The Response of Accretion Disks to Bending Waves: Angular Momentum

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T crto ik rudlow–mass around disks accretion , E tl emulateapj style X INTRODUCTION crto,aceindss—bnre:gnrl—hdoyais— waves hydrodynamics — — sequence general pre–main binaries: stars: — disks accretion accretion, abig,2 lrsnRa,Cmrde B E,U;ct@uco UK; 0EH, CB3 Cambridge, Road, Clarkson 20 Cambridge, h srpyia ora,590000 98Dcme 20 December 1998 509:000–000, Journal, Astrophysical The aoieEJMLJ Terquem E.J.M.L.J. Caroline RESONANCES ,C 56;IacNwo nttt o ahmtclScienc Mathematical for Institute Newton Isaac 95064; CA z, Hub- rnbe nvri´ oehFuirCR,301Grenob 38041 Fourier/CNRS, Université Joseph Grenoble, e ABSTRACT m etrain ihodsmer ihrsetto respect with symmetry odd with perturbations 1 = 1 ugs httepiaywudb oae ta nl of angle an authors at The located as- 0.08. 20 be disk about least would is the at primary AU and the 50 AU of that radius radius suggest 105 a observed is at the binary ratio disk AU, pect projected circumsecondary 340 The is the system of 1998). binary this al. sequence in et separation pre–main (Stapelfeldt the and Tau HST of HK by strik- images given most is optics The noncoplanarity adaptive of such aligned. that for be and evidence necessarily disk ing not the may of orbit plane obser- the the are there that However, indications orbit coplanar. vational 1993 and be disk Papaloizou to the taken & studies, usually these are Lin In (see therein). references stars and binary formation, plane- interacting planet of 1978), and context Tremaine the & (Goldreich in rings studied tary well been has disk rotating (Rod- imaging 1996). direct al. and et of 1996) dier Fuller 1994; study & Simon Mathieu & the Jensen, Guilloteau (Dutrey, through spectra binaries both young received confirmation, recently has observational This Paczyński 1977). 1977; Pringle h ri ftecoesetocpcbnr n hto the of that and that binary suggest spectroscopic close models the some of Cra, orbit Ty the system triple archical h ia ffc fa riigbd nadifferentially a on body orbiting an of effect tidal The ec etrain osntdpn on depend not does perturbations uency luain r eeatt ik around disks to relevant are alculations hstru eut nadces fthe of decrease a in results torque this , ◦ ua oetmte ar epinto deep carry they momentum gular ssrs nyi h icu parameter viscous the if only stress us h rqec ftetdlwvsis waves tidal the of frequency the n oga h ae r apdbefore damped are waves the as long oa to ional 1 bv h ikpae ntecs ftehier- the of case the In plane. disk the above di h ue ein,aefudto found are regions, outer the in ed α . ssrs cigo h background the on acting stress us slre u frsnne the resonance, of Out large. is n h ope otehrzna shear. horizontal the to coupled on pe.I hs velocities these If speed. sound yacmainsa orbiting star companion a by k fsc eoacseit tidal exist, resonances such If oeta,atdltru is torque tidal a potential, g α v lick.org .Teelong–wavelength These g. sgnrlymc larger much generally is , ETMTASOTAND TRANSPORT MENTUM s nvriyof University es, eCdx9 France 9, Cedex le 2 TERQUEM wider binary are not in the same plane (see Corporon, La- structure of the disk was not taken into account in these grange & Beust 1996; Bibo, The & Dawanas 1992). Also, studies. Papaloizou & Lin (1995) performed a full 3 dimen- observations of the pre–main sequence binary T Tau have sional linear analysis of m = 1 bending waves in accretion revealed that two bipolar outflows of very different orien- disks. Comparing analytical results with numerical simu- tations originate from this system (Böhm & Solf 1994). lations, they showed that a vertical averaging approxima- Since it is unlikely that they are both ejected by the same tion can be used when the wavelength of the perturbation star, each of them is more probably associated with a dif- is much larger than the disk semi–thickness. Using the ferent member of the binary. Furthermore, jets are usually WKB approximation, they calculated the dispersion re- thought to be ejected perpendicularly to the disk from lation associated with bending waves taking into account which they originate. These observations then suggest self–gravity, pressure and viscosity (see also Masset & Tag- that disks and orbit are not coplanar in this binary sys- ger 1996 for a study of thick disks). They found that in tem. More generally, observations of molecular outflows in a non self–gravitating thin Keplerian disk in which αv is star forming regions commonly show several jets of differ- small compared to the disk aspect ratio, bending waves ent orientations emanating from unresolved regions of ex- propagate without dispersion at a speed which is about tent as small as a few hundred astronomical units (Davis, half the sound speed. Using these results, Papaloizou & Mundt & Eislöffel 1994). Terquem (1995) calculated the m = 1 bending wave re- We note that the spectral energy distribution of a cir- sponse of an inviscid disk with the rotation axis misaligned cumstellar disk could be significantly affected by the lack with a binary companion’s orbital rotation axis. This of coplanarity in binary or multiple systems (Terquem & work was extended by Larwood et al. (1996) who per- Bertout 1993, 1996). This is because reprocessing of radi- formed nonlinear calculations of the disk response using ation from the central star by a disk depends crucially on smoothed particle hydrodynamics (SPH). This numerical the disk geometry. method was further used by Larwood & Papaloizou (1997) In such noncoplanar systems, both density and bending who analyzed the response of a circumbinary disk subject waves are excited in the disk by the perturbing companion. to the tidal forcing of a binary with a fixed noncoplanar They are respectively of even and odd symmetry with re- circular orbit, and by Larwood (1997) who considered the spect to reflection in the disk mid–plane. Since much work tidal interaction occurring between a disk and a perturber has already been carried out on density waves, which are following a parabolic trajectory inclined with respect to the only waves excited in coplanar systems, we shall focus the disk. here on bending waves. Because their wavelength is larger The present work should be contrasted with earlier than that of density waves, they are expected to propagate studies of purely viscous disks in which pressure and and transport angular momentum further radially into the self–gravity were ignored. The behavior of nonplanar disk. purely viscous accretion disk subject to external torques has been considered previously by a number of authors. Following the analysis of Bardeen & Petterson (1975), 1.2. Bending Waves these studies have been developed in connection with Bending waves were first studied by Hunter & the precessing disks of Her X–1 and SS 433 for instance Toomre (1969) in the context of the Galactic disk. They (e.g., Petterson 1977; Katz 1980a, 1980b), galactic warps calculated the dispersion relation associated with these (e.g., Steiman–Cameron & Durisen 1988), or warped disks waves taking into account self–gravity but ignoring pres- around AGN and in X–ray binaries (Pringle 1996). How- sure and viscosity. Bending waves were further con- ever, the response of a purely viscous disk is very different sidered in the context of galactic disks (Toomre 1983; from that of a viscous gaseous disk, which is a better repre- Sparke 1984; Sparke & Casertano 1988) and planetary sentation of a protostellar disk. In the former case, evolu- rings (Shu, Cuzzi & Lissauer 1983; Shu 1984). In these tion occurs only through viscous diffusion, whereas in the studies, the horizontal motions associated with the ver- latter case pressure effects manifesting themselves through tical displacement of the disk were neglected. However, bending waves can control the disk evolution. When these Papaloizou & Pringle (1983) showed that these motions waves propagate (i.e. when they are not damped), they are important in gaseous Keplerian disks, where they are do so on a timescale comparable to the sound crossing nearly resonant. These authors pointed out that when the time. Since this is much shorter than the viscous diffu- disk is sufficiently viscous so that its response to verti- sion timescale, communication through the disk occurs as cal displacements is diffusive rather than wave–like, these a result of the propagation of these waves. Even when the 2 motions lead to the decay timescale of the warp being αv waves are damped before reaching the disk boundary, a dif- times the viscous timescale tν,v. Here αv is the Shakura & fusion coefficient for warps that is much larger than that Sunyaev (1973) parameter that couples to the vertical produced by the kinematic viscosity may occur because of shear, and tν,v is the disk viscous timescale associated pressure effects (Papaloizou & Pringle 1983). Self–gravity to αv. When αv is smaller than unity, this timescale is may also strongly modify the response of viscous disks, smaller than tν,v, which is the decay timescale of the warp because like pressure it allows bending waves to propagate obtained by previous authors. and its effect is global. For a more detailed discussion of In the context of protostellar disks, Artymowicz (1994) the respective effects of viscosity, pressure and self–gravity, studied the orbital evolution of low–mass bodies crossing see Papaloizou, Terquem & Lin (1998). disks, and Ostriker (1994) calculated the angular momentum exchange between an inviscid disk and a perturber on 1.3. Torque Exerted on the Disk by a Companion on an a near–parabolic noncoplanar orbit outside the disk (see Inclined Orbit also Heller 1995 for numerical simulations). The vertical ACCRETION DISKS AND BENDING WAVES 3

There can be no tidal torque exerted on an inviscid disk tidal torque can exceed the horizontal viscous stress only with reflective boundaries that contains no corotation reso- if the viscous parameter αv that couples to the vertical nance (Goldreich & Nicholson 1989). However, if the disk shear is larger than the parameter αh coupled to the hori- is viscous or has a nonreflective boundary, the compan- zontal shear. Numerical results are presented in § 7, both ion exerts a torque on the disk that leads to an exchange for uniform and nonuniform viscosity. They indicate that, of angular momentum between the disk rotation and the for parameters typical of protostellar disks, bending waves orbital motion. We can distinguish the following three are able to propagate and transport a significant fraction components of the torque: of the negative angular momentum they carry deep into 1. The component along the line of nodes makes the the inner parts of the disk. They also show that, if the disk precess at a rate that can be uniform (see § 5.2 and waves can be reflected at the disk’s inner boundary, res- Kuijken 1991; Papaloizou & Terquem 1995; Papaloizou et onances occur when the frequency of the tidal waves is al. 1998). Although we will calculate in this paper the pre- equal to that of some free normal global bending mode cession frequency induced by the tidal torque, we will not of the disk. Out of resonance, if the disk and the or- discuss the disk precession in detail since this has already bital planes are not too close to being perpendicular, the been done in the papers cited above. torque associated with the zero–frequency perturbing term 2. The component of the torque along the axis per- is found to be much larger than that associated with the pendicular to the line of nodes in the disk plane infinite frequency terms. As long as the waves are damped duces the evolution of the inclination angle of the disk before they reach the center, the torque associated with plane with respect to that of the orbit. This evolution, the finite frequency perturbations does not depend on the which is not necessarily toward coplanarity, occurs on the viscosity, in agreement with theoretical expectation (Gol- disk’s viscous timescale if the warp is linear (Papaloizou & dreich & Tremaine 1982). We summarize and discuss our Terquem 1995; Papaloizou et al. 1998). Since it affects the results in § 8. geometry of the system only on such a long timescale, we will not discuss this effect here. We note that both these 2. PERTURBING POTENTIAL components of the tidal torque modify the direction of the disk angular momentum vector. We consider a binary system in which the primary has 3. The component of the torque along the disk’s spin a mass Mp and the secondary has a mass Ms. The binary axis does not modify the direction but the modulus of the orbit is circular with separation D. We suppose that the disk’s angular momentum vector, thus changing the disk’s primary is surrounded by a disk of radius R ≪ D and accretion rate. This effect is important since it is a source of negligible mass M so that the orbital plane does not of angular momentum transport in the disk, and it will be precess and the secondary describes a prograde Keplerian the main subject of this paper. orbit with angular velocity: So far, angular momentum exchange between a viscous gaseous disk rotation and the orbital motion of a compan- G (Ms + Mp) ω = , ion on an inclined circular orbit has not been considered. 3 r D It is the goal of this paper to present an analysis of this process. In the case of an inviscid disk considered by Pa- about the primary. In the absence of the secondary star, paloizou & Terquem (1995), such angular momentum ex- the disk is nearly Keplerian. We adopt a nonrotating change occurs because the waves are assumed to interact Cartesian coordinate system (x,y,z) centered on the pri- nonlinearly with the background flow before reaching the mary star. The z–axis is chosen to be normal to the initial disk’s inner boundary. Thus, only in-going waves propa- disk mid–plane. We shall also use the associated cylindri- gate through the disk. This induces a lag between the disk cal polar coordinates (r, ϕ, z). The orbit of the secondary response and the perturbation, which enables a torque to star is in a plane that has an initial inclination angle δ be exerted on the disk. In a viscous disk, the lag required with respect to the (x, y) plane. for the torque to be non zero is naturally produced by the We adopt an orientation of coordinates and an origin of viscous shear stress. time such that the line of nodes coincides with, and the secondary is on, the x–axis at time t = 0. We denote the position vector of the secondary star by D with D ≡ |D|. 1.4. Plan of the Paper The total gravitational potential Ψ due to the binary at r The plan of the paper is as follows. In § 2 we derive an a point with position vector is given by: expression for the perturbing potential. In § 3 we outline GM GM GM r · D Ψ= − p − s + s , (1) the basic equations. The model of the unperturbed disk r r D 3 is described in § 4. The disk response is presented in § 5, | | | − | D where we give the linearized equations, study the disk re- where G is the gravitational constant. The first dominant sponse for both zero and finite perturbing frequencies, es- term is due to the primary, while the remaining perturbing tablish the boundary conditions and describe the domain terms are due to the secondary. The last indirect term ac- of validity of this analysis. In § 6 we calculate the torque counts for the acceleration of the origin of the coordinate exerted by the perturber on the disk, and relate it to the system. disk viscosity by using the conservation of angular mo- We are interested in disk warps that are excited by terms mentum. We show that this tidal torque is comparable to in the potential that are odd in z and have azimuthal mode the horizontal viscous stress acting on the background flow number m = 1 when a Fourier analysis in ϕ is carried out. when the perturbed velocities in the disk are on the order The terms in the expansion of the potential that are of the of the sound speed. If these velocities remain subsonic, the required form are given by: 4 TERQUEM

2π n 1/(2n+1) ′ sin ϕ ′ [2K (1+ n)] 1/(2n+1) Ψ = [Ψ (r, ϕ ,z,ωt) −2n H(r)= ΣΩK , (8) 2π 0 Cn Z′ ′ ′ − Ψ (r, ϕ , −z,ωt)] sin ϕ dϕ . (2) − 2 n/(2n+1) We consider a thin disk such that z ≪ r. Since r ≪ D, ρ(r, z)= 2K (1 + n) Cn we then retain only the lowest order terms in r/D and z/D in the above integral, which becomes: z2 n × (ΣΩ )2n/(2n+1) 1 − . (9) K H2 ′ 3 GMs Ψ = − 3 rz [(1 − cos δ) sin δ sin (ϕ +2ωt) 2 4 D Here ΩK is the Keplerian angular velocity given by ΩK = 3 GMp/r , H is the disk semi–thickness, Σ is the surface − (1 + cos δ) sin δ sin (ϕ − 2ωt) + sin 2δ sin ϕ] . (3) mass density deﬁned by:

Because the principle of linear superposition can be H(r) used, the general problem may be reduced to calculating Σ(r)= ρ (r, z) dz, the response to a complex potential of the form: − Z H(r) ′ − Ψ = ifrzei(ϕ Ωpt). (4) and Here the pattern frequency Ω of the perturber is one of 1 p Γ 2 Γ (n + 1) 0, 2ω or −2ω, and f is an appropriate real amplitude. Cn = 1 , Γ 2+ n +1

3. BASIC EQUATIONS where Γ denotes the gamma function. It then follows from the radial hydrostatic equilibrium The response of the disk is determined by the equation equation that the angular velocity is given by: of continuity: 1/(2n+1) 1 [K (1 + n)]2n ∂ρ ∇ v 2 2 + · (ρ )=0, (5) Ω =ΩK + 2 ∂t r ( 2Cn ) and the equation of motion: ∂ 2/(2n+1) v × (ΣΩK ) . (10) ∂ 1 ∂r + (v · ∇) v = (−∇P + Fν ) − ∇Ψ, (6) ∂t ρ In principle, the surfaceh density profilei should be de- where P is the pressure, ρ is the mass density and v is termined self–consistently as a result of viscous diffusion the flow velocity. We allow for the possible presence of a (Lynden–Bell & Pringle 1974). However, since we ignore viscous force Fν but, following Papaloizou & Lin (1995), the effect of viscosity on the unperturbed disk, we are free we shall assume that it does not affect the undistorted to specify arbitrary density profiles for the equilibrium disk axisymmetric disk so that it operates on the perturbed model. For Σ we then take a function that is constant in flow only. This approximation is justified because the disk the main body of the disk with a taper to make it vanish viscous timescale is much longer than the characteristic at the outer boundary: timescales of the processes we are interested in here (in particular, bending waves propagate with a velocity com- σ(r) Σ(r)=Σ0 , (11) parable to the sound speed, so that the waves crossing time [1 + σ(r)q ]1/q is much shorter than the viscous timescale). For simplicity, we adopt a polytropic equation of state: which approximates Σ(r)=Σ0 min [1, σ(r)] with p N 1+1/n r − (R − ∆) P = Kρ , (7) σ(r)= 1 − . ∆ where K is the polytropic constant and n is the polytropic 2 index. The sound speed is then given by cs = dP/dρ. The parameters ∆, which represents the width of the taper interior to the outer boundary, N, q and p are constrained 4. EQUILIBRIUM STRUCTURE OF THE DISK such that the square of the epicyclic frequency, which is defined by: We adopt the same model as in Papaloizou & Terquem (1995). Namely we suppose that the equilibrium 2Ω d r2Ω κ2 = , disk is axisymmetric and in a state of differential rotation r dr such that, in cylindrical coordinates, the flow velocity is given by v = (0, rΩ, 0) . For a barotropic equation of state, is positive everywhere in the disk. For the numerical cal- the angular velocity Ω is a function of r alone. culations we take N = (2n + 1) /2, ∆/R =0.1, q = 4 and For a thin disk under the influence of a point mass, in- p = 1. tegration of the vertical hydrostatic equilibrium equation The polytropic constant K is determined by specifying gives (Papaloizou & Terquem 1995): a maximum value of the relative semi–thickness, H/r, in ACCRETION DISKS AND BENDING WAVES 5 the disk, and the fiducial surface mass density Σ0 is fixed formulation, we implicitly assume that the turbulent vis- by specifying the total mass M of the disk. cosity is not affected by the tidally–produced velocities, Figure 1 shows the surface mass density, Σ, the rela- which is not necessarily true. However, since the wave- tive semi–thickness, H/r, the ratio of the sound speed to length of the tidal waves we consider is much larger than the Keplerian frequency, cs/ΩK ∼ H, and the ratio of the disk semi–thickness, this formulation may be correct. the epicyclic frequency to the Keplerian frequency, κ/ΩK, Papaloizou & Lin (1995) have shown that when the ra- versus the radius r for the equilibrium disk models de- dial wavelength of the perturbations is larger than the −2 scribed above with M = 10 Mp and (H/r)max = 0.1. disk semi–thickness, a vertical averaging approximation, The epicyclic frequency departs from ΩK at the outer edge in whichg ˜ is assumed to be independent of z, can be used. ′ ′ ′ of the disk because of the abrupt decrease of Σ there. Each ofv ˜r andv ˜ϕ are then proportional to z, withv ˜z being independent of z. They have found some support for the 5. procedure in a numerical study of the propagation of dis- THE DISK RESPONSE turbances in disk models that take the vertical structure 5.1. Linear Perturbations fully into account. Linearization of the equation of motion followed by mul- In this paper we consider small perturbations, so that tiplication by ρz and vertical averaging then gives the ve- the basic equations (5) and (6) can be linearized. As the locity perturbations in the form: relevant linearization of the equations has already been discussed by Papaloizou & Lin (1995), we provide only an ′ v˜r abbreviated discussion here. We denote Eulerian pertur- = − (1 − Ωp/Ω)˜g Ωz bations with a prime. Because the perturbing potential, ′ Ψ , is proportional to exp [i (ϕ − Ωpt)] , the ϕ and t de- dg˜ 2 2 (1 − Ωp/Ω) r +˜g (3 − Ωp/Ω)Ω /Ω pendence of the induced perturbations is the same. We + dr p , (14) 2 2 2 denote with a tilde the part of the Eulerian perturbations (1 − Ωp/Ω − iαv) − κ /Ω that depends only on r and z. For instance, the Eule- ′ 2 ′ ′ v˜ i κ v˜ rian perturbed radial velocity is denoted v , and we have ϕ = g˜ + r , (15) ′ ′ r 2 Ωz 1 − Ωp/Ω 2Ω Ωz vr(r,ϕ,z,t)=˜vr(r, z)exp [i (ϕ − Ωpt)]. We define a quantity g (the physical meaning of which v˜′ g˜ will be given at the end of this section) through: z = , (16) rΩ 1 − Ωp/Ω ′ P ′ +Ψ = −irzΩ2g, (12) where αv is a vertical average of the standard Shakura & ρ Sunyaev (1973) parameter that couples to the vertical shear. It is defined by: and we also have g(r,ϕ,z,t)=˜g(r, z)exp [i (ϕ − Ωpt)]. It was first shown by Papaloizou & Pringle (1983) that ∞ ∞ 2 horizontal motions induced in a near Keplerian tilted disk αv = ρνvdz Ω ρz dz . (17) −∞ −∞ are resonantly driven. The main effect of a small viscos- Z Z ity is then to act on the vertical shear in the horizon- ′ ′ As mentioned above, the main effect of viscosity is to tal perturbed velocities ∂vr/∂z and ∂vϕ/∂z. Demianski & damp the resonantly driven horizontal velocities. There- Ivanov (1997) and Ivanov & Illarionov (1997) have recently fore, the viscous terms which do not act directly on the given the relativistic generalization of this effect. Follow- resonance have been neglected. This is consistent with ing Papaloizou & Lin (1995), we then retain only the hor- keeping only the components of the viscous force given by izontal components of the viscous force perturbation: equation (13). Again, this is accurate only when αv ≪ 1. The same procedure applied to the continuity equation ′ ′ gives: ′ ∂ ∂vr ′ ∂ ∂vϕ Fνr = ρνv , Fνϕ = ρνv , (13) 2 ∂z ∂z ∂z ∂z Ω g˜ 2 2 dµ −i (Ω − Ωp) I + r (Ω − Ωp) − Ω J + (Ω − Ωp) dr where νv is the kinematic viscosity that couples to the ver- h i tical shear. This approximation is accurate when α ≪ ′ ′ v 1 ∂ v˜ v˜ϕ 1. The numerical calculations have actually shown that, = − rµ r + iµ , (18) r ∂r z z when αv becomes larger than 0.1, the terms that are neglected in this approximation become comparable to the in which terms that are retained. For this reason, we shall limit our ∞ ∞ ∞ + ρz2 + + ρz numerical calculations to values of αv smaller than 0.1. 2 ˜ ′ J = 2 dz, µ = ρz dz, I = 2 Ψ dz. Here we have assumed that the disk is turbulent and −∞ c −∞ −∞ c that dissipation can be modeled as a turbulent viscosity. Z Z Z The most likely mechanism for producing such a turbulent We note that equation (18) depends on the viscosity only viscosity is the magnetorotational Balbus–Hawley instabil- through the perturbed velocities. ity (Balbus & Hawley 1991). We note however that this To close the system of equations, we need a relation instability may not operate in all the parts of protostel- between P˜′ andρ ˜′. For simplicity, we assume that the lar disks (Balbus & Hawley 1998). Also, it is not clear equation of state is the same for the perturbed and un- that the tidal waves would be damped by the turbulent perturbed flows. Linearization of the equation of state (7) viscosity in the way described by equations (13). In this then gives 6 TERQUEM

where Ω0 = Ω(R) and rin is the disk inner boundary. Here ˜′ 2 ′ P = csρ˜ . (19) the disk has been assumed to precess as a rigid body. This We now give a physical interpretation of g. Equa- is expected if the disk can communicate with itself, ei- tion (16) can be used to expressg ˜ in terms of the vertical ther through wave propagation or viscous diﬀusion (self– gravity being ignored), on a timescale less than the preces- Lagrangian displacement ξ˜z: sion period. As long as αv is smaller than the disk aspect ˜ ratio H/r (which is almost constant throughout most of 2 ξz g˜ = i (1 − Ωp/Ω) , (20) the disk), bending waves may propagate (Papaloizou & r Pringle 1983; Papaloizou & Lin 1995). The ability of ˜ where again we have deﬁned ξz such that ξz(r,ϕ,z,t) = these waves to propagate throughout the disk during a pre- ˜ ξz(r, z)exp [i (ϕ − Ωpt)]. cession time implies approximately that H/r > |ωp| /Ω0. When Ωp ≪ Ω, the physical relative vertical displacement This is the condition for near rigid body precession in an is then given by: inviscid disk (Papaloizou & Terquem 1995). The numerical simulations of Larwood et al. (1996) show that this

ξz condition also holds in a viscous disk in which αv does not Re = Re(˜g) sin (ϕ − Ωpt)+ Im(˜g)cos(ϕ − Ωpt) . exceed H/r. This indicates that in the absence of self– r gravity, as long as α < H/r (which is likely to be satisfied (21) v in protostellar disks), pressure is the factor that controls Thus, when Ω = 0, Re(˜g) and Im(˜g) are the relative ver- p the efficiency of communication throughout the disk and tical displacement along the y and x–axis respectively. We then the precessional behavior of the disk. A more com- note that a constant value ofg ˜ corresponds to a rigid tilt. plete discussion of the precession of warped disks (which 5.2. Zero–Frequency Response includes self–gravity) is given by Papaloizou et al. (1998). The Coriolis force can be taken into account by replac- From equation (3), we have to consider the disk response ing Ψ˜ ′ with: to a secular potential perturbation with zero forcing frequency. When Ωp = 0 and αv ≪ 1, equations (14), (15), (16) and (18) reduce to the single second–order ordinary differential equation forg ˜ (Papaloizou & Lin 1995): ˜ ′ d µ dg˜ iI Ψ +2iωprzΩsinδ. (25) = . (22) 2 2 2 dr " (1 − iαv) − κ /Ω dr # r Here we have assumed that:

′ d µ For the zero–frequency case, Ψ˜ will represent the to- µ ≪ Re r , (23) 2 2 2 tal term (25) and the real amplitude f will represent ( dr "(1 − iαv) − κ /Ω #) f +2ωpΩ sin δ from now on. which is certainly a reasonable approximation for αv ≪ 1. With this new definition of f,g ˜ is then given by: It can be shown that with the term I in its present form, the x–component of the torque is non zero. This torque leads to the precession of the disk, as in a gyroscope. By writing the equation of motion for Ωp = 0 in the frame ′ defined in § 2, we have assumed that the response of the r (1 − iα )2 − κ2/Ω2 r g˜ (r)= − v J (r′′) f (r′′) dr′′dr′, disk appears steady in a nonrotating frame. Because of the µ precessional motion induced by the secular perturbation, Zrin Zrin (26) the response is actually steady in a frame rotating with where the quantity in the outer integral has to be evalu- the precession frequency. The Coriolis force correspond- ′ ated at the radius r . We have supposed thatg ˜ = 0 at ing to this additional motion must then be added in the r = r , but we note that an arbitrary constant may be equation of motion (the centrifugal force can be neglected in added tog ˜. This corresponds to an arbitrarily small rigid since it is of second order in the precession frequency). tilt that we assume eliminated by the choice of coordinate The magnitude of the Coriolis force in the frame in which system. the response appears steady has to be such that the x– The above expression shows that, depending on whether component of the total torque (involving both the Coriolis α is larger or smaller than 1 − κ2/Ω2 ∼ H2/r2, the imag- and the gravitational forces) is zero. v inary part ofg ˜ is larger or smaller than its real part, re- This requirement leads to the expression of the preces- spectively. According to equation (21), this corresponds to sion frequency ω (Papaloizou & Terquem 1995; see also p a vertical displacement produced by the secular response Kuijken 1991 in the context of galactic disks): being mainly along the x– or y–axis, respectively. We finally comment that the above treatment is accu- 2 ωp 3 Ms ω rate only when |ωp| /Ω0 is smaller than the maximum of = − cos δ 2 2 2 2 Ω 4 M + M Ω 1−κ /Ω ∼ H /r and αv. If this is not the case, then the 0 s p 0 R R precession frequency has to be taken into account in the Σ Σ 2 2 2 resonant denominator (1 − iαv) − κ /Ω where it would × 2 dr dr , (24) r (Ω/Ω ) r Ω/Ω0 be the dominant term. Z in 0 ,Z in ACCRETION DISKS AND BENDING WAVES 7

5.3. Finite–Frequency Response 5.4. Boundary Conditions We now consider the response generated in the disk by Equation (27) has a regular singularity at the outer edge those terms in the perturbing potential with finite forcing r = R (see Papaloizou & Terquem 1995). Our outer frequency. When Ωp 6= 0, the second–order differential boundary condition is thus that the solution be regular equation forg ˜ obtained from equations (14), (15), (16) there. and (18) takes on the general form: We take the inner boundary to be perfectly reflective. The radial velocity in spherical polar coordinates, 2 ′ ′ 2 2 1/2 d g˜ dg˜ (rv˜r + zv˜z) / r + z , thus vanishes at the locations A 2 + B + Cg˜ = S, (27) 2 2 2 dr dr (r, z) such that r + z = r , where rin is the disk inner in ′ radius. Using the expression (16) ofv ˜z, this means that with at these locations: ′ r (Ω − Ωp) v˜ Ω˜g A = µ, (28) rz r + =0, (35) D2 z 1 − Ω/Ω p and then, by continuity: 2 Ω2 ′ 1 dlnµ κ Ωp p v˜r Ω˜g B = µΩ 1+ r D1 − + 3 − + =0, (36) D dr 2Ω2 Ω Ω2 ( 2 " # z 1 − Ω/Ωp at the inner boundary. 1 d D rΩ We note that for the largest values of αv, we expect the −D + 1 , (29) 1 Ω dr D waves to be damped before reaching the center of the disk, 2 such that the results are independent of the inner boundary condition. For αv = 0, since there is no dissipation at S = i (Ω − Ωp) I, (30) the boundaries, we expect no torque to be exerted on the disk and no net angular momentum flux to flow through where we have set D1 = 1 − Ωp/Ω and D2 = its boundaries. 2 2 2 (1 − Ωp/Ω − iαv) − κ /Ω . By introducing the function h such that: 5.5. Domain of Validity of This Analysis In this analysis, we have neglected the variation ofg ˜ with 1 dh 1 dA = B − , (31) z. As noted above, Papaloizou & Lin (1995) have shown h dr A dr that this approximation is valid when the scale of variation ofg ˜ with radius is larger than the disk semi–thickness H, equation (27) can be recast in the form: i.e. when d dg˜ 1 dg˜ 1 hA + hCg˜ = hS, (32) < . (37) dr dr g˜ dr H

When this is not satisfied, dispersive effects, which have the solutions of which are given by the Green’s function method: been neglected here, should be taken into account. Since we also consider linear waves, our analysis is valid r as long as the perturbed velocities are smaller than the 1 ′ ′ ′ g˜ (r)= g1 (r) g2 (r ) s (r ) dr sound speed. When they become supersonic, shocks occur W Zrin that damp the waves (Nelson & Papaloizou 1998). How- ever, it is not correct to compare the Eulerian perturbed R velocities given by the equations (14), (15) and (16) with ′ ′ ′ +g2 (r) g1 (r ) s (r ) dr . (33) the sound speed to know whether the waves are linear or Zr # not. This is because the system should be invariant under the addition of a rigid tilt (i.e. a constant) tog ˜. An −2 Here s = h (Ω − Ωp) frΣΩ , W is the constant increase ofg ˜ corresponding to the addition of a rigid tilt could make the Eulerian perturbed velocities larger than dg2 dg1 the sound speed without the waves getting nonlinear. In W = hA g − g , (34) 1 dr dr 2 principle, as pointed out by Papaloizou & Lin (1995), to get perturbed quantities that do not depend on the ad- and g1 and g2 are the solutions of the homogeneous differ- dition of a rigid tilt (i.e. that depend only on the radial ential equation that satisfy the outer and inner boundary gradient ofg ˜), we should use Lagrangian rather than Eu- conditions respectively. In the numerical calculations pre- lerian variables. However, since the Eulerian perturbed sented below, g1 and g2 are calculated directly from equa- radial velocity in spherical polar coordinates depends only tions (14), (15) and (18) using the fourth–order Runge– on the gradient ofg ˜ in the limit of small perturbing fre- Kutta procedure with adaptative stepsize control given by quency, we can compare this quantity to the sound speed Press et al. (1986). Computation of the function h then to decide whether the motion is subsonic. The criterion allowsg ˜ to be calculated from (33). that has to be satisfied is then: 8 TERQUEM

Papaloizou 1993 for example). Because of the conservation of wave action in an inviscid disk, waves excited at ′ ′ dg rvr + zvz Ωzr dr the outer boundary propagate through the disk with an ≃

′ involves J , which, in a Keplerian disk, is equal to ΣΩ−2. −ρξ∗ · ∇Ψ − ρξ∗ · (ξ · ∇) ∇Ψ] dV } =0, (49) We then get where we have defined the linear operator P such that R P (ξ) = ρ∆ (−∇P/ρ) . We note that if the terms on the 3 Tz = π Im (˜g) fΣr dr (44) left hand side of equation (47) had non zero contribution Zrin to equation (49), it would be related to the total time where f is the real amplitude defined by (4) (which has to derivative of the perturbed specific angular momentum of be modified when Ωp = 0 in order to take into account the the disk. disk precession, see § 5.2). In the numerical calculations Using the perturbed equation of state (19) in the form presented below, the torque Tz will be computed from this ∆P/P =(1+1/n) ∆ρ/ρ, it can be shown (Lynden–Bell & expression. Ostriker 1967) that: If JD is the total angular momentum of the disk, the tidal effects we have considered would remove the angular ξ∗ ·P (ξ) dV = B + ξ∗ · O (ξ) dV, (50) momentum content of the disk on a timescale ZV ZV where O is a self–adjoint linear operator, and B is a bound- JD 4 M GMpR ary term: td = ∼ . (45) Tz 5 pTz We shall compare this timescale with the viscous timescale 2 −1 1 ∗ ∗ tν,h ∼ (r/H) Ω(R) /αh where we will take H/r = B = ρ (∇ · ξ) ξ + P (ξ · ∇) ξ · n dS. (51) n (H/r)max (for our disk models, H/r does not vary by more ZS than 20% in the main body of the disk, as indicated in The integration is over the surface S of the disk, and n Fig. 1). Here α is the viscous parameter that couples to h is the unit vector perpendicular to this surface, oriented the horizontal shear. toward the disk exterior. Since P and ρ vanish at r = R We also define the torque integral T (r), which is the z and z = ±H and we consider models for which r ≃ 0, torque exerted between the inner boundary and the ra- in we have B = 0. In addition, since O is self–adjoint, the dius r: integral on the right hand side of equation (50) is real. r The term involving P in equation (49) is then zero. 3 Tz(r)= π Im (˜g) fΣr dr. (46) Using the fact that ∂2Ψ/∂r∂z = ∂2Ψ/∂z∂r, it can also Zrin be shown that the term involving ∇Ψ is zero. Since equation (49) has no ϕ and t–dependence, we can 6.2. Angular Momentum Conservation now use the tilded variables again. We define ξ˜ such that ˜ ˜∗ To get the angular momentum conservation equation, ξ(r,ϕ,z,t) = ξ(r, z)exp [i (ϕ − Ωpt)]. We define ξ simi- which is of second order in the perturbed quantities, we larly. The term involving Ψ˜ ′ in equation (49) can be writ- first take the Lagrangian perturbation of the equation of ten as follows: motion (6) and multiply by the unperturbed mass density. To first order, this leads to (Lynden–Bell & Ostriker 1967): ∗ ′ Im ρξ˜ · ∇Ψ˜ dV V ∂2ξ ∂2ξ ∂2ξ ∂ξ Z ρ +Ω2 + Ω× (Ω×ξ)+2Ω +2Ω× ∂t2 ∂ϕ2 ∂ϕ∂t ∂t ′ ′ ′∗ = Im ∇ · ρξ˜∗Ψ˜ dV +Im Ψ˜ ρ˜ dV , (52) V V ∂ξ 1 ′ ′ Z Z +2ΩΩ× = ρ∆ − ∇P +F −ρ∇Ψ −ρ (ξ · ∇) ∇Ψ. ∂ϕ ρ ν where an integration by parts has been performed and the (47) linearized continuity equation: Here, ξ is the Lagrangian displacement vector, ∆ denotes ′ ∇ ˜ the Lagrangian change operator, and we have used the re- ρ˜ = − · ρξ , (53) lation between Lagrangian and Eulerian perturbations of a quantity Q to first order in ξ: (see, for example, Tassoul 1978) has been used. The first term on the right–hand side of equation (52) is zero be- ∆Q = Q′ + (ξ · ∇) Q. (48) cause ρ vanishes at r = R and z = ±H and rin ≃ 0. Equation (49) then becomes: The vector Ω ≡ Ωkˆ, with kˆ being the unit vector in the z–direction. To derive equation (47), we have used the F 0 ′∗ ′ ˜∗ ˜′ fact that at equilibrium ν = . Im ρ˜ Ψ˜ dV = Im ξ · F dV . (54) We now take the scalar product of equation (47) with ξ∗, ν ZV ZV which is the complex conjugate of ξ. Since ∂/∂t = −iΩp and ∂/∂ϕ = i, all the terms on the left hand side of the From equation (43), we see that this is equivalent to: resulting equation are real. We then take the imaginary R H part of this equation and integrate it over the volume V ˜∗ F˜′ Tz = Im π ξ · ν rdrdz . (55) of the unperturbed disk. This leads to: − Zrin Z H ! ∗ ∗ ′ We now use the relation between the Eulerian velocity Im [ξ ·P (ξ)+ ξ · F ν perturbation and the Lagrangian displacement: ZV 10 TERQUEM

As we shall see, this quantity is negative, which means ˜ ′ ∂ξ that the disk loses angular momentum. ˜v + ξ˜· ∇ v = + (v · ∇) ξ˜ (56) ∂t In the numerical calculations presented below, the torque integral will be computed from equation (46) and F˜′ together with the expression (13) of ν , the relation (17) J(r) will be computed directly from the velocities using and the fact that ρ (±H) = 0 to transform equation (55) equations (14) and (15). into: 7. 2 NUMERICAL RESULTS R |v˜′ /z|2 + v˜′ /z T = −π µα r r ϕ dr We compute the torque exerted on protostellar disks by z v 1 − Ω /Ω rin p a perturber with an orbit that is inclined to the plane of Z the disk. For a coplanar system, one expects the circum- ′∗ ′ R (˜v /z) v˜ /z κ2 primary disk to be truncated by tidal effects in such a − πIm µα r r ϕ 1 − . (57) v 2 2 way that its radius is not greater than about one–third of " rin (1 − Ωp/Ω) 2Ω # Z the separation of the system (Papaloizou & Pringle 1977). In the numerical calculations presented below, the Larwood et al. (1996) have shown that tidal truncation is torque will be computed from equation (44) and compared only marginally affected by the lack of coplanarity. Thus with the result of the above equation. in our models we adopt D ≥ 3R. Since we consider binary In a Keplerian disk, κ ≃ Ω. Then, when Ωp ≪ Ω and mass ratio of unity (Ms = Mp), the perturbing potential αv ≪ 1, equation (15) is well approximated by: is small compared with the potential of the central star, justifying the assumption of small perturbation. ′ v˜ κ2 v˜′ For computational convenience, we normalize the units ϕ ≃ i r , (58) z 2Ω2 z such that Mp = 1, R = 1, and ΩK (R) = 1. Apart for the calculation of the epicyclic frequency, the rotation law so that Tz can be approximated by: in the disk is taken to be Keplerian (the departure from R ′ 2 the Keplerian law, which occurs mainly at the disk outer 3π v˜r Tz ≃− µαvr dr. (59) edge, does not exceed ∼ 3%). The inner boundary of the 2 z disk is r = 10−4, and the outer boundary is taken to be Zrin in r = 0.99 rather than 1 in order to avoid a zero surface As expected, the torque is negative. Here we consider out mass density at the outer edge. linear perturbations such that the perturbed velocities re- We present here the results of some disk response cal- main smaller than the sound speed. We have pointed out culations for an inclination δ = π/4, a polytropic index in § 5.5 that, since the system should be invariant under − n = 1.5 and a disk mass M = 10 2. Results for an arbi- the addition of a rigid tilt tog ˜, it is not in principle correct trary inclination δ <π/2 (prograde orbit) or π/2 <δ<π to compare the Eulerian perturbed velocities to the sound (retrograde orbit) can be obtained straightforwardly since speed to know whether the waves are linear or not. How- ′ Tz (and then 1/td) corresponding to Ωp = ±2ω and 0 is ever, since the dominant term in the expression (14) ofv ˜r 2 2 2 is the term proportional to the gradient ofg ˜, we can com- proportional to (1 ± cos δ) sin δ and sin 2δ, respectively. pare this velocity to the sound speed to decide whether As shown by equation (24), the precession frequency scales the motion is subsonic. Therefore we can get an estimate with cos δ. Since g is independent of M (see equations [14], ′ [15], [16] and [18]), expression (44) shows that Tz ∝ M. of the maximum value of the torque by setting |v˜r| ∼ cs, µ ∼ ρH3 and dropping the integration. This leads to a The quantities ωp, td and tν,h are independent of M. minimum value of td which is equal to tν,h × αh/αv. Only 7.1. Uniform αv if αv were larger than αh could td be smaller than tν,h. An other useful quantity for interpreting our results is In table 1 we summarize the results obtained for dif- the flux of vertical angular momentum in the radial direc- ferent values of the parameters, which are the separation at radius r: tion of the system D, the viscous parameter αv and the maximum value of the relative semi–thickness of the disk (H/r)max. The quantities we compute are the torque Tz H 2π 2 ′ ′ associated with each of the perturbing frequencies 2ω (pro- J(r)= ρr Re vϕ Re (vr) dϕdz, (60) − grade term), −2ω (retrograde term) and 0, the disk pre- Zz= H Zϕ=0 cession frequency ωp and the ratio of the tidal timescale td which can also be written to the viscous timescale tν,h. We have taken here αh = αv. The two values of the torque have been computed using ′ ∗ v˜ v˜′ equations (44) and (57), the result corresponding to equa- J(r)= πµr2Re ϕ r . (61) z z tion (57) being in parentheses. The timescale td is computed from equation (45) with Tz being the sum of the The torque Tz(r) exerted between the radii rin and r re- contributions from each frequency. sults in a change of angular momentum of the disk. Part We note that when either one of the criterions (37) of this angular momentum is advected through the bound- or (38) is not satisfied (cases indicated by a footnote), it aries (this is J(r) − J(rin) ≃ J(r) since rin ≃ 0), and is always close to the disk outer edge. This is because the part is dissipated between these radii. Thus the quantity density varies more rapidly there than in the other parts Tz(r)−J(r) is the amount of angular momentum deposited of the disk. Also the scale of variation ofg ˜ never becomes in the disk between the inner boundary and the radius r. smaller than H by less than a factor of a few. Similarly, ACCRETION DISKS AND BENDING WAVES 11 when the radial velocity in spherical polar coordinates be- the same value when 2ω approaches 0, which probably comes larger than the sound speed, it is only by a factor of means here 2ω/Ω(R) ≪ αv since the torque is controlled a few. We note that the results we present in this case can by viscosity. This is not satisfied even for model 3i which still be used since the perturbed quantities can be reduced has 2ω/Ω(R)=0.09. However, we observe that the differ- by choosing appropriate scaling parameters, like the initial ence between these torques is reduced from a factor 9 to a inclination δ or the mass of the disk M. factor less than 4 when model 3i is run with αv =0.1. 2 2 In all cases the condition |ωp| /Ω(R) < max H /r , αv In Figure 3 the net torque Tz is plotted versus D in −3 is satisfied (see § 5.2). The condition for near rigid body a semi–log representation for αv = 10 (models 3 plus −2 precession, H/R ≫ |ωp| /Ω(R), is also always satisfied. other values of D) and αv = 10 (models 4 plus other val- We note that there is a good agreement between the ues of D) and for Ωp = ±2ω and 0. Before commenting on values of the torque given by equations (44) and (57). In the resonances, we note that the torque associated with the addition, in the cases we computed, we found this latter finite frequency terms out of resonance does not decrease equation to be very well approximated by (59). when the separation increases up to D ∼ 8. This con- + − 0 To simplify the discussion, we note Tz ,Tz and Tz the firms the results of Papaloizou & Terquem (1995). There torque associated respectively with Ωp = 2ω, −2ω and are competing effects acting when the separation of the 0. Furthermore, we separate the δ–dependence by writing system is increased. On the one hand, the wavelength of ± 2 2 0 2 Tz = T±(1 ± cos δ) sin δ and Tz = T0 sin 2δ with T± the tidal waves becomes larger (since it is inversely pro- and T0 being independent of δ. portional to its frequency), which tends to increase the Since part of this section is devoted to the tiltg ˜, we re- torque. On the other hand, the amplitude of the perturb- call thatg ˜ is related to the relative vertical displacement ing potential decreases, which tends to reduce the torque. through equation (21). When Ωp = 0, Re(˜g) and Im(˜g) It seems that the first effect is dominant for D ≤ 8. It are the relative vertical displacement along the y and x– seems like the torque begins to decrease with the separa- axis respectively. tion when the wavelength becomes comparable to the disk radius, which happens for D ∼ 9. This is reasonable since the wavelength cannot increase further. Finite–Frequency Response 7.1.1. In Figure 3 we can see resonances, which occur when Figure 2 shows the real and imaginary parts ofg ˜ versus the frequency of the tidal waves is equal to that of some r for models 2 and Ωp = 2ω (figures for Ωp = −2ω are free normal global bending mode of the disk, and cause similar). We see that the wave–like character of the disk the torque to become very large, even infinite if there is response disappears when αv becomes larger than H/r no dissipation (Papaloizou & Lin 1984). In a resonance, (which is 0.05 in these models). This is in agreement with the torque indeed increases when αv decreases, in contrast Papaloizou & Pringle (1983) and Papaloizou & Lin (1995) to what is observed out of resonance. It seems that there (and also with the relativistic generalization of Demian- is no longer any resonance for D > 10. This would not ski & Ivanov 1997 and Ivanov & Illarionov 1997) who be surprising since the wavelength of the response is then have shown that, in a near Keplerian disk (self–gravitating comparable to the disk radius. A more detailed descrip- or not), the longest wavelength disturbances undergo a tion of the resonance which occurs at D ∼ 8.5 is given in transition between wave–like and diffusive behavior when the Appendix. αv ∼ H/r. We note, however, that when αv ≥ H/r, there Since the resonances depend on the spectrum of the are still some oscillations in the half outer part of the disk. free normal bending modes of the disk, they are model– This is because bending waves have a long wavelength, dependent. In particular, they occur only if the the waves such that they can still penetrate relatively far before be- can be at least partially reflected at the inner boundary, ing diffused out. otherwise there is no free normal global bending mode in The magnitude of Re(˜g) does not vary with αv, since it the disk (see also Hunter & Toomre 1969 for the case of a is controlled mainly by the radial pressure force. On the purely self–gravitating disk). contrary, the magnitude of Im(˜g) is much more sensitive The values of the torque displayed in Figure 3 corre- to the viscosity, although it does not vary monotonically spond to δ = π/4. However, for this particular value of with αv. δ, the perturbed velocities may become larger than the By comparing models 2a and 3c and also models 2c and sound speed close to a resonance. These values of the 4c, which differ only by the value of (H/r)max, we have torque should then be scaled to be used for other values −2 checked that the wavelength of the response is proportional of δ, or for other disk parameters. For αv = 10 , the to H/r. This is expected since bending waves propagate resonances are very weak, indicating that the waves are with a velocity which, being half the sound speed, is pro- almost completely dissipated before reaching the disk in- portional to H/r, and the wavelength is proportional to ner boundary. the wave velocity. We have compared our results with those obtained by In all the cases we have computed, except model 3f, Papaloizou & Terquem (1995) for an inviscid disk in which + − Tz ≥ Tz , but T− ≥ T+, in agreement with Papaloizou & the inner boundary is dissipative. Remembering that they Terquem (1995). In general, the shorter the wavelength had δ = π/2, we found that, in general, we could repro- −2 of the response, the smaller the torque, and the wave cor- duce their results with αv ∼ 10 . This confirms that, −2 responding to the retrograde term has a slightly longer when αv ∼ 10 , the waves are dissipated before reaching wavelength than that corresponding to the prograde term. the disk inner boundary. ± The difference between T+ and T− (and T0) is still sig- Out of resonance, the torque Tz associated with the fi- nificant for the largest values of the separation computed nite frequency terms is proportional to αv as long as this here. We expect all these quantities to converge toward 12 TERQUEM

0 parameter is smaller than some critical value. Then, for we also have T0 ≫ T±, T is always going to be much larger z ◦ ± ± the larger values of αv, Tz is independent of the viscosity. than T for δ not too close to π/2 (typically for δ < 70 This is in agreement with the theoretical expectation (Gol- ◦z ◦ or 110 <δ< 180 ). dreich & Tremaine 1982) and the work of Papaloizou & In contrast to the finite frequency response, the zero– Terquem (1995). We indeed expect the torque to be inde- frequency response is not wave–like, and we then expect pendent of whatever dissipation acts in the disk providing T 0 to decrease continuously with D. This is indeed what the waves are dissipated before reaching the disk inner z we observe in Figure 3. boundary. If the waves are reflected on the disk inner boundary, the situation is more complicated because the angular momentum carried by the waves has a different Angular Momentum Dissipation as a Function of sign depending on whether the waves are in–going or out– 7.1.3. Radius going. Therefore part of the angular momentum lost by the disk while the waves propagate inwards is given back Figures 5 and 6 show Tz(r) − J(r) normalized to unity by the waves propagating outwards. The net amount of versus r for various models. This represents the angu- angular momentum lost by the disk then depends on how lar momentum deposited in the disk between the inner efficiently the waves are reflected. Using this argument, boundary and the radius r. The fact that this quantity is we then find that, when H/r = 0.1 and 0.05, the waves negative means that the disk loses angular momentum. Of are damped before reaching the disk inner boundary when course, |Tz(r) − J(r)| should increase with r. However, we −2 −3 αv is larger than ∼ 5 × 10 and ∼ 5 × 10 , respectively observe in Figure 5 that for D =3andΩ=2ω this func- (for H/r = 0.1, this is a bit larger than the value found tion has a little ’hump’ close to the disk outer edge. We above). Such a dependence of this critical value of αv with interpret it as being a numerical effect, since its amplitude H/r is expected. Indeed, the shorter the wavelength, the is found to depend on the accuracy of the calculations. more easily are the waves dissipated. Since the wavelength We first observe that bending waves are able to trans- is proportional to H/r (see above), we then expect that port a significant fraction of the negative angular momen- for a given αv dissipation is more efficient for small values tum they carry deep into the inner parts of the disk. For ± of H/r. For the same reason, the net torque Tz increases the parameters corresponding to model 1b, Figure 5 shows with H/r (see models 3c and 2a and models 4c and 2c). that the retrograde and prograde waves transport respec- The coupling between the response and the perturbing po- tively 35% and 60% of the total negative angular momen- tential is indeed more efficient when the response has a tum they deposit into the disk at radii smaller than 0.3 and long wavelength, and the integral in equation (44) from 0.4 respectively. In contrast, only 20% of the total negative which the torque is calculated is then larger. angular momentum carried by the zero–frequency perturbation is deposited at radii smaller than 0.4. We note however that, out of resonance, the net amount of angular 7.1.2. Zero–Frequency Response momentum deposited at small radii by the Ωp = 0 per- Figure 4 shows the real and imaginary parts ofg ˜ versus turbation is not necessarily smaller than that deposited r for models 2 and Ωp = 0. As expected, we observe that by the finite frequency perturbations. Also, the fraction the zero–frequency response is not wave–like. of angular momentum deposited close to the disk outer 2 From equation (26) we see that, as long as αv ≪ edge by the finite frequency responses (especially the ret- 2 2 2 2 −2 1−κ /Ω ∼ H /r , which is the case for αv ≤ 10 , Re(˜g) rograde wave) is larger than that deposited by the secular 2 is almost independent of the viscosity. When αv =0.1, αv response. dominates over 1 − κ2/Ω2, and since it appears in the ex- It appears that the secular response gets dissipated very pression of Re(˜g) with the minus sign, Re(˜g) becomes neg- progressively throughout the disk, whereas the prograde ative. The particular behavior of Re(˜g) in the outer parts and retrograde waves are dissipated preferentially at some of the disk is produced by the fact that the surface mass locations in the disk. The position of these locations seems density drops to zero there, thus increasing 1 − κ2/Ω2. As to depend on the wavelength of these waves. expected from equation (26), Im(˜g) is proportional to αv. The resonances do not affect the distribution of dissi- Papaloizou, Korycansky & Terquem (1995) computed pation of angular momentum in the disk, although they the real part ofg ˜ for an inviscid disk with an equilibrium obviously increase the absolute value of the actual amount state similar to that we have set up here and for Ωp = 0 of angular momentum deposited. This is due to the fact and D = 7. We have checked that the results we get for that the magnitude of the response, not its wavelength, is the smallest values of αv are in complete agreement with affected by a resonance. theirs. In Figure 6 we have plotted Tz(r) − J(r) versus r for Equation (44) shows that Tz depends only on Im(˜g), not Ωp = −2ω and different values of αv. We see clearly on 0 on Re(˜g). Since for the secular response Im(˜g) ∝ αv, Tz these plots the transition that occurs when αv ∼ H/r. is also proportional to αv. This is borne out by the numer- When αv becomes larger than H/r, the wave–like response ical results. Also equation (26) indicates thatg ˜ ∝ 1/H2. of the disk disappears. Thus, the curves corresponding to Thus if we vary H/r while keeping Σ constant, we get Ωp = −2ω have a shape similar to those corresponding to 2 from equation (44) that Tz ∝ 1/H . This is confirmed the secular response. by models 3c and 2a and models 4c and 2c, which differ In general, when αv is increased, the fraction of nega- only by the value of (H/r)max. This is in contrast to the tive angular momentum deposited at small radii decreases. finite frequency response, for which the torque increases This is expected since the waves, excited predominantly in with H/r (see above). the outer parts of the disk, get dissipated more easily at 0 ± Figure 3 indicates that Tz ≫ Tz out of resonance. Since large radii when αv is large. ACCRETION DISKS AND BENDING WAVES 13

We observe in Figure 6.a that there is hardly any differ- a long–wavelength response. The effects of perturbations −4 −3 ence between the cases αv = 10 and αv = 10 . This with even symmetry, which occur in the coplanar case, means that whatever the amount of angular momentum may be superposed on the effects of those with the odd transported by the wave, the same fraction is deposited be- symmetry. tween rin and r in both cases. This probably indicates that Since the response of a viscous disk is not in phase with the wave is able to reach the disk inner boundary for these the perturbing potential, a tidal torque is exerted on the values of αv. For larger αv for which the wave is completely disk. When the perturber rotates outside the disk, this dissipated before reaching the inner boundary, no angular torque results in a decrease of the disk angular momen- momentum is deposited at small radii, so that the curves tum. Part of the (negative) angular momentum of the look different. This indicates that, for (H/r)max = 0.1, perturbation is carried by tidal waves away from the lo- the waves do not reach the inner boundary when αv is cation where the torque is exerted, and part is dissipated −2 larger than ∼ 10 . For (H/r)max = 0.05 (Figure 6.b), locally through viscous interaction of the waves with the the critical value of αv above which the waves do not reach background flow. the inner boundary is between 10−3 and 5 × 10−3. These We have shown that the tidal torque is comparable to values are in agreement with those found above. Our cal- the horizontal viscous stress acting on the background flow culations also show that a larger fraction of the negative when the perturbed velocities in the disk are on the order angular momentum is deposited at small radii when H/r of the sound speed cs. If these velocities remain subsonic, is decreased. However, we have checked that this effect the tidal torque can exceed the horizontal viscous stress is less significant when the distance interior to the outer only if the parameter αv which couples to the vertical boundary occupied by the taper of the surface mass den- shear is larger than the parameter αh, which is coupled sity is reduced along with the semi–thickness. to the horizontal shear. We note that, so far, there is no indication about whether these two parameters should be the same or not. Nelson & Papaloizou (1998) have found 7.2. Nonuniform αv that, when the perturbed velocities exceed cs, shocks re- We have run model 1c with a nonuniform viscosity duce the amplitude of the perturbation such that the disk −2 5 −2 5 αv = 10 x and αv = 10 (1 − x) . These models cor- moves back to a state where these velocities are smaller respond to a disk where respectively the inner and outer than cs. When shocks occur, the tidal torque exerted on parts are nonviscous. the disk may become larger than the horizontal viscous Figure 7.a shows the imaginary part ofg ˜ versus r for stress. Ωp = 0 corresponding to these viscosity laws. The uni- We have found that, in protostellar disks, bending waves form αv case is also plotted for comparison. The real part are able to propagate and transport a significant fraction of ofg ˜ is not shown since it does not depend on αv for these the negative angular momentum they carry in the disk in- small values of the viscosity (see discussion above). We ner parts. This is due to their relatively large wavelength. see that when the outer parts of the disk are nonviscous, Therefore, tidal interactions in noncoplanar systems may Im(˜g) in the inner parts is similar to the uniform αv case, not be confined to the regions close to the disk outer edge while it has an almost constant value in the outer parts. where the waves are excited. For the disk models we have When the inner parts of the disk are nonviscous, Im(˜g) set up, which extends down to the stellar surface, the value is almost zero there, while it is significant in the outer of αv above which the waves get dissipated before reaching parts. This behavior can be understood by remembering the disk inner boundary varies between 5 × 10−3 and 10−2 that Im[˜g(r)] is the integral from rin to r of a function for a disk aspect ratio between 0.05 and 0.1 (this critical proportional to αv (see equation [26]). Since Im(˜g) varies value of αv increases with the disk semi–thickness). We ′ less through the disk than in the uniform αv case,v ˜r given note that if we had set up an annulus rather than a disk, by equation (14) is smaller. This, together with a smaller the surface mass density would drop at the inner bound- αv, leads to a smaller net torque given by equation (59). ary. Then, for small αv, the waves would have a tendency When Ωp 6= 0, there is not such a difference between the to become nonlinear and then to dissipate before reaching uniform and nonuniform αv cases. The torque is smaller the inner boundary. Thus, the behavior of bending waves in the nonuniform case, but Im(˜g) does not change signif- could be similar in an annulus with small αv and in a disk icantly from one case to the other. The reason is probably with larger αv. In the limit of small viscosity, dissipation that Im(˜g) does not vary monotonically with αv, as shown could also occur as a result of parametric instabilities (Pa- in Figure 2. paloizou & Terquem 1995; C.F. Gammie, J.J. Goodman & Figure 7.b shows Tz(r)−J(r) versus r for the same mod- G.I. Ogilvie 1998, private communication), the effect of els and Ωp = 0. We observe that, as expected, no angular which may be to lead to a larger effective αv. However, momentum is deposited at radii where αv is very small since the waves we consider here propagate in a nonuni- 5 5 (small radii for αv ∝ x and large radii for αv ∝ (1 − x) ). form medium, it is not clear whether they would be efficiently damped by these instabilities. 8. It the disk model allows at least partial reflection from DISCUSSION the center, the tidal interaction becomes resonant when In this paper, we have calculated the response of a vis- the frequency of the tidal waves is equal to that of some cous gaseous disk to tidal perturbations with azimuthal free normal global bending mode of the disk. For the equi- mode number m = 1 and odd symmetry in z with respect librium disk models we have considered, we have found to the equatorial plane for both zero and finite perturbing that a particularly strong resonance occurs when the sep- frequencies. We concentrated on these types of perturba- aration is about 8.5 times the disk radius. The torque tions because they arise for inclined disks and they lead to 14 TERQUEM

−2 associated with the finite frequency terms then increases αv is larger than but close to 10 , bending waves are still by many orders of magnitude. However the response may able to propagate throughout most of the disk. In addi- be limited by shocks and nonlinear effects. The strength of tion, given that protostellar disks are rather thick, these the resonance is inversely proportional to αv. In our calcu- waves propagate fast (they cross the disk on a time com- lations, the range of separation for which the torque is sig- parable to the sound crossing time). In that case, it may nificantly increased is rather large, which means that the be possible to observe some time–dependent phenomena effects of tidal interaction, like disk truncation, may occur with a frequency equal to twice the orbital frequency. even when the separation of the binary is large. In addition In the case of protostellar disks, comparison between to this strong resonance we have found a few other reso- observations and theory is now becoming possible. A non- nances with different strengths and widths. From these coplanar binary system, HK Tau, has been observed for calculations, it appears that the probability for the inter- the first time very recently (see § 1.1), and subsequent action to be resonant may be significant. However, these work will be devoted to interpreting these observations. results depend on the particular equilibrium disk models In the meantime, we can comment briefly on some obser- we have set up. vations related to X–ray binaries. We note that the disk is expected to be truncated such There is evidence from the light curve of X–ray binaries, that the inner Lindblad (2:1) resonance with the compan- such as Hercules X–1 and SS 433, that their associated ac- ion, which could provide effective wave excitation, is more cretion disks may be in a state of precession in the tidal or less excluded from the disk. field of the binary companion. Katz (1980a, 1980b) has in- Out of resonance, we find that the torque associated dicated that the observed precession periods of these two with the zero–frequency perturbing term is much larger systems are consistent with the precession being induced than that associated with the finite frequency terms. Of by the tidal field of the secondary. course, if the separation of the system is large enough In the case of SS 433, it is interesting to note that an (at least 10 times the disk radius in our models), the dif- additional “nodding” motion with a period half the or- ference between finite frequency and secular terms disap- bital period is observed (see Margon 1984 and references pears. When the secular response is dominant, the tidal therein). Katz et al. (1982) suggested that this nodding torque is proportional to αv in the limit αv ≪ 1. This has motion is produced by the gravitational torque exerted by the consequence that the ratio of the tidal timescale td the companion. In their model, the nodding is viscously (time that would be required for the tidal effects we have transmitted from the outside, where the torque is signifi- considered to remove the angular momentum content of cant, to the interior, where the jets are seen to respond to the disk) to the disk viscous timescale tν,h is proportional the motion. This implies an extremely large viscosity in to αh/αv. the disk. What observable effects would these tidal interactions Here we point out that the observed period of this mo- produce in protostellar disks ? First, as we mentioned tion is consistent with the nodding being produced by in the introduction, they would lead to the precession bending waves. As these waves propagate with the sound of the jets that originate from these disks. The preces- speed and the disk is observed to be very thick (Mar- sion timescale that we can infer from the observation of gon 1984), they could transmit the motion through the jets that are modeled as precessing is consistent with this disk on a timescale comparable to its dynamical timescale. motion being induced by tidal effects in binary systems For the waves to reach the disk interior, αv would have to (Terquem et al. 1998). be smaller than H/r. It is not clear that this condition Also, as we have already pointed out in § 5.5, the sec- is satisfied here. The disk viscosity would then probably ular perturbation produces a tilt the variation of which have to be smaller than that predicted by the model of 2 2 can be up to αv along the line of nodes and H /r along Katz et al. (1982). the direction perpendicular to the line of nodes. Super- We finally observe that a rapid communication time posed on this steady (in the precessing frame) tilt, there through the disk also makes problems with the radiation– is a tilt produced by the finite frequency perturbations, driven warping model proposed by Maloney & Begel- the variation of which across the disk can be up to H/r. man (1997) to explain the precession of the disk in SS 433. Such asymmetries in the outer parts of protostellar disks In this model, the communication between the different could be observed. Because the variations along the line of parts of the disk, which is assumed to occur because of nodes and along the perpendicular to the line of nodes de- viscosity alone, is on a timescale that is characteristic for pend on αv and H/r, respectively, observations of warped mass to flow through the disk. It is therefore very slow protostellar disks have the potential to give important in- since the viscous timescale must be long for the warping formation about the physics of these disks. instability to occur (Pringle 1996). Protostellar disks are believed to be rather thick, i.e. H/r ∼ 0.1. Our calculations show that for such an aspect ratio, the viscosity above which bending waves are damped It is a pleasure to thank John Papaloizou for his ad- −2 before reaching the central star is αv ∼ 10 . Observa- vice, encouragement, and many valuable suggestions and tions seem to indicate that αh in protostellar disks is on discussions. I acknowledge Steve Balbus, Doug Lin, Jim −3 −2 the order of 10 –10 . If αv is smaller than or compa- Pringle and Michel Tagger for very useful comments on an rable to 10−2, resonances may occur, as described above, earlier draft of this paper, and John Larwood for point- providing the disk inner edge allows some reflection of the ing out the “nodding” motion in SS 433. I also thank waves. In that case, we could observe truncated disks with an anonymous referee whose comments helped to improve sharp edges even when the binary separation is large. If the quality of this paper. This work was supported by the Center for Star Formation Studies at NASA/Ames Re- ACCRETION DISKS AND BENDING WAVES 15 search Center and the University of California at Berkeley tute at Cambridge University for support during the final and Santa Cruz. I am grateful to the Isaac Newton Insti- stages of this work.

APPENDIX We describe here in more detail the resonance that occurs at D ∼ 8.5, since it is particularly strong. For Ωp =2ω and −2ω, resonance occurs at D = 8.25 and D = 8.425 respectively. We observe that, for the values of αv considered here, the range of D for which the torque is signiﬁcantly increased is rather large. We see that the strength of the resonance is −1 proportional to αv . This is expected since the strength of the resonance is inversely proportional to the damping factor, which itself is proportional to αv when dissipation is due to viscosity. We also note that the integral of the torque over the range of frequencies in this resonance is independent of αv. This can be understood in the following way. For Ωp = 2ω (the same argument would apply for Ωp = −2ω), the torque at or close to resonance can be written

−Aαv Tz = , (1) 2 2 4(¯ω − ω¯0) + αv whereω ¯ = ω/Ω(R), ω0 is the resonant frequency and A is an amplitude which does not depend on αv. We have calculated −11 that A ≃ 8 × 10 . For the values of αv considered, we have checked that the integral of the torque fromω ¯0 − ∆¯ω to ω¯0 +∆¯ω does not depend very much on whether 2∆¯ω is taken to be the range of frequencies in the resonance or for which expression (1) is valid (the latter being smaller). We then approximate the integral I of the torque over the resonance by using (1), so that 2∆¯ω I = −A Arctan . (2) αv −3 −4 For αv = 10 or 10 , the range of frequencies 2∆¯ω for which (1) is valid is large compared with αv, so that I ≃ −Aπ/2, −2 independent of αv. For αv = 10 , although 2∆¯ω is only a few times αv, this relation is still valid within a factor less than 2 (due to the rapid convergence of the function Arctan).

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Table 1: Torque and Corresponding Tidal Evolution Timescale.

H Label D αv Tz ωp td/tν,h r max 2ω −2ω 0 (10−2) 1a 3 10−4 0.1 −4(3) × 10−12 a −2(0.8) × 10−12 −1(1) × 10−9 b -1.1 5.9 1b – 10−3 – −4(3) × 10−11 a −2(0.7) × 10−11 −1(1) × 10−8 b – – 1c – 10−2 – −3(2) × 10−10 a −1(0.5) × 10−10 −1(1) × 10−7 c – – 1d – 0.1 – −9(8) × 10−10 a −4(2) × 10−10 −1(1) × 10−6 c – –

2a 6 10−3 0.05 −2(2) × 10−11 −3(2) × 10−12 −8(6) × 10−10 -0.14 22.9 2b – 5 × 10−3 – −1(0.7) × 10−10 −1(0.8) × 10−11 −4(3) × 10−9 – 23.1 2c – 10−2 – −2(1) × 10−10 −2(1) × 10−11 −8(6) × 10−9 – 23.2 2d – 5 × 10−2 – −3(2) × 10−10 −3(2) × 10−11 −4(3) × 10−8 – 23.5 2e – 0.1 – −3(2) × 10−10 −3(2) × 10−11 −8(7) × 10−8 – 23.6

3a 4 10−3 0.1 −5(4) × 10−11 −2(1) × 10−11 −2(2) × 10−9 a -0.47 32.3 3b 5 – – −4(3) × 10−11 −8(5) × 10−12 −6(5) × 10−10 a -0.24 118.3 3c 6 – – −7(5) × 10−11 −1(0.8) × 10−11 −2(2) × 10−10 a -0.14 274.9 3d 7 – – −2(1) × 10−10 −1(0.8) × 10−11 −8(6) × 10−11 a -0.087 287.8 3e 8 – – −4(3) × 10−9 −9(6) × 10−11 −4(3) × 10−11 a -0.058 21.9 3f 8.25 – – −7(6) × 10−8 −4(3) × 10−10 −3(2) × 10−11 a -0.053 1.1 3g 8.425 – – −7(5) × 10−9 −4(3) × 10−9 −3(2) × 10−11 a -0.050 7.8 3h 9 – – −3(3) × 10−10 −4(3) × 10−11 −2(1) × 10−11 a -0.041 201.0 3i 10 – – −5(4) × 10−11 −4(3) × 10−12 −1(0.7) × 10−11 a -0.030 1235.1

4a 4 10−2 – −5(4) × 10−10 −1(0.8) × 10−10 −2(2) × 10−8 -0.47 32.4 4b 5 – – −4(3) × 10−10 −7(5) × 10−11 −6(5) × 10−9 -0.24 118.7 4c 6 – – −6(5) × 10−10 −9(6) × 10−11 −2(2) × 10−9 -0.14 280.4 4d 7 – – −2(1) × 10−9 −1(0.7) × 10−10 −8(6) × 10−10 -0.087 315.8 4e 8 – – −7(5) × 10−9 −3(2) × 10−10 −4(3) × 10−10 -0.058 110.8 4f 8.25 – – −7(6) × 10−9 −4(3) × 10−10 −3(2) × 10−10 -0.053 101.2 4g 8.425 – – −6(5) × 10−9 −4(3) × 10−10 −3(2) × 10−10 -0.050 117.1 4h 9 – – −2(2) × 10−9 −2(1) × 10−10 −2(1) × 10−10 -0.041 323.9 4i 10 – – −5(3) × 10−10 −3(2) × 10−11 −1(0.7) × 10−10 -0.030 1368.7

Note.—Listed are the separation of the system D, the viscous parameter αv, the maximum value of H/r, the torque Tz associated with each of the perturbed frequencies 2ω, −2ω and 0, the disk precession frequency ωp and the ratio of the tidal timescale td to the viscous timescale tν,h. aCases where,at the outer edge of the disk, g varies on a scale smaller than H. bCases where,at the outer edge of the disk, the radial spherical velocity is larger than the sound speed. cCases where,at the outer edge of the disk, g varies on a scale smaller than H and the radial spherical velocity is larger than the sound speed. ACCRETION DISKS AND BENDING WAVES 17

.003

.08

.002 H(r)/r .06

.001

.04

0 0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1

.05 1

.04 .8

.03

.6 .02

.01 .4

0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1 r r

Fig. 1.— Surface mass density (top left), relative semithickness (top right), ratio of the sound speed to the Keplerian frequency (bottom left) and ratio of the epicyclic frequency to the Keplerian frequency (bottom right) vs. r for the −2 equilibrium disk models. The parameters are n =1.5,M = 10 Mp and (H/r)max =0.1. 18 TERQUEM

.002

a) Real part of g

-.002 Re(g) -.004

-.006

-.008 0 .2 .4 .6 .8 1 r

b) Imaginary part of g

.0005 Im(g)

-.0005

0 .2 .4 .6 .8 1 r

Fig. 2.— Real (panel a) and imaginary (panel b) parts ofg ˜ vs. r for models 2 and Ωp = 2ω. The diﬀerent curves −2 −2 −3 correspond to αv = 0.1 (solid line), 5 × 10 (dotted line), 10 (short dashed line), 5 × 10 (long dashed line) and −3 10 (dot–short dashed line). The parameters are D = 6 and (H/r)max = 0.05. The wavelike character of the response disappears when αv ≥ H/r. ACCRETION DISKS AND BENDING WAVES 19

3 4 5 6 7 8 9 10 11 D

Fig. 3.— Net torque |Tz| vs. D in a semilog representation for (H/r)max = 0.1. The panels a and b correspond to −3 −2 αv = 10 (models 1b and 3) and αv = 10 (models 1c and 4) respectively. The diﬀerent curves correspond to Ωp =2ω (solid line), −2ω (dotted line) and 0 (dashed line). These plots show several resonances. 20 TERQUEM

.002

-.002 Re(g)

-.004 a) Real part of g

-.006

0 .2 .4 .6 .8 1 r

-.05 Im(g)

-.1

b) Imaginary part of g

-.15 0 .2 .4 .6 .8 1 r

Fig. 4.— Real (panel a) and imaginary (panel b) parts ofg ˜ vs. r for models 2 and Ωp = 0. The diﬀerent curves correspond −2 −2 −3 −3 to αv =0.1 (solid line), 5 × 10 (dotted line), 10 (short dashed line), 5 × 10 (long dashed line) and 10 (dot–short dashed line). The parameters are D = 6 and (H/r)max =0.05. The zero–frequency response is not wave–like. ACCRETION DISKS AND BENDING WAVES 21

-.2

-.4

-.6

-.8

-1

0 .2 .4 .6 .8 1 r

−3 Fig. 5.— Tz(r) − J(r) (angular momentum dissipated between rin and r) normalized to unity vs. r for α = 10 , D =3 (model 1b) and Ωp = 0 (solid line), 2ω (dotted line) and −2ω (dashed line). 22 TERQUEM

-.2

-.4

-.6

-.8

-1

0 .2 .4 .6 .8 1 r

-.2

0.1 -.4

-.6

-.8

-1

0 .2 .4 .6 .8 1 r

Fig. 6.— Tz(r) − J(r) (angular momentum dissipated between rin and r) normalized to unity vs. r for Ωp = −2ω. The −4 −3 −2 panel (a) corresponds to models 1, i.e. D =3, (H/r)max =0.1 and α = 10 (solid line), 10 (dotted line), 10 (short −3 dashed line) and 0.1 (long dashed line). The panel (b) corresponds to models 2, i.e. D =6,(H/r)max =0.05 and α = 10 (solid line), 5 × 10−3 (dotted line), 10−2 (short dashed line), 5 × 10−2 (long dashed line) and 0.1 (dot–short dashed line). ACCRETION DISKS AND BENDING WAVES 23

-.01 Im(g)

-.02

a) Imaginary part of g

-.03 0 .2 .4 .6 .8 1 r

-.2

-.4

-.6

-.8

-1

0 .2 .4 .6 .8 1 r

Fig. 7.— Imaginary part ofg ˜ (panel a) and Tz(r) − J(r) (angular momentum dissipated between rin and r) normalized −2 to unity (panel b) vs. r for Ωp = 0 and a nonuniform α. The diﬀerent curves correspond to α = 10 (solid line), −2 5 −2 5 10 (1 − x) (dotted line) and 10 x (dashed line). The parameters are D = 3 and (H/r)max =0.1.