Keiji Ohtsuki

Local N-Body Simulations for the Distribution and Evolution of Particle Velocities in Planetary Rings

Distribution and evolution of particle velocities in planetary rings of one- and two-size components are investigated by a local N-body simulation with periodic boundary conditions, which includes collisions and gravitational interactions between particles. Evolution of rms eccentricities and inclinations for a system of low optical depth is found to be well predicted by the numerical results of three-body orbit calculations. The Kolmogorov-Smirnov test is performed to examine the particle velocity distribution, and it is found that the distribution of orbital eccentricities and inclinations of particles in a disk with low optical depth can be well approximated by a Rayleigh distribution when the restitution coefficient of particles is small enough (0.6) to achieve an equilibrium state, whereas excess of high-velocity particles is found in more elastic cases in which velocity dispersion increases monotonically. When the optical depth is larger and the disk of particles becomes gravitationally unstable, however, effects of collective wakes become important. Theoretical results based on three-body orbit calculations fail to predict the evolution in this case, and eccentricities and inclinations deviate from the Rayleigh distribution.

Dynamics of Saturn's Dense Rings

The Cassini mission to Saturn opened a new era in the research of planetary rings, bringing data in unprecedented detail, monitoring the structure and properties of Saturns ring system. The question of ring dynamics is to identify and understand underlying physical processes and to connect them to the observations in terms of mathematical models and computer simulations. For Saturns dense rings important physical processes are dissipative collisions between ring particles, their motion in Saturns gravity field, their mutual self-gravity, and the gravitational interaction with Saturns moons, exterior to or embedded in the rings.

Growth of the earth in nebular gas

Abstract Taking into account newly found gas drag effect on a gravitating body, we simulated complete growth of the Earth in a gaseous solar nebula—from planetesimals to the fullsize Earth—by numerically solving coupled growth equations for the largest body and smaller planetesimals. Numerical results show that planetesimals in the Earth zone with initial masses of 10 18 g grow to the body with the present Earth mass, 6 × 10 27 g, within such short time scale as 1 × 10 7 years, which is reduced by a factor of about 2 compared with the results obtained by using the drag law for gravitation-free bodies. The growth is further accelerated if dissipation due to inelastic collision is taken into account. The growth of a planet at the final stage, if its coalescence cross section becomes large enough, is subjected to the rate of mass supply into its accretion zone—an inward flow of planetesimals is caused by the gas drag. Some problems included in the present calculation are discussed.

Evolution of random velocities of planetesimals in the course of accretion

Abstract The role of accretion in the random velocity evolution of planetesimals is examined. By combining the equation for velocity change caused by accretion and the coagulation equation for the mass distribution function, we derive an equation for random velocity evolution in the course of accretion, which is described in terms of eccentricity and inclination of planetesimal orbits in the framework of the Hills approximations. We find that dissipation of random velocity due to accretion has the tendency to lead to equipartition of kinetic energy of random motion between planetesimals with different masses. We also compare the three effects on velocity evolution, that is, gas drag of the solar nebula, gravitational scattering, and accretion. At the Earths orbit, all these three effects seem to contribute to velocity evolution at early stages of planetesimal accumulation. However, when the typical size of planetesimals becomes larger than 10 20 g, the effect of accretion on velocity evolution becomes less effective, and it is considered that planetesimals have certain equilibrium velocities determined by the balance between gravitational scattering and gas drag. At this, or at a slightly later stage, the onset of runaway growth of several protoplanets seems more probable.

Artificial acceleration in accumulation due to coarse mass-coordinate divisions in numerical simulation

Abstract The effects of coarse mass-coordinate divisions in numerical computations of the coagulation equation are examined. The numerical accuracy is essential to the estimate of the growth times of dust and planets. Three different logarithmic divisions with common ratios of √2, 2, and 8 are used to numerically solve the coagulation equation in cases of simple coalescence rates, where analytic solutions are known. The comparison of numerical solutions and analytic ones shows appreciable discrepancies between them and the discrepancies are quite large in the cases of larger common ratios; the coagulation is considerably accelerated. Comparison between the simulations of planetary growth with these three types of divisions is also made to show that growth time in early stages differs by a factor of about 10 between the cases of common ratios √2 and 8. This suggests that the previous numerical simulations of the coagulation process using coarse divisions with common ratios of 2 or 8 were very likely affected by this artificial acceleration.

Runaway planetary growth with collision rate in the solar gravitational field

Abstract The effect of collision rate between planetesimals in the solar gravitational field on runaway planetary growth is intensively examined. First, the characteristics of the collision rate in the solar gravitational field, which were obtained recently by a large number of orbital calculations (S. Ida and K. Nakazawa, 1989, Astron. Astrophys. 224, 303–315) , are investigated through comparison with those in the two-body approximation neglecting the Sun. They are qualitatively explained with the aid of two-body approximation neglecting the sun. They are qualitatively explained with the aid of two-body collision rates in three-dimensional space in a high-velocity region and in two-dimensional space in a low-velocity region; however, there still remains a particular feature which cannot be explained by analogy of the two-body approximation and is closely related to the occurrence of runaway growth. Next, competitive growth of several protoplanets is numerically simulated with a simple model. The results show that the collision rate in the solar gravitational field enhances the possibility of occurrence of runaway growth in a certain low-velocity region (but suppresses it in a very low-velocity region), compared with that in the two-body approximation. However, whether runaway growth occurs or not sensitively depends on the random velocity of planetesimals: its magnitude, the ratio of eccentricity to inclination, mass dependence, and distribution around its mean value, all of which should be also determined including the effect of the solar gravitational field.

Collisional disruption of gravitational aggregates in the tidal environment

The degree of disruption in collisions in free space is determined by specific impact energy, and the mass fraction of the largest remnant is a monotonically decreasing function of impact energy. However, it has not been shown whether such a relationship is applicable to collisions under the influence of a planets tidal force, which is important in ring dynamics and satellite accretion. Here we examine the collisional disruption of gravitational aggregates in the tidal environment by using local N-body simulations. We find that outcomes of such a collision largely depend on the impact velocity, the direction of impact, and the radial distance from the planet. In the case of a strong tidal field corresponding to Saturns F ring, collisions in the azimuthal direction are much more destructive than those in the radial direction. Numerical results of collisions sensitively depend on the impact velocity, and a complete disruption of aggregates can occur even in impacts with velocity much lower than their escape velocity. In such low-velocity collisions, the deformation of colliding aggregates plays an essential role in determining collision outcomes, because the physical size of the aggregate is comparable to its Hill radius. On the other hand, the dependence of collision outcomes on impact velocity becomes similar to the case in free space when the distance from the planet is sufficiently large. Our results are consistent with Cassini observations of the F ring, which suggest ongoing creation and disruption of aggregates within the ring.

Accretion Rates of Planetesimals by Protoplanets Embedded in Nebular Gas

Abstract When protoplanets growing by accretion of planetesimals have atmospheres, small planetesimals approaching the protoplanets lose their energy by gas drag from the atmospheres, which leads them to be captured within the Hill sphere of the protoplanets. As a result, growth rates of the protoplanets are enhanced. In order to study the effect of an atmosphere on planetary growth rates, we performed numerical integration of orbits of planetesimals for a wide range of orbital elements and obtained the effective accretion rates of planetesimals onto planets that have atmospheres. Numerical results are obtained as a function of planetesimals’ eccentricity, inclination, planet’s radius, and non-dimensional gas-drag parameters which can be expressed by several physical quantities such as the radius of planetesimals and the mass of the protoplanet. Assuming that the radial distribution of the gas density near the surface can be approximated by a power-law, we performed analytic calculation for the loss of planetesimals’ kinetic energy due to gas drag, and confirmed agreement with numerical results. We confirmed that the above approximation of the power-law density distribution is reasonable for accretion rate of protoplanets with 1–10 Earth masses, unless the size of planetesimals is too small. We also calculated the accretion rates of planetesimals averaged over a Rayleigh distribution of eccentricities and inclinations, and derived a semi-analytical formula of accretion rates, which reproduces the numerical results very well. Using the obtained expression of the accretion rate, we examined the growth of protoplanets in nebular gas. We found that the effect of atmospheric gas drag can enhance the growth rate significantly, depending on the size of planetesimals.

High-accuracy statistical simulation of planetary accretion: I. Test of the accuracy by comparison with the solution to the stochastic coagulation equation

The object of this series of studies is to develop a highly accurate statistical code for describing the planetary accumulation process. In the present paper, as a first step, we check the validity of the method proposed by Wetherill and Stewart (1989) by comparing the results obtained by their method with the analytical solution to the stochastic coagulation equation (or to a well-evaluated numerical solution). As the collisional probability Ai j between bodies with masses of im1 and jm1 (m1 being the unit mass), we consider the two cases: one is Ai j∝ i × j and another is Ai j ∝ min(i, j )(i 1/3 + j1/3)(i + j ). In both cases, it is known that runaway growth occurs. The latter case corresponds to a simplified model of the planetesimal accumulation. We assumed that a collision of two bodies leads to their coalescence. Wetherill and Stewart’s method contains some parameters controlling the practical numerical computation. Among these, two parameters are important: the mass division parameter δ, which determines the mass ratio of the adjacent mass batches, and the time division parameter ∈, which controls the size of a time step in numerical integration. Through a number of numerical simulations for the case of Ai j = i × j, we find that when δ ≤ 1.6 and ∈ ≤ 0.03 the numerical simulation can reproduce the analytical solution within a certain level of accuracy independently of the size of the body system. For the case of the planetesimal accumulation, it is shown that the simulation with δ ≤ 1.3 and ∈ ≤ 0.04 can describe precisely runaway growth. Because the accumulation process is stochastic, in order to obtain reliable mean values it is necessary to take the ensemble mean of the numerical results obtained with different random number generators. It is also found that the number of simulations, Nc, demanded to obtain the reliable mean value is about 500 and does not strongly depend on the functional form of Ai j. From the viewpoint of the numerical handling, the above value of δ(≤ 1.3) and Nc(∼500) are reasonable and, hence, we conclude that the numerical method proposed by Wetherill and Stewart is a valid and useful method for describing the planetary accumulation process. The real planetary accumulation process is more complex since it is coupled with the velocity evolution of the planetesimals. In the subsequent paper, we will complete the high-accuracy statistical code which simulate the accumulation process coupled with the velocity evolution and test the accuracy of the code by comparing with the results of N -body simulation.

Equilibrium velocities in planetary rings with low optical depth

Abstract We examine the equilibrium random velocity in planetary rings of one and two particle size components with low optical depth, where the epicyclic approximation for particle motions can be applied. First, assuming that the random velocity is large enough to allow particle gravity to be neglected, we evaluate the rate of random velocity evolution due to collisions. In the case that the restitution coefficient of the particles is independent of impact velocity, we derive analytic expressions for the rate of evolution and find that collisions lead to a state e ⋍ 2i (e and i being orbital eccentricity and inclination). We also estimate the equilibrium random velocity, using the restitution coefficient of icy particles obtained by impact experiments. Next, we include particle gravity with the aid of orbital calculations of the three-body problem. For random velocity smaller than the escape velocity, not only direct collisions but also gravitational scattering contribute to rapid enhancement of random velocity, and eccentricity grows much faster than inclination. For random velocity comparable to or larger than escape velocity, the rate smoothly tends toward the above analytic estimates for collisions without particle gravity. Finally, we examine the equilibrium velocities of two-component systems for arbitrary mass ratios. We find that the smaller particles will have equilibrium velocity determined by the velocity-dependent restitution coefficient, while velocity of larger particles seems to tend toward energy balance with smaller particles if the mass ratio is not large and the mass of large particles does not exceed a certain value. If these conditions are violated, the velocity of large particles is also determined by the velocity-dependent restitution coefficient.