Planetesimal Accumulation Research Articles

Influence of the circumbinary disk gravity on planetesimal accumulation in the Kepler–16 system

Recent observations from NASA's Kepler mission detected the first planets in circumbinary orbits. The question we try to answer is where these planets formed in the circumbinary disk and how far inside they migrated to reach their present location. We investigate the first and more delicate phase of planet formation when planetesimals accumulate to form planetary embryos. We use the hydrodynamical code FARGO to study the evolution of the disk and of a test population of planetesimals embedded in it. With this hybrid hydrodynamical--N--body code we can properly account for the gas drag force on the planetesimals and for the gravitational force of the disk on them. The numerical simulations show that the gravity of the eccentric disk on the planetesimal swarm excites their eccentricities to values much larger than those induced by the binary perturbations only within 10 AU from the stars. Moreover, the disk gravity prevents a full alignment of the planetesimal pericenters. Both these effects lead to large impact velocities, beyond the critical value for erosion. Planetesimals accumulation in circumbinary disks appears to be prevented close to the stellar pair by the gravitational perturbations of the circumbinary disk. The observed planets possibly formed in the outer regions of the disk and then migrated inside by tidal interaction with the disk.

On the isotopic fractionation of terrestrial xenon by gravitational separation inside porous planetesimals with size distribution

Taking account of the size distribution of planetesimals inferred from recent theoretical studies of planetesimal accumulation, we examine gravitational separation of noble gases inside large porous planetesimals in the solar nebula. The size distribution is characterized by the value of the minimum planetesimal size R0 for a given power-law distribution, which also determines the maximum planetesimal size for a given total mass of planetesimals. Both enrichment and isotopic fractionation of noble gases strongly depend on the value of R0, as it determines the number of large planetesimals inside which gravitational separation of noble gases is significant. We find the ranges of the values of R0 that can produce the enrichment and the isotopic fractionation of Xe similar to the terrestrial values separately, but the two ranges are found not to overlap. We also find significant isotopic fractionation of Kr for the parameter value that can produce the terrestrial isotopic fractionation of Xe, in contrast to the observation. These results suggest that the isotopic fractionation of terrestrial Xe cannot be explained if we only consider the gravitational separation inside porous planetesimals with the size distribution.

Terrestrial planet formation in exoplanetary systems with a giant planet on an external orbit

The Epsilon Eridani and 47 UMa extrasolar systems both have moderately massive planets orbiting relatively far away from the central star (at 3.3 and beyond 2.1 AU, respectively). This peculiarity makes them possible candidates for harboring terrestrial planets in their inner regions since the external giant planet might not have inhibited planet growth in the neighborhood of the star. Here we numerically investigate how the accretion of terrestrial planets may have been affected by the presence of the detected giant planets in external orbits. The uncertain timing of the giant planet formation with respect to the planetesimal accumulation process lead us to consider two possible scenarios: 1) the perturbing massive planet was fully formed when the planetesimals in the inner zone were still in their early phases of accumulation, and 2) the giant planet reached its final mass only later on when lunar-sized embryos were already present in the inner disk. In the first case we have used a deterministic numerical algorithm to compute the relative velocity distribution in a swarm of planetesimals affected by the gravitational perturbations of the giant planet and, possibly, by gas drag. The second scenario is explored by using the symplectic algorithm by Chambers (1999) to simulate the late stage of terrestrial planet formation. The Epsilon Eridani system turns out to be a very hostile environment for planetary formation in the terrestrial region when the giant planet formed early. The planet's high eccentricity (0.608) induces very large relative velocities between planetesimals in inner orbits and collisions result mostly in disruption rather than accretion. On the other hand, if large embryos had enough time to form before the completion of the giant planet, they can accrete each other in giant impacts and coalesce into a few terrestrial planets within 0.8 AU from the star. For the 47 UMa system the situation is quite similar. The large mass of the inner giant planet excites large eccentricities among the planetesimals in the terrestrial region preventing accumulation. However, the damping effect on the relative velocities caused by gas drag opens a chance for accretion in a small region around the star inside 0.3 AU. The second and more distant planet recently discovered around 47 Uma does not significantly affect the encounter velocity distribution in the inner regions of the system. As for Epsilon Eridani, orderly growth of lunar-sized embryos takes place within 0.8 AU in presence of a late-formed giant planet. In either case, beyond 0.8 AU planetesimals or planetary embryos are rapidly removed by the massive planet perturbations.

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.

Mass Spectrum and Velocity Dispersions during Planetesimal Accumulation. II. Fragmentation

The increase of the velocity dispersions which occurs during the growth of the planetesimals (P. Barge and R. Pellat, 1991, Icarus 93, 270-287) strongly suggested that fragmentation could come at work in the final stages of the accumulation scenario.Fragmentation has been modeled assuming that mass and energy are redistributed in a simple way between the various fragments; it is found to play a role whose importance is greater than previously believed and depends on the average characteristic size of the primordial planetesimals but not on their mass distribution. In the small size range the mass spectrum is strongly modified with the formation of a small bodies tail but, on the other hand, the growth of the most massive objects is significantly slowed down. With the recent fragmentation model of K. R. Housen, R. M. Schmidt, and K. R. Holsapple (1991, Icarus 94, 180-190) it is found that 10-km-sized objects can grow into planetary embryos (size of the order of 1000 km) in a time scale of the order of 105 or 107 years (depending on the fragmentation properties of the planetesimals) whereas 1-km-sized objects cannot on a reasonable time scale. The formation of planetary embryos much more massive than the rest of the swarm is delayed; nevertheless, the time scale problem for the growth of the outer planets is not revived. The elasticity of the collisions is found to play a negligible role in the investigated size range.

Evolution of random velocities of planetesimals in the course of accretion

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 Hill's 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 Earth's 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.

Mass spectrum and velocity dispersions during planetesimal accumulation: I. Accretion

The accumulation of the planetesimals has been investigated taking account of the simultaneous evolution of mass distribution and velocity dispersions. Under the assumption that collision frequencies are much smaller than the orbital frequency ω the velocity dispersions are determined from a previously developed formalism ( Hornung et al. 1985 , Barge and Pellat 1990) which works as long as the velocity dispersions remain approximately greater than the “Hill velocity” ωR H; this formalism is extended to the lower velocities through a model based on the numerical computations of Ida (1990). On the other hand the evolution of the mass distribution is derived from the simplified Oort-Van de Hulst formulation of the coagulation equation, which is sufficient to investigate the main trends of the accumulation of the planetesimals. It It is found that: • —A runaway growth of the largest bodies occurs with the formation of massive embryos (the mass can grow by a factor of one million on a time scale of the order of 0.1 million years); this is in contrast with the results of the Moscow “school” (Safronov and others) which assumes that mass and velocity are decoupled. • —Velocity dispersions increase with growing planetesimals; this is in contrast with the numerical simulation of Greenberg et al. (1978) and with the more recent work of Wetherill and Stewart (1989) whose computations are outside the range of validity of the formalism. These conclusions result from the yet unsuspected importance of viscous stirring of gravitational encounters which plays the major role in the collisional evolution of planetesimals by the transfer of significant amounts of orbital energy into random motions. The increase of the velocity dispersions reveals also that the fragmentation could play an important role in the final evolution of the planetesimals toward the planetary stage.

Comparison of analytical and physical modeling of planetesimal accumulation

In recent work on the accumulation of planetesimals to form planetary embryos, a physical model for the evolution of the mass distribution was used ( Wetherill and Stewart, 1989, Icarus 77, 330–357). In this paper the mathematical validity of this technique is tested by comparison with the three cases for which analytic solutions to the coagulation equation are known: coagulation rate constant, rate proportional to the sum of the masses of the two colliding bodies, and rate proportional to the product of their masses. The first two cases correspond to orderly growth and the third to runaway growth. In all cases excellent agreement is found between the results of numerical physical modeling and the corresponding analytic solutions. Full treatment of the runaway case requires an extension of the work of Trubnikov (1971, Doklady Akad. Nauk USSR 196, 1316–1319) and leads to development of an analytic solution for the growth of both the runaway body and the residual swarm. Additional calculations for collision probabilities of the form ( m 1 + m 2) λ , where m 1 and m 2 are the colliding masses, confirm the expectation that runaway occurs when λ > 1, and orderly growth when λ ≤ 1.

Accumulation of a swarm of small planetesimals

The stage of planetesimal accumulation in which ∼ 10-km planetesimals in the vicinity of 1 AU grow into “planetary embryos” greater than 10 25 to 10 27 g in mass has been studied using the methods of gas dynamics. Particular attention is given to identifying the circumstances for which runaway growth results in the formation of a relatively small number (e.g., 30) of massive (∼4 × 10 26g) embryos in the terretrial planet region on a time scale of 10 5 to 10 6 years. It is found that under the assumptions made by the Moscow (Safronov and others) and Kyoto (Hayashi and others) “schools,” no runaways are found, in agreement with the conclusions of these investigators. In contrast, when more plausible physical processes are included, e.g., the importance of equipartition of energy in gravitational encounters, the presence of the “seeds” in the initial distribution, enhancement of the gravitational cross section above the two-body value at low velocity, and fragmentation, runaways in the terrestrial planet region are found to be very probable on a time scale of about 10 5 years. The final stage of planetary accumulation may then consist of the accumulation of these embryos into the present planets on a time scale of 10 7 to 10 8 years. At larger heliocentric distances the planetesimal evolution could be different; circumstances may have existed in which runaway growth of Jupiter prevented runaways in the asteroid belt.

Origin of the Solar System

The current status of the classical model of solar-system formation is surveyed, reviewing the results of recent observational and theoretical investigations. Topics addressed include interstellar clouds, the collapse of interstellar gas, the primitive solar nebula, the formation of the sun, planetesimal accumulation, planetary accumulation, major planetary collisions, the development of planetary atmospheres, and comets. The relative merits of conflicting theories on many key problems are indicated, with reference to more detailed reviews in the literature.