Disc tearing: implications for black hole accretion and AGN variability
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Disc tearing: implications for black hole accretion and AGN variability
A. Raj and C. J. Nixon School of Physics and Astronomy, University of Leicester, Leicester, LE1 7RH, UK
ABSTRACTAccretion discs around black holes power some of the most luminous objects in the Universe. Discsthat are misaligned to the black hole spin can become warped over time by Lense-Thirring precession.Recent work has shown that strongly warped discs can become unstable, causing the disc to breakinto discrete rings producing a more dynamic and variable accretion flow. In a companion paper,we present numerical simulations of this instability and the resulting dynamics. In this paper, wediscuss the implications of this dynamics for accreting black hole systems, with particular focus onthe variability of Active Galactic Nuclei (AGN). We discuss the timescales on which variability mightmanifest, and the impact of the observer orientation with respect to the black hole spin axis. When thedisc warp is unstable near the inner edge of the disc, we find quasi periodic behaviour of the inner discwhich may explain the recent quasi periodic eruptions observed in, for example, the Seyfert 2 galaxyGSN 069 and in the galactic nucleus of RX J1301.9+2747. These eruptions are thought to be similarto the ‘heartbeat’ modes observed in some X-ray binaries (e.g. GRS 1915+105 and IGR J17091-3624).When the instability manifests at larger radii in the disc, we find that the central accretion rate canvary on timescales that may be commensurate with, e.g., changing-look AGN. We therefore suggestthat some of the variability properties of accreting black hole systems may be explained by the discbeing significantly warped, leading to disc tearing.
Keywords:
Accretion, accretion discs — Hydrodynamics — Instabilities — Black hole physics INTRODUCTIONAccreting black holes come in different sizes, from stel-lar mass black holes in X-ray binaries (e.g. Verbunt 1993;Remillard & McClintock 2006) to supermassive blackholes in the centres of galaxies (e.g. Kormendy & Ho2013). In all cases, the accreting matter is in the formof a disc (Pringle 1981). Disc formation is inevitableas the matter possesses angular momentum which itcannot easily shed, whereas the excess energy associ-ated with orbital eccentricity can be readily lost; colli-sions between gas orbits turn orbital energy into heatwhich can be radiated away. Once in the form of adisc, additional energy may be extracted from the gasorbits. Disc viscosity, usually ascribed to turbulencein the disc (Shakura & Sunyaev 1973), leads to torqueswhich transport angular momentum outwards throughthe disc allowing most of the matter to spiral slowlyinwards. This liberates gravitational potential energy.Accretion on to black holes extracts large amounts of [email protected] energy per unit mass accreted, and the high accretionrates that are sustainable on to supermassive black holesin galaxy centres makes them the most luminous contin-ually emitting objects in the Universe.In simple accretion discs, for example those foundin dwarf novae, the standard accretion disc model(Shakura & Sunyaev 1973) provides a good fit to theobserved disc spectrum (e.g. Hamilton et al. 2007) andthe thermal-viscous disc instability model provides agood description of the time-dependence (e.g. Lasota2001). However, even in similar types of system, e.g.the nova-like variables, the observed disc emission candeviate from the standard model and requires addi-tional physical effects to be included (see, for example,Nixon & Pringle 2019). In the dwarf novae and nova-like variables, the central accretor is a white dwarf. Itcomes as no surprise that accretion discs around blackholes are more complex and often times provide moreextreme behaviour. In some cases this is due to environ-mental effects, such as the surrounding matter respon-sible for creating broad emission lines in AGN spectraand producing the obscuring ‘torus’ (Antonucci 1993;Ho 2008), or to additional physical processes such as
Raj & Nixon relativistic jets and outflows that routinely accompanyobservations of high luminosity black hole accretion atany mass scale (Russell & Fender 2010; Fabian 2012;King & Pounds 2015). These, and many other pro-cesses, imply that application of the standard disc modelto the observed spectra of accreting black hole systemsis often inadequate.Recently, Lawrence (2018, see also Antonucci 2018and references therein) highlighted a potentially moreserious issue concerning the timescale of large ampli-tude luminosity variations in AGN. He points out (andwhich Antonucci 2018 argues was already well-known;cf., e.g., the discussion of Alloin et al. 1985) that thetimescale on which a standard accretion disc is ex-pected to vary—roughly the viscous timescale of theouter disc regions—is orders of magnitude larger thanthe observed variability timescales, and thus the stan-dard disc model (comprising planar and circular orbits,with a local viscosity) is incompatible with the observedshort-timescale, large-amplitude variability. Lawrence(2018) concludes that standard viscous disc theory beabandoned in favour of either (1) an extreme repro-cessing scenario in which all the energy emerges froma quasi-point source and is reprocessed by a passive discor external media, or (2) non-local processes capable ofproviding significantly shorter timescales by e.g. largescale magnetic stresses. The latter possibility is similarto the ideas put forth by King et al. (2004, see also thediscussion in Cannizzaro et al. 2020), but these have notyet been applied to AGN disc parameters. The formerscenario appears somewhat consistent with the standarddisc model, in which the energy dissipation rate per unitarea is proportional to R − , and thus the total energy re-leased by the disc at each radius is proportional to 1 /R .This, quite generally, arises as the gravitational poten-tial scales as 1 /R and it is gravitational potential energythat is the source of the accretion luminosity. Thus, forexample, in the standard model (and accounting for thezero torque inner boundary condition) ≈ / R ISCO , and10 R ISCO .Re-processing of this energy flux in the outer disc re-gions is then required to explain the large optical/UVfluxes observed in AGN, as a large emitting area is re-quired. So, combining the large flux and large emittingarea, it seems likely that large scale re-processing is nec- essary . Such reprocessing of the central flux may occurmost readily in the regions where the disc is stronglywarped (cf. Natarajan & Pringle 1998), as the centralregions then ‘see’ a larger surface area of the disc. How-ever, these arguments, which focus on the disc spectralproperties, have little to say about the appropriatenessor not of the standard disc model for the dynamics ofAGN discs. Instead, as noted by Lawrence (2018), it isthe timescales on which the lightcurves vary which mostcleanly provides information about the underlying dy-namics and thus is most useful in constraining disc mod-els. What the energy arguments (and subsequent repro-cessing) do tell us, is that if the central regions can vary(which is of course where the timescales are generallyshortest), then we can expect the bulk emission proper-ties to vary with it – and with lags between wavebandssimilar to those observed (De Rosa et al. 2015). A sim-ilar conclusion was reached by Sniegowska et al. (2020)who argue that the source of the variability lies in insta-bilities operating in the accretion disc (see also, for ex-ample, Alloin et al. 1985). Recently Ricci et al. (2020)showed that the changing-look AGN phenomenon canbe associated with rapid evolution of the inner disc re-gions.In this paper we discuss the possibility that disc tear-ing (Nixon et al. 2012a), which occurs in warped accre-tion discs that achieve a sufficient warp amplitude to beunstable to the disc breaking instability (Doˇgan et al.2018; Do˘gan & Nixon 2020), plays a role in producingvariable accretion flows around black holes. For discus-sion of other possible mechanisms we refer the readerto the discussion in Cannizzaro et al. (2020, see alsoSniegowska et al. 2020). In Section 2 we provide anoverview of the expected dynamics in unstable warpeddiscs. In Section 3 we give the timescales on which theresulting variability can be expected to manifest. In Sec-tion 4 we explore the impact of the system orientation Heating of the outer disc is also a required addition to the stan-dard disc model in its application to the lightcurves of soft X-raytransients (SXTs), where the outbursts exhibit long exponentialdecay (King & Ritter 1998). In all X-ray binaries, it is well-known that reprocessing of inner disc emission is required to ex-plain the optical to X-ray flux ratios (van Paradijs & McClintock1994). In AGN, the peak temperature of a standard accretiondisc typically corresponds to rest-frame wavelengths in the ex-treme ultra-violet, which, if to be seen (at low-redshift) typicallyhas to be reprocessed into the optical/UV. It is worth noting thatan irradiated disc is expected to have T ( R ) ∝ R − / , in contrastto the typical T ( R ) ∝ R − / , and by modelling the broadbandspectra of 23 Seyfert 1 galaxies, Cheng et al. (2019) find that thetemperature profiles are best fit with power-law indices rangingfrom − / − /
4. The suggestion that irradiation and repro-cessing of central emission plays an important role is not new(see, for example, Collin-Souffrin 1991). isc tearing DYNAMICS OF DISC TEARINGWarped discs have been found through direct andindirect methods in a variety of astrophysical sys-tems. The most direct evidence now exists from spa-tially resolved observations of protoplanetary discs (e.g.Andrews 2020), and recent observations in this areahave connected observed disc structures with disc tear-ing (Kraus et al. 2020). For discs around black holes,the most compelling evidence of disc warps comesfrom water masers (Miyoshi et al. 1995; Greenhill et al.1995, 2003, see also Maloney 1997 for discussion).Further evidence is derived from, for example, thelong X-ray periods in X-ray binaries (e.g. the mod-els of Wijers & Pringle 1999; Ogilvie & Dubus 2001)and quasi-periodic oscillations that are fit by the Rel-ativistic Precession Model (RPM; Stella & Vietri 1998;Motta et al. 2014). As a specific example, warped discmodels are used to explain the time-dependence of theX-ray flux in the X-ray binary Her X-1 (Scott et al.2000; Leahy 2002). In the case of discs around blackholes, when the disc is misaligned to the spin plane of aKerr black hole the disc orbits precess due to the Lense-Thirring effect. The rate of precession depends on theradius of the disc orbits from the black hole, and thusover time the disc acquires a differential twist and thusbecomes warped.The dynamics of a warped disc differ from that of aplanar disc primarily due to a resonance between the or-bital motion and the radial pressure gradient producedby the warped disc shape (Papaloizou & Pringle 1983;i.e. the midplanes, and thus the regions of high pres-sure, between two neighbouring rings are misaligned; seee.g. Fig. 10 of Lodato & Pringle 2007). The radial pres-sure gradient, which oscillates around the orbit, inducesepicyclic motion, which in a near-Keplerian disc also os-cillates on the orbital timescale. The resonance leadsto strong in-plane motions which communicate the discwarp radially and are damped by the disc turbulence.For the case of weak damping (referred to as “wave-like” with α < H/R ) the result is a propagating warpwave (Papaloizou & Lin 1995; Pringle 1999), while forthe case of strong damping (referred to as “diffusive”with α > H/R ) the disc warp evolves following a dif-fusion equation (Pringle 1992; Ogilvie 1999). Here, wewill consider the diffusive case as black hole discs aretypically expected to be thin and viscous. For a reviewof warped disc dynamics, see Nixon & King (2016).Numerical simulations of warped discs have revealedthat they can break into discrete rings that can sub- sequently precess effectively independently (see, for ex-ample, Nixon et al. 2012a). The simulation behaviour,which sees rings of matter break off from the rest of thedisc in regions where the warp amplitude is high, hintsat an underlying instability. Ogilvie (2000, Section 3)presents a stability analysis of the warped disc equa-tions, finding that they can becomes viscously unstable.Doˇgan et al. (2018) explore this instability in detail andshow that for α . . α ) at which thedisc becomes unstable, and further they connect the in-stability with the disc tearing behaviour observed in thenumerical simulations, arguing that it provides a phys-ical mechanism underpinning the dynamical behaviour.Do˘gan & Nixon (2020) further explore the properties of Nixon et al. (2012a) used Lagrangian (SPH) hydrodynamics(see also Larwood et al. 1996; Larwood & Papaloizou 1997;Nixon et al. 2013; Do˘gan et al. 2015; Kraus et al. 2020; Raj et al.2021). This behaviour has also been found in numeri-cal simulations employing Eulerian (grid) hydrodynamics (e.g.Fragner & Nelson 2010), and in simulations that model the ef-fects of magnetic fields (Liska et al. 2020). The warp amplitude, ψ = R | ∂ ℓ /∂R | where ℓ is the unit angularmomentum vector pointing normal to the orbital plane at radius R , is a dimensionless measure of the strength of the disc warp. Raj & Nixon the instability for the case of non-Keplerian rotationthat is relevant to discs around black holes. The basic reason for instability is that as warp am-plitude increases the internal stresses weaken causingthe disc to become less able to hold itself together(Ogilvie 1999; Nixon & King 2012). In a companion pa-per (Raj et al. 2021) we present numerical simulations ofdiscs with various parameters including the disc thick-ness, viscosity parameter and inclination with respectto the black hole spin. We found that the properties ofthe simulated discs were consistent with, for example,the predictions for the critical warp amplitude at whichthe disc becomes unstable. We showed that discs witha lower viscosity parameter become unstable at lowerwarp amplitudes, and that discs which are thinner andhave a larger misalignment with respect to the blackhole spin generally reach larger warp amplitudes and It is worth noting that essentially all of the analytical work on thedynamics of warped discs makes use of a Navier-Stokes (local andisotropic) viscosity (but see, for example, Appendix C of Nixon2015). The numerical work generally makes the same assumptionby applying a Navier-Stokes viscosity to model the physical vis-cosity in a disc (but see, for example, Liska et al. 2020). Pringle(1992) notes that this may not be right in real discs where theangular momentum transport and dissipation arises through acombination of turbulent and magnetic processes. It has beenshown by Torkelsson et al. (2000, see also Ogilvie 2003) that hy-dromagnetic turbulence produced by the magnetorotational in-stability (MRI) leads to the dissipation of motions induced by awarp at a rate that is consistent with an isotropic viscosity. Sim-ilarly, Zhuravlev et al. (2014) note that while the viscosity theymeasure from their GRMHD simulations is anisotropic, they findthat “the effects of anisotropic viscosity on the evolution [of thedisc structure] may be rather small”. Nealon et al. (2016) es-tablish that the results of numerical simulations that employ aNavier-Stokes viscosity provide results that are indistinguishablefrom simulations that model MRI driven turbulence explicitly.So, to the extent that numerical magnetohydrodynamics is ca-pable of modelling the angular momentum transport and energydissipation in real discs, we can expect the disc models basedupon a Navier-Stokes viscosity to provide a reasonable descrip-tion of the disc dynamics. King et al. (2007) and Martin et al.(2019) have argued (see also the discussion in Pjanka & Stone2020) that current MHD models are inadequate for explainingthe observed behaviour in accreting systems, particularly withrespect to the magnitude of the angular momentum transportand the nature of the energy dissipation, with the latter in nu-merical MHD models typically controlled by numerical dissipa-tion. Thus our understanding of accretion disc dynamics is notyet so well established that we can be sure that the disc tearingbehaviour seen in numerical simulations (now found with bothLagrangian and Eulerian methods, and with viscosity modelledas either Navier-Stokes or explicit MHD turbulence) can occur inreal discs. However, all that is physically needed for the disc tobe unstable in this manner is that as the disc warp grows, the discbecomes less and less able to hold itself together. More explicitly,that the internal torque attempting to keep the disc locally flatdecreases sufficiently with increasing warp amplitude. This seemsreasonable to expect from a physical disc, and this has potentiallybeen seen in the protostellar system GW Ori (Kraus et al. 2020). are thus more likely to become unstable. Once a discbecomes unstable it breaks into two or more discreterings, joined by a low-density transition region in whichthe orbital plane varies sharply with radius. As the warpis produced by the gravitational potential, which causesthe orbits to precess, each of the individual rings subse-quently precess with only weak interaction with neigh-bouring rings. However, over time the misalignment be-tween neighbouring rings grows and thus their velocityfields become partially opposed. Thus any subsequentspreading of the rings, or perturbations to the orbits,results in collisions of the gas. This promotes shocksand the loss of rotational support for the gas . Thisputs some of the gas on eccentric orbits (with apocentreat the radius of the original ring), which upon reach-ing pericentre have either accreted on to the central ob-ject, or collide with other gas orbits to circularise ata smaller radius. As the precession timescale must beshorter than the standard viscous timescale in the discto produce a substantial warp (Nixon et al. 2012a), thisprocess enhances the accretion rate on to the centralobject by delivering matter to small radii faster thanviscous torques could do so. The efficiency of this pro-cess is determined by the misalignment angle betweenthe disc and the black hole. The maximum angle cre-ated between the disc orbits is twice the angle to theblack hole spin vector. For small angles, the accretionrate may be enhanced by a factor of order unity. Whilefor larger angles the gas can fall a considerable distancein radius and increase the accretion rate substantially.As two neighbouring rings precess their mutual mis-alignment angle oscillates between zero and twice theirinitial angle to the black hole spin vector . Collisions be-tween matter in the rings leads to shocks as the orbitalvelocity is significantly larger than the sound speed for adisc. The resulting dynamics depends on the local cool-ing rate of the shocked gas. The relevant timescales are It may appear that angular momentum is not conserved as thegas has experienced a reduction in angular momentum. However,the total angular momentum, including the angular momentumtransferred to the black hole through the back reaction of theLense-Thirring effect, is conserved through this process. Essen-tially, the disc orbits have borrowed angular momentum from thehole in order to change their orbital plane and thus allow internalcancellations within the disc. The maximum angle is somewhat reduced by the rings also slowlyaligning with the black hole spin vector, and in general simula-tions show that the inner ring is closer to alignment than theouter ring. Thus, when the disc is close to the line of stability,i.e. the warp amplitude is not far from the critical value for in-stability and the growth rates of the instability are small, theinternal angle is much less than the maximal value. However,for regions of strong instability and growth rates of order thedynamical timescale, the angle is close to the maximal one. isc tearing ∼ H/c s (where here the soundspeed c s is that of the shocked gas) and the (typicallyfree-free) cooling timescale (Pringle & Savonije 1979). Ifit can cool efficiently, then the cooling gas can fall in-wards to the new circularisation radius (as above). How-ever, if the cooling is inefficient, as might be expectedin the innermost regions of the disc or if the shocksinvolve only a small amount of matter, the shockedgas takes on a more quasi-spherical distribution andcan be supported somewhat against infall by pressure(cf. Stone et al. 1999). This shocked gas may resem-ble the X-ray corona (as suggested by Nixon & Salvesen2014), and its subsequent dynamics is yet to be exploredfully but may resemble a radiatively inefficient accretionflow (see, for example, Stone et al. 1999; Inayoshi et al.2018).An alternative possibility is that in these central re-gions the energy dissipated in the strong warping ofthe disc is not readily radiated away, and thus on thetimescales on which the viscous instability of the warpgrows the disc structure may be changed; specifically thedisc thickness may increase due to the additional dissi-pation of energy. This may have a stabilising effect onthe viscous instability, but may also lead to thermal in-stability and would also have a significant impact on thecentral accretion rate due to the dependence of the vis-cous timescale on disc temperature. We plan to explorethis dynamics in subsequent work. TIMESCALESIn this section we provide a discussion of the rel-evant timescales on which we can expect the disc toshow variability. We limit the discussion to AGN discs.For discussion of the types of variability and the rel-evant timescales from disc warping and disc tearingin X-ray binaries we refer to Nixon & Salvesen (2014).Variability may also be produced by other mechanisms(e.g. Sniegowska et al. 2020), and the timescales maybe affected by the inclusion of additional physics (e.g.Dexter & Begelman 2019), but here we employ what weconsider to be standard AGN disc structure and considerthe effect of disc warping alone (for a broad discussion ofseveral other mechanisms we refer the reader to Section4 of Cannizzaro et al. 2020).AGN discs are expected to be limited in radial extentby their own self-gravity to a size of order R sg = 0 .
02 pc,almost independent of disc or black hole parameters(Goodman 2003; Levin 2007; King & Pringle 2007).Discs that form or grow beyond this radius are typicallygravitationally unstable, with Toomre Q = c s κ/πG Σ < R > R sg (Gammie2001). This is consistent with the disc of stars observedin our own galaxy centre which orbit just outside thisradius (Genzel et al. 2003; Ghez et al. 2005). For suffi-ciently large SMBH, with M & × M ⊙ , the self-gravity radius can be of the order of the gravitationalradius of the black hole and thus the formation of a sta-ble accretion disc is likely to be prohibited (King 2016,see also Natarajan & Treister 2009). For black holes be-low this limit, but with high mass, say M ≫ M ⊙ , thedisc extent is limited to 10s to 100s of gravitational radii,and thus the entire disc may be in the regime where ra-diation pressure provides the dominant vertical support,i.e. determines the disc scale height . While for typicalblack hole masses of 10 − M ⊙ , the self gravity ra-dius is large enough that radiation pressure there is notimportant.The basic timescale on which an accretion disc evolvesis the viscous timescale on which angular momentum istransported and matter diffuses through the disc, givenby (Pringle 1981) t ν = R ν (1)where R is the disc radius, and ν the kinematic vis-cosity. The viscosity in an accretion disc is usu-ally modelled as ν = αc s H (Shakura & Sunyaev1973), and is assumed to originate from hydromag-netic turbulence (see e.g. Martin et al. 2019, for dis-cussion). In black hole accretion discs outside of qui-escence, the accretion rate is typically high enoughthat the disc is sufficiently ionised for the magnetorota-tional instability to provide a source of disc turbulence(Balbus & Hawley 1991). Application of the disc struc-ture presented by Collin-Souffrin & Dumont (1990), forthe case where radiation pressure is not important,yields (King & Pringle 2007) t ν ( R sg ) = 1 . × (cid:16) α . (cid:17) − / (cid:16) ǫ . (cid:17) / (2) × (cid:18) L . L Edd (cid:19) − / (cid:18) M M ⊙ (cid:19) − / yr , where ǫ is the radiative efficiency of accretion and L Edd is the Eddington luminosity. Thus the timescale of vari-ability associated with the accretion timescale of the discitself is long, and commensurate with the timescaleson which AGN global properties appear to ‘flicker’(Schawinski et al. 2015; King & Nixon 2015). However, The actual structure of the radiation pressure dominated regionsis uncertain, as the standard disc model including a Shakura-Sunyaev viscosity is unstable there (Lightman & Eardley 1974;Shakura & Sunyaev 1976).
Raj & Nixon this timescale is not consistent with short-timescale,large-amplitude variability observed in AGN lightcurves(as has been pointed out by, for example, Lawrence2018). If this short-timescale variability arises due toviscous inflow of matter, i.e. the luminosity variationsare produced by the standard energy dissipation in anaccretion disc due to viscous torques, then some mech-anism is required for creating time-dependent fluctua-tions in the mass flow rate at smaller radii.For smaller radii, we need to account for the ef-fects of radiation pressure on the disc vertical structure(Shakura & Sunyaev 1973; Novikov & Thorne 1973). Inthis case, the viscous timescale can be written as (e.g.Dexter & Begelman 2019) t ν ≈ (cid:16) α . (cid:17) − (cid:18) M M ⊙ (cid:19) (cid:18) ˙ m . (cid:19) − (cid:18) R R g (cid:19) / yr , (3)where ˙ m is the accretion rate in Eddington units, andwe have assumed electron scattering opacity. From thiswe can see that if mass can be fed in a time-dependentmanner to radii of order 10 − R g , then the correspond-ing viscous timescales (of weeks to years) are capable ofproviding the necessary timescales to generate the ob-served luminosity variations.For the disc tearing scenario described in Section 2above, the other basic timescale is the time taken fordisc orbits that are misaligned to the black hole spin toprecess around the black hole spin vector – the nodal (orLense-Thirring) precession timescale. For a test particleorbiting the black hole, this is given by t np ( R ) = 1Ω np = c R GJ h = 12 a (cid:18) RR g (cid:19) GMc (4)= 0 . (cid:16) a . (cid:17) − (cid:18) R R g (cid:19) (cid:18) M M ⊙ (cid:19) days Note that close to the black hole this formula is strictly only validfor small black hole spin. For example, for a ≈ . where Ω np is the (nodal) precession frequency, R is thedisc radius, J h is the black hole angular momentum, a the black hole spin, and R g = GM/c the gravitationalradius of the black hole.Equation 4 gives the timescale on which a single nar-row ring of gas orbiting a black hole undergoes nodalprecession. However, in general, precessing rings of gasare accompanied by other precessing rings or a warpedand precessing outer disc. For two freely-precessingrings separated by a radial distance of ∆ R , the differ-ential precession timescale on which the rings precessapart is given by δt np ( R ) = 1Ω np ( R ) − Ω np ( R + ∆ R ) ≈ R R t np ( R ) , (5)where in the last step we have assumed that ∆ R ≪ R . For rings produced by disc tearing, we expect ∆ R & H when the disc is strongly unstable and can have ∆ R ≫ H when it is only weakly unstable.As discussed above, the disc dynamics resulting fromthe instability of the disc warp can produce differentscenarios. If the innermost regions of the disc are un-stable, then we expect the variability to manifest onthe precession timescale of the inner disc regions. Inparticular, in this case we expect variability on the dif-ferential precession timescale (5), and the disc mattermay plunge directly into the hole. If, instead, the discis unstable further out such that the mass flow from theunstable region does not reach the ISCO, then the mat-ter circularises at a new smaller radius . In this casethere are several competing timescales for the variabil-ity. The (differential) precession timescale determinesthe timescale for shocks between rings and the inflowof matter. The orbits circularise at their new smallerradius efficiently on the local orbital time (due to addi-tional shocks at smaller radii; cf. Nixon et al. 2012b).Then, once the material is circularised, a strong increasein disc luminosity follows over a timescale of order the We note that if ∆ R ∼ R , then this patch of the disc no longerfollows the precession timescale of a particular disc location, butinstead follows the angular momentum weighted average preces-sion timescale (e.g. Larwood & Papaloizou 1997). Typically, the new orbits here are close to alignment with theblack hole spin as the reduction in angular momentum of thematerial comes from the misaligned component of the originalorbits. However, a small residual misalignment is possible andthe orbits are initially eccentric. Note that the system as a wholeconserves angular momentum, but each ring of gas in the discborrows angular momentum from the black hole via the Lense-Thirring effect in order to be misaligned to neighbouring rings.Subsequent interaction between the misaligned rings can reducethe orbital angular momentum of both (cf. Eqn 5 of Hall et al.2014). isc tearing M ⊙ black hole, which canbe rescaled to any mass black hole by multiplying thetime axis by M/ M ⊙ . The black line correspondsto the accretion rate in the simulation with α = 0 . H/R = 0 .
02, and θ = 10 ◦ , where θ is the initial an-gle between the disc and black hole spin. The red linecorresponds to the accretion rate for the same param-eters, except for this time θ = 60 ◦ . For the 10 ◦ case,the disc attains a mild warp which does not create anoticeable impact on the disc structure or evolution andthus the accretion rate closely resembles the accretionfor a planar disc. In the Figure we see the accretionrate slowly decline with time as the simulated disc losesmass through the inner boundary at R ISCO and spreadsto larger radii. In contrast, for the 60 ◦ case the disc un-dergoes disc tearing in the inner regions at early times(the first ≈
10 days of evolution), and over time the un-stable region moves outwards. After t ≈
10 days, pre-cessing rings at larger radii (10s of R g ) feed matter intoan aligned disc at radii between R ISCO and a few R ISCO .Thus the variability in the accretion rate becomes bothlonger-timescale and lower-amplitude as the innermostaligned disc’s viscous timescale regulates the accretion.This shows that short-timescale (weeks to months) andlarge-amplitude (factors of several) variability in the ac-cretion rate on to supermassive black holes could becaused by disc tearing. We expect that this variabil-ity would be reflected in the time-dependent emission ofsuch systems as the accretion rate is a reasonable proxyfor the rate at which energy can be extracted from thedisc orbits. IMPACT OF SYSTEM ORIENTATIONWe have discussed in the previous section thetimescales on which the disc structure is expected tovary, through either accretion (equation 1) or preces-sion (equations 4 & 5). In this section we discuss the t (M days) l og M . Figure 1.
Example accretion rates versus time from thesimulations presented in Raj et al. (2021). The two curvesrepresent the accretion rate in the simulations with α = 0 . H/R = 0 .
02 and θ = 10 ◦ (black line) and θ = 60 ◦ (red line).The accretion rate is plotted on a log scale, and is in arbitraryunits. The time axis has been scaled to days for a 10 M ⊙ black hole, which can be rescaled to any mass by multiplyingthe time axis by M/ M ⊙ . The 10 ◦ simulation achieves amild warp which does not strongly impact the disc structure,and thus the accretion rate is similar to a planar disc withthe slow decline caused by the disc accretion and spreadingto larger radii with time (the simulations did not model asteady disc with mass input over time). The 60 ◦ simulationachieves a strong warp which results in disc tearing. At earlytimes the inner disc is unstable, resulting in periodic accre-tion of rings of matter. At later times the unstable regionmoves to larger radii, and the inner disc is aligned to theblack hole spin. This results in lower-amplitude and longer-timescale variability. Note that the properties of the disctearing events, e.g. the radius and thus timescale at whichthey occur, depends on the disc properties and also the blackhole spin. Therefore this plot should be taken as represen-tative of the kind of behaviour that may occur, and not aprediction of an accretion rate for any particular system. impact of the orientation of the system with respect toan observer.In general, the angle between the black hole spin andthe accretion disc angular momentum is expected to de-crease over time through dissipation between neighbour- Raj & Nixon ing rings ; Lense-Thirring precession of each ring main-tains the inclination angle with respect to the black hole,and subsequent interaction between neighbouring ringsthat occupy different planes (e.g. ones that have pre-cessed apart) reduces the inclination to the black holespin. This means that a line-of-sight that is closer tothe black hole spin axis than the original disc misalign-ment always has a clear view of the disc centre. In thiscase, any observed variability is caused by either emis-sion from the shocks occurring between precessing ringsin unstable regions of the disc or from the variable accre-tion rate caused by the time-dependent mass flow ratesfrom the unstable regions.However, if the line-of-sight to the black hole passesthrough the disc (i.e. θ obs > θ disc ) then additional vari-ability due to obscuration effects is possible. If thedisc warp (i.e. the outer disc regions that do not un-dergo disc tearing) move into (or out of) the line-of-sightthen the emission properties vary significantly as the in-ner regions of the disc are blocked (revealed), and thetimescales for such changes are long – relying upon pre-cession at large radii (cf. equation 4) or slow decay ofthe disc warp. However, if the interloper is instead aring of matter in the unstable region of the disc, thenwe can expect the precession timescale at that radius tobe imprinted on the emission as the ring blocks some ofthe flux from the central regions reaching the observer.The presence of multiple such rings with different incli-nations blocking different parts of the disc at differenttimes may erase a simple periodic signal, but the generictimescale for variability from a ring at a given radius isgoverned by equation 4. In Fig. 2 we show an exam-ple of the effect different orientations may have on thesystem properties. Each panel in the figure is the samedisc model ( α = 0 . H/R = 0 .
02 and θ = 60 ◦ ) viewedat the same time, but from different orientations. Theblack hole spin axis is in the z -direction, and the ref-erence views are of the x - y plane (the top and bottomleft-most panels; which are the same) and thus the blackhole spin points out of the page in the left-most panels.The top row of panels, from left to right, are views of the However, there are exceptions to this. For example, if one in-cludes the effects of radiation warping, it is possible for thedisc to achieve a shape in which the entire sky—as seen fromthe black hole—is covered by the disc surface (Pringle 1997).Alternatively, if the total disc angular momentum dominatesthe angular momentum of the black hole and the disc is ini-tially closer to counteralignment (i.e. the disc-black hole angleis θ > ◦ ) then while the inner disc regions initially counter-align ( θ → ◦ ) with the black hole spin, the outer regions(on a longer timescale) align ( θ → ◦ ) with the black hole spin(King et al. 2005; Lodato & Pringle 2006) and thus any line ofsight may become blocked over time. disc starting with the x - y plane, which are subsequentlyrotated by an angle X around the x -axis. The bottompanels are the same, but now the rotation is performedby an angle Y around the y -axis. In some configurationsthe central regions are clearly visible, and in others theyare largely blocked from view – either by the outer discor a precessing ring which is crossing the line-of-sight tothe disc centre.Finally, we note that while the geometry may be im-portant for determining which parts of the disc are seenby the observer, it is also important for determiningwhich parts of the disc are available to be seen by thematerial that makes up the broad line region (typicallyassumed to be clouds orbiting near the outer disc re-gions). As the matter comprising the broad line regionis typically expected to orbit close to the original discplane, it is plausible that precessing rings in the unstableregion of the disc may act to block the central disc emis-sion from fully illuminating the broad line regions. Thusan observer looking unobstructed at the disc central re-gions may see a constant flux, but the flux arriving atthe broad line regions may be time variable. This couldcause a disconnect between the continuum and emissionlines, and may thus offer a potential cause for the re-cently observed ‘broad line holidays’ (Goad et al. 2016,2019). DISCUSSIONIn the previous sections we have discussed the dynam-ics of disc tearing, the timescales on which variabilitymight manifest in the observable properties, and howthe disc geometry can affect what we see. Here we dis-cuss the possible connection between disc tearing andrapid flaring variability observed in some AGN (referredto as quasi-periodic eruptions), which may also be re-lated to the ‘heartbeat’ modes of some X-ray binaries.In the left hand panel of Fig 3 we provide the accretionrate with time of the simulation reported in Raj et al.(2021) with parameters α = 0 . H/R = 0 .
05 and θ = 60 ◦ performed with 10 particles. In this simulationthe disc tearing behaviour was restricted to the inner-most regions of the disc (Fig. 4). The disc inner edgewarps with time, and then, once the warp amplitude islarge enough, the innermost ring (with radial thickness∆ R ∼ H ) breaks off and begins to precess. Once a largeangle is reached with the next ring of the disc, shocksoccur and the innermost ring is robbed of rotational sup-port and accretes dynamically on to the black hole. Thisprocess repeats, and thus quasi-periodicity is imprintedon the accretion rate shown in Fig. 3. We anticipatethat this signal is imprinted on the observable proper-ties of the system, primarily through radiating away the isc tearing X = 0
X = 30
X = 60
X = 90
X = 120
X = 150
Y = 0
Y = 30
Y = 60
Y = 90
Y = 120
Y = 150
Figure 2.
3D renderings of the gas distribution in disc tearing simulation with α = 0 . H/R = 0 .
02 and θ = 60 ◦ (Raj et al.2021). Each panel depicts the same simulation at the same time, viewed from different orientation. The left-hand top andbottom row panels both show the same view—that of the x - y plane—where the black hole spin axis, which coincides with the z -axis points out of the page. Then, from left to right, the top row shows the disc view with the disc rotated by an angle X around the x -axis where the value of X in each case is given in the panel. In contrast, the bottom row is rotated from the x - y plane by an angle Y around the y -axis. In some cases the majority of the disc is visible and pointed predominantly towards the‘observer’. While in other cases, the inner disc regions are highly inclined, or obscured by either the warped outer disc or aninterloping ring of matter in the unstable region. energy generated by the shocks between the innermostrings. Thus we expect the high-energy flux (e.g. X-rays)to broadly follow the same time-dependence as the ac-cretion rate in the simulation.Recently Miniutti et al. (2019) have reported quasi-periodic eruptions in XMM-Newton and
Chandra obser-vations of the Seyfert 2 galaxy GSN 069. The eruptionsoccur approximately every nine hours and are seen inobservations spanning several months. The 0.4-2 keVflux increases by a factor of order 10-100 for a dura-tion of approximately one hour. Comparison of theseeruptions with the accretion rate shown in Fig. 3, per-haps also with an additional X-ray emitting componentto provide a base-level of flux (e.g. the disc corona),are again suggestive. Applying the black hole mass of4 × M ⊙ to the simulation data results in a recurrenceperiod of ∼ . . Similar behaviour has also beenreported for the galactic nucleus of RX J1301.9+2747(Giustini et al. 2020). Here the black hole mass is re-ported to be 0 . − . × M ⊙ , and with the eruptionsseparated by ∼
20 ks these eruptions are also consistentwith the timescale shown in Fig. 3 with variations inthe disc structure shown in Fig. 4. We therefore suggest We note that no attempt has been made to survey different disc-black hole parameters to achieve a better fit. Such an effort is notcurrently useful, as the black hole mass is not constrained to highenough precision. Miniutti et al. (2019) estimate the uncertaintyat the factor of a few level. They further estimate the blackhole mass from the fundamental plane of black hole accretion(Shu et al. 2018a) and find a value of 2 × M ⊙ , in which casethe simulated period would be ∼ . that these quasi periodic eruptions may be the result ofthe disc inner regions undergoing disc tearing, with theresulting shocks that occur when broken-off rings collideproviding the additional energy dissipation to power theeruptions. This behaviour may appear as if the innerdisc temperature briefly rises due to the additional dis-sipation of energy. If the black hole mass in GSN 069is of the order of 4 × M ⊙ , then the nine hour erup-tions are consistent with the free precession timescale(eq. 4) at R ≈ R g , but due to the precession of neigh-bouring material (eq. 5) the location is more likely tobe at or close to the disc inner edge . a few × R ISCO .It is also interesting to note that the eruptions do notseem to have been present while the source was at higherluminosity (see e.g. Shu et al. 2018b). This may indi-cate that, as suggested by the simulations of Raj et al.(2021), variations in disc parameters (in this case thedisc thickness, which is expected to be higher for higherEddington ratio) are responsible for generating changesin the variability of the accretion flow. For the disctearing scenario discussed here, the disc parameters areresponsible for determining the location and strength ofthe instability and thus the timescale and amplitude ofthe variability.As noted by Miniutti et al. (2019), the time-dependence is reminiscent of the ‘heartbeat’ modeof some X-ray binaries (e.g. GRS 1915+105 andIGR J17091-3624; Belloni et al. 1997; Altamirano et al.2011), and recently reported in an X-ray source in thegalaxy NGC 3621 (Motta et al. 2020). In these systemsthe timescales on which the beats recur is 10-100 s. Scal-0
Raj & Nixon ing the simulation data to a black hole of mass 10 M ⊙ ,the recurrence time is ∼ . a ≈ .
5, and while a higherspin value might reduce some timescales (by moving theISCO closer to the black hole horizon) it may also movethe location of the tearing region (where the warp ampli-tude is highest) outwards and thus increase the preces-sion timescale on which the periodic behaviour occurs.Similarly different values of α or H/R would affect thelocation of the instability in the disc and thus the typeand properties of the subsequent variability. An addi-tional possibility in these discs is that the location of the thin , i.e. radiatively efficient, disc inner edge is truncatedat a radius larger than the ISCO, with a radiatively in-efficient flow inside. If disc tearing occurs at the inneredge of the thin component of the disc, then the trunca-tion radius determines the fastest precession timescales.Increasing the radius of warping instability by a factor of4-5 would bring the timescales into agreement for, e.g.,GRS 1915+105. CONCLUSIONSWe have discussed the dynamics of disc tearingwhich Nixon et al. (2012a) suggest may provide asource of variability in black hole accretion, andNixon & Salvesen (2014) discuss in the context of X-raybinaries. Disc tearing is caused by an instability of thedisc warp, and occurs most prominently when the discviscosity is weak ( α . .
1) and the warp amplitude islarge (Ogilvie 2000; Doˇgan et al. 2018; Do˘gan & Nixon2020). In a companion paper, Raj et al. (2021), wepresent the results of numerical simulations of disc tear-ing around black holes. Here we have discussed the re-sulting dynamics, and the timescales on which we expectblack hole discs to show variability. We have suggestedthat the large amplitude, short timescale variability ex-hibited by AGN may be explained by the disc undergo-ing such dynamics, and that this can result in intrinsicchanges to the disc (both geometric and energy dissipa-tion/accretion rate changes) and also, in some cases, tochanges to the observable properties of the disc throughtime-dependent obscuration. These effects may accountfor the short timescale evolution observed in, for exam-ple, changing-look AGN. We have also shown that forsome disc parameters, disc tearing can occur predomi-nantly at the disc inner edge, and that these cases showsimilarity with the quasi periodic eruptions observedin GSN 069 and RX J1031.9+2747 and the ‘heartbeat’mode observed in some X-ray binaries. In future work,we will develop more sophisticated simulation models toconnect in more detail with the available observationaldata. t (ks) M
50 100 15005×10 -8 -7 -7 . Figure 3.
The accretion rate on to the black hole in thesimulation presented by Raj et al. (2021) with α = 0 . H/R = 0 .
05 and θ = 60 ◦ with the disc modelled with 10 particles. The time axis has been scaled such that the unitsare kiloseconds for a 4 × M ⊙ black hole. We note that thetimescale on which the behaviour occurs depends on wherein the disc the instability occurs (here it occurs at the in-ner disc edge, see Fig. 4 below, but this can be scaled tolonger timescales if the behaviour occurs at larger radii; cf.equation 4). The accretion rate is in arbitrary units. In thiscase the disc tearing behaviour is restricted to the disc innerregions, see Fig 4. The inner disc behaviour is cyclic; theinnermost ring of matter is torn off, which then precessesuntil it interacts strongly with the neighbouring ring and isrobbed of angular momentum, loses rotational support andaccretes dynamically on to the black hole. The accretionrate is representative of the dissipation rate in the inner discregions, and therefore may reflect the rate of generation ofenergy from the accretion flow. This should be comparedto Figure 1 of Miniutti et al. (2019) for GSN 069, Figure1 of Giustini et al. (2020) for RX J1301.9+2747, or for the‘heartbeat’ mode in X-ray binaries to (for example) Figure1 of Zoghbi et al. (2016) for GRS 1915+105. isc tearing y x-50 0 50-1000100 t=56.3 ks x-50 0 50t=60.3 ks x-50 0 50t=62.3 ks Figure 4.
The disc structures from the simulation accretionrate depicted in Fig 3 at three times just before, during andafter a peak in the accretion rate. The axes are in units ofgravitational radii ( R g = GM/c ), and the colour denotesthe column density with white the highest and dark blue thelowest. The times in the plots have been scaled to representa 4 × M ⊙ black hole (same as Fig. 3). The left hand panelshows the disc in a warped state, the middle panel shows theinnermost ring broken off from the disc, and the right handpanels shows a return to the warped state once the ring hasbeen accreted. splash (Price 2007)for the figures.REFERENCES Alloin, D., Pelat, D., Phillips, M., & Whittle, M. 1985,ApJ, 288, 205Altamirano, D., Belloni, T., Linares, M., et al. 2011, ApJL,742, L17Andrews, S. M. 2020, ARA&A, 58, 483Antonucci, R. 1993, ARA&A, 31, 473—. 2018, Nature Astronomy, 2, 504Balbus, S. A., & Hawley, J. F. 1991, ApJ, 376, 214Belloni, T., M´endez, M., King, A. R., van der Klis, M., &van Paradijs, J. 1997, ApJL, 479, L145Cannizzaro, G., Fraser, M., Jonker, P. G., et al. 2020,MNRAS, 493, 477Cheng, H., Yuan, W., Liu, H.-Y., et al. 2019, MNRAS, 487,3884Collin-Souffrin, S. 1991, A&A, 249, 344Collin-Souffrin, S., & Dumont, A. M. 1990, A&A, 229, 292De Rosa, G., Peterson, B. M., Ely, J., et al. 2015, ApJ, 806,128Dexter, J., & Begelman, M. C. 2019, MNRAS, 483, L17Do˘gan, S., Nixon, C., King, A., & Price, D. J. 2015,MNRAS, 449, 1251Do˘gan, S., & Nixon, C. J. 2020, MNRAS, 495, 1148Doˇgan, S., Nixon, C. J., King, A. R., & Pringle, J. E. 2018,MNRAS, 476, 1519Fabian, A. C. 2012, ARA&A, 50, 455Fragner, M. M., & Nelson, R. P. 2010, A&A, 511, A77 Gammie, C. F. 2001, ApJ, 553, 174Genzel, R., Sch¨odel, R., Ott, T., et al. 2003, ApJ, 594, 812Ghez, A. M., Salim, S., Hornstein, S. D., et al. 2005, ApJ,620, 744Giustini, M., Miniutti, G., & Saxton, R. D. 2020, A&A,636, L2Goad, M. R., Korista, K. T., De Rosa, G., et al. 2016, ApJ,824, 11Goad, M. R., Knigge, C., Korista, K. T., et al. 2019,MNRAS, 486, 5362Goodman, J. 2003, MNRAS, 339, 937Greenhill, L. J., Jiang, D. R., Moran, J. M., et al. 1995,ApJ, 440, 619Greenhill, L. J., Booth, R. S., Ellingsen, S. P., et al. 2003,ApJ, 590, 162Hall, P. B., Noordeh, E. S., Chajet, L. S., Weiss, E., &Nixon, C. J. 2014, MNRAS, 442, 1090Hamilton, R. T., Urban, J. A., Sion, E. M., et al. 2007,ApJ, 667, 1139Ho, L. C. 2008, ARA&A, 46, 475Inayoshi, K., Ostriker, J. P., Haiman, Z., & Kuiper, R.2018, MNRAS, 476, 1412King, A. 2016, MNRAS, 456, L109King, A., & Nixon, C. 2015, MNRAS, 453, L46King, A., & Pounds, K. 2015, ARA&A, 53, 115 Raj & Nixon
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