D. N. C. Lin

On the tidal interaction between protoplanets and the protoplanetary disk. III - Orbital migration of protoplanets

The tidal interaction between a protoplanet and a gaseous protoplanetary disk is investigated, and the dynamical evolution of the disk and the orbital migration of the protoplanet in a self-consistent manner is considered. It is shown that the orbital migration of a protoplanet does not suppress the tendency for tidal truncation in the vicinity of its orbit. If the necessary condition for tidal truncation is satisfied, the protoplanet induces a tidal feedback mechanism that regulates the rate of angular momentum transfer between the protoplanet and the disk. Significant orbital migration can only occur on the viscous evolution time scale of the disk.

The Occurrence and Mass Distribution of Close-in Super-Earths, Neptunes, and Jupiters

Closing in on Extraterrestrial Earths With close to 500 extrasolar planets discovered to date, researchers are starting to estimate the occurrence of low-mass planets to help our understanding of how planets form and evolve. Based on observations of 166 nearby stars with the Keck Telescope, Howard et al. (p. 653) report the occurrence of short-period planets around Sun-like stars as a function of planet mass. Planet formation models predicted that planet occurrence would increase with decreasing mass, such that satellites with masses similar to that of Neptune, and less, would be more common than gas-giant planets like Jupiter. Contrary to predictions, there is no dearth of planets with masses 5 to 30 times that of Earth, implying that the models may need revision. Nevertheless, observations suggest that 23% of Sun-like stars may be orbited by a close-in, terrestrial mass planet. About one-quarter of observed Sun-like stars harbors a close-in terrestrial-mass planet. The questions of how planets form and how common Earth-like planets are can be addressed by measuring the distribution of exoplanet masses and orbital periods. We report the occurrence rate of close-in planets (with orbital periods less than 50 days), based on precise Doppler measurements of 166 Sun-like stars. We measured increasing planet occurrence with decreasing planet mass (M). Extrapolation of a power-law mass distribution fitted to our measurements, df/dlogM = 0.39 M−0.48, predicts that 23% of stars harbor a close-in Earth-mass planet (ranging from 0.5 to 2.0 Earth masses). Theoretical models of planet formation predict a deficit of planets in the domain from 5 to 30 Earth masses and with orbital periods less than 50 days. This region of parameter space is in fact well populated, implying that such models need substantial revision.

Toward a Deterministic Model of Planetary Formation. II. The Formation and Retention of Gas Giant Planets around Stars with a Range of Metallicities

The apparent dependence of detection frequency of extrasolar planets on the metallicity of their host stars is investigated with Monte Carlo simulations using a deterministic core-accretion planet formation model. According to this model, gas giants formed and acquired their mass Mp through planetesimal coagulation followed by the emergence of cores onto which gas is accreted. These protoplanets migrate and attain their asymptotic semimajor axis a through tidal interaction with their nascent disk. Based on the observed properties of protostellar disks, we generate an Mp-a distribution. Our results reproduce the observed lack of planets with intermediate mass Mp = 10-100 M⊕ and a 3 AU and with large mass Mp 103 M⊕ and a 0.2 AU. Based on the simulated Mp-a distributions, we also evaluate the metallicity dependence of the fraction of stars harboring planets that are detectable with current radial velocity surveys. If protostellar disks attain the same fraction of heavy elements as contained in their host stars, the detection probability around metal-rich stars would be greatly enhanced because protoplanetary cores formed in them can grow to several Earth masses prior to their depletion. These large masses are required for the cores to initiate rapid gas accretion and to transform into giant planets. The theoretically extrapolated metallicity dependence is consistent with the observations. This correlation does not arise naturally in the gravitational-instability scenario. We also suggest other metallicity dependences of the planet distributions that can be tested by ongoing observations.

Tidally Induced Gap Formation in Protostellar Disks: Gap Clearing and Suppression of Protoplanetary Growth

We present the results of numerical simulations of protostellar accretion disks that are perturbed by a protoplanetary companion that has a much smaller mass than the central object. We consider the limiting cases where the companion is in a coplanar, circular orbit and is initially embedded in the disk. Three independent numerical schemes are employed, and generic features of the flow are found in each case. In the first series of idealized models, the secondary companion is modeled as a massless, orbiting sink hole able to absorb all matter incident upon it without exerting any gravitational torque on the disk. In these simulations, accretion onto the companion induces surface density depression and gap formation centered on its orbital radius. After an initial transitory adjustment, the accretion rate onto the sink hole becomes regulated by the rate at which viscous evolution of the disk can cause matter to diffuse into the vicinity of the sink hole orbit, and thus the sink hole grows on a disk viscous timescale. In the second series of comprehensive models, the companions gravity is included. When the tidal torque exerted by the companion on the disk becomes important, the angular momentum exchange rate between the companion and the disk induces the protoplanetary accretion to reduce markedly below that in the idealized sink hole models. Whether this process is effective in inhibiting protoplanetary accretion depends on the equation of state and disk model parameters. For polytropic or isothermal equations of state, we find, in basic agreement with earlier work, that when the mean Roche lobe radius of the companion exceeds the disk thickness and when the mass ratio, q, between the companion and the central object exceeds ~40/, where is the effective Reynolds number, a clean deep gap forms. Although precise estimation is rendered difficult due to the limitation of numerical schemes in dealing with large density contrasts, the generic results of three series of simulations indicate that accretion onto sufficiently massive protoplanets can become ineffective over the expected disk lifetimes in their neighborhood. We suggest that such a process operates during planetary formation and is important in determining the final mass of giant planets.

On the tidal inflation of short-period extrasolar planets

We examine the consequences of tidal interaction between eccentric short-period extrasolar planets and their host stars and of secular perturbations between planets in a given system. If the planet is within 0.05 AU of the star, the dissipation of the stellar tidal disturbance within the planet provides a significant energy source, which causes the planet to inflate as it adjusts to a thermal equilibrium. We determine the planetary size as a function of the tidal dissipation rate with or without the presence of a core. Inflation intensifies the star-planet tidal interaction and accelerates the pace of the planets spin synchronization and orbital circularization. We apply our results to three systems with short-period planets: HD 209458, Ups And, and Tau Boo.

Toward a Deterministic Model of Planetary Formation. IV. Effects of Type I Migration

In a further development of a deterministic planet formation model (Ida & Lin), we consider the effect of type I migration of protoplanetary embryos due to their tidal interaction with their nascent disks. During the early phase of protostellar disks, although embryos rapidly emerge in regions interior to the ice line, uninhibited type I migration leads to their efficient self-clearing. But embryos continue to form from residual planetesimals, repeatedly migrate inward, and provide a main channel of heavy-element accretion onto their host stars. During the advanced stages of disk evolution (a few Myr), the gas surface density declines to values comparable to or smaller than that of the minimum mass nebula model, and type I migration is no longer effective for Mars-mass embryos. Over wide ranges of initial disk surface densities and type I migration efficiencies, the surviving population of embryos interior to the ice line has a total mass of several M⊕. With this reservoir, there is an adequate inventory of residual embryos to subsequently assemble into rocky planets similar to those around the Sun. However, the onset of efficient gas accretion requires the emergence and retention of cores more massive than a few M⊕ prior to the severe depletion of the disk gas. The formation probability of gas giant planets and hence the predicted mass and semimajor axis distributions of extrasolar gas giants are sensitively determined by the strength of type I migration. We suggest that the distributions consistent with observations can be reproduced only if the actual type I migration timescale is at least an order of magnitude longer than that deduced from linear theories.

Tidal Dissipation in Rotating Giant Planets

Many extrasolar planets orbit sufficiently close to their host stars that significant tidal interactions can be expected, resulting in an evolution of the spin and orbital properties of the planets. The accompanying dissipation of energy can also be an important source of heat, leading to the inflation of short-period planets and even mass loss through Roche lobe overflow. Tides may therefore play an important role in determining the observed distributions of mass, orbital period, and eccentricity of the extrasolar planets. In addition, tidal interactions between gaseous giant planets in the solar system and their moons are thought to be responsible for the orbital migration of the satellites, leading to their capture into resonant configurations. Traditionally, the efficiency of tidal dissipation is simply parameterized by a quality factor Q, which depends, in principle, in an unknown way on the frequency and amplitude of the tidal forcing. In this paper we treat the underlying fluid dynamical problem with the aim of determining the efficiency of tidal dissipation in gaseous giant planets such as Jupiter, Saturn, or the short-period extrasolar planets. Efficient convection enforces a nearly adiabatic stratification in these bodies, which may or may not contain solid cores. With some modifications, our approach can also be applied to low-mass stars with extended convective envelopes. In cases of interest, the tidal forcing frequencies are typically comparable to the spin frequency of the planet but are small compared to its dynamical frequency. We therefore study the linearized response of a slowly and possibly differentially rotating planet to low-frequency tidal forcing. Convective regions of the planet support inertial waves, which possess a dense or continuous frequency spectrum in the absence of viscosity, while any radiative regions support generalized Hough waves. We formulate the relevant equations for studying the excitation of these disturbances and present a set of illustrative numerical calculations of the tidal dissipation rate. We argue that inertial waves provide a natural avenue for efficient tidal dissipation in most cases of interest. In the presence of a solid core, the excited disturbance tends to be localized on a web of rays rather than resembling a smooth eigenfunction. The resulting value of Q depends, in principle, in a highly erratic way on the forcing frequency, but we provide analytical and numerical evidence that the frequency-averaged dissipation rate may be asymptotically independent of the viscosity in the limit of small Ekman number. For a smaller viscosity, the tidal disturbance has a finer spatial structure and individual resonances are more pronounced. In short-period extrasolar planets, tidal dissipation via inertial waves becomes somewhat less efficient once they are spun down to a synchronous state. However, if the stellar irradiation of the planet leads to the formation of a radiative outer layer that supports generalized Hough modes, the tidal dissipation rate can be enhanced, albeit with significant uncertainty, through the excitation and damping of these waves. The dissipative mechanisms that we describe offer a promising explanation of the historical evolution and current state of the Galilean satellites, as well as the observed circularization of the orbits of short-period extrasolar planets.

TIDAL DISSIPATION IN ROTATING SOLAR-TYPE STARS

We calculate the excitation and dissipation of low-frequency tidal oscillations in uniformly rotating solar-type stars. For tidal frequencies smaller than twice the spin frequency, inertial waves are excited in the convective envelope and are dissipated by turbulent viscosity. Enhanced dissipation occurs over the entire frequency range rather than in a series of very narrow resonant peaks and is relatively insensitive to the effective viscosity. Hough waves are excited at the base of the convective zone and propagate into the radiative interior. We calculate the associated dissipation rate under the assumption that they do not reflect coherently from the center of the star. Tidal dissipation in a model based on the present Sun is significantly enhanced through the inclusion of the Coriolis force but may still fall short of that required to explain the circularization of close binary stars. However, the dependence of the results on the spin frequency, tidal frequency, and stellar model indicate that a more detailed evolutionary study including inertial and Hough waves is required. We also discuss the case of higher tidal frequencies appropriate to stars with very close planetary companions. The survival of even the closest hot Jupiters can be plausibly explained provided that the Hough waves they generate are not damped at the center of the star. We argue that this is the case because the tide excited by a hot Jupiter in the present Sun would marginally fail to achieve nonlinearity. As conditions at the center of the star evolve, nonlinearity may set in at a critical age, resulting in a relatively rapid inspiral of the hot Jupiter.

Toward a Deterministic Model of Planetary Formation. V. Accumulation Near the Ice Line and Super-Earths

We address two outstanding issues in the sequential accretion scenario for gas giant planet formation, the retention of dust grains in the presence of gas drag and that of cores despite type I migration. The efficiency of these processes is determined by the disk structure. Theoretical models suggest that planets form in protostellar disk regions with an inactive neutral dead zone near the midplane, sandwiched together by partially ionized surface layers where magnetorotational instability is active. Due to a transition in the abundance of dust grains, the active layers thickness decreases abruptly near the ice line. Over a range of modest accretion rates (~10−9 to 10−8 -->M☉ yr−1), the change in the angular momentum transfer rate leads to local surface density and pressure distribution maxima near the ice line. The azimuthal velocity becomes super-Keplerian and the grains accumulate in this transition zone. This barrier locally retains protoplanetary cores and enhances the heavy-element surface density to the critical value needed to initiate efficient gas accretion. It leads to a preferred location and epoch of gas giant formation. We simulate and reproduce the observed frequency and mass-period distribution of gas giants around solar-type stars without having to greatly reduce the type I migration strength. The mass function of the short-period planets shows an enhanced population of super-Earths relative to hot Jupiters, and it can be utilized to calibrate the efficiency of type I migration and to extrapolate the fraction of stars with habitable terrestrial planets.

Grain Retention and Formation of Planetesimals near the Snow Line in MRI-driven Turbulent Protoplanetary Disks

The first challenge in the formation of both terrestrial planets and the cores of gas giants is the retention of grains in protoplanetary disks. In most regions of these disks, gas attains sub-Keplerian speeds as a consequence of a negative pressure gradient. Hydrodynamic drag leads to orbital decay and depletion of the solid material in the disk, with characteristic timescales as short as only a few hundred years for meter-sized objects at 1 AU. In this Letter, we suggest a particle retention mechanism that promotes the accumulation of grains and the formation of planetesimals near the water sublimation front or snow line. This model is based on the assumption that, in the regions most interesting for planet formation, the viscous evolution of the disk is due to turbulence driven by the magnetorotational instability (MRI) in the surface layers of the disk. The depth to which MRI effectively generates turbulence is a strong function of grain size and abundance. A sharp increase in the grain-to-gas density ratio across the snow line reduces the column depth of the active layer. As the disk evolves toward a quasi-steady state, this change in the active layer creates a local maximum in radial distribution of the gas surface density and pressure, causing the gas to rotate at super-Keplerian speed and halting the inward migration of grains. This scenario presents a robust process for grain retention that may aid in the formation of proto-gas giant cores preferentially near the snow line.