GW Ori: Interactions between a Triple-star System and Its Circumtriple Disk in Action

Figure 1(a) shows the continuum map with spatial resolution of 0098 (∼39 au). We identify three dust rings with different apparent shapes in the disk at ∼46, 188, and 338 au (hereafter the inner, middle, and outer ring). The location of the inner ring coincides with the predicted cavity size from SED modeling (Fang et al. 2017). The continuum flux densities of the inner, middle, and outer ring are 42 ± 4, 95 ± 10, and 58 ± 6 mJy, respectively. To our knowledge, the outer ring is the largest ever found in protoplanetary disks.

The three rings harbor an enormous amount of solid material. We estimate the dust (solid) mass M_dust of the rings with the equation provided in Hildebrand (1983)

$$\begin{eqnarray}&&{M}_{\mathrm{dust}}=\displaystyle \frac{{F}_{\nu }{d}^{2}}{{B}_{\nu }({T}_{\mathrm{dust}}){\kappa }_{\nu }},\end{eqnarray} \tag{ 1 }$$

where F_ν is the continuum surface brightness at a submillimeter frequency ν, d is the distance from the observer to the source, B_ν(T_dust) is the Planck function at the dust temperature T_dust, and κ_ν is the dust opacity. The dust temperature is estimated using a fitting function provided by Dong et al. (2018b)

$$\begin{eqnarray}&&{T}_{\mathrm{dust}}=30{\left(\displaystyle \frac{{L}_{\star }}{38{L}_{\odot }}\right)}^{1/4}{\left(\displaystyle \frac{r}{100{\rm{au}}}\right)}^{-1/2},\end{eqnarray} \tag{ 2 }$$

where L_⋆ is the total stellar luminosity, and r is the location of the ring. The stellar luminosity modified by the distance provided by GAIA DR2 is 49.3 ± 7.4 L_⊙ (Calvet et al. 2004; Gaia Collaboration et al. 2018). We assume a dust grain opacity of 10 cm² g⁻¹ at 1000 GHz with a power-law index of 1 (Beckwith et al. 1990). We estimate the dust masses of the rings to be 74 ± 8, 168 ± 19, and 245 ± 28 M_⊕, respectively, with the uncertainties incorporating the uncertainties in the surface brightness of the rings, source distance, stellar luminosity, and radial location of the rings.

Figure 1(b) shows the first-moment map of ¹²CO J = 2 − 1 emission (with the zeroth-moment map provided in Appendix A). For regular Keplerian rotating disks, we expect a well-defined butterfly-like pattern in the first-moment map. However, we find a twisted pattern inside ∼02, which may result from a misalignment between the inner and outer parts of the disk (i.e., having different inclinations and orientations; Rosenfeld et al. 2014), as has been found in the disks around, e.g., HD 142527 (Casassus et al. 2015; Marino et al. 2015) and HD 143006 (Benisty et al. 2018; Pérez et al. 2018).

We fit the dust continuum map assuming that there are three dust rings in the disk with Gaussian radial profiles of surface brightness

$$\begin{eqnarray}&&{F}_{i}(r)={F}_{0,i}{e}^{\tfrac{-{\left(r-{r}_{i}\right)}^{2}}{2{\sigma }_{i}^{2}}},\end{eqnarray} \tag{ 3 }$$

where F_i is the surface brightness as a function of the distance to the center r, with i = 1, 2, 3 denoting parameters for the inner, middle, and outer ring, respectively. F_0,i is the peak surface brightness, r_i is the radius of the ring (i.e., where the ring has the highest surface brightness), and σ_i is the standard deviation.

Initially, we assume all three rings are intrinsically circular when viewed face-on, and their different apparent shapes entirely originate from different inclinations. For each ring, we assume an independent set of peak surface brightness, center location, radius, width, inclination, and position angle as the model parameters. We call this combination of assumptions Model 1.

After projecting the rings according to their position angles and inclinations, we calculate the synthetic visibility of the models using galario (Tazzari et al. 2018), and launch MCMC parameter surveys to derive posterior distribution of model parameters using emcee (Foreman-Mackey et al. 2013). In the MCMC parameter surveys, the likelihood function L is defined as

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{ln}L & = & -\displaystyle \frac{1}{2}\sum _{j=1}^{N}{m}_{j}\times \left[{\left({{ReV}}_{\mathrm{obs},j}-{{ReV}}_{\mathrm{mod},j}\right)}^{2}\right.\\ & & +\ \left.{\left({{ImV}}_{\mathrm{obs},j}-{{ImV}}_{\mathrm{mod},j}\right)}^{2}\right],\end{array}\end{eqnarray} \tag{ 4 }$$

where V_obs is the visibility data from ALMA observations, V_mod is the synthetic model visibility, N is the total number of visibility data points in V_obs, and m_j is the weight of each visibility data point in V_obs. The prior function is set to guarantee the surface brightness, ring radius, and ring width do not go below zero, the position angle does not go beyond (−90, 90) degrees, and the inclination does not go beyond (0, 90) degrees. For each model, there are 144 chains spread in the hyperspace of parameters. Each chain has 15,000 iterations including 10,000 burn-in iterations. The results of the parameter surveys are listed in Table 1.

Table 1.  The Complete MCMC Result of Dust Continuum Visibility Modeling

(a) Model 1
	Inner Ring	Middle Ring	Outer Ring
R.A Offset [arcsecond]	$1.89\times {10}^{-2}{\,}_{-9.49\times {10}^{-5}}^{+9.47\times {10}^{-5}}$	$-2.44\times {10}^{-3}{\,}_{-2.08\times {10}^{-4}}^{+1.82\times {10}^{-4}}$	$-4.24\times {10}^{-3}{\,}_{-3.91\times {10}^{-4}}^{+4.64\times {10}^{-4}}$
decl. Offset [arcsecond]	$-1.32\times {10}^{-2}{\,}_{-1.01\times {10}^{-4}}^{+1.09\times {10}^{-4}}$	$-2.26\times {10}^{-2}{\,}_{-2.06\times {10}^{-4}}^{+2.22\times {10}^{-4}}$	$-1.21\times {10}^{-2}{\,}_{-4.67\times {10}^{-4}}^{+6.57\times {10}^{-4}}$
Ring Radius [arcsecond]	$1.15\times {10}^{-1}{\,}_{-1.48\times {10}^{-4}}^{+1.46\times {10}^{-4}}$	$4.68\times {10}^{-1}{\,}_{-2.92\times {10}^{-4}}^{+3.07\times {10}^{-4}}$	$8.40\times {10}^{-1}{\,}_{-1.15\times {10}^{-3}}^{+1.03\times {10}^{-3}}$
Ring Width [arcsecond]	$4.97\times {10}^{-2}{\,}_{-6.24\times {10}^{-4}}^{+3.16\times {10}^{-4}}$	$1.74\times {10}^{-1}{\,}_{-7.56\times {10}^{-4}}^{+6.69\times {10}^{-4}}$	$3.32\times {10}^{-1}{\,}_{-2.78\times {10}^{-3}}^{+1.84\times {10}^{-3}}$
Surface Brightness [Jy/pixel]	$2.49\times {10}^{-4}{\,}_{-1.56\times {10}^{-6}}^{+1.80\times {10}^{-6}}$	$3.96\times {10}^{-5}{\,}_{-1.69\times {10}^{-7}}^{+8.67\times {10}^{-8}}$	$1.13\times {10}^{-5}{\,}_{-5.52\times {10}^{-7}}^{+2.77\times {10}^{-8}}$
Inclination [degree]	$22.24{\,}_{-0.31}^{+0.23}$	$32.62{\,}_{-0.11}^{+0.07}$	$37.93{\,}_{-0.08}^{+0.09}$
Position Angle [degree]	$-60.75{\,}_{-0.56}^{+1.06}$	$-7.43{\,}_{-0.12}^{+0.19}$	$-3.57{\,}_{-0.12}^{+0.15}$
(b) Model 2
	Inner Ring	Middle Ring	Outer Ring
R.A Offset [arcsecond]		$1.77\times {10}^{-2}{\,}_{-1.61\times {10}^{-4}}^{+1.69\times {10}^{-4}}$
decl. Offset [arcsecond]		$-2.22\times {10}^{-2}{\,}_{-1.99\times {10}^{-4}}^{+1.95\times {10}^{-4}}$
Ring Radius [arcsecond]	$1.17\times {10}^{-1}{\,}_{-1.35\times {10}^{-4}}^{+1.36\times {10}^{-4}}$	$4.68\times {10}^{-1}{\,}_{-2.71\times {10}^{-4}}^{+2.88\times {10}^{-4}}$	$8.40\times {10}^{-1}{\,}_{-9.53\times {10}^{-4}}^{+9.48\times {10}^{-4}}$
Ring Width [arcsecond]	$4.97\times {10}^{-2}{\,}_{-3.91\times {10}^{-4}}^{+3.16\times {10}^{-4}}$	$1.74\times {10}^{-1}{\,}_{-6.03\times {10}^{-4}}^{+6.66\times {10}^{-4}}$	$3.30\times {10}^{-1}{\,}_{-1.96\times {10}^{-3}}^{+1.68\times {10}^{-3}}$
Surface Brightness [Jy/pixel]	$2.51\times {10}^{-4}{\,}_{-1.55\times {10}^{-6}}^{+1.53\times {10}^{-6}}$	$3.96\times {10}^{-5}{\,}_{-1.01\times {10}^{-7}}^{+8.20\times {10}^{-8}}$	$1.13\times {10}^{-5}{\,}_{-3.37\times {10}^{-8}}^{+2.65\times {10}^{-8}}$
Inclination [degree]	$23.15{\,}_{-0.23}^{+0.22}$	$32.64{\,}_{-0.07}^{+0.07}$	$37.91{\,}_{-0.07}^{+0.08}$
Position Angle [degree]	$-55.67{\,}_{-0.50}^{+0.61}$	$-7.44{\,}_{-0.12}^{+0.14}$	$-3.60{\,}_{-0.11}^{+0.13}$
Apoapsis Angle [degree]	$65.04{\,}_{-0.49}^{+0.50}$	⋯	⋯
Eccentricity	$0.21{\,}_{-1.43\times {10}^{-3}}^{+1.75\times {10}^{-3}}$	⋯	⋯
(c) Model 3
	Inner Ring	Middle Ring	Outer Ring
Ring Radius [arcsecond]	$1.16\times {10}^{-1}{\,}_{-1.49\times {10}^{-4}}^{+1.20\times {10}^{-4}}$	$4.68\times {10}^{-1}{\,}_{-2.98\times {10}^{-4}}^{+2.72\times {10}^{-4}}$	$8.37\times {10}^{-1}{\,}_{-1.06\times {10}^{-3}}^{+8.95\times {10}^{-4}}$
Ring Width [arcsecond]	$4.99\times {10}^{-2}{\,}_{-3.86\times {10}^{-4}}^{+3.06\times {10}^{-4}}$	$1.73\times {10}^{-1}{\,}_{-5.98\times {10}^{-4}}^{+6.33\times {10}^{-4}}$	$3.39\times {10}^{-1}{\,}_{-1.85\times {10}^{-3}}^{+1.90\times {10}^{-3}}$
Surface Brightness [Jy/pixel]	$2.47\times {10}^{-4}{\,}_{-1.42\times {10}^{-6}}^{+1.64\times {10}^{-6}}$	$3.94\times {10}^{-5}{\,}_{-1.12\times {10}^{-7}}^{+8.55\times {10}^{-8}}$	$1.13\times {10}^{-5}{\,}_{-3.33\times {10}^{-8}}^{+2.63\times {10}^{-8}}$
Inclination [degree]	$20.63{\,}_{-0.29}^{+0.23}$	$32.86{\,}_{-0.07}^{+0.06}$	$37.96{\,}_{-0.08}^{+0.09}$
Position Angle [degree]	$-60.37{\,}_{-0.65}^{+0.81}$	$-7.26{\,}_{-0.13}^{+0.13}$	$-3.49{\,}_{-0.12}^{+0.13}$
Apoapsis Angle [degree]	$121.39{\,}_{-0.25}^{+0.26}$	⋯	⋯
Eccentricity	$0.19{\,}_{-6.46\times {10}^{-4}}^{+8.55\times {10}^{-4}}$	⋯	⋯

Note. The radius of each ring is the location of the peak in our model in Section 4, and the width is the full width at half maximum (FWHM) of the profile. The center offsets for Model 1 and Model 2 are relative to the center in Model 3, which is the location of GW Ori provided by GAIA DR2 (ICRS R.A. = 5^h29^m08390 and decl. = 11°52'12661). The position angles and apoapsis angles are measured east of north. The inclination is defined in the range from 0° to 90°, with 0° denoting face-on. The pixel size in the unit of surface brightness is determined internally by galario.

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The fitting result of Model 1, listed in Table 1(a), suggests that the three rings have statistically different centers. Particularly, the center of the inner ring differs from the centers of the outer two by ∼20% of the inner ring's radius. This nonconcentricity indicates nonzero intrinsic eccentricities in the rings, particularly the inner ring (see Section 5.1).

We explore the nonzero intrinsic eccentricity in the inner ring with two models. In both models, the outer two rings are intrinsically circular and concentric. Their center coincides with one of the two foci of the inner ring. In Model 2, that center is set free, while in Model 3 it is assumed to coincide with the stellar position provided by GAIA DR2 (Gaia Collaboration et al. 2018). In those two models, we introduce two more parameters for the intrinsic eccentricity and apoapsis angle of the inner ring. The position angle only indicates the direction to the ascending node on the axis along which the ring is inclined. The fitting results are listed in Tables 1(b) and (c), and the following calculations are based on the result of Model 3.

Figures 1(c) and (e) show the model image and the residual map of Model 3, respectively. The residual map is produced by subtracting model from data in the visibility plane, and then imaging the results in the same way used for the observations. We interpret the residuals as additional substructures on top of the ideal model (e.g., a warp within the ring; Huang et al. 2020).

All three models yield roughly consistent inclinations and position angles of each ring. However, we cannot determine the mutual inclinations between them (i.e., the angles between their angular momentum vectors) from dust emission modeling alone, due to the unknown direction of orbital motion.

Following the prescription and parameter values used to fit low resolution CO isotopologue data of GW Ori (Fang et al. 2017), we set up a gas surface density model using a power-law profile with an exponential tail

$$\begin{eqnarray}&&{\rm{\Sigma }}(r)={{\rm{\Sigma }}}_{c}{\left(\displaystyle \frac{r}{{r}_{c}}\right)}^{-\gamma }{e}^{\left[-{\left(r-{r}_{c}\right)}^{2-\gamma }\right]},\end{eqnarray} \tag{ 5 }$$

and the aspect ratio h/r parameterized as

$$\begin{eqnarray}&&\displaystyle \frac{h}{r}={\left(\displaystyle \frac{h}{r}\right)}_{c}{\left(\displaystyle \frac{r}{{r}_{c}}\right)}^{\psi },\end{eqnarray} \tag{ 6 }$$

where Σ_c and (h/r)_c are corresponding values at the characteristic scaling radius r_c. The disk mass is taken as 0.12 M_⊙, corresponding to Σ_c = 3 g cm⁻² for r_c = 320 au, with γ = 1.0, (h/r)_c = 0.18, and ψ = 0.1. The dust surface density profile is set by assuming a gas-to-dust ratio of 100, and decreasing the dust surface density by a factor of 1000 inside the derived gap radii: inside 37 au, from 56 to 153 au, and from 221 to 269 au. The ¹²CO channel maps are then computed and ray-traced by the physical-chemical modeling code dali (Bruderer 2013), which simultaneously solves the heating-cooling balance of the gas and chemistry to determine the gas temperature, molecular abundances, and molecular excitation for a given density structure.

Similar to Walsh et al. (2017), we model the misaligned disk with an inner cavity and three annuli each with its own inclination and position angle,¹⁹ as listed in Table 2. The channel map is run through the ALMA simulator using the settings of the ALMA observations. The resulting first-moment map is shown in Figure 1(d). In Figure 1(f) we show the simulated first-moment map for another model as a comparison, in which the disk is coplanar with an inclination of 379 and a position angle of −5°.

The models show that the ¹²CO J = 2 − 1 first-moment map in the ALMA observation cannot be reproduced by a coplanar disk. Instead, the misaligned disk model described in Table 2 matches the observed first-moment map better, indicating the presence of misalignment in the GW Ori disk.

The A–B binary and the C component can be dynamically viewed as an AB–C binary. The eccentricity of the circumbinary disk may increase through resonant interactions with the binary (Papaloizou et al. 2001). In the case of no binary-disk misalignment, the binary's perturbing gravitational potential on the midplane of the disk is given by Lubow (1991). The coupling of this perturbing potential with the imposed eccentricity of the disk excites density waves at the 1:3 outer eccentric Lindblad resonance, which lead to angular momentum removal in the inner parts of the disk. As no energy is removed along with the angular momentum in this process, the disk orbit cannot remain circular (Papaloizou et al. 2001). In the case of GW Ori, the inner dust ring is the most susceptible to this effect, which could explain why its center in Model 1 is more deviated from those of the other two rings.

Our dust and gas observations alone cannot break the degeneracy in the mutual inclination between different parts in the disk due to the unknown on-sky projected orbital direction of the disk. Previous studies indicate that the on-sky projected gas motion is likely to be clockwise (Czekala et al. 2017), same as the orbital motion of GW Ori C given by astrometric observations (Berger et al. 2011). Given the inclination and longitude of ascending node of the AB-C binary orbit being 150 ± 7 and 282 ± 9 degrees (Czekala et al. 2017), we assume that the entire disk has the same clockwise on-sky projected orbital direction. Following Fekel (1981), we find out that the binary-disk misalignments at 46 au (the inner ring),  100 au (a gap), 188 au (the middle ring), and 338 au (outer ring) are 11 ± 6, ∼28,²⁰ 35 ± 5, and 40 ± 5 degrees, respectively. A schematic diagram of our disk model is displayed in Figure 2. Therefore, the inner ring and the AB-C binary plane are close to being coplanar, and there is a monotonic trend of binary-disk misalignment from ∼10° at ∼50 au to ∼40° at ∼340 au, consistent with the expected outcome of the disk misalignment (see Section 5.3).

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Several mechanisms could produce an initial binary-disk misalignment, such as turbulence in the star-forming gas clouds (Bate 2012), binary formation in the gas cloud whose physical axes are misaligned to the rotation axis (Bonnell & Bastien 1992), and accretion of cloud materials with misaligned angular momentum with respect to the binary after the binary formation (Bate 2018).

A test particle orbiting a binary on a misaligned orbit undergoes nodal precession due to gravitational perturbations from the binary (Nixon et al. 2011; Facchini et al. 2018). For a protoplanetary disk in the bending-wave regime (i.e., where the aspect ratio is higher than the α-prescription of viscosity; Shakura & Sunyaev 1973), disk parts at different radii shall undergo global precession like a rigid body with possibly a small warp (Smallwood et al. 2019). Therefore, the timescale of radial communication of disk materials (i.e., for pressure-induced bending waves propagating at half of the sound speed) and the timescale of global precession determine whether the disk can develop a misalignment inside.

Assuming an inner radius at 32 au (3–4 times the AB-C binary semimajor axis; Czekala et al. 2017; Kraus et al. 2020) and an outer radius at 1300 au (size of the gas disk; Fang et al. 2017), the global precession timescale of the entire disk is ∼0.83 Myr, and the radial communication timescale is ∼0.06 Myr (see Appendix D.1 and D.2 for detailed calculations). The radial communication is able to prevent the disk from breaking or developing a significant warp, and we would not expect the observed large deviations in the inclination and position angle between the inner and middle ring. Therefore, we propose that the gap between the inner and middle ring is deep enough to break the disk into two parts (hereafter the inner and outer disk), undergoing nodal precession independently, due to another mechanism.

Due to the viscous dissipation, the precessing disk is torqued toward either polar alignment (i.e., the binary-disk misalignment becomes 90°; Martin & Lubow 2017; Zanazzi & Lai 2018), or coplanar alignment/counteralignment. The minimum critical initial binary-disk misalignment for which a disk moves toward polar alignment is ∼63° in the limit of zero disk mass. Since a higher disk mass will lead to a larger critical angle (Martin & Lubow 2019), the GW Ori disk is most likely moving toward coplanar alignment. As we propose that the disk breaks into two parts undergoing global precession independently, they are also aligning to the binary independently on different timescales.

Assuming the radial communication is blocked at 60 au, we estimate the alignment timescales to be ∼1 Myr for the inner disk and above 100 Myr for the outer disk (see Appendix D.3). This is consistent with the observed significantly smaller inclination of the inner ring with respect to the binary than those of the outer rings. The latter are likely inherited from birth and have not evolved much given the system age.

If the radial communication is also blocked at the gap between the middle and outer rings (e.g., ∼250 au), the nodal precession timescales for the middle and outer rings would be ∼0.6 Myr and ∼120 Myr, respectively, and the two rings are likely to develop significantly different position angles. Thus we propose that the gap between the middle and outer rings does not cut off the radial communication, and the two rings precess roughly as a rigid body with only a small warp between them.

The analytical results suggest that the radial communication is able to prevent the binary from breaking the disk (e.g., Nixon et al. 2013). As a result, we propose a break at ∼60 au that is due to other mechanisms in order to explain the observed structures. We carry out a demonstrative smoothed particle hydrodynamic (SPH) simulation with the phantom code (Lodato & Price 2010; Price & Federrath 2010; Price et al. 2018) to test the nonbreaking hypothesis in the nonlinear regime. The results are shown in Figure 3.

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We model the triple-star system as the outer binary in order to speed up the simulation. The simulation consists of 10⁶ equal mass Lagrangian SPH particles initially distributed from r_in = 40 au to r_out = 400 au. The initial truncation radius of the disk does not affect the simulation significantly, since the material moves inwards quickly due to the short local viscous timescale. A smaller initial outer truncation radius r_out than what is observed is chosen in order to better resolve the disk. The binary begins at apastron with e_b = 0.22 and a_b = 9.2 au (Czekala et al. 2017). The accretion radius of each binary component is 4 au. Particles within this radius are accreted, and their mass and angular momentum are added to the star. We ignore the effect of self-gravity since it has no effect on the nodal precession rate of flat circumbinary disks.

The initial surface density profile is taken by

$$\begin{eqnarray}&&{\rm{\Sigma }}(r)={{\rm{\Sigma }}}_{0}{\left(r/{r}_{0}\right)}^{-3/2},\end{eqnarray} \tag{ 7 }$$

where Σ₀ is the density normalization at r₀ = 40 au, corresponding to a total disk mass of 0.1 M_⊙. We take a locally isothermal disk with a constant aspect ratio h/r = 0.05, where h is the scale height. The Shakura & Sunyaev (1973) α parameter varies in the range 0.008–0.013 over the disk. The SPH artificial viscosity α_AV = 0.31 mimics a disk with

$$\begin{eqnarray}&&\alpha \approx \displaystyle \frac{{\alpha }_{{\rm{av}}}}{10}\displaystyle \frac{\langle H\rangle }{h}\end{eqnarray} \tag{ 8 }$$

(Lodato & Price 2010), where $\langle H\rangle$ is the mean smoothing length on particles in a cylindrical ring at a given radius and we take β_AV = 2. The disk is resolved with average smoothing length per scale height of 0.32.

The evolution of surface density, binary-disk misalignment, and longitude of the ascending node at different radii suggest that the disk does not show any sign of breaking in 3000 binary's orbital periods (∼0.04 Myr), which is sufficiently long to tell if the disk would break or not since the radial communication timescale in the simulation is ∼0.01 Myr. Instead, the disk presents a global warp. The warp is not taken into account in the analytic estimates. The small outer truncation radius in the simulation leads to a faster precession timescale than that predicted by the analytic model, comparable to the radial communication timescale. The simulations suggests that unless the disk is very cool (i.e., low aspect ratio) and in the viscous regime, some other mechanism, e.g., a companion, is needed to break the disk at the gap between the inner and middle rings. This mechanism may also be responsible for producing the observed misalignment in the disk.

A disk with a lower aspect ratio or a higher α value, such that it falls into the viscous regime (h/r < α), may break due to the binary's torque (Nixon et al. 2013). However, observations have suggested lower α values than our adaption here (e.g., α ≲ 10⁻³, Flaherty et al. 2017). A lower viscosity leads to a larger binary truncation radius (Artymowicz & Lubow 1994), and a longer global precession timescale (Equation (D4)). Therefore we expect even less warping (and no break) in the GW Ori disk than in our simulation.

For a disk around a binary system of separation a_b, its radial communication timescale t_c can be estimated by Lubow & Martin (2018)

$$\begin{eqnarray}&&{t}_{c}\approx \displaystyle \frac{8}{5{{\rm{\Omega }}}_{b}{\left(h/r\right)}_{\mathrm{out}}}{\left(\displaystyle \frac{{r}_{\mathrm{out}}}{{a}_{{\rm{b}}}}\right)}^{3/2},\end{eqnarray} \tag{ D1 }$$

where r_out is the outer radius within which the disk is in good radial communication, and (h/r)_out is the aspect ratio at r_out, calculated based on the estimated temperature in the disk (Equation (2)). The radial communication timescale of the entire disk (i.e., r_out = 1300 au) is estimated to be ∼0.06 Myr.

If there is no radial communication in the disk, each part of the disk shall undergo differential precession with its local precession angular frequency ω_n,local given by Smallwood et al. (2019)

$$\begin{eqnarray}&&{\omega }_{{\rm{n}},\mathrm{local}}=k{\left(\displaystyle \frac{{a}_{{\rm{b}}}}{r}\right)}^{7/2}{{\rm{\Omega }}}_{{\rm{b}}},\end{eqnarray} \tag{ D2 }$$

where

$$\begin{eqnarray}&&k=\displaystyle \frac{3}{4}\sqrt{1+3{e}_{{\rm{b}}}^{2}-4{e}_{{\rm{b}}}^{4}}\,\displaystyle \frac{{M}_{1}{M}_{2}}{{\left({M}_{1}+{M}_{2}\right)}^{2}}\end{eqnarray} \tag{ D3 }$$

is a constant depending on the eccentricity of binary's orbit e_b, and the primary and secondary mass of the binary M₁ and M₂. However, for a protoplanetary disk in the bending-wave regime, where radial communication is active and prompt, the disk parts at different radii shall undergo global precession with the angular frequency ω_n,global given by Smallwood et al. (2019)

$$\begin{eqnarray}&&{\omega }_{{\rm{n}},\mathrm{global}}=k\,\left\langle {\left(\displaystyle \frac{{a}_{{\rm{b}}}}{r}\right)}^{7/2}\right\rangle \,{{\rm{\Omega }}}_{{\rm{b}}},\end{eqnarray} \tag{ D4 }$$

and

$$\begin{eqnarray}&&\left\langle {\left(\displaystyle \frac{{a}_{{\rm{b}}}}{r}\right)}^{7/2}\right\rangle =\displaystyle \frac{{\int }_{{r}_{\mathrm{in}}}^{{r}_{\mathrm{out}}}{\rm{\Sigma }}{r}^{3}{\rm{\Omega }}{\left({a}_{{\rm{b}}}/r\right)}^{7/2}{\rm{d}}r}{{\int }_{{r}_{\mathrm{in}}}^{{r}_{\mathrm{out}}}{\rm{\Sigma }}{r}^{3}{\rm{\Omega }}{\rm{d}}r}\end{eqnarray} \tag{ D5 }$$

is the angular momentum weighted averaging term, in which Ω(r) is the angular frequency at a given radius r, Σ(r) is the disk surface density with a radial dependence of r^−3/2, and r_in and r_out are inner and outer radii of the disk. Assuming r_in = 32 au and r_out = 1300 au, we take t_n,global = 2π/ω_n,global and estimate the global precession timescale of the entire GW Ori disk to be ∼0.83 Myr.

The alignment timescale t_a is given by Lubow & Martin (2018) and Bate et al. (2000)

$$\begin{eqnarray}&&{t}_{{\rm{a}}}=\displaystyle \frac{{\left(h/r\right)}^{2}\,{{\rm{\Omega }}}_{{\rm{b}}}}{\alpha \,{\omega }_{{\rm{n}}}^{2}},\end{eqnarray} \tag{ D6 }$$

where α = 0.01, h/r is defined in Equation (6), and the angular frequency of global precession (Equation (D4)) is used for ω_n.