Effects of Disk Warping on the Inclination Evolution of Star-Disk-Binary Systems
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 8 October 2018 (MN L A TEX style file v2.2)
Effects of Disk Warping on the Inclination Evolution ofStar-Disk-Binary Systems
J. J. Zanazzi (cid:63) , and Dong Lai Cornell Center for Astrophysics and Planetary Science, Department of Astronomy, Cornell University, Ithaca, NY 14853, USA
ABSTRACT
Several recent studies have suggested that circumstellar disks in young stellar binariesmay be driven into misalignement with their host stars due to secular gravitationalinteractions between the star, disk and the binary companion. The disk in such systemsis twisted/warped due to the gravitational torques from the oblate central star and theexternal companion. We calculate the disk warp profile, taking into account of bendingwave propagation and viscosity in the disk. We show that for typical protostellar diskparameters, the disk warp is small, thereby justifying the “flat-disk” approximationadopted in previous theoretical studies. However, the viscous dissipation associatedwith the small disk warp/twist tends to drive the disk toward alignment with thebinary or the central star. We calculate the relevant timescales for the alignment. Wefind the alignment is effective for sufficiently cold disks with strong external torques,especially for systems with rapidly rotating stars, but is ineffective for the majority ofstar-disk-binary systems. Viscous warp driven alignment may be necessary to accountfor the observed spin-orbit alignment in multi-planet systems if these systems areaccompanied by an inclined binary companion.
Key words: hydrodynamics - planets and satellites: formation - protoplanetary discs- stars: binaries: general
Circumstellar disks in young protostellar binary systemsare likely to form with an inclined orientation relativeto the binary orbital plane, as a result of the complexstar/binary/disc formation processes (e.g. Bate, Bonnell &Bromm 2003; McKee & Ostriker 2007; Klessen 2011). In-deed, many misaligned circumstellar disks in protostellar bi-naries have been found in recent years (e.g. Stapelfeldt et al.1998, 2003; Neuh¨auser et al. 2009; Jensen & Akeson 2014;Williams et al. 2014; Brinch et al. 2016; Fern´andez-L´opez,Zapata & Gabbasov 2017; Lee et al. 2017). Such misaligneddisks experience differential gravitational torques from thebinary companion, and are expected to be twisted/warpedwhile undergoing damped precession around the binary(e.g. Lubow & Ogilvie 2000; Bate et al. 2000; Foucart &Lai 2014). On the other hand, a spinning protostar has arotation-induced quadrupole, and thus exerts a torque onthe disk (and also receives a back-reaction torque) when thestellar spin axis and the disk axis are misaligned. This torquetends to induce warping in the inner disk and drives mutualprecession between the stellar spin and disk. In the presenceof both torques on the disk, from the binary and from the (cid:63)
Email: [email protected] central star, how does the disk warp and precess? What isthe long-term evolution of the disk and stellar spin in suchstar-disk-binary systems? These are the questions we intendto address in this paper.Several recent studies have examined the secular dy-namics of the stellar spin and circumstellar disk in the pres-ence of an inclined binary companion (Batygin 2012; Baty-gin & Adams 2013; Lai 2014a; Spalding & Batygin 2014,2015). These studies were motivated by the observationsof spin-orbit misalignments in exoplanetary systems con-taining hot Jupiters, i.e., the planet’s orbital plane is oftenmisaligned with the stellar rotational equator (see Winn &Fabrycky 2015 and Triaud 2017 for recent reviews). It wasshown that significant “primordial” misalignments may begenerated while the planetary systems are still forming intheir natal protoplanetary disks through secular star-disk-binary gravitational interactions (Batygin & Adams 2013;Lai 2014a; Spalding & Batygin 2014, 2015). In these studies,various assumptions were made about the star-disk inter-actions, and uncertain physical processes such as star/diskwinds, magnetic star-disk interactions, and accretion of diskangular momentum onto the star were incorporated in a pa-rameterized manner. Nevertheless, the production of spin-orbit misalignments seems quite robust.In Zanazzi & Lai (2018b), we showed that the formation c (cid:13) a r X i v : . [ a s t r o - ph . E P ] A p r J. J. Zanazzi and Dong Lai of hot Jupiters in the protoplanetary disks can significantlysuppress the excitation of spin-orbit misalignment in star-disk-binary systems. This is because the presence of suchclose-in giant planets lead to strong spin-orbit coupling be-tween the planet and its host star, so that the spin-orbitmisalignment angle is adiabatically maintained despite thegravitational perturbation from the binary companion. How-ever, the formation of small planets or distant planets (e.g.warm Jupiters) do not affect the generation of primordialmisalignments between the host star and the disk.A key assumption made in all previous studies on mis-alignments in star-disk-binary systems (Batygin & Adams2013; Lai 2014a; Spalding & Batygin 2015) is that the diskis nearly flat and behaves like a rigid plate in response to theexternal torques from the binary and from the host star. Therationale for this assumption is that different regions of thedisk can efficiently communicate with each other through hy-drodynamical forces and/or self-gravity, such that the diskstays nearly flat. However, to what extent this assumption isvalid is uncertain, especially because in the star-disk-binarysystem the disk experiences two distinct torques from theoblate star and from the binary which tend to drive the disktoward different orientations (see Tremaine & Davis 2014 forexamples of non-trivial disk warps when a disk is torquedby different forces). Moreover, the combined effects of diskwarps/twists (even if small) and viscosity can lead to non-trivial long-term evolution of the star-disk-binary system.Previous works on warped disks in the bending wave regimehave considered a single external torque, such as an ext bi-nary companion (Lubow & Ogilvie 2000; Bate et al. 2000;Foucart & Lai 2014), an inner binary (Facchini, Lodato, &Price 2013; Lodato & Facchini 2013; Foucart & Lai 2014;Zanazzi & Lai 2018a), magnetic torques from the centralstar (Foucart & Lai 2011), a central spinning black hole(Demianski & Ivanov 1997; Lubow, Ogilvie, & Pringle 2002;Franchini, Lodato, & Facchini 2016; Chakraborty & Bhat-tacharyya 2017), and a system of multiple planets on nearlycoplanar orbits (Lubow & Ogilvie 2001). In this paper, wewill focus on the hydrodynamics of warped disks in star-disk-binary systems, and will present analytical calculationsfor the warp amplitudes/profiles and the rate of evolution ofdisk inclinations due to viscous dissipation associated withthese warps/twists.This paper is organized as follows. Section 2 describesthe setup and parameters of the star-disk-binary system westudy. Section 3 presents all the technical calculations of ourpaper, including the disk warp/twist profile and effect of vis-cous dissipation on the evolution of system. Section 4 exam-ines how viscous dissipation from disk warps modifies thelong-term evolution of star-disk-binary systems. Section 5discusses theoretical uncertainties of our work. Section 6contains our conclusions.
Consider a central star of mass M (cid:63) , radius R (cid:63) , rotationrate Ω (cid:63) , with a circumstellar disk of mass M d , and innerand outer truncation radii of r in and r out , respectively. Thisstar-disk system is in orbit with a distant binary companionof mass M b and semimajor axis a b . The binary compan- ion exerts a torque on the disk, driving it into differentialprecession around the binary angular momentum axis ˆ l b .Averaging over the orbital period of the disk annulus andbinary, the torque per unit mass is T db = − r Ω ω db ( ˆ l · ˆ l b ) ˆ l b × ˆ l , (1)where Ω( r ) (cid:39) (cid:112) GM (cid:63) /r is the disk angular frequency, ˆ l = ˆ l ( r, t ) is the unit orbital angular momentum axis of a disk“ring” at radius r , and ω db ( r ) = 3 GM (cid:63) a Ω (2)is the characteristic precession frequency of the disk“ring” at radius r . Similarly, the rotation-induced stellarquadrapole drives the stellar spin axis ˆ s and the disk ontomutual precession. The stellar rotation leads to a differencein the principal components of the star’s moment of inertiaof I − I = k q M (cid:63) R (cid:63) ¯Ω (cid:63) , where k q (cid:39) . T ds ( r, t ) = − r Ω ω ds ( ˆ s · ˆ l ) ˆ s × ˆ l , (3)where ω ds ( r ) = 3 G ( I − I )2 r Ω = 3 Gk q M (cid:63) R (cid:63) ¯Ω (cid:63) r Ω (4)is the characteristic precession frequency of the disk ring atradius r . Since ω db and ω ds both depend on r , the disk wouldquickly lose coherence if there were no internal coupling be-tween the different “rings.”We introduce the following rescaled parameters typicalof protostellar systems:¯ M (cid:63) = M (cid:63) (cid:12) , ¯ R (cid:63) = R (cid:63) (cid:12) , ¯Ω (cid:63) = Ω (cid:63) (cid:112) GM (cid:63) /R (cid:63) , ¯ M d = M d . (cid:12) , ¯ r in = r in (cid:12) , ¯ r out = r out
50 au , ¯ M b = M b (cid:12) , ¯ a b = a b
300 au . (5)The rotation periods of T Tauri stars vary from P (cid:63) ∼ −
10 days (Bouvier 2013), corresponding to ¯Ω (cid:63) ∼ . − . (cid:63) to be 0 .
1, corresponding toa stellar rotation period of P (cid:63) = 3 . r in is motivated by typical values of a T Tauri star’s mag-netospheric radius r m , set by the balance of magnetic andplasma stresses (see Lai 2014b for review) r in ≈ r m = η (cid:18) µ (cid:63) GM (cid:63) ˙ M (cid:19) / = 7 . η (cid:18) B (cid:63) (cid:19) / (cid:18) − M (cid:12) / yr˙ M (cid:19) / ¯ R / (cid:63) ¯ M / (cid:63) R (cid:12) . (6)Here, µ (cid:63) = B (cid:63) R (cid:63) is the stellar dipole moment, B (cid:63) is thestellar magnetic field, ˙ M is the accretion rate onto the cen-tral star (e.g. Rafikov 2017), and η is a parameter of orderunity. We note that we take the stellar radius to be fixed,in contrast to the models of Batygin & Adams (2013) andSpalding & Batygin (2014, 2015), but we argue this will notchange our results significantly. We are primarily concerned c (cid:13) , 000–000 isk Warp in Star-Disk-Binary Systems with the effects of viscous dissipation from disk warping, anda changing stellar radius will not affect the viscous torquecalculations to follow.We parameterize the disk surface density Σ = Σ( r, t ) asΣ( r, t ) = Σ out ( t ) (cid:16) r out r (cid:17) p . (7)We take p = 1 unless otherwise noted. The disk mass M d isthen (assuming r in (cid:28) r out ) M d = (cid:90) r out r in π Σ r d r (cid:39) π Σ out r − p . (8)The disk angular momentum vector is L d = L d ˆ l d (assum-ing a small disk warp), and stellar spin angular momentumvector is S = S ˆ s , where ˆ l d and ˆ s are unit vectors, and L d = (cid:90) r out r in π Σ r Ωd r (cid:39) − p / − p M d √ GM (cid:63) r out , (9) S = k (cid:63) M (cid:63) R (cid:63) Ω (cid:63) . (10)Here k (cid:63) (cid:39) . L b = L b ˆ l b . Because typical star-disk-binary systems satisfy L b (cid:29) L d , S , we take ˆ l b to be fixed for the remainder of thiswork. When α (cid:46) H/r ( α is the Shakura-Sunyaev viscosity param-eter, H is the disk scaleheight), which is satisfied for pro-tostellar disks (e.g. Rafikov 2017), the main internal torqueenforcing disk rigidity and coherent precession comes frombending wave propagation (Papaloizou & Lin 1995; Lubow& Ogilvie 2000). As bending waves travel at 1/2 of the soundspeed, the wave crossing time is of order t bw = 2( r/H )Ω − .When t bw is longer than the characteristic precession times ω − or ω − from an external torque, significant disk warpscan be induced. In the extreme nonlinear regime, disk break-ing may be possible (Larwood et al. 1996; Do˘gan et al. 2015).To compare t bw with ω − and ω − , we adopt the disk soundspeed profile c s ( r ) = H ( r )Ω( r ) = h out (cid:114) GM (cid:63) r out (cid:16) r out r (cid:17) q = h in (cid:114) GM (cid:63) r in (cid:16) r in r (cid:17) q , (11)where h in = H ( r in ) /r in and h out = H ( r out ) /r out . Passivelyheated disks have q ≈ . − . q ≈ / t bw ω ds = 4 . × − (cid:18) . h in (cid:19) (cid:18) k q . (cid:19) ¯ R (cid:63) ¯ r (cid:18) rr in (cid:19) q − / , (12) t bw ω db = 1 . × − (cid:18) . h out (cid:19) ¯ M b ¯ r ¯ M (cid:63) ¯ a (cid:18) rr out (cid:19) q +3 / . (13)Thus, we expect the small warp approximation to be valideverywhere in the disk. This expectation is confirmed byour detailed calculation of disk warps presented later in thissection.Although the disk is flat to a good approximation, the interplay between the disk warp/twist and viscous dissipa-tion can lead to appreciable damping of the misalignmentbetween the disk and the external perturber (i.e., the oblatestar or the binary companion). In particular, when an ex-ternal torque T ext (per unit mass) is applied to a disk in thebending wave regime (which could be either T db or T ds ), thedisk’s viscosity causes the disk normal to develop a smalltwist, of order ∂ ˆ l ∂ ln r ∼ αc T ext . (14)The detailed derivation of Eq. (14) is contained in Sec-tions 3.1-3.3. Since T ext ∝ ˆ l ext × ˆ l ( ˆ l ext is the axis aroundwhich ˆ l precesses), where the viscous twist interacts withthe external torque, effecting the evolution of ˆ l over viscoustimescales. To an order of magnitude, we have (cid:12)(cid:12)(cid:12)(cid:12) d ˆ l d t (cid:12)(cid:12)(cid:12)(cid:12) visc ∼ (cid:28)(cid:18) αc (cid:19) T r Ω (cid:29) ∼ (cid:28) αc ( r Ω) ω (cid:29) , (15)where ω ext is either ω ds or ω db , and (cid:104)· · · (cid:105) implies properaverage over r .We now study the disk warp and viscous evolutionquantitatively, using the formalism describing the structureand evolution of circular, weakly warped disks in the bend-ing wave regime. The relevant equations have been derivedby a number of authors (Papaloizou & Lin 1995; Demianski& Ivanov 1997; Lubow & Ogilvie 2000). We choose the for-malism of Lubow & Ogilvie (2000) and Lubow, Ogilvie, &Pringle (2002) (see also Ogilvie 2006 when | ∂ ˆ l /∂ ln r | ∼ r Ω ∂ ˆ l ∂t = Σ T ext + 1 r ∂ G ∂r , (16) ∂ G ∂t = (cid:18) Ω − κ (cid:19) ˆ l × G − α Ω G + Σ c r Ω4 ∂ ˆ l ∂r , (17)where T ext is the external torque per unit mass acting on thedisk, κ = (2Ω /r ) ∂ ( r Ω) /∂r | z =0 is the epicyclic frequency,and G is the internal torque, which arises from slightlyeccentric fluid particles with velocities sheared around thedisk mid-plane (Demianski & Ivanov 1997). Eq. (16) is the2D momentum equation generalized to non-coplanar disks.Eq. (17) is related to how internal torques generated fromdisk warps are communicated across the disk under the influ-ence of viscosity and precession from non-Keplarian epicyclicfrequencies. See Nixon & King (2016) for a qualitative dis-cussion and review of Eqs. (16)-(17).We are concerned with two external torques acting ondifferent regions of the disk. For clarity, we break up ourcalculations into three subsections, considering disk warpsproduced by individual torques before examining the com-bined effects. The torque from an external binary companion is given byEq. (1). The companion also gives rise to a non-Keplarianepicyclic frequency, given byΩ − κ
2Ω = ω db P ( ˆ l · ˆ l b ) , (18)where P l are Legendre polynomials.To make analytic progress, we take advantage of our c (cid:13)000
2Ω = ω db P ( ˆ l · ˆ l b ) , (18)where P l are Legendre polynomials.To make analytic progress, we take advantage of our c (cid:13)000 , 000–000 J. J. Zanazzi and Dong Lai − Radius (au) − . − . − . − . . . . . ˜ τ b − Radius (au) − . − . − . − . − . . . . ˜ V b − Radius (au) − . − . . . . . . ˜ W bb Figure 1.
The rescaled radial functions [see Eq. (34) for rescaling] ˜ τ b [Eq. (30)], ˜ V b [Eq. (31)], and ˜ W bb [Eq. (32)]. We take ( p, q ) values[Eq. (7) and (11)] of p = 0 . p = 1 . p = 1 . q = 0 . q = 0 . V b ) and warp (˜ τ b , ˜ W bb ) profiles of thedisk due to the gravitational torque from the binary companion. p q U b V b W bb Table 1.
Dimensionless coefficients U b [Eq. (35)], V b [Eq. (36)],and W bb [Eq. (37)] tabulated for various p and q values [Eqs. (7)and (11)]. All the parameter values are canonical [Eq. (5)]. Whenvarying q , we fix h out = 0 . expectation that | ∂ ˆ l /∂ ln r | (cid:28) t bw ω db (cid:28) ˆ l ( r, t ) = ˆ l d ( t ) + l ( r, t ) + . . . , (19) G ( r, t ) = G ( r, t ) + G ( r, t ) + . . . , (20)where | l | (cid:28) | ˆ l d | = 1. Here, G ( r, t ) is the internal torquemaintaining coplanarity of ˆ l d ( t ), G ( r, t ) is the internaltorque maintaining the leading order warp profile l ( r, t ),etc. To leading order, Eq. (16) becomesΣ r Ω d ˆ l d d t = − Σ r Ω ω db ( ˆ l d · ˆ l b ) ˆ l b × ˆ l d + 1 r ∂ G ∂r . (21)Integrating (21) over r d r , and using the boundary condition G ( r in , t ) = G ( r out , t ) = 0 , (22)we obtain d ˆ l d d t = − ˜ ω db ( ˆ l d · ˆ l b ) ˆ l b × ˆ l d , (23)where ˜ ω db is given by˜ ω db = 2 πL d (cid:90) r out r in ω db Σ r Ωd r (cid:39) / − p )4(4 − p ) (cid:18) M b M (cid:63) (cid:19) (cid:18) a b r out (cid:19) (cid:115) GM (cid:63) r . (24) The physical meaning of ˆ l d thus becomes clear: ˆ l d is the unittotal angular momentum vector of the disk, or ˆ l d ≡ πL d (cid:90) r out r in Σ r Ω ˆ l ( r, t )d r. (25)Using Eqs. (22) and (23), we may solve Eq. (21) for G ( r, t ): G ( r, t ) = g b ( r )( ˆ l d · ˆ l b ) ˆ l b × ˆ l d , (26)where g b ( r ) = (cid:90) rr in ( ω db − ˜ ω db )Σ r (cid:48) Ωd r (cid:48) . (27)Using Eqs. (26) and (17), and requiring that l not con-tribute to the total disk angular momentum vector, or (cid:90) r out r in Σ r Ω l ( r, t )d r = 0 , (28)we obtain the leading order warp l ( r, t ): l ( r, t ) = − ˜ ω db τ b ( ˆ l d · ˆ l b ) ˆ l b × ( ˆ l b × ˆ l d ) − W bb ( ˆ l d · ˆ l b ) P ( ˆ l d · ˆ l b ) ˆ l d × ( ˆ l b × ˆ l d )+ V b ( ˆ l d · ˆ l b ) ˆ l b × ˆ l d , (29)where τ b ( r ) = (cid:90) rr in g b Σ c r (cid:48) Ω d r (cid:48) − τ b0 , (30) V b ( r ) = (cid:90) rr in αg b Σ c r (cid:48) d r (cid:48) − V b0 , (31) W bb ( r ) = (cid:90) rr in ω db g b Σ c r (cid:48) Ω d r (cid:48) − W bb0 , (32)and the constants X of the functions X ( r ) (either τ b ( r ), V b ( r ), or W bb ( r )) are determined by requiring (cid:90) r out r in Σ r Ω X d r = 0 . (33)Notice the radial functions τ b , V b , and W bb trace out thedisk’s warp profile | l ( r ) | due to the binary companion’sgravitational torque. Because the magnitudes for the radialfunctions (2 π/ Myr) τ b , V b , and W bb are much smaller than c (cid:13) , 000–000 isk Warp in Star-Disk-Binary Systems p q U s V s W ss Table 2.
Dimensionless coefficients U s [Eq. (50)], V s [Eq. (51)],and W ss [Eq. (52)], for different values of p and q [Eqs. (7)and (11)]. All other parameter values are canonical [Eq. (5)].When varying q , we fix h out = 0 . unity everywhere [see Eqs. (35)-(37)], we define the re-scaledradial function ˜ X ( r ) = ˜ τ b , ˜ V b , and ˜ W bb as˜ X ( r ) ≡ X ( r ) (cid:46)(cid:2) X ( r out ) − X ( r in ) (cid:3) . (34)Figure 1 plots the dimensionless radial functions ˜ τ b , ˜ V b , and˜ W bb for the canonical parameters of the star-disk-binary sys-tem [Eq. (5)]. The scalings of the radial functions evaluatedat the outer disk radius are τ b ( r out ) − τ b ( r in ) = − . × − U b × (cid:18) . h out (cid:19) ¯ M b ¯ r / ¯ M / (cid:63) ¯ a Myr2 π , (35) V b ( r out ) − V b ( r in ) = − . × − V b × (cid:16) α . (cid:17) (cid:18) . h out (cid:19) ¯ M b ¯ r ¯ M (cid:63) ¯ a , (36) W bb ( r out ) − W bb ( r in ) = − . × − W bb × (cid:18) . h out (cid:19) ¯ M ¯ r ¯ M (cid:63) ¯ a . (37)Equations (35)-(37) provide an estimate for the mis-alignment angle between the disk’s inner and outer or-bital angular momentum vectors, or | X ( r out ) − X ( r in ) | ∼| ˆ l ( r out , t ) × ˆ l ( r in , t ) | , where X = (2 π/ Myr) τ b , V b , and W bb .The dimensionless coefficients U b , V b , and W bb dependweakly on the parameters p , q , and r in /r out . Table 1 tab-ulates U b , V b , and W bb for values of p and q as indicated,with the canonical value of r in /r out [Eq. (5)]. The torque on the disk from the oblate star is given byEq. (3). The stellar quadrupole moment also gives rise to anon-Keplarian epicyclic frequency given byΩ − κ
2Ω = ω ds P ( ˆ l · ˆ s ) . (38)Equations (16)-(17) are coupled with the motion of the hoststar’s spin axis: S d ˆ s d t = − (cid:90) r out r in (cid:104) π Σ r Ω ω ds ( ˆ s · ˆ l ) ˆ l × ˆ s (cid:105) d r, (39)Expanding ˆ l and G according to Eqs. (19) and (20), inte-grating Eq. (16) over r d r , and using the boundary condi-tion (22), we obtain the leading order evolution equationsd ˆ s d t = − ˜ ω sd ( ˆ s · ˆ l d ) ˆ l d × ˆ s , (40) d ˆ l d d t = − ˜ ω ds ( ˆ l d · ˆ s ) ˆ s × ˆ l d , (41)where (assuming r in (cid:28) r out )˜ ω ds = 2 πL d (cid:90) r out r in ω ds Σ r Ωd r (cid:39) / − p ) k q p ) R (cid:63) ¯Ω (cid:63) r − p out r p in (cid:115) GM (cid:63) r , (42)˜ ω sd = ( L d /S )˜ ω ds (cid:39) − p ) k q p ) k (cid:63) (cid:18) M d M (cid:63) (cid:19) ¯Ω (cid:63) (cid:112) GM (cid:63) R (cid:63) r − p out r p in . (43)With d ˆ l d / d t and d ˆ s / d t determined, Eq. (16) may be inte-grated to obtain the leading order internal torque: G ( r, t ) = g s ( r )( ˆ l d · ˆ s ) ˆ s × ˆ l d , (44)where g s ( r ) = (cid:90) rr in ( ω ds − ˜ ω ds )Σ r (cid:48) Ωd r (cid:48) . (45)Similarly, the leading order warp profile is l ( r, t ) = − ˜ ω sd τ s ( ˆ l d · ˆ s ) ( ˆ l d × ˆ s ) × ˆ l d − ˜ ω ds τ s ( ˆ l d · ˆ s ) ˆ s × ( ˆ s × ˆ l d ) − W ss ( ˆ l d · ˆ s ) P ( ˆ l d · ˆ s ) ˆ l d × ( ˆ s × ˆ l d )+ V s ( ˆ l d · ˆ s ) ˆ s × ˆ l d , (46)where τ s ( r ) = (cid:90) rr in g s Σ c r (cid:48) Ω d r (cid:48) − τ s0 , (47) V s ( r ) = (cid:90) rr in αg s Σ c r (cid:48) d r (cid:48) − V s0 , (48) W ss ( r ) = (cid:90) rr in ω ds g s Σ c r (cid:48) Ω d r (cid:48) − W ss0 . (49)In Figure 2, we plot the rescaled radial functions ˜ τ s , ˜ V s , and˜ W ss for various p and q values, tracing out the re-scaled warpprofile across the radial extent of the disk due to the oblatestar’s torque. The radial function differences evaluated atthe disk’s outer and inner truncation radii are τ s ( r out ) − τ s ( r in ) = 2 . × − U s (cid:18) . h out (cid:19) (cid:18) k q . (cid:19) × (cid:18) r out ¯ r in (cid:19) p − ¯ R (cid:63) ¯ r / ¯ r ¯ M / (cid:63) (cid:18) ¯Ω (cid:63) . (cid:19) Myr2 π , (50) V s ( r out ) − V s ( r in ) = 1 . × − V s × (cid:16) α . (cid:17) (cid:18) . h in (cid:19) (cid:18) k q . (cid:19) ¯ R (cid:63) ¯ r (cid:18) ¯Ω (cid:63) . (cid:19) , (51) W ss ( r out ) − W ss ( r in ) = 4 . × − W ss × (cid:18) k q . (cid:19) (cid:18) . h in (cid:19) ¯ R (cid:63) ¯ r (cid:18) ¯Ω (cid:63) . (cid:19) . (52)Equations (50)-(52) provide an estimate for the misalign-ment angle between the disk’s outer and inner orbital an-gular momentum unit vectors | ˆ l ( r out , t ) × ˆ l ( r in , t ) | due to theoblate star’s torque. The dimensionless coefficients U s , V s ,and W ss depend weakly on the parameters p , q , and r in /r out .In Table 2, we tabulate U s , V s , and W ss for the p and q valuesindicated, with r in /r out taking the canonical value [Eq. (5)]. c (cid:13)000
2Ω = ω ds P ( ˆ l · ˆ s ) . (38)Equations (16)-(17) are coupled with the motion of the hoststar’s spin axis: S d ˆ s d t = − (cid:90) r out r in (cid:104) π Σ r Ω ω ds ( ˆ s · ˆ l ) ˆ l × ˆ s (cid:105) d r, (39)Expanding ˆ l and G according to Eqs. (19) and (20), inte-grating Eq. (16) over r d r , and using the boundary condi-tion (22), we obtain the leading order evolution equationsd ˆ s d t = − ˜ ω sd ( ˆ s · ˆ l d ) ˆ l d × ˆ s , (40) d ˆ l d d t = − ˜ ω ds ( ˆ l d · ˆ s ) ˆ s × ˆ l d , (41)where (assuming r in (cid:28) r out )˜ ω ds = 2 πL d (cid:90) r out r in ω ds Σ r Ωd r (cid:39) / − p ) k q p ) R (cid:63) ¯Ω (cid:63) r − p out r p in (cid:115) GM (cid:63) r , (42)˜ ω sd = ( L d /S )˜ ω ds (cid:39) − p ) k q p ) k (cid:63) (cid:18) M d M (cid:63) (cid:19) ¯Ω (cid:63) (cid:112) GM (cid:63) R (cid:63) r − p out r p in . (43)With d ˆ l d / d t and d ˆ s / d t determined, Eq. (16) may be inte-grated to obtain the leading order internal torque: G ( r, t ) = g s ( r )( ˆ l d · ˆ s ) ˆ s × ˆ l d , (44)where g s ( r ) = (cid:90) rr in ( ω ds − ˜ ω ds )Σ r (cid:48) Ωd r (cid:48) . (45)Similarly, the leading order warp profile is l ( r, t ) = − ˜ ω sd τ s ( ˆ l d · ˆ s ) ( ˆ l d × ˆ s ) × ˆ l d − ˜ ω ds τ s ( ˆ l d · ˆ s ) ˆ s × ( ˆ s × ˆ l d ) − W ss ( ˆ l d · ˆ s ) P ( ˆ l d · ˆ s ) ˆ l d × ( ˆ s × ˆ l d )+ V s ( ˆ l d · ˆ s ) ˆ s × ˆ l d , (46)where τ s ( r ) = (cid:90) rr in g s Σ c r (cid:48) Ω d r (cid:48) − τ s0 , (47) V s ( r ) = (cid:90) rr in αg s Σ c r (cid:48) d r (cid:48) − V s0 , (48) W ss ( r ) = (cid:90) rr in ω ds g s Σ c r (cid:48) Ω d r (cid:48) − W ss0 . (49)In Figure 2, we plot the rescaled radial functions ˜ τ s , ˜ V s , and˜ W ss for various p and q values, tracing out the re-scaled warpprofile across the radial extent of the disk due to the oblatestar’s torque. The radial function differences evaluated atthe disk’s outer and inner truncation radii are τ s ( r out ) − τ s ( r in ) = 2 . × − U s (cid:18) . h out (cid:19) (cid:18) k q . (cid:19) × (cid:18) r out ¯ r in (cid:19) p − ¯ R (cid:63) ¯ r / ¯ r ¯ M / (cid:63) (cid:18) ¯Ω (cid:63) . (cid:19) Myr2 π , (50) V s ( r out ) − V s ( r in ) = 1 . × − V s × (cid:16) α . (cid:17) (cid:18) . h in (cid:19) (cid:18) k q . (cid:19) ¯ R (cid:63) ¯ r (cid:18) ¯Ω (cid:63) . (cid:19) , (51) W ss ( r out ) − W ss ( r in ) = 4 . × − W ss × (cid:18) k q . (cid:19) (cid:18) . h in (cid:19) ¯ R (cid:63) ¯ r (cid:18) ¯Ω (cid:63) . (cid:19) . (52)Equations (50)-(52) provide an estimate for the misalign-ment angle between the disk’s outer and inner orbital an-gular momentum unit vectors | ˆ l ( r out , t ) × ˆ l ( r in , t ) | due to theoblate star’s torque. The dimensionless coefficients U s , V s ,and W ss depend weakly on the parameters p , q , and r in /r out .In Table 2, we tabulate U s , V s , and W ss for the p and q valuesindicated, with r in /r out taking the canonical value [Eq. (5)]. c (cid:13)000 , 000–000 J. J. Zanazzi and Dong Lai − Radius (au) − . − . − . − . − . . . . ˜ τ s − Radius (au) − . − . − . − . − . . . ˜ V s − Radius (au) − . − . − . − . − . . ˜ W ss Figure 2.
The rescaled radial functions [see Eq. (34) for rescaling] ˜ τ s [Eq. (47)], ˜ V s [Eq. (48)], and ˜ W ss [Eq. (49)]. We take ( p, q ) values[Eq. (7) and (11)] of p = 0 . p = 1 . p = 1 . q = 0 . q = 0 . V s ) and warp (˜ τ s , ˜ W ss ) profiles of thedisk due to the gravitational torque from the oblate star. p q W bs W sb Table 3.
Dimensionless coefficients W bs [Eq. (56)] and W sb [Eq. (57)] for values of p and q as indicated [Eqs. (7) and (11)].All parameter values are canonical [Eq. (5)]. When varying q , wefix h out = 0 . The combined torques from the distant binary and oblatestar are given by Eqs. (1) and (3), and the non-Keplarianepicyclic frequencies are given by Eqs. (18) and (38). Usingthe same procedure as Sections 3.1-3.2, the leading ordercorrection to the disk’s warp is l ( r, t ) = ( l ) bin + ( l ) star − ˜ ω ds τ b ( ˆ l d · ˆ s ) (cid:104) ( ˆ s × ˆ l d ) · ˆ l b (cid:105) ˆ l b × ˆ l d − ˜ ω ds τ b ( ˆ l d · ˆ l b )( ˆ l d · ˆ s ) ˆ l b × ( ˆ s × ˆ l d ) − ˜ ω db τ s ( ˆ l d · ˆ l b ) (cid:104) ( ˆ l b × ˆ l d ) · ˆ s (cid:105) ˆ s × ˆ l d − ˜ ω db τ s ( ˆ l d · ˆ s )( ˆ l d · ˆ l b ) ˆ s × ( ˆ l b × ˆ l d ) − W sb ( ˆ l d · ˆ l b ) P ( ˆ l d · ˆ s ) ˆ l d × ( ˆ l b × ˆ l d ) − W bs ( ˆ l d · ˆ s ) P ( ˆ l d · ˆ l b ) ˆ l d × ( ˆ s × ˆ l d ) , (53)where ( l ) bin is Eq. (29), ( l ) star is Eq. (46), τ b and τ s aregiven in Eqs. (30) and (47), and W bs ( r ) = (cid:90) rr in ω db g s Σ c r (cid:48) Ω d r (cid:48) − W bs0 , (54) W sb ( r ) = (cid:90) rr in ω ds g b Σ c r (cid:48) Ω d r (cid:48) − W sb0 . (55)Notice l is not simply the sum l = ( l ) bin + ( l ) star . Thecross ω ds τ b ( ω db τ s ) terms come from the motion of the in- ternal torque resisting T ds ( T db ) induced by T db ( T ds ). Thecross W bs ( W sb ) terms come from the internal torque re-sisting T ds ( T db ) twisted by the non-Keplarian epicyclic fre-quency induced by the binary [Eq. (18)] [star, Eq. (38)]. InFigure 3, we plot the re-scaled radial functions ˜ W bs and ˜ W sb for various p and q values, tracing out the warp profile acrossthe radial extent of the disk due to the combined binary andstellar torques. The radial functions W bs and W sb evaluatedat the disk’s outer and inner truncation radii are W bs ( r out ) − W bs ( r in ) = − . × − W bs (cid:18) . h out (cid:19) × (cid:18) k q . (cid:19) (cid:18) r out ¯ r in (cid:19) p − ¯ M b ¯ R (cid:63) ¯ r ¯ M (cid:63) ¯ a ¯ r (cid:18) ¯Ω (cid:63) . (cid:19) , (56) W sb ( r out ) − W sb ( r in ) = 1 . × − W sb × (cid:18) . h out (cid:19) (cid:18) k q . (cid:19) ¯ M b ¯ R (cid:63) ¯ r out ¯ M (cid:63) ¯ a (cid:18) ¯Ω (cid:63) . (cid:19) . (57)These provide an estimate for the misalignment angle be-tween the disk’s outer and inner orbital angular momentumunit vectors | ˆ l ( r out , t ) × ˆ l ( r in , t ) | due to the binary and stellartorques. The dimensionless coefficients W bs and W sb dependon the parameters p , q , and r in /r out . Table 3 tabulates W bs and W sb for several p and q values, with r in /r out taking thecanonical value [Eq. (5)]. In the previous subsections, we have derived semi-analyticexpressions for the disk warp profiles due to the combinedtorques from the oblate host star and the binary companion.Our general conclusion is that the warp is quite small acrossthe whole disk. We illustrate this conclusion with a few ex-amples (Figs. 4-5). We define the disk misalignment angle β = β ( r, t ) as the misalignment of the disk’s local angularmomentum unit vector ˆ l ( r, t ) bysin β ( r, t ) ≡ (cid:12)(cid:12) ˆ l ( r, t ) × ˆ l d ( t ) (cid:12)(cid:12) , (58)where ˆ l d is unit vector along the total angular momentumof the disk [Eq. (25)]. c (cid:13) , 000–000 isk Warp in Star-Disk-Binary Systems − Radius (au) − . − . − . − . − . . ˜ W bs − Radius (au) − . − . − . − . . . . . ˜ W sb Figure 3.
The rescaled radial functions [see Eq. (34) for rescaling] ˜ W bs [Eq. (54)], and ˜ W sb [Eq. (55)]. We take ( p, q ) values [Eq. (7)and (11)] of p = 0 . p = 1 . p = 1 . q = 0 . q = 0 . W bs , ˜ W sb ) profiles of the disk due to the interaction betweenthe binary companion and oblate star torques (see text for discussion). − Radius r (au) . . . . . . . β ( d e g r ee s ) h in = 0 . h in = 0 . Figure 4.
Disk misalignment angle β [Eq. (58)] as a function ofradius r , for the h in [Eq. (11)] values indicated, all for h out = 0 . M d = 0 . (cid:12) (solid) and M d =0 .
01 M (cid:12) (dashed), with p = 1 [Eq. (7)], α = 0 . a b = 300 au,and ˆ s , ˆ l d , and ˆ l b lying in the same plane with θ sd = θ db = 30 ◦ . Figures 4-5 that the disk warp angle is less than a fewdegrees for the range of parameters considered. When h in =0 .
05, the binary’s torque has the strongest influence on thedisk’s warp profile. As a result, the disk warp ( ∂β/∂ ln r ) isstrongest near the disk’s outer truncation radius ( r (cid:38)
10 au).When h in = 0 .
01, the spinning star’s torque has a stronginfluence on the disk’s warp profile, and the warp becomeslarge near the inner truncation radius ( r (cid:46) M d = 0 . (cid:12) , solid lines) and low disk-mass ( M d =0 . (cid:12) , dashed lines) are marginal. This is because onlythe precession rate of the star around the disk ˜ ω sd [Eq. (43)]depends on the disk mass, and it enters the disk warp profileonly through the term ˜ ω sd τ s [see Eq. (46)]. Because the disk’s − Radius r (au) . . . . . . . . . . β ( d e g r ee s ) h in = 0 . h in = 0 . Figure 5.
Same as Fig. 4, except a b = 200 au. internal torque from bending waves is purely hydrodynami-cal, the other terms in the disk warp profile are independentof the disk mass. As noted above, when a hydrodynamical disk in the bendingwave regime is torqued externally, viscosity causes the diskto develop a small twist, which exerts a back-reaction torqueon the disk. When torqued by a central oblate star and adistant binary, the leading order viscous twist in the disk is( l ) visc = V b ( ˆ l b · ˆ l d ) ˆ l b × ˆ l d + V s ( ˆ s · ˆ l d ) ˆ s × ˆ l d , (59)where V b and V s are defined in Eqs. (31) and (48). All otherterms in Eq. (53) are non-dissipative, and do not contributeto the alignment evolution of the disk. Inserting ( l ) visc into c (cid:13)000
Same as Fig. 4, except a b = 200 au. internal torque from bending waves is purely hydrodynami-cal, the other terms in the disk warp profile are independentof the disk mass. As noted above, when a hydrodynamical disk in the bendingwave regime is torqued externally, viscosity causes the diskto develop a small twist, which exerts a back-reaction torqueon the disk. When torqued by a central oblate star and adistant binary, the leading order viscous twist in the disk is( l ) visc = V b ( ˆ l b · ˆ l d ) ˆ l b × ˆ l d + V s ( ˆ s · ˆ l d ) ˆ s × ˆ l d , (59)where V b and V s are defined in Eqs. (31) and (48). All otherterms in Eq. (53) are non-dissipative, and do not contributeto the alignment evolution of the disk. Inserting ( l ) visc into c (cid:13)000 , 000–000 J. J. Zanazzi and Dong Lai p q Γ b Γ s Γ (bs) Table 4.
Dimensionless viscosity coefficients Γ b [Eq. (69)], Γ s [Eq. (70)], and Γ (bs) [Eq. (71)], for various p and q values. Allother parameter values are canonical [Eq. (5)].
200 400 800 a b (au) − − − − − − γ b ( π / M y r ) Efficient Alignment
Figure 6.
The damping rate γ b [Eq. (69)] as a function of thebinary semi-major axis a b . We take the p [Eq. (7)] value to be p = 0 . p = 1 . p = 1 . q [Eq. (11)] value of q = 0 . q = 0 . q , we fix h out = 0 .
05 [Eq. (11)]. When the damping rate γ b (cid:38) . π/ Myr), viscous torques from disk warping may significantlydecrease the mutual disk-binary inclination θ db [Eq. (79)] over thedisk’s lifetime. Eqs. (16) and (39), and integrating over 2 πr d r , we obtain (cid:18) d L d d t (cid:19) visc = L d γ b ( ˆ l d · ˆ l b ) ˆ l b × ( ˆ l b × ˆ l d )+ L d γ s ( ˆ l d · ˆ s ) ˆ s × ( ˆ s × ˆ l d )+ L d γ (bs) ( ˆ l d · ˆ l b )( ˆ l d · ˆ s ) ˆ l b × ( ˆ s × ˆ l d )+ L d γ (bs) ( ˆ l d · ˆ s )( ˆ l d · ˆ l b ) ˆ s × ( ˆ l b × ˆ l d ) , (60) (cid:18) d S d t (cid:19) visc = − L d γ s ( ˆ l d · ˆ s ) ˆ s × ( ˆ s × ˆ l d ) − L d γ (bs) ( ˆ l d · ˆ s )( ˆ l d · ˆ l b ) ˆ s × ( ˆ l b × ˆ l d ) , (61)where γ b ≡ πL d (cid:90) r out r in αg Σ c r d r = − πL d (cid:90) r out r in Σ r Ω( ω db − ˜ ω db ) V b d r, (62) .
05 0 . . ¯Ω ? − − − − − γ s d ( π / M y r ) Efficient Alignment
Figure 7.
The damping rate γ sd [Eq. (72)] as a function of thenormalized stellar rotation frequency ¯Ω (cid:63) [Eq. (5)]. We take the p [Eq. (7)] values to be p = 0 . p = 1 . p = 1 . q [Eq. (11)] values of q = 0 . q = 0 . q , we fix h in = 0 .
03 [Eq. (11)]. When thedamping rate γ sd (cid:38) . π/ Myr), viscous torques from disk warp-ing may significantly decrease the mutual star-disk inclination θ sd [Eq. (77)] over the disk’s lifetime. γ s ≡ πL d (cid:90) r out r in αg Σ c r d r = − πL d (cid:90) r out r in Σ r Ω( ω ds − ˜ ω ds ) V s d r, (63) γ (bs) ≡ πL d (cid:90) r out r in αg b g s Σ c r d r = − πL d (cid:90) r out r in Σ r Ω( ω ds − ˜ ω ds ) V b d r = − πL d (cid:90) r out r in Σ r Ω( ω db − ˜ ω db ) V s d r. (64)When deriving Eqs. (60) and (61), we have neglected termsproportional to l · ˆ s or l · ˆ l b , as these only modify the dy-namics by changing the star-disk and disk-binary preces-sional frequencies, respectively. Usingd ˆ l d d t = 1 L d (cid:18) d L d d t − ˆ l d d L d d t (cid:19) , (65)d ˆ s d t = 1 S (cid:18) d S d t − ˆ s d S d t (cid:19) , (66)the leading order effect of viscous disk twisting on the timeevolution of ˆ l d and ˆ s is (cid:18) d ˆ l d d t (cid:19) visc = γ b ( ˆ l d · ˆ l b ) ˆ l d × ( ˆ l b × ˆ l d )+ γ s ( ˆ l d · ˆ s ) ˆ l d × ( ˆ s × ˆ l d )+ γ (bs) ( ˆ l d · ˆ l b )( ˆ l d · ˆ s ) ˆ l d × ( ˆ l b × ˆ l d )+ γ (bs) ( ˆ l d · ˆ s )( ˆ l d · ˆ l b ) ˆ l d × ( ˆ s × ˆ l d ) , (67) c (cid:13) , 000–000 isk Warp in Star-Disk-Binary Systems (cid:18) d ˆ s d t (cid:19) visc = − L d S γ s ( ˆ l d · ˆ s ) ˆ s × ( ˆ s × ˆ l d ) − L d S γ (bs) ( ˆ l d · ˆ s )( ˆ l d · ˆ l b ) ˆ s × ( ˆ l b × ˆ l d ) . (68)The four terms in (d ˆ l d / d t ) visc [Eq. (67)] arises from fourdifferent back-reaction torques of the disk in response to T ds [Eq. (3)] and T db [Eq. (1)]. To resist the influence of the twoexternal torques T ds and T db , the disk develops two twists( ∂ ˆ l /∂ ln r ) ds and ( ∂ ˆ l /∂ ln r ) db , given by Eqs. (46) and (29).The terms in Eqs. (67)-(68) proportional to γ s arise from theback reaction of ( ∂ ˆ l /∂ ln r ) ds to T ds , and works to align ˆ s with ˆ l d . The term in Eq. (67) proportional to γ b arises fromthe back reaction of T db to ( ∂ ˆ l /∂ ln r ) db , and works to align ˆ l d with ˆ l b . Because γ (bs) <
0, the terms in Eqs. (67)-(68)proportional to γ (bs) have different effects than the termsproportional to γ s and γ b . One of the terms in Eqs. (67)-(68) proportional to γ (bs) arises from the back reaction of T ds to ( ∂ ˆ l /∂ ln r ) db , and works to drive ˆ l d perpendicularto ˆ s , while the other arises from the back-reaction of T db to ( ∂ ˆ l /∂ ln r ) ds , and works to drive ˆ l d perpendicular to ˆ l b .Although typically | γ s | > | γ (bs) | or | γ b | > | γ (bs) | (so thedynamical effect of γ (bs) may be absorbed into γ b and γ s ),the magnitude of γ (bs) is not negligible compared to γ s and γ b . For completeness, we include the effects of the γ (bs) termsin the analysis below.The damping rates (62)-(64) may be evaluated andrescaled to give γ b = 1 . × − Γ b (cid:16) α . (cid:17) (cid:18) . h out (cid:19) × ¯ M ¯ r / ¯ a ¯ M / (cid:63) (cid:18) π yr (cid:19) , (69) γ s = 2 . × − Γ s (cid:16) α . (cid:17) (cid:18) . h in (cid:19) (cid:18) r out ¯ r in (cid:19) p − × (cid:18) k q . (cid:19) ¯ M / (cid:63) ¯ R (cid:63) ¯ r ¯ r / (cid:18) ¯Ω (cid:63) . (cid:19) (cid:18) π yr (cid:19) , (70) γ (bs) = − . × − Γ (bs) (cid:16) α . (cid:17) (cid:18) . h out (cid:19) (cid:18) r out ¯ r in (cid:19) p − × (cid:18) k q . (cid:19) ¯ M b ¯ R (cid:63) ¯ r / ¯ M / (cid:63) ¯ a ¯ r (cid:18) ¯Ω (cid:63) . (cid:19) (cid:18) π yr (cid:19) , (71)where h in = ( r in /r out ) q − / h out . The rescaling above hasremoved the strongest dependencies of the damping rates on p , q , and r in /r out . Table 4 lists values of the dimensionlessviscous coefficients Γ b , Γ s , and Γ (bs) , varying p and q .Note that there are “mixed” terms in Eqs. (67)-(68):the counter-aligment rate of ˆ l d and ˆ l b depends on ˆ s , whilethe counter-alignment rate of ˆ l d and ˆ s depends on ˆ l b . Alsonote that net spin-disk alignment rate is given by γ sd = (cid:18) L d S (cid:19) γ s . (72)Assuming L d (cid:29) S , γ sd evaluates to be γ sd (cid:39) . × − (2 − p )Γ s / − p (cid:16) α . (cid:17) (cid:18) . h in (cid:19) (cid:18) r out ¯ r in (cid:19) p − × (cid:18) k q k (cid:63) (cid:19) (cid:18) k q . (cid:19) ¯ M d ¯ R / (cid:63) ¯ M / (cid:63) ¯ r ¯ r out (cid:18) ¯Ω (cid:63) . (cid:19) (cid:18) π yr (cid:19) . (73)Figure 6 plots the disk-binary damping rate γ b as a function of the binary semi-major axis a b . In agreementwith Foucart & Lai (2014), we find the damping rate tobe small, and weakly dependent on the power-law surfacedensity and sound-speed indices p and q . This is becausethe torque from the binary companion is strongest around r ∼ r out . The properties of the disk near r out are “global,”since the amount of inertia of disk annuli near r out is setmainly by the total disk mass rather than the surface den-sity profile, and the disk sound-speed does not vary greatlyaround r ∼ r out . We conclude that viscous torques fromdisk warping are unlikely to significantly decrease the mu-tual disk-binary inclination θ db unless a b (cid:46)
200 au.Figure 7 plots the star-disk alignment rate γ sd as a func-tion of the normalized stellar rotation frequency ¯Ω (cid:63) . Un-like the disk-binary alignment rate γ b (Fig. 6), γ sd dependsstrongly on the surface density and sound-speed power-lawindices p and q . The alignment rate of a circumbinary diskwith its binary orbital plane has a similarly strong depen-dence on p and q (Foucart & Lai 2013, 2014; Lubow & Mar-tin 2018). This strong dependence arises because the torqueon the inner part of a disk from an oblate star or binaryis strongest near r in . The disk properties near r ∼ r in arevery local (both the amount of inertia for disk annuli anddisk sound-speed), and hence will depend heavily on p and q . Despite this uncertainty, Figure 7 shows that there arereasonable parameters for which viscous torques from diskwarping can significantly reduce the star-disk inclination θ sd [when γ sd (cid:38) . π/ Myr)], especially when the stellar rota-tion rate is sufficiently high ( ¯Ω (cid:63) (cid:38) . This section investigates the evolution of star-disk-binarysystems under gravitational and viscous torques:d ˆ s d t = − ˜ ω sd ( ˆ s · ˆ l d ) ˆ l d × ˆ s + (cid:18) d ˆ s d t (cid:19) visc , (74)d ˆ l d d t = − ˜ ω ds ( ˆ l d · ˆ s ) ˆ s × ˆ l d − ˜ ω db ( ˆ l d · ˆ l b ) ˆ l b × ˆ l d + (cid:18) d ˆ l d d t (cid:19) visc . (75)The viscous terms are given by Eqs. (67)-(68). As in Batygin& Adams (2013) and Lai (2014a), we assume the disk’s massis depleted according to M d ( t ) = M d0 t/t v , (76)where M d0 = 0 . (cid:12) and t v = 0 . ˆ s and ˆ l d and secular resonance (˜ ω sd ∼ ˜ ω db )when viscous dissipation from disk warping is neglected.The effect of the γ s term on the dynamical evolution of ˆ s over viscous timescales depends on the precessional dynam-ics of the star-disk-binary system. If ˜ ω sd (cid:29) ˜ ω db , ˆ s rapidlyprecesses around ˆ l d , and the γ s term works to align ˆ s with ˆ l d .If ˜ ω sd (cid:28) ˜ ω db , ˆ s cannot “follow” the rapidly varying ˆ l d , andeffectively precesses around ˆ l b . In the latter case, becauseof the rapid variation of ˆ l d around ˆ l b , ˆ s is only effected bythe secular ˆ l d . As a result, γ s works to drive θ sb to θ db . The c (cid:13) , 000–000 J. J. Zanazzi and Dong Lai θ s d , θ s b , θ db ( d e g r ee s ) α = 0 θ sd θ sb θ db θ s d , θ s b , θ db ( d e g r ee s ) α = 0 . , h in = 0 . θ sd θ sb θ db Time (Myr) θ s d , θ s b , θ db ( d e g r ee s ) α = 0 . , h in = 0 . θ sd θ sb θ db Time (Myr) ˜ ω s d , ˜ ω db ( π / M y r ) ˜ ω sd ˜ ω db Figure 8.
Inclination evolution of star-disk-binary systems. The top panels and bottom left panel plot the time evolution of theangles θ sd [Eq. (77)], θ sb [Eq. (78)], and θ db [Eq. (79)], integrated using Eqs. (74) and (75), with values of α and h in [Eq. (11)]as indicated. The bottom right panel shows the precession frequencies ˜ ω sd [Eq. (43)] and ˜ ω db [Eq. (24)]. We take θ db (0) = 60 ◦ , θ sd (0) = 5 ◦ , and h out = 0 .
05 [Eq. (11)]. The damping rates are γ b = 5 . × − (2 π/ yr) [Eq. (69)], γ sd (0) = 2 . × − (2 π/ yr)[Eq. (72)], and γ bs = − . × − (2 π/ yr) [Eq. (71)] for h in = 0 .
05, and γ b = 7 . × − (2 π/ yr), γ sd (0) = 1 . × − (2 π/ yr), and γ (bs) = − . × − (2 π/ yr) for h in = 0 . effect of the γ b term is simpler: it always works to align ˆ l d with ˆ l b .Figure 8 shows several examples of the evolution of star-disk-binary systems. The top panels and bottom left panelof Fig. 8 show the time evolution of the angles θ sd = cos − ( ˆ s · ˆ l d ) , (77) θ sb = cos − ( ˆ s · ˆ l b ) , (78) θ db = cos − ( ˆ l d · ˆ l b ) , (79)from integrating Eqs. (74)-(75), while the bottom right panelplots the characteristic precession frequencies ˜ ω sd and ˜ ω db .The top left panel of Fig. 8 does not include viscous torques( α = 0). Because the damping rates γ b [Eq. (69)] and γ sd [Eq. (72)] are much less than 0 . π/ Myr) over most of thesystem’s lifetime (10 Myr), viscous torques have a negligibleeffect on the evolution of θ sd , θ sb , and θ db . The bottom leftpanel of Fig. 8 shows the evolution of θ sd , θ sb , and θ db with α = 0 .
01 and h in = 0 .
01. Because the inner edge of thedisc has a much smaller scaleheight, the oblate star warpsthe inner edge of the disk more [Eq. (14)], resulting in γ sd taking a value larger than 0 . π/ Myr). This increase in γ sd causes a much tighter coupling of ˆ s to ˆ l d before secularresonance (˜ ω sd (cid:38) ˜ ω db ), evidenced by the damped oscillationsin θ sd . After secular resonance (˜ ω sd (cid:46) ˜ ω db ), the γ s termdamps ˆ s toward ˆ l b . Notice θ sb approaches θ db because ofthe rapid precession of ˆ l d around ˆ l b after secular resonance,not θ sb → β between the disk’s outer and inner or-bital angular momentum unit vectors:sin ∆ β ( t ) = (cid:12)(cid:12) ˆ l ( r out , t ) × ˆ l ( r in , t ) (cid:12)(cid:12) (cid:39) (cid:12)(cid:12) [ l ( r out , t ) − l ( r in , t )] × ˆ l d ( t ) (cid:12)(cid:12) (80)Figure 9 plots ∆ β as a function of time, for the examplesconsidered in Fig. 8. We see even when viscous torques fromdisk warping significantly alter the star-disk-binary systemdynamics (e.g. α = 0 .
01 and h in = 0 . β < . ◦ over thedisk’s lifetime, indicating a high degree of disk coplanaritythroughout the system’s evolution. c (cid:13) , 000–000 isk Warp in Star-Disk-Binary Systems Time (Myr) . . . . . . . ∆ β ( d e g r ee s ) Figure 9.
Total disk warp ∆ β [Eq. (80)] for the integra-tions of Fig. 8. The blue curve denotes the integration where( h in , α ) = (0 . , . h in , α ) = (0 . , . h in , α ) = (0 . , . β < . ◦ , indicating thedisk remains highly coplanar throughout the system’s evolution.Notice ∆ β (cid:28) ◦ when α = 0 (blue, hugs the x-axis). Figure 10 is identical to Fig. 8, except we take a b =200 au instead of a b = 300 au. Since γ b is greater than0 . π/ Myr), ˆ l d aligns with γ b over the disk’s lifetime. Inthe top right panel of Fig. 10, γ sd is less than 0 . π/ Myr)for most of the disk’s lifetime, so ˆ s stays misaligned withboth ˆ l d and ˆ l b . At the end of the disk’s lifetime, ˆ s precessesaround ˆ l b , which is aligned with ˆ l d . In the bottom left panel,both γ b and γ sd are greater than 0 . π/ Myr) for most ofthe disk’s lifetime. This results in alignment of ˆ l d , ˆ s , and ˆ l b over 10 Myr. Figure 11 shows the evolution of disk misalign-ment angles for the examples considered in Fig. 10. We see∆ β < . ◦ for all examples considered, indicating the diskremains highly co-planar throughout the system’s evolution. Our study of warped disks in star-disk-binary systems reliescritically on the warp evolution equations derived in Lubow& Ogilvie (2000) for disks in the bending wave regime ( α (cid:46) H/r ), assuming a small disk warp ( | ∂ ˆ l /∂ ln r | (cid:28) θ db (cid:38) ◦ . Lidov-Kozai oscillations may besuppressed by the disk’s self-gravity when (Fu, Lubow, &Martin 2015b) M d (cid:38) . M b (cid:18) r out a b (cid:19) , (81)and by the disk’s pressure gradients when (Zanazzi & Lai2017; Lubow & Ogilvie 2017) a b (cid:38) . r out (cid:18) M b M (cid:63) (cid:19) / (cid:18) h out . (cid:19) − / . (82)For our canonical parameters [Eq. (5)], the Lidov-Kozai ef-fect is unlikely to be relevant unless a b (cid:46) r out . In our companion work (Zanazzi & Lai 2018b), we show thatthe formation of a short-period (orbital periods less than 10days) massive planet in many instances significantly reducesor completely suppresses primordial misalignments gener-ated by the gravitational torque from an inclined binarycompanion. Primordial misalignments are still robustly gen-erated in protostellar systems forming low-mass ( ∼ ⊕ )multiple planets, and systems with cold (orbital periodsgreater than one year) Jupiters. On the other hand, ob-servations suggest that most Kepler compact multi-planetsystems have small stellar obliquities (e.g. Albrecht et al.2013; Winn et al. 2017). A major goal of this work was toexamine if viscous torques from disk warping may reduceor suppress the generation of primordial misalignments instar-disk-binary systems. We find that for some parameters,the star-disk inclination damping rate can be significant (seeFig. 7); in particular, the star-disk misalignment may be re-duced when the disk is sufficiently cold with strong externaltorques (Figs. 8 & 10).Observational evidence is mounting which suggests hotstars (effective temperatures (cid:38) ◦ K) have higher obliq-uities than cold stars (Winn et al. 2010; Albrecht et al. 2012;Mazeh et al. 2015; Li & Winn 2016). Since all damping ratesfrom viscous disk-warping torques in star-disk-binary sys-tems are inversely proportional to the disk’s sound-speedsquared [see Eqs. (69)-(72)], a tempting explanation for thiscorrelation is that hot stars have hot disks with low damp-ing rates which remain misaligned, while cold stars havecold disks with high damping rates which have star-diskmisalignments significantly reduced over the disk’s lifetime.However, we do not believe this is a likely explanation, sincethe protostellar disk’s temperature should not vary stronglywith the T-Tauri stellar mass. If a disk is passively heatedfrom irradiation by its young host star (Chiang & Goldreich1997), low mass ( (cid:46) (cid:12) ) pre-main sequence stars have ef-fective temperatures which are not strongly correlated with c (cid:13)000
Total disk warp ∆ β [Eq. (80)] for the integra-tions of Fig. 8. The blue curve denotes the integration where( h in , α ) = (0 . , . h in , α ) = (0 . , . h in , α ) = (0 . , . β < . ◦ , indicating thedisk remains highly coplanar throughout the system’s evolution.Notice ∆ β (cid:28) ◦ when α = 0 (blue, hugs the x-axis). Figure 10 is identical to Fig. 8, except we take a b =200 au instead of a b = 300 au. Since γ b is greater than0 . π/ Myr), ˆ l d aligns with γ b over the disk’s lifetime. Inthe top right panel of Fig. 10, γ sd is less than 0 . π/ Myr)for most of the disk’s lifetime, so ˆ s stays misaligned withboth ˆ l d and ˆ l b . At the end of the disk’s lifetime, ˆ s precessesaround ˆ l b , which is aligned with ˆ l d . In the bottom left panel,both γ b and γ sd are greater than 0 . π/ Myr) for most ofthe disk’s lifetime. This results in alignment of ˆ l d , ˆ s , and ˆ l b over 10 Myr. Figure 11 shows the evolution of disk misalign-ment angles for the examples considered in Fig. 10. We see∆ β < . ◦ for all examples considered, indicating the diskremains highly co-planar throughout the system’s evolution. Our study of warped disks in star-disk-binary systems reliescritically on the warp evolution equations derived in Lubow& Ogilvie (2000) for disks in the bending wave regime ( α (cid:46) H/r ), assuming a small disk warp ( | ∂ ˆ l /∂ ln r | (cid:28) θ db (cid:38) ◦ . Lidov-Kozai oscillations may besuppressed by the disk’s self-gravity when (Fu, Lubow, &Martin 2015b) M d (cid:38) . M b (cid:18) r out a b (cid:19) , (81)and by the disk’s pressure gradients when (Zanazzi & Lai2017; Lubow & Ogilvie 2017) a b (cid:38) . r out (cid:18) M b M (cid:63) (cid:19) / (cid:18) h out . (cid:19) − / . (82)For our canonical parameters [Eq. (5)], the Lidov-Kozai ef-fect is unlikely to be relevant unless a b (cid:46) r out . In our companion work (Zanazzi & Lai 2018b), we show thatthe formation of a short-period (orbital periods less than 10days) massive planet in many instances significantly reducesor completely suppresses primordial misalignments gener-ated by the gravitational torque from an inclined binarycompanion. Primordial misalignments are still robustly gen-erated in protostellar systems forming low-mass ( ∼ ⊕ )multiple planets, and systems with cold (orbital periodsgreater than one year) Jupiters. On the other hand, ob-servations suggest that most Kepler compact multi-planetsystems have small stellar obliquities (e.g. Albrecht et al.2013; Winn et al. 2017). A major goal of this work was toexamine if viscous torques from disk warping may reduceor suppress the generation of primordial misalignments instar-disk-binary systems. We find that for some parameters,the star-disk inclination damping rate can be significant (seeFig. 7); in particular, the star-disk misalignment may be re-duced when the disk is sufficiently cold with strong externaltorques (Figs. 8 & 10).Observational evidence is mounting which suggests hotstars (effective temperatures (cid:38) ◦ K) have higher obliq-uities than cold stars (Winn et al. 2010; Albrecht et al. 2012;Mazeh et al. 2015; Li & Winn 2016). Since all damping ratesfrom viscous disk-warping torques in star-disk-binary sys-tems are inversely proportional to the disk’s sound-speedsquared [see Eqs. (69)-(72)], a tempting explanation for thiscorrelation is that hot stars have hot disks with low damp-ing rates which remain misaligned, while cold stars havecold disks with high damping rates which have star-diskmisalignments significantly reduced over the disk’s lifetime.However, we do not believe this is a likely explanation, sincethe protostellar disk’s temperature should not vary stronglywith the T-Tauri stellar mass. If a disk is passively heatedfrom irradiation by its young host star (Chiang & Goldreich1997), low mass ( (cid:46) (cid:12) ) pre-main sequence stars have ef-fective temperatures which are not strongly correlated with c (cid:13)000 , 000–000 J. J. Zanazzi and Dong Lai θ s d , θ s b , θ db ( d e g r ee s ) α = 0 θ sd θ sb θ db θ s d , θ s b , θ db ( d e g r ee s ) α = 0 . , h in = 0 . θ sd θ sb θ db Time (Myr) θ s d , θ s b , θ db ( d e g r ee s ) α = 0 . , h in = 0 . θ sd θ sb θ db Time (Myr) ˜ ω s d , ˜ ω db ( π / M y r ) ˜ ω sd ˜ ω db Figure 10.
Same as Figure 8, except a b = 200 AU. The damping rates are γ b = 5 . × − (2 π/ yr), γ sd (0) = 2 . × − (2 π/ yr), and γ (bs) = − . × − (2 π/ yr) for h in = 0 .
05, and γ b = 8 . × − (2 π/ yr), γ sd (0) = 1 . × − (2 π/ yr), and γ (bs) = − . × − (2 π/ yr)for h in = 0 . Time (Myr) . . . . . ∆ β ( d e g r ee s ) Figure 11.
Same as Fig. 9, except for the examples consideredin Fig. 10. All examples considered have ∆ β < . ◦ , indicatingthe disk remains highly coplanar throughout the disk’s lifetime. their masses (Hayashi 1961). If the disk is actively heatedby turbulent viscosity (Lynden-Bell & Pringle 1974), thedisk’s accretion rate does not vary enough between differenthost star masses to create a difference in disk temperature(Rafikov 2017).Even in systems where viscous torques from disk warp-ing alter the dynamics of the star-disk-binary system overthe disk’s lifetime (Figs. 8 & 10), we find the misalignmentangle between the outer and inner disk orbital angular mo-mentum unit vectors to not exceed a few degrees (Figs. 9& 11). Therefore, it is unlikely that the disk warp profileplays a role in setting the mutual inclinations of formingexoplanetary systems with inclined binary companions. We have studied how disk warps and the associated vis-cous dissipation affect the evolution of star-disk inclinationsin binary systems. Our calculation of the disk warp pro-file shows that when the circumstellar disk is torqued byboth the exterior companion and the central oblate star,the deviation of the disk angular momentum unit vectorfrom coplanarity is less than a few degrees for the entire c (cid:13) , 000–000 isk Warp in Star-Disk-Binary Systems parameter space considered (Figs. 9 & 11). This indicatesthat disk warping in star-disk-binary systems does not al-ter exoplanetary architectures while the planets are form-ing in the disk. We have derived analytical expressions forthe viscous damping rates of relative inclinations (Sec. 3.5),and have examined how viscous dissipation affects the in-clination evolution of star-disk-binary systems. Because thestar-disk [Eq. (72), Fig. 7] and disk-binary [Eq. (69), Fig. 6]alignment timescales are typically longer than the proto-planetary disk’s lifetime ( (cid:46)
10 Myrs), viscous dissipationfrom disk warping does not significantly modify the long-term inclination evolution of most star-disk-binary systems(Fig. 8, top left panel). However, in sufficiently cold disks(small
H/r ) with strong external torques from the oblatestar or inclined binary companion, the star-disk-binary evo-lution may be altered by viscous dissipation from disk warp-ing, reducing the star-disk misalignment generated by star-disk-binary interactions (Figs. 8 & 10). In particular, we findwhen the stellar rotatation rate is sufficiently high (rotationperiods (cid:46)
ACKNOWLEDGEMENTS
We thank the referee, Christopher Spalding, for many com-ments which improved the presentation and clarity of thiswork. JZ thanks Re’em Sari and Yoram Lithwick for helpfuldiscussions. This work has been supported in part by NASAgrants NNX14AG94G and NNX14AP31G, and NSF grantAST- 1715246. JZ is supported by a NASA Earth and SpaceSciences Fellowship in Astrophysics.
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