Three-body Orbital Integration Research Articles

ORBITAL CHARACTERISTICS OF PLANETESIMALS CAPTURED BY CIRCUMPLANETARY GAS DISKS

ABSTRACT Sufficiently massive growing giant planets have circumplanetary disks, and the capture of solid bodies by the disks would likely influence the growth of the planets and formation of satellite systems around them. In addition to dust particles that are supplied to the disk with inflowing gas, recent studies suggest the importance of capture of planetesimals whose motion is decoupled from the gas, but the orbital evolution of captured bodies in circumplanetary gas disks has not been studied in detail. In the present work, using three-body orbital integration and analytic calculations, we examine orbital characteristics and subsequent dynamical evolution of planetesimals captured by gas drag from circumplanetary gas disks. We find that the semimajor axes of the planet-centered orbits of planetesimals at the time of permanent capture are smaller than about one-third of the planet's Hill radius in most cases. Typically, captured bodies rapidly spiral into the planet, and the rate of the orbital decay is faster for the retrograde orbits due to the strong headwind from the circumplanetary gas. When a planetesimal captured in a retrograde orbit suffers from sufficiently strong gas drag before spiraling into the planet, its orbit turns to the prograde direction at a radial location that can be explained using the Stokes number. We also find that those captured in certain types of orbits can survive for a long period of time even under gas drag both in the prograde and retrograde cases, which may be important for the origin of irregular satellites of giant planets.

CAPTURE OF PLANETESIMALS BY WANING CIRCUMPLANETARY GAS DISKS

ABSTRACT Circumplanetary disks will form when gas-giant protoplanets grow sufficiently massive. While solid bodies captured by the circumplanetary disks likely contribute to the growth of the planets and regular satellites around them, some of the captured bodies will remain in planet-centered orbits after the dispersal of the disk. We examine the capture and subsequent orbital evolution of planetesimals in waning circumplanetary gas disks using three-body orbital integration. We find that some of the captured planetesimals can survive in the circumplanetary disk for a long period of time under such weak gas drag. Captured planetesimals have semimajor axes smaller than about one-third of the planet's Hill radius. The distributions of their eccentricities and inclinations after disk dispersal depend on the strength of gas drag and the timescale of disk dispersal, and initially strong gas drag and quick disk dispersal facilitate capture and survival of planetesimals. However, in such cases, the final orbital eccentricities and inclinations of captured bodies remain rather large. Although our results suggest that some of the present irregular satellites of gas-giant planets with small semimajor axes would have been captured by gas drag, other mechanisms are required to fully explain their current orbital characteristics.

ACCRETION RATES OF MOONLETS EMBEDDED IN CIRCUMPLANETARY PARTICLE DISKS

We examine the gravitational capture probability of colliding particles in circumplanetary particle disks and accretion rates of small particles onto an embedded moonlet, using analytic calculation, three-body orbital integrations, and N-body simulations. Expanding our previous work, we take into account the Rayleigh distribution of particles' orbital eccentricities and inclinations in our analytic calculation and orbital integration and confirm agreement between them when the particle velocity dispersion is comparable to or larger than their mutual escape velocity and the ratio of the sum of the physical radii of colliding particles to their mutual Hill radius () is much smaller than unity. As shown by our previous work, the capture probability decreases significantly when the velocity dispersion is larger than the escape velocity and/or . Rough surfaces of particles can enhance the capture probability. We compare the results of three-body calculations with N-body simulations for accretion of small particles by an embedded moonlet and find agreement at the initial stage of accretion. However, when particles forming an aggregate on the moonlet surface nearly fill the Hill sphere, the aggregate reaches a quasi-steady state with a nearly constant number of particles covering the moonlet, and the accretion rate is significantly reduced compared to the three-body results.

TEMPORARY CAPTURE OF PLANETESIMALS BY A PLANET FROM THEIR HELIOCENTRIC ORBITS

When planetesimals encounter a planet, they can be temporarily captured by the planet's gravity and orbit about it for an extended period of time before escaping from the planet's vicinity. Such a process may have played an important role in the origin of irregular satellites or the dynamical evolution of short-period comets. Using three-body orbital integration, we study the temporary capture of planetesimals by a planet from their heliocentric eccentric orbits. We examine the dependence of the orbital characteristics during temporary capture as well as the rate of capture on the pre-capture heliocentric orbital parameters. We find that typical orbital size and direction of revolution around the planet change depending on planetesimals' initial eccentricity and energy. When initial eccentricity is so small that Kepler shear dominates the relative velocity between planetesimals and the planet, temporary capture typically occurs in the retrograde direction in the vicinity of the planet's Hill sphere, while large retrograde capture orbits outside the Hill sphere are predominant for large eccentricities. Long prograde capture occurs in a very narrow range of planetesimal eccentricity and energy. We obtain the rate of temporary capture of planetesimals and find that the rate of long capture increases with increasing eccentricity at low and high eccentricities, but decreases with increasing eccentricity in intermediate values of eccentricity. We also examine the dependence of capture rate on the duration of capture and find an approximate power-law dependence.

Dynamical behaviour of planetesimals temporarily captured by a planet from heliocentric orbits: basic formulation and the case of low random velocity

Planetesimals encountering with a planet cannot be captured permanently unless energy dissipation is taken into account, but some of them can be temporarily captured in the vicinity of the planet for an extended period of time. Such a process would be important for the origin and dynamical evolution of irregular satellites, short-period comets, and Kuiper-belt binaries. In this paper, we describe the basic formulation for the study of temporary capture of planetesimals from heliocentric orbits using three-body orbital integration, such as the definition of the duration and rate of temporary capture, and present results in the case of low random velocity of planetesimals. In the case of planetesimals initially on circular orbits, we find that planetesimals undergo a close encounter with the planet before they become temporarily captured. When planetesimals are scattered by the planet into the vicinity of one of periodic orbits around the planet, the duration of temporary capture tends to be extended. Typically, these capture orbits are in the retrograde direction around the planet. We evaluate the rate of temporary capture of planetesimals, and find that the ratio of this rate to their collision rate on to the planet increases with increasing semimajor axis of the planet. Similar results are obtained for planetesimals with non-zero but small random velocities, as long as Kepler shear dominates the relative velocity between the planet and planetesimals. For larger initial random velocities of planetesimals, temporary capture in both prograde and retrograde directions with much longer duration becomes possible.

Rotation rate and velocity dispersion of planetary ring particles with size distribution II. Numerical simulation for gravitating particles

We examine rotation rates of gravitating particles in low optical depth rings, on the basis of the evolution equation of particle rotational energy derived by Ohtsuki [Ohtsuki, K., 2006. Rotation rate and velocity dispersion of planetary ring particles with size distribution. I. Formulation and analytic calculation. Icarus 183, 373–383]. We obtain the rates of evolution of particle rotation rate and velocity dispersion, using three-body orbital integration that takes into account distribution of random velocities and rotation rates. The obtained stirring and friction rates are used to calculate the evolution of velocity dispersion and rotation rate for particles in one- and two-size component rings as well as those with a narrow size distribution, and agreement with N-body simulation is confirmed. Then, we perform calculations to examine equilibrium rotation rates and velocity dispersion of gravitating ring particles with a broad size distribution, from 1 cm up to 10 m. We find that small particles spin rapidly with 〈 ω 2 〉 1 / 2 / Ω ≳ 10 2 – 10 3 , where ω and Ω are the particle rotation rate and its orbital angular frequency, respectively, while the largest particles spin slowly, with 〈 ω 2 〉 1 / 2 / Ω ≲ 1 . The vertical scale height of rapidly rotating small particles is much larger than that of slowly rotating large particles. Thus, rotational states of ring particles have vertical heterogeneity, which should be taken into account in modeling thermal infrared emission from Saturn's rings.

On the rotation of a moonlet embedded in planetary rings

We examine the rotation of a small moonlet embedded in planetary rings caused by impacts of ring particles, using analytic calculation and numerical orbital integration for the three-body problem. Taking into account the Rayleigh distribution of particles' orbital eccentricities and inclinations, we evaluate both systematic and random components of rotation, where the former arises from an average of a large number of small impacts and the latter is contribution from large impacts. Calculations for parameter values corresponding to inner parts of Saturn's rings show that a moonlet would spin slowly in the prograde direction if most impactors are small particles whose velocity dispersion is comparable to or smaller than the moonlet's escape velocity. However, we also find that the effect of the random component can be significant, if the velocity dispersion of particles is larger and/or impacts of large particles comparable to the moonlet's size are common: in this case, both prograde and retrograde rotations can be expected. In the case of a small moonlet embedded in planetary rings of equal-sized particles, we find that the systematic component dominates the moonlet rotation when m / M ≪ 0.1 ( m and M are the mass of a particle and a moonlet, respectively), while the random component is dominant when m / M ≳ 0.3 . We derive the condition for the random component to dominate moonlet rotation on the basis of our results of three-body orbital integration, and confirm agreement with N-body simulation.

Semi-Analytic Formulas of Velocity Stirring Rates in Particle Disks

We have obtained viscous stirring and dynamical friction rates of planetesimals based on three-body orbital integrations, taking into account the Rayleigh distribution of eccentricities and inclinations. We have found that the evolution of root mean square eccentricities and inclinations calculated by using these rates agrees with N-body simulations quite well.

Radial diffusion rate of planetesimals due to gravitational encounters

We study the rate of radial diffusion of planetesimals due to mutual gravitational encounters under Hill’s approximations in the three-body problem. Planetesimals orbiting a central star radially migrate inward and outward as a result of mutual gravitational encounters and transfer angular momentum. We calculate the viscosity in a disk of equal-sized planetesimals due to their mutual gravitational encounters using three-body orbital integrations, and obtain a semianalytic expression that reproduces the numerical results. We find that the viscosity is independent of the velocity dispersion of planetesimals when the velocity dispersion is so small that Kepler shear dominates planetesimals’ relative velocities. On the other hand, in high-velocity cases where random velocities dominate the relative velocities, the viscosity is a decreasing function of the velocity dispersion, and is found to agree with previous estimates under the two-body approximation neglecting the solar gravity. We also calculate the rate of radial diffusion of planetesimals due to gravitational scattering by a massive protoplanet. Using these results, we discuss a condition for formation of nonuniform radial surface density distribution of planetesimals by gravitational perturbation of an embedded protoplanet.

Evolution of Planetesimal Velocities Based on Three-Body Orbital Integrations and Growth of Protoplanets

We obtain the viscous stirring and dynamical friction rates of planetesimals with a Rayleigh distribution of eccentricities and inclinations, using three-body orbital integration and the procedure described by Ohtsuki (1999, Icarus 137, 152), who evaluated these rates for ring particles. We find that these rates based on orbital integrations agree quite well with the analytic results of Stewart and Ida (2000, Icarus 143, 28) in high-velocity cases. In low-velocity cases where Kepler shear dominates the relative velocity, however, the three-body calculations show significant deviation from the formulas of Stewart and Ida, who did not investigate the rates for low velocities in detail but just presented a simple interpolation formula between their high-velocity formula and the numerical results for circular orbits. We calculate evolution of root mean square eccentricities and inclinations using the above stirring rates based on orbital integrations, and find excellent agreement with N-body simulations for both one- and two-component systems, even in the low-velocity cases. We derive semi-analytic formulas for the stirring and dynamical friction rates based on our numerical results, and confirm that they reproduce the results of N-body simulations with sufficient accuracy. Using these formulas, we calculate equilibrium velocities of planetesimals with given size distributions. At a stage before the onset of runaway growth of large bodies, the velocity distribution calculated by our new formulas are found to agree quite well with those obtained by using the formulas of Stewart and Ida or Wetherill and Stewart (1993, Icarus 106, 190). However, at later stages, we find that the inclinations of small collisional fragments calculated by our new formulas can be much smaller than those calculated by the previously obtained formulas, so that they are more easily accreted by larger bodies in our case. The results essentially support the previous results such as runaway growth of protoplanets, but they could enhance their growth rate by 10–30% after early runaway growth, where those fragments with low random velocities can significantly contribute to rapid growth of runaway bodies.

Evaluation of collision and stirring rates in circumplanetary particle disks based on three-body orbital integrations

We study collision and velocity stirring rates of particles in a circumplanetary disk with low optical depth where the epicyclic approximations can be applied to particle motions. We use orbital integrations in the three-body problem where both direct collisions and gravitational encounters between particles are taken into account. Using these numerical results together with the previously derived analytic results for nongravitating particles (Ohtsuki, 1999. Icarus 137, 152–177), we examine the dependence of collision and stirring rates on eccentricities and inclinations in the case of gravitating particles to calculate the averaged rates for a Rayleigh distribution of particles’ eccentricities and inclinations. Numerical results show that the dynamical friction rates for nongravitating ring particles can approximate the numerical results of three-body orbital integrations fairly well for a wide range of parameters. However, the effects of self-gravity substantially changes the collision and viscous stirring rates in low velocity cases. On the basis of the numerical results of orbital integrations in such low velocity cases and the analytic results for nongravitating cases, we derive semi-analytic formulae for collision and stirring rates for gravitating particles which can be used for the entire range of velocities. We calculate the evolution of root mean square eccentricities and inclinations of ring particles using these formulae, and find excellent agreement with the results obtained by N-body simulations.

Orbital Resonances and Poynting-Robertson Drag

Grains orbiting a star experience orbital decay due to Poynting-Robertson drag and can become trapped in commensurability resonances with a planet. We examine conditions for trapping in the circular restricted three-body problem, and derive criteria for trapping in terms of the radiation pressure factor β, planetary mass, resonance location, and the grain's orbital eccentricity. These analytic criteria show good agreement with results of three-body orbital integrations and more elaborate multiplanet simulations by other researchers. We show that resonances under Poynting-Robertson drag are "metastable" because trapped grains inevitably acquire eccentricities large enough to cause their orbits to cross that of the planet.

Stirring and dynamical friction rates of planetesimals in the solar gravitational field

The random velocity (eccentricity and inclination) change of planetesimals due to mutual gravitational scattering is investigated, taking into account the solar gravitational field. Viscous stirring and dynamical friction rates are presented with three-body formalism (using eccentricity and inclination), and calculated through a large number ∼10 7−8) of three-body orbital integrations. The results are summarized as follows: (1) The role of distant encounters is not important. (2) In the low energy cases where Keplerian shear dominates, eccentricity grows much more rapidly than inclination, and dynamical friction is not effective. The viscous stirring time scale for eccentricity is much greater than the two-body relaxation time. (3) In the high energy cases where random motion dominates, a 2:1 ratio of eccentricity to inclination is spontaneously attained. Both stirring and dynamical friction time scales are almost given by the two-body relaxation time, and dynamical friction is effective for the larger planetesimals, which should lead to “runaway growth” of protoplanets.