > . (17)The above inequalities can be rewritten as q − > × − ˜ h − L M − , q − > × − ˜ h − L M − / ˜ ω − / − and M < × ˜ ω − − . (18)In the most realistic case q − > × − ˜ h − L M − / ˜ ω − / − , which corresponds to β ∗ > β max (19)86 P. B. Ivanov et al and the duration of this final stage is∆ t gw ≈ q − − M − / ˜ ω − / − yr . (20)During this period the frequency of gravitational waves increases from ω min to ω ∗ ,where ω ∗ = ω max , if M < × − ˜ ω − − (which corresponds to β min >
3) (21)or ω ∗ = 3 × − M − Hz (the frequency corresponding to r = r st = 3 r g ) , if 3 × − ˜ ω − − < M < × ˜ ω − − (corresponding to β min < < β max ) . (22) In this note we briefly reviewed results obtained in IPPP on the interaction of aretrograde circular binary with a coplanar accretion disk. We discussed the fol-lowing results.1) When the mass ratio q is small, but larger than ∼ . H/r p ) a gap in thevicinity of the perturber opens due to increase of radial velocity of the gas in thisregion. Its size smaller than the orbital distance r p in this limit.2) For such systems assuming that perturber’s mass is larger than a typical diskmass at distances ∼ r p the disk structure outside the gap is close to a quasi-stationary one. The inner disk has nearly zero angular momentum flux, while theouter disk has angular momentum flux equal to the mass flux times the binary spe-cific angular momentum. The orbital distance evolution timescale t ev = M p / (2 ˙ M )is determined by the law of conservation of angular momentum. Note that thispicture differs from the prograde case with similar parameters, where there is apronounced cavity instead of the inner disk and the orbital evolution is somewhatfaster.3) When the orbital evolution is determined by the interaction with the disk themass flux onto the more massive component ∼ ˙ M , while the average mass flux ontothe perturber is smaller ∼ q / ˙ M .
However, the latter exhibits strong variabilityon timescales on the order of the orbital period. The mass flux to the perturbercan increase significantly during the late stages of the inspiral of SBBH when theemission of gravitational waves controls the orbital evolution.4) When the binary is sufficiently eccentric and the disk is sufficiently thin, theopening of a ’conventional’ cavity within the disk is also possible due to the pres-ence of Lindblad resonances.Additionally, we estimated a time duration for which the emitted gravitationalwaves would have sufficient amplitude for detection by a space-borne interferomet-ric gravitational wave antenna with realistic parameters. This is given by eq. (20)as well as the appropriate range of frequencies as a function of the primary blackhole mass.Note that all these results have been obtained under the assumption that thebinary orbit and the disk are coplanar. This may break down at late times since retrograde binary embedded in accretion disk t evev