Radiation-Driven Warping of Circumbinary Disks Around Eccentric Young Star Binaries

# Radiation-Driven Warping of Circumbinary Disks Around Eccentric Young Star Binaries

Kimitake Hayasaki11affiliation: Korea Astronomy and Space Science Institute, Daedeokdaero 776, Yuseong, Daejeon 305-348, Korea , Bong Won Sohn11affiliation: Korea Astronomy and Space Science Institute, Daedeokdaero 776, Yuseong, Daejeon 305-348, Korea 22affiliation: Department of Astronomy and Space Science, University of Science and Technology, 217 Gajeong-ro, Daejeon, Korea , Atsuo T. Okazaki33affiliation: Faculty of Engineering, Hokkai-Gakuen University, Toyohira-ku, Sapporo 062-8605, Japan , Taehyun Jung11affiliation: Korea Astronomy and Space Science Institute, Daedeokdaero 776, Yuseong, Daejeon 305-348, Korea , Guangyao Zhao11affiliation: Korea Astronomy and Space Science Institute, Daedeokdaero 776, Yuseong, Daejeon 305-348, Korea , and Tsuguya Naito44affiliation: Faculty of Management Information, Yamanashi Gakuin University, Kofu, Yamanashi 400-8575, Japan
###### Abstract

accretion, accretion disks - hydrodynamics - masers - protoplanetary disks - stars: massive - stars: low-mass - stars: formation - (stars:) binaries: general
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## 1 Introduction

About 60 of main sequence stars are considered to be born as binary or multiple systems (Duquennoy & Mayor, 1991). Indeed, the direct imaging of the circumbinary disk was successfully done for a few young binary star systems with the binary mass comparable to a solar mass, including GG Tau (Dutrey et al., 1994) and UY Aurigae (Duvert et al., 1998) by the Plateau de Bure Interferometer (PdBI). It is confirmed by numerical simulations that the circumbinary disk is formed around such young low mass binary stars embedded in the dense molecular gas (Artymowicz & Lubow, 1994; Bate & Bonnell, 1997; Günther & Kley, 2002, 2004).

The existence of a significant number of OB eclipsing binaries (Hilditch et al., 2005) suggests that most known massive stars are also born as binary or multiple systems (Sana et al., 2012). Such binaries are expected to be accompanied by circumbinary disks at the early stage of the massive star formation. Contrary to the case of T Tauri stars, however, the massive star formation is still poorly understood because the observation of massive star forming regions is challenging. This is because they are obscured by a dusty environment and the chance to observe massive young stellar objects is small because of their short lives. Araya et al. (2010) suggested that observed periodic maser emissions are originated from periodic accretion onto the binary stars on a significantly eccentric orbit from the circumbinary disk (Artymowicz & Lubow, 1996), although there are two other scenarios: the colliding wind binary scenario (van der Walt, 2011) and the scenario of pulsation of protostars growing via high mass accretion rate (Inayoshi et al., 2013).

It is natural that the circumstellar and circumbinary disks in the star forming regions have a warped structure. Krist et al. (2005) suggested that there is observational evidence for the disk warping in GG Tau system. The tidal effects of close encounters between stars and a disk surrounding it can make the disk outer parts misaligned or warped (Moeckel&Bally, 2006). If the circumbinary disk is originally misaligned with the binary orbital plane, it should be warped by the tidal alignment, although the disk tilt secularly decays by the disk viscosity (Facchini et al., 2013; Lodato&Facchini, 2013). The origin of the misalignment and disk warping is, however, still a matter of debate.

Hayasaki et al. (2014a) (hereafter, Paper I) derived the condition that a circumbinary disk subject to both the tidal torque and the torque due to the radiation emitted from two accretion disks around the individual supermassive black holes is unstable to the radiation-driven warping in the context of observed warped maser disks in active galactic nuclei (e.g., Kuo et al. 2011; Kormendy&Ho 2013). They assumed that the binary is on a circular orbit and the irradiation luminosity is proportional to the mass accretion rate. There is, however, observational evidence that young binaries have a significant orbital eccentricity (Mathieu et al., 1990). In addition, the circumbinary disk is mainly irradiated by two young stars in a binary in the current problem. In order to apply our previous model to those binaries, therefore, we need to relax the above assumptions adopted in Paper I and make the binary mass scale down to the stellar mass from supermassive black hole mass.

In this paper, we study the warping instability of a circumbinary disk around young binary stars on an eccentric orbit. In section 2, we describe the external torques acting on a circumbinary disk. We consider both the tidal torques originating from a time-dependent binary potential and the radiative torques from the binary stars. In section 3, we examine the evolution of a slightly tilted circumbinary disk subject to those two torques, and derive the warping condition and timescale of the local precession of the linear warping mode. Finally, section 4 is devoted to summary and discussion of our scenario.

## 2 External Torques acting on the circumbinary disk

Let us consider the torques from the binary potential acting on the circumbinary disk surrounding two stars in a binary on an eccentric orbit. Figure 1 illustrates a schematic picture of the setting of our model; binary stars orbiting each other are surrounded by a misaligned circumbinary disk. The binary is put on the - plane with its center of mass being at the origin in the Cartesian coordinate. The masses of the primary and secondary stars are represented by and , respectively, and . We put a circumbinary disk around the origin. The unit vector of specific angular momentum of the circumbinary disk is expressed by (e.g. Pringle 1996)

 \boldmathl=cosγsinβ\boldmathi+sinγsinβ\boldmathj+cosβ\boldmathk, (1)

where is the tilt angle between the circumbinary disk plane and the binary orbital plane, and is the azimuth of tilt. Here, , , and are unit vectors in the , , and , respectively. The position vector of the circumbinary disk can be expressed by

 \boldmathr=r(cosϕsinγ+sinϕcosγcosβ)% \boldmathi+r(sinϕsinγcosβ−cosϕcosγ)\boldmath% j−rsinϕsinβ\boldmathk (2)

where the azimuthal angle is measured from the descending node. The only difference from the Paper I is the position vector of each star, which is given by

 \boldmathri=ricosfi\boldmathi+risinfi\boldmathj(i=1,2), (3)

where is the true anomaly and is written as

 ri=ξia(1−e2)1+ecosfi (4)

with and . Here, is the binary orbital eccentricity, is the binary mass ratio, and is the binary semi-major axis with and . These and other model parameters are listed in Table 1.

### 2.1 Gravitational Torques

The gravitational force on the unit mass at position on the circumbinary disk can be written by

 \boldmathFgrav=−2∑i=1GMi|\boldmathr−\boldmathri|3(\boldmathr−% \boldmathri) (5)

The corresponding torque is given by

 \boldmathtgrav=\boldmathr×% \boldmathFgrav=2∑i=1GMi|\boldmathr−\boldmathri|3(\boldmathr×\boldmathri) (6)

We consider the tidal warping/precession with timescales much longer than local rotation period of the circumbinary disk. This allows us to use the torque averaged in the azimuthal direction and over the orbital period:

 ⟨\boldmathTgrav⟩ ≈ 14π2∫2π0∫2π0\boldmathtgravdϕd(Ωorbt)=38ξ1ξ2(ar)2GMr (7) × [(1−e2)sinγsin2β\boldmathi−(1+4e2)cosγsin2β\boldmathj+5e2sin2γsin2β\boldmathk],

where is the angular frequency of mean binary motion. Here, we used for the integration the following relationship:

 d(Ωorbt)=(1−e2)3/2(1+ecosfi)2df (8)

and the approximations:

 |\boldmathr−\boldmathri|−3 ≈ r−3[1+3\boldmathr⋅\boldmathrir2+O((ri/r)2)], (1+ecosfi)−4 ≈ 1−4ecosfi+10e2cos2fi+O(e3). (9)

Note that equation (7) is derived in the same manner as in equation (7) of Hayasaki et al. (2014b).

For a small tilt angle , equation (7) is reduced to

 ⟨\boldmathTgrav⟩≈34ξ1ξ2(ar)2GMr[(1−e2)ly\boldmathi−(1+4e2)lx\boldmathj], (10)

where and can be written from equation (1) as and . The magnitude of the specific tidal torque with is then given by

 |⟨\boldmathTgrav⟩|=34ξ1ξ2(ar)2GMrβ√5e2(3e2+2)cos2γ+(e2−1)2. (11)

The tidal torque tends to align the tilted circumbinary disk with the orbital plane (c.f. Bate et al. 2000). The tidal alignment timescale for is given by

 τtid=|\boldmathJ|sinβ|⟨\boldmathTgrav⟩|≈83π1√5e2(3e2+2)cos2γ+(e2−1)2(1/4ξ1ξ2)(ra)7/2Porb, (12)

where with the disk angular frequency and are the specific angular momentum and binary orbital period, respectively. Equation (12) is reduced to equation (8) of Paper I for . Note that the tidal alignment timescale depends on the azimuth of tilt for . Since the inner edge of the circumbinary disk is estimated to be (Artymowicz & Lubow, 1994), the tidal alignment timescale is longer than the binary orbital period.

The circumbinary disk around young binary stars can be mainly illuminated by light emitted from each star. The re-radiation from the circumbinary-disk surface, which absorbs photons emitted from these stars, causes a reaction force. This is the origin of the radiative torques. Here, the central two stars are can be regarded as point irradiation sources, because each star is much smaller than the size of circumbinary disk. Circumstellar disks in binary systems have been observed by many authors in the past (e.g. Mayama et al. 2010). These circumstellar disks could be other irradiation sources. We estimate their luminosities as follows: the maximum luminosity of each disk is given by with , where the Eddington luminosity is given by . Since it is substantially smaller than the stellar luminosity, the effect of the circumstellar-disk irradiation is negligible.

Since the surface element on the circumbinary disk is given in the polar coordinates by

 \boldmathdS=∂\boldmathr∂r×∂\boldmathr∂ϕdrdϕ=[\boldmathl−\boldmathr(−∂β∂rsinϕ+∂γ∂rcosϕsinβ)]rdrdϕ, (13)

the radiative flux at is given by

 dL=14π2∑i=1Li|\boldmathr−\boldmathri|2|(\boldmathr−\boldmathri)⋅d\boldmathS||\boldmathr−\boldmathri|, (14)

where is the sum of luminosities of the radiation emitted from the primary star, , and that from the secondary star, . Here, we assume that the surface element is not shadowed by other interior parts of the circumbinary disk. If we ignore limb darkening, the force acting on the disk surface by the radiation reaction has the magnitude of and is antiparallel to the local disk normal (Pringle, 1996). The total radiative force on can then be written by

Consequently, the total radiative torque acting on a ring of radial width is given by

where holds for and , and the first term, which we call , of the right-hand side of equation (16) corresponds to equation (2.15) of Pringle (1996):

 d\boldmathT0=L6c(r∂ly∂r\boldmathi−r∂lx∂r% \boldmathj)dr (17)

and the second term, which we call , is originated from the orbital motion of the binary.

Here, we consider the radiation-driven warping/precession with timescales much longer than the orbital period, as in the case of tidally driven warping/precession. The orbit-average of the torque is then given by

where is a binary irradiation parameter ( by definition), and is reduced to for .

From equation (18), the specific radiative torque averaged over azimuthal angle and orbital phase is given by

where and are the growth speed of a warping mode induced by the radiative torque and disk surface density, respectively. The growth timescale of the warping mode for an optically thick gas disk can be estimated to be

 (20)

Since it is clear that the growth timescale for is much longer that the orbital period, our assumption for the orbit-averaged radiative torque is ensured.

## 3 Tilt angle evolution of circumbinary disks

In this section, we examine the response of the circumbinary disk to the external forces for case. The evolution equation for disk tilt is given by(Pringle, 1996)

 ∂\boldmathl∂t+[vr−ν1Ω′Ω−12ν2(r3ΩΣ)′r3ΩΣ]∂\boldmathl∂r=∂∂r(12ν2∂\boldmathl∂r)+12ν2∣∣∣∂\boldmathl∂r∣∣∣2\boldmathl+\boldmathTex, (21)

where and are respectively the horizontal and vertical shear viscosities, the latter of which tends to reduce disk tilt, and is the term to which the external torques contribute. For a geometrically thin disk, is approximately given by with the Shakura-Sunyaev viscosity parameter (Shakura & Sunyaev, 1973), the isothermal sound speed , the temperature , the gas constant , and the molecular weight . The primes indicate differentiation with respect to . We adopt for the circumbinary disk structure the following assumptions that ,

 Σ(r) = Σ0(rr0)n(n<0), (22) T(r) = T0(rr0)s(s<0), (23)

where and are a constant, and , , and are the fiducial radius, fiducial surface density, and fiducial temperature, respectively. Equation (21) can be then reduced to

 ∂\boldmathl∂t=12ν2∂2\boldmathl∂r2+12(n+s+3)ν2r∂\boldmathl∂r+%\boldmath$T$ex, (24)

where is used. Here, with the ratio of vertical to horizontal viscosities: and is written as

We look for solutions of equation (24) of the form , with . Replacing with , with , and with , we have the following set of linearized equations:

 [iω+ν2k2/2−(ik/2)(n+s+3)(ν2/r)−(A+ikB)(C+ikD)iω+ν2k2/2−(ik/2)(n+s+3)(ν2/r)](lxly)=0, (26)

where

 A = (1−e2)Ωp,circ, B = Γ[1−38ζ(ar)2(4+e2)], C = (1+4e2)Ωp,circ, D = Γ[1−38ζ(ar)2(4+11e2)], (27)

and, represents the magnitude of the local precession frequency of the liner warping mode for the case of the standard disk model and (see equation (35) of Paper I):

The determinant of the coefficient matrix on the left hand side of equation (26) must vanish because of . The local dispersion relation is then obtained as

 ω=i[ν2k22±1√2(√X2+Y2−X)1/2]±1√2(√X2+Y2+X)1/2, (29)

where

 X = AC−k2BD≈(1+3e2)Ω2p,circ−(kΓ)2[1−3ζ(1+32e2)(ar)2], Y = AD+BC≈−2kΓΩp,circ[(1+32e2)−32ζ(1+3e2)(ar)2], X2+Y2 ≈ {(1+3e2)Ω2p,circ+(kΓ)2[1−3ζ(1+32e2)(ar)2]}2. (30)

Here, we adopt the approximation that the terms proportional to or are negligible in comparison with the other terms. The dispersion relation is then rewritten in the following simple form:

 ω=i{ν2k22±kΓ[1−3ζ(1+32e2)(ar)2]1/2}±√1+3e2Ωp,circ+12kν2r(n+s+3) (31)

The imaginary part of corresponds to the excitation or damping of oscillation, whereas the real part provides the local precession frequency due to the external torques.

In order for the perturbation to grow, must be negative. The growth condition is given by

 0

In terms of , the growth timescale of the warping mode induced by the radiative torques in the binary system is given by

Figure 2 shows the dependence of on the orbital eccentricity and in a circumbinary disk with , , , , and . Each growth timescale is normalized by . The growth timescale is longer than that of circular binary case in the range of . This is because the averaged incident flux normal to the disk surface per binary orbit is lower than that of the circular binary case. Note that is approximately equal to for . We therefore treat as the growth timescale of the binary star case in what follows.

We focus our attention on a perturbation with , where is the radial wavelength of the perturbation. The condition that the circumbinary disk is unstable to the warping mode can be, then, rewritten as

 r≤112π2ηαΣcs2(Lc)[1−3ζ(1+32e2)(ar)2]1/2. (34)

Substituting equations (22) and (23) into the above equation, we obtain

 rr0⎧⎪ ⎪⎨⎪ ⎪⎩≥(rwarp/r0)[1−3ζ(1+32e2)(ar)2]1/(2(n+s+1))(n+s+1<0)≤(rwarp/r0)[1−3ζ(1+32e2)(ar)2]1/(2(n+s+1))(n+s+1>0), (35)

where shows the marginally stable warping radius for a single star case:

 rwarpr0=[12π2ηαr0Σ0RgμT0cL]−1/(n+s+1)≈[6π21αr0Σ0RgμT0cL]−1/(n+s+1), (36)

where in deriving the second equation, we have used the relationship between and : for (Ogilvie, 1999). In the case of , no unstable solution exists except for very special combination of parameters.

The radiative torques work only for a region of optically thick to irradiation emitted from the central stars. In order for the circumbinary disk to be optically thick, the surface density must be higher than , where the electron scattering opacity is assumed for simplicity. This condition is rewritten as

 rr0≤(ΣminΣ0)1/n, (37)

using equation (22). The equality is held at the radius where the circumbinary disk changes from optically thick to optically thin:

 ropr0=(ΣminΣ0)1/n. (38)

In order for radiation-driven warping to be a possible mechanism for disk warping, the marginally stable warping radius must be less than the outer radius of the optically thick region, . This gives the upper (lower) limit of for as follows:

 Σ0⎧⎪ ⎪⎨⎪ ⎪⎩≤Σcrit[1−3ζ(1+32e2)(ar)2]−n/(2(s+1))(n+s+1<0)≥Σcrit[1−3ζ(1+32e2)(ar)2]−n/(2(s+1))(n+s+1>0), (39)

where is the critical surface density given by

 Σcrit=Σ(n+s+1)/(s+1)min[12π2ηαr0RgμT0cL]n/(s+1). (40)

Figure 3 shows the possible range in , for the circumbinary disk to be unstable to warping modes, as a function of and . The upper two panels give the range for case, whereas the lower two panels for case. From the figure, we can express as

 Σ0=fΣcrit[1−3ζ(ar)2(1+32e2)]−n/(2(s+1)) (41)

with a factor in the range for and for . Substituting equation (41) into equation (36) with equation (35), we can obtain the marginally stable warping radius as

 rwarp,bin=rwarp[1−3ζ(1+32e2)(ar)2]1/(2(s+1)), (42)

where can be rewritten as

 rwarp = f−1/(n+s+1)[6π21αr0ΣminRgμT0cL]−1/(s+1)r0≃10−4/(s+1)[AU](1f)1/(n+s+1) (43) × [(α0.01)(r01AU)s(Σmin1gcm−2)−1(T0100K)−1(LL⊙)]1/(s+1).

Adopting for equation (43) , which corresponds to the power law index of the radial temperature profile of an optically thick region in the standard disk (Shakura & Sunyaev, 1973), the marginally stable warping radius is very sensitive to the values of , , , , and . It is clear from equations (42) and (43) that a circumbinary disk around young binary stars with and is stable for radiation driven warping, because the marginally stable warping radius is of the order of for and of even for . This shows that the circumbinary disks of a classical T Tauri star system such as GG Tau (Dutrey et al., 1994) and UY Aurigae (Duvert et al., 1998) are not warped by radiation-driven warping instability.

On the other hand, it is likely that a circumbinary disk around binary stars with much larger luminosities is unstable to radiation-driven warping. Figure 4 shows the dependence of the marginally stable warping radius in such a disk on the semi-major axis. Here, we adopt , , , , , , , , , and . While panels (a) and (c) are for , panels (b) and (d) are for . We also adopt and in panels (a) and (b), and and in panels (c) and (d). In these four panels, the black solid line and orange dashed line show and in units of , respectively. The red dotted line shows the radius where the growth timescale of the radiation-driven warping mode, , equals the timescale for the disk to align with the orbital plane by the tidal torque, . This tidal alignment radius is given by

 (44)

from equations (12), (20) and (22). The growth of a finite-amplitude warping mode induced by the radiative torque can be significantly suppressed by the tidal torque in the region inside the tidal alignment radius. The blue dot-dashed line show the inner radius of the circumbinary disk. Here, the inner radius is assumed to be equal to the tidal/resonant truncation radius, where the tidal/resonant torque is balanced with the horizontal viscous torque of the circumbinary disk (Artymowicz & Lubow, 1994). In the case of eccentric binaries with a nearly equal mass ratio, the tidal/resonant truncation radius is estimated to be . The shaded area shows the region where the circumbinary disk is unstable to the warping modes induced by the radiative torques.

The local precession frequency of the linear warping mode is obtained from the dispersion relation by