Disk Warp in Star-Disk-Binary Systems

# Effects of Disk Warping on the Inclination Evolution of Star-Disk-Binary Systems

## Abstract

Several recent studies have suggested that circumstellar disks in young stellar binaries may be driven into misalignement with their host stars due to secular gravitational interactions between the star, disk and the binary companion. The disk in such systems is twisted/warped due to the gravitational torques from the oblate central star and the external companion. We calculate the disk warp profile, taking into account of bending wave propagation and viscosity in the disk. We show that for typical protostellar disk parameters, the disk warp is small, thereby justifying the “flat-disk” approximation adopted in previous theoretical studies. However, the viscous dissipation associated with the small disk warp/twist tends to drive the disk toward alignment with the binary or the central star. We calculate the relevant timescales for the alignment. We find the alignment is effective for sufficiently cold disks with strong external torques, and is ineffective for the majority of star-disk-binary systems. Viscous warp driven alignment may be necessary to account for the observed spin-orbit alignment in multi-planet systems if these systems are accompanied by an inclined binary companion.

###### keywords:
hydrodynamics - planets and satellites: formation - protoplanetary discs - stars: binaries: general

## 1 Introduction

Circumstellar disks in young protostellar binary systems are likely to form with an inclined orientation relative to the binary orbital plane, as a result of the complex star/binary/disc formation processes (e.g. Bate, Bonnell & Bromm 2003; McKee & Ostriker 2007; Klessen 2011). Indeed, many misaligned circumstellar disks in protostellar binaries have been found in recent years (e.g. Stapelfeldt et al. 1998, 2003; Neuhäuser et al. 2009; Jensen & Akeson 2014; Williams et al. 2014; Brinch et al. 2016; Fernández-López, Zapata & Gabbasov 2017; Lee et al. 2017). Such misaligned disks experience differential gravitational torques from the binary companion, and are expected to be twisted/warped while 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 a rotation-induced quadrupole, and thus exerts a torque on the disk (and also receives a back-reaction torque) when the stellar spin axis and the disk axis are misaligned. This torque tends to induce warping in the inner disk and drives mutual precession between the stellar spin and disk. In the presence of both torques on the disk, from the binary and from the central star, how does the disk warp and precess? What is the long-term evolution of the disk and stellar spin in such star-disk-binary systems? These are the questions we intend to address in this paper.

Several recent studies have examined the secular dynamics of the stellar spin and circumstellar disk in the presence of an inclined binary companion (Batygin, 2012; Batygin & Adams, 2013; Lai, 2014; Spalding & Batygin, 2014, 2015). These studies were motivated by the observations of spin-orbit misalignments in exoplanetary systems containing hot Jupiters, i.e., the planet’s orbital plane is often misaligned with the stellar rotational equator (see Winn & Fabrycky 2015 and Triaud 2017 for recent reviews). It was shown that significant “primordial” misalignments may be generated while the planetary systems are still forming in their natal protoplanetary disks through secular star-disk-binary gravitational interactions (Batygin & Adams, 2013; Lai, 2014; Spalding & Batygin, 2014, 2015). In these studies, various assumptions were made about the star-disk interactions, and uncertain physical processes such as star/disk winds, magnetic star-disk interactions, and accretion of disk angular momentum onto the star were incorporated in a parameterized manner. Nevertheless, the production of spin-orbit misalignments seems quite robust.

In Zanazzi & Lai (2017b), we showed that the formation of hot Jupiters in the protoplanetary disks can significantly suppress the excitation of spin-orbit misalignment in star-disk-binary systems. This is because the presence of such close-in giant planets lead to strong spin-orbit coupling between the planet and its host star, so that the spin-orbit misalignment angle is adiabatically maintained despite the gravitational perturbation from the binary companion. However, the formation of small planets or distant planets (e.g. warm Jupiters) do not affect the generation of primordial misalignments between the host star and the disk.

A key assumption made in all previous studies on misalignments in star-disk-binary systems (Batygin & Adams, 2013; Lai, 2014; Spalding & Batygin, 2015) is that the disk is nearly flat and behaves like a rigid plate in response to the external torques from the binary and from the host star. The rationale for this assumption is that different regions of the disk can efficiently communicate with each other through hydrodynamical forces and/or self-gravity, such that the disk stays nearly flat. However, to what extent this assumption is valid is uncertain, especially because in the star-disk-binary system the disk experiences two distinct torques from the oblate star and from the binary which tend to drive the disk toward different orientations (see Tremaine & Davis 2014 for examples of non-trivial disk warps when a disk is torqued by different forces). Moreover, the combined effects of disk warps/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 regime have considered a single external torque, such as an ext binary 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, 2018), magnetic torques from the central star (Foucart & Lai, 2011), a central spinning black hole (Demianski & Ivanov, 1997; Lubow, Ogilvie, & Pringle, 2002; Franchini, Lodato, & Facchini, 2016; Chakraborty & Bhattacharyya, 2017), and a system of multiple planets on nearly coplanar orbits (Lubow & Ogilvie, 2001). In this paper, we will focus on the hydrodynamics of warped disks in star-disk-binary systems, and will present analytical calculations for the warp amplitudes/profiles and the rate of evolution of disk inclinations due to viscous dissipation associated with these warps/twists.

This paper is organized as follows. Section 2 describes the setup and parameters of the star-disk-binary system we study. Section 3 presents all the technical calculations of our paper, including the disk warp/twist profile and effect of viscous dissipation on the evolution of system. Section 4 examines how viscous dissipation from disk warps modifies the long-term evolution of star-disk-binary systems. Section 5 discusses theoretical uncertainties of our work. Section 6 contains our conclusions.

## 2 Star-Disk-Binary System and Gravitational Torques

Consider a central star of mass , radius , rotation rate , with a circumstellar disk of mass , and inner and outer truncation radii of and , respectively. This star-disk system is in orbit with a distant binary companion of mass and semimajor axis . The binary companion exerts a torque on the disk, driving it into differential precession around the binary angular momentum axis . The torque per unit mass is

 Tdb=−r2Ωωdb(^l⋅^lb)^lb×^l, (1)

where is the disk angular frequency, is the unit orbital angular momentum axis of a disk “ring” at radius , and

 ωdb(r)=3GM⋆4a3bΩ (2)

is the characteristic precession frequency of the disk “ring” at radius . Similarly, the rotation-induced stellar quadrapole drives the stellar spin axis and the disk onto mutual precession. The stellar rotation leads to a difference in the principal components of the star’s moment of inertia of , where for fully convective stars (Lai, Rasio, & Shapiro, 1993). The torque on the disk from the oblate star is

 Tds(r,t)=−r2Ωωds(^s⋅^l)^s×^l, (3)

where

 ωds(r)=3G(I3−I1)2r5Ω=3GkqM⋆R2⋆¯Ω2⋆2r5Ω (4)

is the characteristic precession frequency of the disk ring at radius . Since and both depend on , the disk would quickly lose coherence if there were no internal coupling between the different “rings.”

We introduce the following rescaled parameters typical of protostellar systems:

 Missing dimension or its units for \hskip ¯Md=Md0.1M⊙,¯rin=rin8R⊙,¯rout=rout50au, Missing dimension or its units for \hskip (5)

The cannonical value of is , corresponding to a stellar rotation period of . The other canonical values in Eq. (5) are unity, except the disk mass, which can change significantly during the disk lifetime.

We parameterize the disk surface density as

 Σ(r,t)=Σout(t)(routr)p. (6)

We take unless otherwise noted. The disk mass is then (assuming )

 Md=∫routrin2πΣrdr≃2πΣoutr2out2−p. (7)

The disk angular momentum vector is (assuming a small disk warp), and stellar spin angular momentum vector is , where and are unit vectors, and

 Ld =∫routrin2πΣr3Ωdr≃2−p5/2−pMd√GM⋆rout, (8) S =k⋆M⋆R2⋆Ω⋆. (9)

Here for fully convective stars. The binary has orbital angular momentum . Because typical star-disk-binary systems satisfy , we take to be fixed for the remainder of this work.

## 3 Disk Warping

When ( is the Shakura-Sunyaev viscosity parameter, is the disk scaleheight), which is satisfied for protostellar disks, the main internal torque enforcing disk rigidity and coherent precession comes from bending wave propagation (Papaloizou & Lin, 1995; Lubow & Ogilvie, 2000). As bending waves travel at 1/2 of the sound speed, the wave crossing time is of order . When is longer than the characteristic precession times or from an external torque, significant disk warps can be induced. In the extreme nonlinear regime, disk breaking may be possible (Larwood et al., 1996; Doğan et al., 2015). To compare with and , we adopt the disk sound speed profile

 cs(r) =H(r)Ω(r)=hout√GM⋆rout(routr)q =hin√GM⋆rin(rinr)q, (10)

where and . Passively heated disks have (Chiang & Goldreich, 1997), while actively heated disks have (Lynden-Bell & Pringle, 1974). We find

 tbwωds Missing or unrecognized delimiter for \left (11) tbwωdb =1.7×10−2(0.1hout)¯Mb¯r3out¯M⋆¯a3b(rrout)q+3/2. (12)

Thus, we expect the small warp approximation to be valid everywhere in the disk. This expectation is confirmed by our detailed calculation of disk warps presented later in this section.

Although the disk is flat to a good approximation, the interplay between the disk warp/twist and viscous dissipation can lead to appreciable damping of the misalignment between the disk and the external perturber (i.e., the oblate star or the binary companion). In particular, when an external torque (per unit mass) is applied to a disk in the bending wave regime [which could be either or ], the disk’s viscosity causes the disk normal to develop a small twist, of order

 ∂^l∂lnr∼4αc2sText. (13)

This twist interacts with the external torque, affecting the evolution of over viscous timescales. To an order of magnitude, we have

 Missing or unrecognized delimiter for \left (14)

where is either or , and implies proper average over .

We now study the disk warp and viscous evolution quantitatively, using the formalism describing the structure and evolution of circular, weakly warped disks in the bending wave regime. The relevant equations have been derived by a number of authors (Papaloizou & Lin, 1995; Demianski & Ivanov, 1997; Lubow & Ogilvie, 2000). We choose the formalism of Lubow & Ogilvie (2000) and Lubow, Ogilvie, & Pringle (2002) (see also Ogilvie 2006 when ), where the evolution of the disk is governed by

 Σr2Ω∂^l∂t =ΣText+1r∂G∂r, (15) ∂G∂t =(Ω2−κ22Ω)^l×G−αΩG+Σc2sr3Ω4∂^l∂r, (16)

where is the external torque per unit mass acting on the disk, is the epicyclic frequency, and is the internal torque, which arises from slightly eccentric fluid particles with velocities sheared around the disk mid-plane (Demianski & Ivanov, 1997).

We are concerned with two external torques acting on different regions of the disk. For clarity, we break up our calculations into three subsections, considering disk warps produced by individual torques before examining the combined effects.

### 3.1 Disk Warp Induced by Binary Companion

The torque from an external binary companion is given by Eq. (1). The companion also gives rise to a non-Keplarian epicyclic frequency, given by

 Ω2−κ22Ω=ωdbP2(^l⋅^lb), (17)

where are Legendre polynomials. Equations (15)-(16) become

 Σr2Ω∂^l∂t =−Σr2Ωωdb(^l⋅^lb)^lb×^l+1r∂G∂r, (18) ∂G∂t =ωdbP2(^l⋅^l%b)^l×G−αΩG+Σc2sr3Ω4∂^l∂r. (19)

To make analytic progress, we take advantage of the fact that [see Eq. (12)]. Specifically, we take

 ^l(r,t) =^ld(t)+l1(r,t)+…, (20) G(r,t) =G0(r,t)+G1(r,t)+…, (21)

where . Here, is the internal torque maintaining coplanarity of , is the internal torque maintaining the leading order warp profile , etc. To leading order, Eq. (18) becomes

 Σr2Ωd^lddt=−Σr2Ωωdb(^ld⋅^lb)^lb×^ld+1r∂G0∂r. (22)

Integrating (22) over , and using the boundary condition

 G0(rin,t)=G0(rout,t)=0, (23)

we obtain

 d^lddt=−~ωdb(^ld⋅^lb)^lb×^ld, (24)

where is given by

 ~ωdb =2πLd∫routrinωdbΣr3Ωdr ≃3(5/2−p)4(4−p)(MbM⋆)(abrout) ⎷GM⋆r3out. (25)

The physical meaning of thus becomes clear: is the unit total angular momentum vector of the disk, or

 ^ld≡2πLd∫routrinΣr3Ω^l(r,t)dr. (26)

Using Eqs. (23) and (24), we may solve for :

 G0(r,t)=gb(r)(^ld⋅^lb)^lb×^ld, (27)

where

 gb(r)=∫rrin(ωdb−~ωdb)Σr′3Ωdr′. (28)

Using Eqs. (27) and (19), and requiring that not contribute to the total disk angular momentum vector, or

 ∫routrinΣr3Ωl1(r,t)dr=0, (29)

we obtain the leading order warp :

 l1(r,t)= −~ωdbτb(^ld⋅^lb)2^lb×(^lb×^ld) −Wbb(^ld⋅^lb% )P2(^ld⋅^lb)^ld×(^lb×^ld) +Vb(^ld⋅^lb)^lb×^ld, (30)

where

 τb(r) =∫rrin4gbΣc2sr′3Ωdr′−τb0, (31) Vb(r) =∫rrin4αgbΣc2sr′3dr′−Vb0, (32) Wbb(r) =∫rrin4ωdbgbΣc2sr′3Ωdr′−Wbb0, (33)

and the constants of the functions , , or are determined by requiring

 ∫routrinΣr3ΩXdr=0. (34)

We may rescale the radial functions by defining

 ~X(r)≡X(r)/[X(rout)−X(rin)]. (35)

In Figure 1, we plot the dimensionless radial functions , , and for the canonical parameters of the star-disk-binary system [Eq. (5)]. The scalings of the radial functions evaluated at the outer disk radius are

 τb (rout)−τb(rin)=−1.82×10−5Ub ×(0.1hout)2¯Mb¯r9/2out¯M3/2⋆¯a6bMyr2π, (36) Vb (rout)−Vb(rin)=−1.54×10−3Vb ×(α0.01)(0.1hout)2¯Mb¯r3out¯M⋆¯a3b, (37) Wbb (rout)−Wbb(rin)=−8.93×10−5Wbb ×(0.1hout)2¯M2b¯r6out¯M2⋆¯a6b. (38)

The dimensionless coefficients , , and depend weakly on the parameters , , and . In Figure 2, we plot , , and as a function of with . In Table 1, we tabulate , , and for values of and as indicated, with the canonical value of [Eq. (5)].

### 3.2 Disk Warp Indued by Oblate Star

The torque on the disk from the oblate star is given by Eq. (3). The stellar quadrupole moment also gives rise to a non-Keplarian epicyclic frequency given by

 Ω2−κ22Ω=ωdsP2(^l⋅^s). (39)

Equations (15)-(16), coupled with the time evolution of the stellar spin, become

 Sd^sdt =−∫routrin[2πΣr3Ωωds(^s⋅^l)^l×^s]dr, (40) Σr2Ω∂^l∂t =−Σr2Ωωds(^l⋅^s)^s×^l+1r∂G∂r, (41) ∂G∂t =ωdbP2(^l⋅^s)^l×G−αΩG+Σc2sr3Ω4∂^l∂r. (42)

Expanding and according to Eqs. (20) and (21), integrating Eq. (41) over , and using the boundary condition (23), we obtain the leading order evolution equations

 d^sdt =−~ωsd(^s⋅^ld)^ld×^s, (43) d^lddt =−~ωds(^ld⋅^s)^s×^ld, (44)

where (assuming )

 ~ωds =2πLd∫routrinωdsΣr3Ωdr ≃3(5/2−p)kq2(1+p)R2⋆¯Ω2⋆r1−poutr1+pin ⎷GM⋆r3out, (45) ~ωsd =(Ld/S)~ωds ≃3(2−p)kq2(1+p)k⋆(M%dM⋆)¯Ω⋆√GM⋆R3⋆r2−poutr1+pin. (46)

With and determined, Eq. (41) may be integrated to obtain the leading order internal torque:

 G0(r,t)=gs(r)(^ld⋅^s)^s×^ld, (47)

where

 gs(r)=∫rrin(ωds−~ωds)Σr′3Ωdr′. (48)

Similarly, the leading order warp profile is

 l1(r,t)= −~ωsdτs(^ld⋅^s)2(^ld×^s)×^ld −~ωdsτs(^ld⋅^s)2^s×(^s×^ld) −Wss(^ld⋅^s)P2(^ld⋅^s)^ld×(^s×^ld) +Vs(^ld⋅^s)^s×^ld, (49)

where

 τs(r) =∫rrin4gsΣc2sr′3Ωdr′−τs0, (50) Vs(r) =∫rrin4αgsΣc2sr′3dr′−Vs0, (51) Wss(r) =∫rrin4ωdsgsΣc2sr′3Ωdr′−Wss0. (52)

In Figure 3, we plot the rescaled radial functions , , and for various and values. These radial functions may be rescaled giving

 τs (rout)−τs(rin)=2.21×10−6Us(0.1hout)2(kq0.1) ×(1358¯rout¯rin)p−1¯R2⋆¯r3/2out¯r2in¯M1/2⋆(¯Ω⋆0.1)2Myr2π, (53) Vs (rout)−Vs(rin)=1.13×10−3Vs ×(α0.01)(0.1hin)2(kq0.1)¯R2⋆¯r2in(¯Ω⋆0.1)2, (54) Wss (rout)−Wss(rin)=4.39×10−7Wss ×(kq0.1)2(0.1hin)2¯R4⋆¯r4in(¯Ω⋆0.1)2. (55)

The dimensionless coefficients , , and depend weakly on the parameters , , and . In Figure 4, we plot , , and as a function of with . In Table 2, we tabulate , , and for the and values indicated, with taking the canonical value [Eq. (5)].

### 3.3 Disk Warps Induced by Combined Torques

The combined torques from the distant binary and oblate star are given by Eqs. (1) and (3), and the non-Keplarian epicyclic frequencies are given by Eqs. (17) and (39). Using the same procedure as Sections 3.1-3.2, the leading order correction to the disk’s warp is

 l1(r,t)= (l1)bin+(l1)star −~ωdsτb(^ld⋅^s)[(^s×^ld)⋅^lb]^lb×^ld −~ωdsτb(^ld⋅^lb)(^ld⋅^s)^lb×(^s×^ld) −~ωdbτs(^ld⋅^lb)[(^lb×^ld)⋅^s]^s×^ld −~ωdbτs(^ld⋅^s)(^ld⋅^lb)^s×(^lb×^ld) −Wsb(^ld⋅^lb% )P2(^ld⋅^s)^ld×(^lb×^ld) −Wbs(^ld⋅^s)P2(^ld⋅^lb)^ld×(^s×^ld), (56)

where is Eq. (30), is Eq. (49), and are given in Eqs. (31) and (50), and

 Wbs(r) =∫rrin4ωdbgsΣc2sr′3Ωdr′−Wbs0, (57) Wsb(r) =∫rrin4ωdsgbΣc2sr′3Ωdr′−Wsb0. (58)

In Figure 5, we plot the re-scaled radial functions and as a function of , for various and values. Notice is not simply the sum . The cross () terms come from the motion of the internal torque resisting () induced by (). The cross () terms come from the internal torque resisting () twisted by the non-Keplarian epicyclic frequency induced by the binary [Eq. (17)] [star, Eq. (39)]. The radial functions and may be rescaled to give

 Wbs (rout)−Wbs(rin)=−7.23×10−6Wbs(0.1hout)2 ×(kq0.1)(1358¯rout¯rin)p−1¯Mb¯R2⋆¯r3out¯M