Astrophys Space Sci (2011) 335:125–129 DOI 10.1007/s10509-011-0601-5
O R I G I N A L A RT I C L E
Gaseous flows in the inner part of the circumbinary disk of the T Tauri star A.M. Fateeva · D.V. Bisikalo · P.V. Kaygorodov · A.Y. Sytov
Received: 18 October 2010 / Accepted: 6 January 2011 / Published online: 27 January 2011 © Springer Science+Business Media B.V. 2011
Abstract We present results of 3D numerical simulations of the matter flow in the disk of a binary T Tauri star. It is shown that two bow-shocks caused by the supersonic motion of the binary components in the gas of the disk are formed in the system having parameters typical for T Tauri stars. These bow-shocks significantly change the flow pattern. In particular, for systems with circular orbits they determine the size and shape of the inner gap. We also show that the redistribution of the angular momentum due to the bow-shocks leads to occurrence of two matter flows propagating from the inner edge of the circumbinary disk to the components. Further redistribution of this matter between the components is considered. Keywords Binary stars · T Tauri · Accretion disks · Gap formation · Gas dynamic simulations
1 Introduction T Tauri stars (TTSs) are pre-main sequence stars aged 106 – 107 years. They are at the intermediate stage between protostars and young low mass (1–2 M ) stars of the main sequence. It is known that the most of TTSs are binary and even multiple systems (Mathieu et al. 1989). In this paper we consider only binary TTSs. These objects are commonly surrounded by a circumbinary disk, a remnant of the protostellar cloud. For many years those disks have been observed in the UV (Brooks and Costa 2003; Carmona 2010), scattered visual starlight (McCaughrean and O’dell 1996; Watson et al. 2007), infrared (Duvert A.M. Fateeva () · D.V. Bisikalo · P.V. Kaygorodov · A.Y. Sytov Institute of Astronomy of the RAS, Moscow, Russia e-mail:
[email protected]
et al. 1998; Close et al. 1998; Roddier et al. 1996; Sargent and Beckwith 1991) and at millimeter wavelengths (Carmona 2010). Unfortunately, the spatial resolution of modern observational facilities has not been sufficient enough to resolve and directly map flow details in inner regions of the disk. There are only few wide binary TTSs where researchers recently managed to resolve the inner structure of the disks using the Subaru telescope (Mayama et al. 2010). Numerical simulations of gas accretion processes in young binaries have been performed by several groups since 1994: Artymowicz and Lubow (1994, 1996), Bate and Bonnell (1997), Sotnikova and Grinin (2007), Ochi et al. (2005), Hanawa et al. (2010), Matsuda (2010). The simulations and comparison of their results with observational data allowed obtaining information about various gas dynamic details. However, there is still no general view concerning the flow structure in inner regions of binary TTSs. For example, we still have no answer for the crucial question concerning the evolution of those stars, which component, primary or secondary accretes more. First simulations of the gas accretion onto a binary system were performed by Artymowicz and Lubow in 1994 using the smoothed-particle hydrodynamics (SPH) technique. Despite recognized disadvantages of the SPH technique (high numerical viscosity, low spatial resolution with a reasonable number of particles, heavily smeared-out shock waves and contact discontinuities, etc.) the main flow elements including a gap in the inner region of the disk, matter flows through the gap and circumstellar disks around the components were identified in a number of studies (Artymowicz and Lubow 1994, 1996; Bate and Bonnell 1997; Sotnikova and Grinin 2007). Since 2005 numerical simulations of gas dynamics in TTSs based on total variation diminishing (TVD) schemes have been performed by Ochi et al. (2005), Hanawa et al.
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(2010), Matsuda (2010). Results obtained with the TVD methods have the higher spatial resolution and describe shock waves more precisely. The flow structure details found in these simulations were observed by the Subaru coronograph (Mayama et al. 2010). Despite the progress achieved during last few years there are a number of unanswered crucial questions on physical processes running in these systems. For example, we still need to reveal a mechanism of the gap formation; explain reasons of formation of observationally found flows from the inner edge of the circumbinary disk to the components; make it clear, which component accretes more. This paper is dedicated to investigations of the matter flow morphology in inner regions of the circumbinary disk.
2 The model In this paper we consider a binary T Tauri star, having the following parameters: components’ masses are M1 = 0.6 M and M2 = 1 M ; binary separation is A = 0.17 AU; orbital period is equal to 19.65 days. We assume that the orbits of the components are circular. The system is surrounded by a circumbinary disk which is in the hydrostatic equilibrium and has a Keplerian velocity distribution. Adopted density ρ0 = 10−11 g/cm3 and temperature in the disk Tdisk = 2500 K are typical for stars of this type. Real circumbinary disks have radii of few hundreds of AU but in this study we simulate only a limited inner region having a radius Rext ∼ 1 AU or ∼ 6 separations. It is important that in the considered system the orbital velocity of the components is 20 times higher than the sound speed. Furthermore, simple estimates show that in TTSs with mass ratios q = M2 /M1 from 0.1 to 1 and orbital periods up to few hundred years, orbital velocities of the components are supersonic. To describe the flow structure we use the system of equations of gravitational gas dynamics closed by the equation of state of the ideal gas as it is done in papers of Bisikalo et al. (2000, 2003): ⎧ ∂ρ ⎪ ∂t + div ρv = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ρv + div(ρv ⊗ v) + grad P = −ρ grad − 2ρ[ × v] ∂t
⎪ ∂ρ(ε+|v|2 /2) ⎪ ⎪ + div ρv(ε + P /ρ + |v|2 /2) = −ρv grad ⎪ ∂t ⎪ ⎪ ⎪ ⎩ P = (γ − 1)ρε (1) Here ρ is the density, v = (u, v, w) is the velocity vector, P is the pressure, ε is the thermal energy and is the angular velocity of the system rotation. To mimic the system with radiative losses, we accept the value of the adiabatic index
γ = 1.01, that corresponds to the case close to isothermal one (Sawada et al. 1987). The system is solved numerically using the Roe-OsherEinfeld finite-difference scheme. The scheme has the first order of approximation in time and third order in space. On the cylindrical outer boundary the hydrostatic density distribution is adopted. The velocity in boundary cells on the sidesurface of the cylinder is put equal to the local keplerian velocity relatively to the center of mass of the system. The radial component of the velocity is set to conserve the constant rate of the matter inflow into the disk M˙ = 10−8 M /yr through the entire side-surface of the cylinder. The vertical component of the velocity is equal to zero. In planes corresponding to the upper and lower boundaries of the computational domain (z = 1.2A and z = −1.2A) the constant boundary conditions were adopted with the density ρbnd = 10−6 ρ0 , temperature T = Tdisk , and velocity equal to zero in the laboratory frame. On the surface of the stars a free inflow boundary condition was defined i.e. all the material which eventually appears in the star cells is supposed to be captured by the stars. A Keplerian velocity distribution relatively to the center of mass of the system is set for the entire computational domain as an initial condition. It is known, that proper magnetic fields of pre-main sequence stars lie in a range 102 ÷ 103 Gs (Johns-Krull et al. 1999; Guenther et al. 1999). In this case a radius of the magnetosphere is comparable with the stellar radius. At the same time the last stable orbit of the circumstellar disk is about 10 Rstar (Paczynski 1977). Thus, the magnetic field influence becomes significant only in a small domain around the binary components and is negligible in the gap, so in the frame of the considered problem we can neglect it with no accuracy loss. The simulations are performed in a coordinate frame rotating with the binary. The center of the frame coincides with the center of the secondary. The primary’s coordinates are (A, 0, 0). The stars were defined as spheres both having radii of R = 1 R . The accepted radius is less than the radius of the last stable orbit of the circumstellar disk, then this assumption will not influence the solution, since all physical processes (formation of the circumstellar disks, accretion processes, etc.) will be taken into account in the model. The spin rotation of the stars is synchronized with the rotation of the system. The rotation axes of both the stars are aligned with the axis of the orbital rotation. The computational domain size is 12A × 12A × 2.4A (2.04 AU × 2.04 AU × 0.408 AU). Equations (1) are solved in a non-uniform Cartesian grid with the size 480 × 480 × 224 cells. The grid becomes denser in the central region of 1.5A × 1.5A × 0.2A and has there the constant resolution equal to 0.01 separation per cell. The computations are performed using the MVS-100K supercomputer of Joint Supercomputer Center.
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Fig. 1 Bird-eye view of the 3D density logarithm distribution at the angle of 45◦ . Black points denote the components. The primary component is the right-most and the secondary is in the left. The system rotates counter-clockwise Fig. 3 The matter flux distribution and velocity field in the central region of the system. The bold dashed lines denotes the marginal lines separating the streams directed to and from the components. The flow lines corresponding to the main matter streams in the system are also shown
Fig. 2 The density distribution and velocity field over the central region of the equatorial plane of the system. The Roche equipotentials are depicted by the dashed line. White circles denote the components. The primary component is the right-most and the secondary is in the left. The system rotates counter-clockwise
3 Results The morphology of the flow pattern in the considered systems is shown in Figs. 1 and 2. Figure 1 represents the 3D density distribution over the entire computational domain and Fig. 2 shows the density distribution over the equatorial plane in inner regions of the system. In figures the primary component is the right-most and the secondary is in the left. Analysis of the density and velocity distributions plotted in Figs. 1 and 2 shows that the flow structure is rather complicated and includes a number of details. In particular, a gap with the specific size ∼ 2.4A containing rarefied gas is formed. The solution clearly demonstrates presence of circumstellar disks surrounding the components, bow-shocks
before both the components and “bridge” connecting the circumstellar accretion disks. By recent times it had been accepted that the main mechanism of the gap formation was that of the paper of Artymowicz and Lubow (1994). According to the mechanism the gap size is determined by a position of the strongest Lindblad resonance. But as it is seen in Fig. 2 gas velocities in the rarefied region near the binary are significantly nonKeplerian. This means that the flow pattern in this region is mainly governed by gas dynamic processes. 2D numerical simulations described in Kaigorodov et al. (2010) show that in the system with circular orbits a radius of the gap formed by the bow-shocks is larger than the gap radius calculated using positions of the resonances. Besides the “bow-shock radius” fits observations better. Results of the 3D simulations proved the conclusion that the bow-shocks occurring due to the supersonic motion of the components is the gas of the disk drastically change the flow pattern. In particular, they govern the size and shape of the gap at least in systems with the zero eccentricity. Analysis of the fluxes shown in Fig. 3 demonstrates that the re-distribution of the angular momentum in the envelope due to the bow-shocks leads to occurrence of two distinguishable flows propagating from the inner edge of the circumbinary disk to the components. Let us consider streams moving along the bow-shocks. Points of the head-on collision are located at rather large distances from the components near the L3 point for the primary and L2 for the secondary. The matter in the gap splits into two streams when colliding with the bow-shock. The first portion of matter (stream A in Fig. 3) loses its angular momentum at the shock and starts to move toward the circumstellar disk. It forms
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Fig. 4 The density distribution and velocity field in the vicinity of the inner Lagrangian point. Dashed lines denote the Roche lobes of the components
a distinguishable spiral starting from the head-on collision point. The second portion of matter (stream B in Fig. 3) moves from the head-on collision point along the bow-shock to the circumbinary disk. This stream carries the excess of angular momentum and finally mixes with matter of the disk. It is interesting to note that the matter stream moving to the circumbinary disk determines the gap size. The matter from the inner edge of the circumbinary disk starts to flow inward the gap since it is pushed by the pressure gradient.1 After the half-orbit this process is interrupted by the upcoming bow-shock of the other component. Thus, the gap size and its density distribution are governed by the bow-shocks. The presented analysis of the streams allows us to make certain conclusions on the rates of accretion onto each component of the system. If q < 1 the secondary moves faster than the primary, so its bow-shock is stronger. It means that the gas colliding with this shock loses more of its angular momentum than the gas colliding with the shock of the primary and it must lead to the higher flux onto the secondary. Furthermore, the secondary is located closer to the edge of the circumbinary disk where the gas density is higher and, thus, the flux onto this component must increase even more. Indeed, analysis of the fluxes shown in Fig. 3 demonstrates that starting from the head-on collision point the matter flux along the secondary bow-shock notably exceeds the same flux along the primary shock. Let us again consider the streams propagating toward the components after the angular momentum re-distribution at the shocks. A circumstellar accretion disk may accept 1 The
pressure gradient occurs due to the large difference between circumbinary disk and gap densities (three orders of magnitude) while the additional heating of the gas in the gap is relatively small (less than one order of magnitude).
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not more matter then it is allowed by the viscosity. A portion of the gas having lost its angular momentum replenishes the circumstellar accretion disk of the corresponding component, and the rest of the matter turns the disk and collides with the same stream of another component. As a result, a bridge-like stationary shock between the accretion disks of the components is formed (see Fig. 4). Due to this collision the streams lose more of the angular momentum, since it is partially annihilated. Thus, the gas obtains an additional opportunity to be accreted by one of the components. If we account for a higher mass (and as a rule larger radius of the gravitational capture) of the primary we conclude that the matter having lost its angular momentum is mainly accreted by the primary. As we see in Fig. 4 the “bridge” is significantly tilted. While it sets its rightmost end to the circumstellar disk of the primary, its opposite end is almost tangent to the circumstellar disk of the secondary. This effect occurs due to the higher orbital velocity of the secondary that allows the incident flow (part of the stream A) to push the shock farther. As a consequence, the part of the stream moving along the “bridge” toward the primary directly collides with its circumstellar disk. This is similar to the “hot line” shock in close binaries occurring due to the collision of the stream from the L1 point and the circumdisk halo (Boyarchuk et al. 2002; Bisikalo et al. 2003). The oblique collision of two streams must lead to the system of two divergent shocks and contact discontinuity between them. When the matter passes through these shocks it loses more of the angular momentum. These additional losses lead to the higher rate of the accretion onto the primary. Thus, despite the higher matter flux from the circumbinary disk to the secondary, the rate of accretion onto the primary must be higher.
4 Conclusions Due to the supersonic motion of the components of TTSs two bow-shocks are formed in the circumbinary disks of these stars. The shocks lead to re-distribution of the angular momentum in the system. In the inner region of the system the velocity distribution is far from the Keplerian one and the flow structure is mainly governed by the bow-shocks. Presence of the stationary bow-shocks leads to formation of a gap, rarefied region in the circumbinary disk having a radius ∼ 2.4A (for systems with the zero eccentricity). The gas from the inner edge of the circumbinary disk starts to move inward the gap and finally collide with the bow shock of the secondary component. At the front of this bow-shock the gas splits into two streams. One of them moves to the circumstellar disk of the component, another one moves outwards, to the inner edge of the circumbinary disk removing the exceeding angular momentum.
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A rate of the accretion onto a star is limited by the rate of the angular momentum transfer in its circumstellar accretion disk. In the considered model efficiency of the angular momentum transfer is not high enough for the disk to accept all the falling matter. As it is shown in Figs. 3 and 4 the exceeding matter turns the disks and moves to the region between them. There the colliding streams form a stationary shock having a shape of a bridge between the disks. The collision of matter streams in the region between the disks leads to mutual annihilation of their angular momentum and makes most of the matter fall onto the accretion disk of the primary. The collision of the stream with the circumstellar disk of the primary leads to occurrence of a system of shock waves and, thus, to additional losses of the angular momentum. It increases the efficiency of the angular momentum transfer in the disk. All these mechanisms finally lead to the increase of the rate of accretion onto the more massive binary component. As it follows from the performed analysis shock waves and especially bow-shocks play an important role in T Tauri stars. The gas behind the bow-shocks heat up to temperatures of few thousand Kelvins and its emission in the UV band becomes stronger. Further progress in investigations of TTSs is concerned with opportunities to observe them in the UV. Thus the launch of the WSO UV observatory must lead to a breakthrough in our understanding of physics of such interesting objects as T Tauri stars. Acknowledgements This work was supported by the Basic Research Program of the Presidium of the Russian Academy of Sciences, Russian Foundation for Basic Research (projects 08-02-00371, 09-0200064), Federal Targeted Program “Science and Science Education for Innovation in Russia 2009–2013”.
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