Stephen H. Lubow

Dynamics of binary-disk interaction. 1: Resonances and disk gap sizes

We investigate the gravitational interaction of a generally eccentric binary star system with circumbinary and circumstellar gaseous disks. The disks are assumed to be coplanar with the binary, geometrically thin, and primarily governed by gas pressure and (turbulent) viscosity but not self-gravity. Both ordinary and eccentric Lindblad resonances are primarily responsible for truncating the disks in binaries with arbitrary eccentricity and nonextreme mass ratio. Starting from a smooth disk configuration, after the gravitational field of the binary truncates the disk on the dynamical timescale, a quasi-equilibrium is achieved, in which the resonant and viscous torques balance each other and any changes in the structure of the disk (e.g., due to global viscous evolution) occur slowly, preserving the average size of the gap. We analytically compute the approximate sizes of disks (or disk gaps) as a function of binary mass ratio and eccentricity in this quasi-equilibrium. Comparing the gap sizes with results of direct simulations using the smoothed particle hydrodynamics (SPH), we obtain a good agreement. As a by-product of the computations, we verify that standard SPH codes can adequately represent the dynamics of disks with moderate viscosity, Reynolds number R approximately 10(exp 3). For typical viscous disk parameters, and with a denoting the binary semimajor axis, the inner edge location of a circumbinary disk varies from 1.8a to 2.6a with binary eccentricity increasing from 0 to 0.25. For eccentricities 0 less than e less than 0.75, the minimum separation between a component star and the circumbinary disk inner edge is greater than a. Our calculations are relevant, among others, to protobinary stars and the recently discovered T Tau pre-main-sequence binaries. We briefly examine the case of a pre-main-sequence spectroscopic binary GW Ori and conclude that circumbinary disk truncation to the size required by one proposed spectroscopic model cannot be due to Linblad resonances, even if the disk is nonviscous.

Mass Flow through Gaps in Circumbinary Disks

We demonstrate through smoothed particle hydrodynamics simulations that a circumbinary disk can supply mass to the central binary through gas streams that penetrate the disk gap without closing it. The conditions for an efficient flow typically require the disk thickness-to-radius ratio z/r 0.05, if the turbulent viscosity parameter ? is greater than 0.01. This mass flow may be important for both the individual systems and their statistics. It occurs preferentially onto the lower mass object and acts toward equalization of component masses. The less massive component may be more luminous and easier to detect, owing to its larger accretion luminosity. For eccentric binaries, the mass flow is strongly modulated in time, providing diagnostics for both the disk and the binary. In the protostellar disks, the flow could be detected as shock emission phased with the binary orbit, resulting from stream impact with the circumstellar disks and/or the young stars. In the (super)massive black hole binaries in nuclei of galaxies, the flow may result from the surrounding interstellar medium and produce nearly periodic emission, as observed in quasar OJ 287. For star-planet-disk systems, our results show that the opening of a gap around a planet is not always sufficient for the termination of its growth. This suggests that planets supplied by gas streams from protoplanetary disks may outgrow Jupiter to become superplanets with properties heretofore reserved for stars.

Disk Accretion onto High-Mass Planets

We analyze the nonlinear, two-dimensional response of a gaseous, viscous protoplanetary disk to the presence of a planet of one Jupiter mass (1 MJ) and greater that orbits a 1 M☉ star by using the ZEUS hydrodynamics code with high resolution near the planets Roche lobe. The planet is assumed to be in a circular orbit around the central star and is not allowed to migrate. A gap is formed about the orbit of the planet, but there is a nonaxisymmetric flow through the gap and onto the planet. The gap partitions the disk into an inner (outer) disk that extends inside (outside) the planets orbit. For a 1 MJ planet and typical disk parameters, the accretion through the gap onto the planet is highly efficient. That is, the rate is comparable to the accretion rate toward the central star that would occur in the absence of the planet (at the location of the planet). For typical disk parameters, the mass-doubling timescale is less than 105 yr, considerably shorter than the disk lifetime. Following shocks near the L1 and L2 Lagrangian points, disk material enters the Roche lobe in the form of two gas streams. Shocks occur within the Roche lobe as the gas streams collide, and shocks lead to rapid inflow toward the planet within much of planets Roche lobe. Shocks also propagate in the inner and outer disks that orbit the star. For higher mass planets (of order 6 MJ), the flow rate onto the planet is considerably reduced, which suggests an upper mass limit to planets in the range of 10 MJ. This rate reduction is related to the fact that the gap width increases relative to the Roche (Hill sphere) radius with increasing planetary mass. The flow in the gap affects planetary migration. For the 1 MJ planet case, mass can penetrate from the outer disk to the inner disk, so that the inner disk is not depleted. The results suggest that most of the mass in gas giant planets is acquired by flows through gaps.

A model for tidally driven eccentric instabilities in fluid disks

Observations and simulations of superhump phenomena in close binary star systems and observations and previous studies of eccentric planetary rings, such as the rings of Uranus, indicate that eccentric instabilities occur in disks and rings. The eccentric stability of a fluid disk or ring in the presence of the tidal field of a comparison object which is in circular orbit is investigated. The disk can have pressure, viscosity, and self-gravity.

The effect of an external disk on the orbital elements of a central binary

Using smoothed particle hydrodynamics, the quasi-stationary structure of a circumbinary disk is found and its gravitational interaction with the central binary system is evaluated. The resulting rate of change of the orbital elements of the binary is examined. It is demonstrated that, in general, the semimajor axis decreases, the eccentricity grows, and the orbit precesses in the prograde sense. Numerical results for one particular model and a qualitative explanation for the effects found, mainly that of eccentricity driving, are provided. A protobinary or newly formed binary star embedded in a disk may undergo a significant orbital evolution before the disk dispersal. The application of the results to the interpretation of the observed statistics of binary eccentricities is discussed. 32 refs.

Simulations of tidally driven eccentric instabilities with application to superhumps

The tidally induced eccentric evolution of a fluid disk having pressure and viscosity is numerically analyzed. The effects of the 3:1 eccentric inner Lindblad resonance on disks in close binaries are studied, as a model for superhumps, using a two-dimensional SPH code. The numerical results for the m = 3 component of the perturbing potential are presented

Evolution of Giant Planets in Eccentric Disks

We investigate the interaction between a giant planet and a viscous circumstellar disk by means of high-resolution, two-dimensional hydrodynamic simulations. We consider planetary masses that range from 1 to 3 Jupiter masses (MJ) and initial orbital eccentricities that range from 0 to 0.4. We find that a planet can cause eccentricity growth in a disk region adjacent to the planets orbit, even if the planets orbit is circular. Disk-planet interactions lead to growth in a planets orbital eccentricity. The orbital eccentricities of a 2MJ and a 3MJ planet increase from 0 to 0.11 within about 3000 orbits. Over a similar time period, the orbital eccentricity of a 1MJ planet grows from 0 to 0.02. For a case of a 1MJ planet with an initial eccentricity of 0.01, the orbital eccentricity grows to 0.09 over 4000 orbits. Radial migration is directed inward but slows considerably as a planets orbit becomes eccentric. If a planets orbital eccentricity becomes sufficiently large, e 0.2, migration can reverse and so be directed outward. The accretion rate toward a planet depends on both the disk and the planetary orbital eccentricity and is pulsed over the orbital period. Planetary mass growth rates increase with planetary orbital eccentricity. For e ~ 0.2, the mass growth rate of a planet increases by ~30% above the value for e = 0. For e 0.1, most of the accretion within the planets Roche lobe occurs when the planet is near the apocenter. Similar accretion modulation occurs for flow at the inner disk boundary, which represents accretion toward the star.

Saturation of the Corotation Resonance in a Gaseous Disk

We determine the torque exerted in a steady state by an external potential on a three-dimensional gaseous disk at a non-coorbital corotation resonance. Our model accounts for the feedback of the torque on the surface density and vorticity in the corotation region and assumes that the disk has a barotropic equation of state and a nonzero effective viscosity. The ratio of the torque to the value given by the formula of Goldreich & Tremaine depends essentially on a single dimensionless parameter, which quantifies the extent to which the resonance is saturated. We discuss the implications for the eccentricity evolution of young planets.

On the tilting of protostellar disks by resonant tidal effects

We consider the dynamics of a protostellar disk surrounding a star in a circular-orbit binary system. Our aim is to determine whether, if the disk is initially tilted with respect to the plane of the binary orbit, the inclination of the system will increase or decrease with time. The problem is conveniently formulated in the binary frame in which the tidal potential of the companion star is static. We may then consider a steady, flat disk that is aligned with the binary plane and investigate its linear stability with respect to tilting or warping perturbations. The dynamics is controlled by the competing effects of the m = 0 and m = 2 azimuthal Fourier components of the tidal potential. In the presence of dissipation, the m = 0 component causes alignment of the system, while the m = 2 component has the opposite tendency. We find that disks that are sufficiently large, in particular those that extend to their tidal truncation radii, are generally stable and will therefore tend to alignment with the binary plane on a timescale comparable to that found in previous studies. However, the effect of the m = 2 component is enhanced in the vicinity of resonances where the outer radius of the disk is such that the natural frequency of a global bending mode of the disk is equal to twice the binary orbital frequency. Under such circumstances, the disk can be unstable to tilting and acquire a warped shape, even in the absence of dissipation. The outer radius corresponding to the primary resonance is always smaller than the tidal truncation radius. For disks smaller than the primary resonance, the m = 2 component may be able to cause a very slow growth of inclination through the effect of a near resonance that occurs close to the disk center. We discuss these results in the light of recent observations of protostellar disks in binary systems.

Three-dimensional Waves Generated at Lindblad Resonances in Thermally Stratified Disks

We analyze the linear, three-dimensional response to tidal forcing of a disk that is thin and thermally stratified in the direction normal to the disk plane. We model the vertical disk structure locally as a polytrope that represents a disk of high optical depth. We solve the three-dimensional gasdynamics equations semianalytically in the neighborhood of a Lindblad resonance. These solutions match asymptotically onto those valid away from resonances (previously obtained by Korycansky & Pringle) and provide solutions valid at all radii r. We obtain the following results: (1) A variety of waves are launched at the resonance, including r-modes and g-modes. However, the f-mode carries more than 95% of the torque exerted at the resonance. (2) These three-dimensional waves collectively transport exactly the amount of angular momentum predicted by the standard two-dimensional resonant torque formula. (3) Near resonance, the f-mode behaves compressibly and occupies the full vertical extent of the disk. Away from resonance, the f-mode behaves incompressibly, becomes confined near the surface of the disk, and, in the absence of other dissipation mechanisms, damps via shocks. In general, the radial length scale for this process is roughly rL/m (for resonant radius rL and azimuthal tidal forcing wavenumber m), independent of the disk thickness H. This wave-channeling process is due to the variations of physical quantities in r and is not due to wave refraction. (4) However, the inwardly propagating f-mode launched from an m = 2 inner Lindblad resonance experiences relatively minor channeling (accompanied by about a factor of 5 increase in nonlinearity), all the way to the radial center of the disk. We conclude that for binary stars, tidally generated waves at Lindblad resonances in highly optically thick circumbinary disks are subject to strong nonlinear damping by the channeling mechanism, while those in circumstellar accretion disks are subject to weaker nonlinear effects. We also apply our results to waves excited by young planets for which m ≈ r/H and conclude that the waves are likely damped on the scale of a few H.