Minidisk dynamics in accreting, spinning black hole binaries: Simulations in full general relativity
DDraft version February 17, 2021
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MINIDISK DYNAMICS IN ACCRETING, SPINNING BLACK HOLE BINARIES: SIMULATIONS IN FULLGENERAL RELATIVITY
Vasileios Paschalidis , , Jane Bright , Milton Ruiz , Roman Gold Department of Astronomy, University of Arizona, Tucson, AZ 85726, USA Department of Physics, University of Arizona, Tucson, AZ 85726, USA Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 CP3 origins | Southern Denmark University (SDU) Campusvej 55, Odense, Denmark
Draft version February 17, 2021
ABSTRACTWe perform magnetohydrodynamic simulations of accreting, equal-mass binary black holes in fullgeneral relativity focusing on the impact of black hole spin on the dynamical formation and evolution ofminidisks. We find that during the late inspiral the sizes of minidisks are primarily determined by theinterplay between the tidal field and the effective innermost stable orbit around each black hole. Ourcalculations support that a minidisk forms when the Hill sphere around each black hole is significantlylarger than the black hole’s effective innermost stable orbit. As the binary inspirals, the radius ofthe Hill sphere decreases, and minidisk sconsequently shrink in size. As a result, electromagneticsignatures associated with minidisks may be expected to gradually disappear prior to merger whenthere are no more stable orbits within the Hill sphere. In particular, a gradual disappearance of a hardelectromagnetic component in the spectrum of such systems could provide a characteristic signature ofmerging black hole binaries. For a binary of given total mass, the timescale to minidisk “evaporation”should therefore depend on the black hole spins and the mass ratio. We also demonstrate that accretingbinary black holes with spin have a higher efficiency for converting accretion power to jet luminosity.These results could provide new ways to estimate black hole spins in the future.
Subject headings: black hole physics—gamma-ray burst: general—gravitation—gravitational waves—stars: neutron INTRODUCTION
Supermassive black hole (SMBH) binaries are expectedto form in gas rich environments as the result of galaxymergers (Rodriguez et al. 2009). These systems formunique sources for multi-messenger signals. SMBH bi-nary evolution is particularly promising as potentiallystrong sources of gravitational and various forms of elec-tromagnetic emission. Gravitational waves from SMBHbinaries are expected to be detectable by Pulsar TimingArrays or by future space-based gravitational wave ob-servatories such as the Laser Interferometer Space An-tenna (LISA) (Hobbs et al. 2010; Amaro-Seoane et al.2017). Recently, observational efforts searching for suchsystems have yielded an unprecedented number of ex-citing candidate sources based on various diagnostics in-cluding periodicities in light curves spectral features suchas double-horned emission lines, and (lensing) flares (seee.g. Drake et al. (2009); Graham et al. (2015); Charisiet al. (2016); Hu et al. (2019)). A main challenge inthese efforts is our lack of understanding of the uniquefeatures the electromagnetic emission from these systemshas. As a result, much of the theoretical work on accret-ing binary black holes to-date has focused on identifyinginteresting characteristic electromagnetic signatures thataccompany their gravitational wave signals.In this paper we investigate the conditions under whichminidisks form around each individual black hole in thelate stages of the binary black hole inspiral. It has beenargued that minidisks are responsible for the emission ofthe hardest part of the electromagnetic spectrum dueto shock heating (Farris et al. 2015). Emission from the hot minidisks could make up for any “notch” in thehigh frequency spectrum that might be expected due tolack of gas in the disk cavity during the inspiral phase.Minidisks could also be responsible for providing mod-ifications to broad emission-line profiles (Nguyen et al.2019). The potentially observable electromagnetic (EM)emission originating from minidisks as well as the pos-sibility to use them as “smoking-gun” features to iden-tify SMBH binaries, makes understanding the conditionsunder which minidisks form and maintain a persistentstructure toward the late stages of the inspiral a vitalgoal. Minidisks have been seen in some recent Newtonianstudies (e.g. Farris et al. (2014, 2015); Bowen et al. (2017,2018), but not in previous relativistic studies (e.g. Far-ris et al. (2012); Gold et al. (2014a,b)), highlighting theneed to investigate the conditions for minidisk formationin relativistic gravitation. Some more recent studies us-ing approximate spacetime metrics with focus on nearequal mass systems (Bowen et al. 2017, 2018, 2019), ini-tialize the simulations with minidisks in addition to thecircumbinary disk, but do not report persistent, steady-state minidisks. Instead, those works report cycles ofdepletion and replenishment as well as sloshing of massbetween the two minidisks.Previous theoretical work on circumbinary disks in-corporating varying degrees of relativistic gravity andmagnetic fields include (Giacomazzo et al. 2012; Nobleet al. 2012; Farris et al. 2012; Farris et al. 2014; Goldet al. 2014a,b; Kelly et al. 2017; Bowen et al. 2018; Khanet al. 2018; Bowen et al. 2019; Armengol et al. 2021),but only a subset of these focus on circumbinary accre- a r X i v : . [ a s t r o - ph . H E ] F e b Paschalidis, Bright, Ruiz, & Goldtion, see (Gold 2019) for a recent review. Gold et al.(2014b) proposed that circumbinary disks form minidiskswhenever there are stable circular orbits within the Hillsphere around each black hole. In this work we performsimulations in full general relativity (GR) to test thishypothesis. Our approach makes no approximation forthe spacetime metric and as a result we do not have toexcise parts of the domain and/or impose ad hoc inflowboundary conditions. Instead, the black holes are re-solved objects on our computational grid. The durationof our simulations are significantly shorter than in someearlier treatments. However, the regime of minidisks thatis probed here exhibits much shorter relaxation times.The analysis we follow is guided in part by the re-stricted 3-body Newtonian problem, where one can de-fine regions in which gravity is dominated by each ofthe binary components, as quantified by the Hill sphere r Hill = 0 . q/ / d . Here q is the mass ratio and d the binary separation. While the systems studied hereare not in the Newtonian regime, it is still expectedthat many Newtonian aspects carry over to the rela-tivistic regime with some appropriate corrections. Onthe other hand, it is well-known that stable circular or-bits (and hence disks) around general relativistic blackholes can exist only outside the innermost stable circularorbit (ISCO) r ISCO = r ISCO ( χ ) where χ is the dimen-sionless black hole spin parameter satisfying | χ | ≤ r Hill (cid:29) r ISCO ( χ ), so that stable orbits around eachblack hole can exist within the Hill sphere (Gold et al.2014b). However, in the later stages of the inspiral when r Hill ∼ r ISCO ( χ ) no stable orbits exist within the Hillsphere and matter is quickly accreted as soon as it en-ters the Hill sphere. For an equal mass, non-spinning sys-tem, we can perform a simple Newtonian estimate whichyields r Hill = 0 . q/ / d ∼ . d and r ISCO = 3 M ,setting these equal gives a threshold separation for mini-disks to exist d thres (cid:39) . M . In particular, one wouldrequire d (cid:29) d thres for persistent minidisks to exist. Asthe binary inspirals d approaches d thres , and any persis-tent minidisks should evaporate.Astrophysical black holes are expected to havespin (Lynden-Bell 1969). X-ray reflection measurementtechniques have been used in over two dozen AGN sys-tems to determine the spins of the SMBHs, and havefound that the majority of these systems are rapidly ro-tating with χ ≥ .
9, while the most massive SMBHs( (cid:38) M (cid:12) ) have slower, but still significant spins in therange χ ∼ . − . q = 1 binary black holes with spins both (anti-)aligned with the orbital angular momentum, and onealigned and the other anti-aligned we demonstrate explic-itly the impact of spin in allowing the formation of mini-disks, thereby confirming the expectations from Goldet al. (2014b) that minidisks should form only when sta-ble orbits exist within the Hill sphere around each blackhole. In addition, we demonstrate the importance of spin on jet outflows as we find that spinning binaries havestronger jet luminosities.In the fully relativistic circumbinary accretion studiesof Farris et al. (2012); Gold et al. (2014a,b) the launch-ing of jets from the interaction of magnetic fields withblack hole horizons, even for nonspinning binary blackholes, had already highlighted the importance of full GRin discovering features that are entirely missed in New-tonian studies, and studies that include varying degreesof relativity, but still do not capture black hole horizons.Minidisk evaporation as discussed in this work highlightsyet another effect in which GR plays a key role in deter-mining even the qualitative behavior.The rest of the paper is structured as follows: In Sec-tion 2 we present the initial data, grids and numericalmethods we adopt for our evolutions. In Section 3 wepresent our results, and summarize in Section 4. We de-note the total binary mass M and both the primary andsecondary mass m . We set G=c=1. METHODS AND INITIAL DATA
Our methods for solving Einstein’s field equations forthe gravitational field, and the equations of ideal mag-netohydrodynamics in curved spacetime have been de-scribed previously in (Farris et al. 2012; Gold et al.2014a,b), where we refer the reader for more details.As in (Khan et al. 2018) we use puncture initial datafor the spacetime metric adopting the
TwoPunctures code (Ansorg et al. 2004), but set the black holes ini-tially on a quasicircular orbit at a larger coordinate sep-aration of 20M. We also treat black hole spin for thefirst time in full GR calculations of circumbinary ac-cretion. We consider equal-mass binaries in four spinconfigurations: χ = χ = 0 (nonspinning case labeled χ ), χ = χ = 0 .
75 (case χ ++ ), χ = χ = − . χ −− ), χ = − χ = 0 .
75 (case χ + − ). Where χ , χ are the dimensionless spins of each black hole, and +( − ) sign indicates spin aligned (anti-aligned) with the or-bital angular momentum. The matter and magnetic fieldinitial data are identical to those in (Gold et al. 2014a)and the same for all binary configurations we study, withthe equation of state corresponding to a Γ = 4 / χ , and 13 levels for cases with non-zero spin. We place the outer boundary at 3072 M , andthe resolution on the coarsest refinement level to 48 M .The finest level resolution is (cid:39) M/
85 ( (cid:39) M/
43) for caseswith (without) spin. The black hole apparent horizon di-ameters are resolved by (cid:38)
51 ( (cid:38)
41) grid points in theirsmallest dimension for the spinning (non-spinning) cases.The half-side lengths of the refinement boxes are givenby 3072 / n , n = 0 , , . . .
12, where n indexes the levelsof refinement with n = 0 denoting the coarsest level. RESULTS
Non-spinning black holes have an ISCO (areal) radiusof r ISCO = 6 m = 3 M . Prograde (retrograde) orbitsaround black holes with spin χ = 0 .
75 have smaller(larger) r ISCO = 3 . m = 1 . M ( r ISCO = 8 . m =4 . M ), thereby allowing more (less) space for stable or-imulations of accreting, spinning binary black holes in full GR 3bits within the black hole Hill sphere, making it easier(more difficult) for minidisks to form. We use these radiias coordinate radii in our figures to indicate the locationof the ISCO around each black hole. We point out thatthis is neither a gauge invariant, nor is it a precise mea-sure for relativistic binaries. We only use these radii asan approximate way to visualize the ISCO in our figures.We use the rest mass density profiles, the mass con-tained within the Hill sphere of each black holes, and theaccretion rates onto the black holes (as defined in Farriset al. (2010)) as diagnostics for studying minidisk forma-tion and evolution. The Hill sphere radius is computedbased on the Newtonian formula r Hill = 0 . q/ / d ,within which we integrate the total rest mass. This defi-nition of the Hill sphere is based on the coordinate radius,it is not gauge invariant, and is used only as a means toapproximate the mass contained in minidisks. However,since the binary is symmetric this coordinate radius de-fines a volume around each black hole that is the same,which roughly coincides with the outer edges of the mini-disks that form in our simulations as we show below. Wealso investigate the temperatures of the minidisks as wellas the efficiency of converting accretion power to lumi-nosity in each of our models.The initial data were designed to probe the interplayof tidal forces and black hole spins to determine theirimpact on minidisks. In our earlier work, we did notfind evidence for persistent minidisks. Minidisks wouldform and quickly be depleted. In those works the blackholes were non-spinning ( r ISCO = 6 m = 3 M ) and theinitial separation was d = 10 M , and hence a Hill sphereof r Hill ∼ . M , leaving almost no room for stable orbitswithin the Hill sphere. By contrast in this study, theinitial separation is d ∼ M , providing a larger Hillsphere of r Hill (cid:39) M , which leaves plenty of room forstable orbits for χ = 0 .
75 black holes, some room in the χ = 0 black holes, and little room in the χ = − . χ = 0 .
75 black holes, but not so much around χ = − . Rest-Mass Density
As in our previous studies the matter undergoes a tran-sient phase as the initial data is relaxed, and the systemsbegin to settle after a time period ∆ t (cid:39) M , based onthe saturation of the mass accretion rate onto the blackholes ˙ M . Minidisk structures form even before the accre-tion rate saturates and then settle down to a quasi-steadystate around the same time.In Fig. 1 we show equatorial rest-mass density snap-shots from case χ + − . This case is the most unambiguousdemonstration of the effect of spin on minidisk forma-tion. For the χ = +0 .
75 black hole r Hill is significantlylarger than r ISCO , while for χ = − .
75 black hole r ISCO is comparable to r Hill . As a result, we observe that thepositive spin black hole develops a minidisk while thenegative spin black hole does not . For the χ = − . r Hill (cid:29) r ISCO , re-solving the question under what conditions minidisks canexist during the late stages of the inspiral.In addition, we investigate our other models, χ , χ ++ ,and χ −− to further support the findings of the χ + − model. As expected from our earlier arguments, we findthat cases χ and χ ++ form persistent minidisks, whilecase χ −− ( r ISCO = 8 . m = 4 . M ) does not, and ex-hibits the same behavior as the χ = − .
75 black holein the χ + − model, with spiral streams entering the Hillsphere and crossing the ISCO, and thereby plunging intothe black hole. In Fig 2, we show a equatorial rest-massdensity snapshot after 5 binary orbits have been com-pleted for the χ ++ , and χ −− , demonstrating the above. Rest-mass within Hill spheres and Accretion Rates
We next investigate the rest mass contained within theHill sphere of each black hole. The top left panel of Fig 3shows the rest mass within the individual Hill spheres forthe χ + − case normalized to the time-averaged value ofthe total rest-mass within the Hill spheres of the χ case.The χ = +0 .
75 black hole clearly contains more rest massthan the χ = − .
75 one, demonstrating in an anotherway that the matter plunges into the χ = − .
75 blackhole as soon as it enters the Hill sphere because it crossesthe ISCO. A similar picture is painted by the top rightpanel in the same figure, which shows the total rest masswithin the Hill spheres normalized to the time-averagedvalue of the total rest-mass within the Hill spheres ofthe χ case for cases χ , χ ++ and χ −− . As is clearfrom the image the total mass within the Hill spheres inthe χ ++ is about twice that of the χ model, which isabout twice that of the χ −− model. However, the plotsalso demonstrate that the amount of rest-mass withinthe Hill sphere of even the χ = +0 .
75 black holes, whichform persistent minidisks, oscillates around a mean valuein agreement with the non-spinning simulations reportedin (Bowen et al. 2017).In the bottom left panel of Fig 3 we show the rest-massaccretion rate onto each black hole of the χ + − modelnormalized by the time-averaged total accretion rate ofthe χ case. The plot clearly shows that the χ = − . χ = +0 . χ < χ > χ case for cases χ , χ ++ and χ −− , with the hor-izontal lines indicating the time-averaged accretion ratefor each case. As expected the time-averaged rest-massaccretion rate in the χ −− is larger than that in the χ case, which in turn is larger than in χ ++ case. The χ , χ ++ , and χ −− cases reinforce our findings from the χ + − model. The accretion rates exhibit clear periodicities,which will be the focus of a future paper of ours.All these findings are consistent with the expectationthat the location of the ISCO has a significant impact onthe systems ability to maintain mass within the Rochelobes and form minidisks toward the late stages of the Paschalidis, Bright, Ruiz, & Gold Fig. 1.—
Rest-mass density in the equatorial plane for the χ + − model. A persistent minidisk quickly forms around the χ = +0 .
75 blackhole, but no disk forms around the χ = − .
75 black hole. The Hill spheres (black dashed circles) and the ISCO radii (white circles) areshown around each black hole (assuming each BH was in isolation). For the χ = +0 .
75 black hole the Hill sphere is significantly largerthan the ISCO, but for the χ = − .
75 they are more comparable in size.
Fig. 2.—
Comparison of the rest-mass density in the equatorial plane for χ −− (left panel), and χ ++ (right panel), taken followingcompletion of 5 orbits in both cases. Persistent minidisks are seen only in the χ ++ case as the Hill sphere (black dashed circles) issignificantly larger than ISCO radius (white circles). inspiral. Jet Luminosities and Temperatures
We find that jets are launched from both spinningand non-spinning systems as illustrated in Fig. 4, whichshows a 3D rendering of the rest-mass density of the χ ++ model with white lines indicating the magnetic field linesanchored to the black holes. The magnetic field lines aremore twisted than in the non-spinning cases we reportedin earlier work (Gold et al. 2014a,b; Khan et al. 2018) – a result of black hole spin. This combined effect leads toa dual jet structure close to the black holes that mergeto form a single jet structure at larger height.We calculate the Poynting luminosity associated withthe collimated jet outflow on the surface of coordinatespheres S as L EM = (cid:72) S T r, (EM) dS , where T µν, (EM) is theEM stress-energy tensor. L EM / ˙ M eq is the efficiency forconverting accretion power to EM jet luminosity, where˙ M eq is the time-averaged accretion rate after the accre-tion rate has settled ( t (cid:38) M ). We plot the efficiencyimulations of accreting, spinning binary black holes in full GR 5 Fig. 3.—
Top Left:
Rest mass within the Hill sphere of each black hole in the χ + − model as a function of time, normalized to the averagevalue for the total rest mass in both Hill spheres for the χ case. Top Right:
Total rest mass within the Hill spheres of both black holesin each model listed as a function of time, with the same normalization as in the top left.
Bottom Left:
Rest mass accretion rate onto eachblack hole for the χ + − model as a function of time, normalized to the time-averaged value of the total rest-mass accretion rate onto bothblack holes in the χ case. Bottom Right:
Total rest-mass accretion rate onto both black holes in each model listed as a function of timewith the same normalization as in the bottom left. The horizontal lines indicate the time-averaged value for each model. M ≈ M M (cid:12) )sec. as a function of time in Fig. 5 for the χ , χ ++ , and χ + − models. We note that it takes time for the out-flow to develop and propagate out the EM luminosityextraction radius of 150 M , which is why although theflow around the black holes can relax, it takes longer forthe EM luminosity to relax. The evolution of the χ −− model was long enough for the accretion rate to relax,but not long enough for the EM luminosity to do so. Asa result we do not include this model here. The figureclearly demonstrates that spin plays a significant role inthe efficiency of the luminosity output, with the great-est efficiency achieved in the χ ++ model and the lowestefficiency in the χ model. This is the expected out-come resulting from the combination of the Blandford-Znajek (BZ) effect, which describes outgoing luminosityproduced from a single spinning black hole (Blandford &Znajek 1977) and an “orbital BZ effect (Palenzuela et al.2010).Finally, we compute the effective disk temperaturebased on the assumption of radiation pressure dominance(consistent with the the Γ = 4 / ρ (cid:15) = aT . We scale the total binary mass to10 M (cid:12) and scale the maximum rest mass density suchthat the average accretion rate equals the Eddington ac-cretion rate with efficiency 10%. Doing this providesa characteristic temperature in the minidisks of around T ∼ (cid:16) ˙ M avg ˙ M Edd (cid:17) / (cid:16) M M (cid:12) (cid:17) − / K. We find this to beconsistent across each of the models. The minidisks havea higher temperature than the circumbinary disk, which is itself hottest at the inner edges and cooler in the bulkof the disk. CONCLUSIONS AND DISCUSSION
We have demonstrated that minidisks in binary blackholes form if the Hill sphere is significantly larger thanthe ISCO radius. We further showed that the size of theminidisks (on average) traces the approximate Hill spherearound each black hole, which implies that the sizes ofminidisks are tidally truncated and therefore a simple,linear function of the binary separation only. This find-ing could be important if the size of mini disks couldbe inferred from observations. One of the challenges in-volved is that source size depends on observation fre-quency, but one may hope that correlations between theactual minidisk size and its photosphere can be found inthe future.Our work establishes a common evolutionary sequencethat similar systems are expected to follow: At largeenough separations when the Hill sphere is larger thanthe ISCO radius, minidisks are expected to be present.In this phase the accretion rate is quasi-periodic and thesize and separation of the minidisks are linear functionsof the binary separation. Since r Hill = 0 . q/ / d de-pends on the binary separation, as the binary inspirals itwill reach a threshold where r Hill (cid:39) r ISCO and persistentminidisks will no longer be able to exist and will “evapo-rate” because the tidally stripped accretion streams fromthe circumbinary disk are accreted on a short dynamicaltime scale as they plunge into the black hole following Paschalidis, Bright, Ruiz, & Gold
Fig. 4.—
3D view showing rest-mass density (color coded) for the χ ++ model and magnetic field lines (white) anchored on the black holehorizons. Twin jets are visible above and below each black hole (see inset for a zoomed in view). The black spheres indicate the resolvedblack hole apparent horizons in our simulations. Fig. 5.—
The efficiency for converting accretion power to EM lu-minosity L EM / ˙ M eq as a function of time. Here L EM is the Poynt-ing luminosity and ˙ M eq is the average total rest-mass accretionrate for each model after t = 1500 M when the accretion rate hassettled, and M ≈ M M (cid:12) ) sec. ISCO crossing. This leads to more complicated vari-ability in the mass accretion rate until merger and anyEM signature associated with persistent minidisks is ex-pected to become fainter. Note that the onset of thistransition depends on black hole spin (through the ISCOradius). We term this anticipated dimming of EM emis-sion from minidisks “minidisk evaporation”. Followingminidisk evaporation toward the late stages of the in-spiral, the qualitative evolution then proceeds in accordwith our previously obtained results (Farris et al. 2012;Gold et al. 2014a,b; Khan et al. 2018). Our results also confirm that black hole binaries with prograde spin main-tain minidisks for a longer timescale than non-spinningand retrograde spin black holes.These finding could in principle serve as a new diag-nostic to probe black hole spins observationally whencombined with information from the gravitational wavesignal. In particular, when LISA is operating or in theevent that Pulsar Timing Arrays detect an individualsupermassive black hole binary, the merger time can bepredicted from the gravitational wave signal. Therefore,the difference of merger time to the time when the mini-disk signature fades away should open a new avenue toprobe black hole spins observationally. For this strat-egy to work out additional source modeling and betterpredictions from theory will be invaluable. Our workhere has shown that when r Hill (cid:29) r ISCO , minidisks canform, but additional studies to probe the epoch of mini-disk evaporation where r Hill (cid:38) r ISCO will be importantto make this a viable and useful diagnostic.In addition to the minidisk dynamics here we alsofound that jets arising from circumbinary accretion ontobinary black holes toward the late stages of the inspi-ral are significantly more powerful when spinning blackholes (even with moderately high dimensionless spin of χ = 0 .
75) are involved. With proper theoretical model-ing this finding also paves a new way to probe black holespin from future EM jet observations of these systems.Finally, apart from the observational implications, ourresults have important consequences for future relativis-tic simulations of these systems. In particular, the choseninitial orbital separation should respect the criterion forminidisk formation in order to properly compute accre-tion rates and electromagnetic signals, and to faithfullyimulations of accreting, spinning binary black holes in full GR 7represent the expected structure of the circumbinary sys-tem.This work was in part supported by NSF Grant PHY-1912619 to the University of Arizona, NSF Graduate Re-search Fellowship Grant DGE-1746060, and NSF GrantPHY-1662211 and NASA Grant 80NSSC17K0070 to the University of Illinois at Urbana-Champaign. Computa-tional resources were provided by the Extreme Scienceand Engineering Discovery Environment (XSEDE) undergrant number TG-PHY190020. XSEDE is supported bythe NSF grant No. ACI-1548562. Simulations were per-formed on
Comet , and
Stampede2 , which is funded by theNSF through award ACI-1540931., which is funded by theNSF through award ACI-1540931.