共查询到20条相似文献，搜索用时 15 毫秒
1.
Gravitational perturbations in semimajor axis, eccentricity, and inclination resulting from close planetesimal encounters (near 1 AU) out to 10 Tisserand sphere of influence radii were calculated by two- and three-dimensional numerical integration. These are compared with the results of treating the encounter as a two-body problem, as is customary in Monte Carlo calculations of orbital evolution and in numerical and analytical studies of planetary accumulation. It is found that for values of , the two-body body approximation fails to describe the outcome of individual encounters. In this low-velocity region, the two-body “gravitational focusing” cross section is no longer valid; “anomalous gravitational focusing” often leads to bodies on distant unperturbed trajectories becoming close encounters and vice versa. In spite of these differences, average perturbations given by the two-body approximation are valid within a factor of 2 when . In this same velocity range the “Arnold extrapolation,” whereby a few very close encounters are used to estimate the effect of many more distant encounters, is found to be a useful approximation. 相似文献
2.
We have developed a semi-analytic method of calculating the changes in heliocentric Keplerian orbital elements due to gravitational scattering by a protoplanet as a three-body problem. In encounters with high incident velocities, either the gravity of the protoplanet or the solar gravity can be regarded as perturbation force. In close encounters, by taking into account the solar gravity as a perturbation, we modified the two-body gravitational scattering. On the other hand, in slightly distant encounters, we apply the perturbing force of the protoplanet to the heliocentric Keplerian orbit of planetesimals. As a result, as for high-velocity encounters, the three-body problem is semi-analytically solvable. Our semi-analytic method can reproduce the numerical result of the orbital changes of individual planetesimals for the broad range of high-energy encounters with surprising high accuracy. We found that our method is valid under the conditions (i)b0? 2 and (ii) (e20+i20− b20)1/2? 4, wheree0andi0are eccentricity and inclination of relative motion normalized by the reduced Hill radius andb0is the difference between semimajor axes normalized by the Hill radius. Though our method needs some numerical procedure, its cpu time is negligibly short compared with that of the direct orbital integration. In simulation of orbital evolution of planetesimals around a protoplanet in the gas, which we will perform in the subsequent paper, most encounters can be calculated by the semi-analytic method. This makes it possible to perform the long term (∼105years) orbital calculation of ∼103–4planetesimals. 相似文献
3.
G.P. Horedt 《Icarus》1985,64(3):448-470
We derive first-order differential equations for the late stages of planetary accretion (planetesimal mass >1013 g). The effect of gravitational encounters, energy exchange, collisions, and gas drag has been included. Two simple models are discussed, namely, (i) when all planetesimals have the same mass and (ii) when there is one large planetesimal and numerous small planetesmals. Gravitational two-body encounters are modeled according to Chandrasekhar's classical theory from stellar dynamics. It is shown that the velocity increase due to mutual encounters can be modeled according to the simple theory of random flights. We find analytical equations for the average velocity decrease due to collisions. Gas drag, if present, is modeled in averaged form up to the first order in the eccentricities and inclinations of the planetesimals. Characteristic time scales for the formation of the terrestrial planets are found for the most favorable models to be of order 108 year. The calculated mass of rock and ice of the giant planets is too low as compared to the observed one. This difficulty of our model could be overcome by assuming a several times larger surface density, an enlarged accretion cross section, and gas accretion during the final stages of accretion of the solid cores of the giant planets. Analytical and numerical results are presebted, the evolutionary tracks showing satisfactory agreement with observations for some models. 相似文献
4.
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. 相似文献
5.
Stephen J. Kortenkamp 《Icarus》2005,175(2):409-418
Numerical simulations of the gravitational scattering of planetesimals by a protoplanet reveal that a significant fraction of scattered planetesimals can become trapped as so-called quasi-satellites in heliocentric 1:1 co-orbital resonance with the protoplanet. While trapped, these resonant planetesimals can have deep low-velocity encounters with the protoplanet that result in temporary or permanent capture onto highly eccentric prograde or retrograde circumplanetary orbits. The simulations include solar nebula gas drag and use planetesimals with diameters ranging from ∼1 to ∼1000 km. Initial protoplanet eccentricities range from ep=0 to 0.15 and protoplanet masses range from 300 Earth-masses (M⊕) down to 0.1M⊕. This mass range effectively covers the final masses of all planets currently thought to be in possession of captured satellites—Jupiter, Saturn, Neptune, Uranus, and Mars. For protoplanets on moderately eccentric orbits (ep?0.1) most simulations show from 5-20% of all scattered planetesimals becoming temporarily trapped in the quasi-satellite co-orbital resonance. Typically, 20-30% of the temporarily trapped quasi-satellites of all sizes came within half the Hill radius of the protoplanet while trapped in the resonance. The efficiency of the resonance trapping combined with the subsequent low-velocity circumplanetary capture suggests that this trapped-to-captured transition may be important not only for the origin of captured satellites but also for continued growth of protoplanets. 相似文献
6.
John Chambers 《Icarus》2006,180(2):496-513
A new semi-analytic model for the oligarchic growth phase of planetary accretion is developed. The model explicitly calculates damping and excitation of planetesimal eccentricities e and inclinations i due to gas drag and perturbations from embryos. The effects of planetesimal fragmentation, enhanced embryo capture cross sections due to atmospheres, inward planetesimal drift, and embryo-embryo collisions are also incorporated. In the early stages of oligarchic growth, embryos grow rapidly as e and i fall below their equilibrium values. The formation of planetesimal collision fragments also speeds up embryo growth as fragments have low-e, low-i orbits, thereby optimizing gravitational focussing. At later times, the presence of thick atmospheres captured from the nebula aids embryo growth by increasing their capture cross sections. Planetesimal drift due to gas drag can lead to substantial inward transport of solid material. However, inward drift is greatly reduced when embryo atmospheres are present, as the drift timescale is no longer short compared to the accretion timescale. Embryo-embryo collisions increase embryo growth rates by 50% compared to the case where growth is solely due to accretion of planetesimals. Formation of 0.1-Earth-mass protoplanets at 1 AU and 10-Earth-mass cores at 5 AU requires roughly 0.1 and 1 million years respectively, in a nebula where the local solid surface density is 7 g cm−2 at each of these locations. 相似文献
7.
Planetesimals orbiting a protostar in a circumstellar disk are affected by gravitational interaction among themselves and by gas drag force due to disk gas. Within the Kyoto model of planetesimal accretion, the migration rate is interpreted as the inverse of the planetary formation time scale. Here, we study time scales of gravitational interaction and gas drag force and their influence on planetesimal migration in detail. Evaluating observations of 86 T Tauri stars (Beckwithet al., 1990), we find the mean radial temperature profile of circumstellar disks. The disk mass is taken to be 0.01M
in accordance with minimum mass models and observed T Tauri disks. The time scale of gravitational interaction between planetesimals is studied analogously to Chandrasekhar's stellar dynamics. Hence, Chandrasekhar's coefficient , defined as the fraction between the mean separation of planetesimals and the impact parameter, plays an important role in determining the migration rate. We find ln to lie between 5 and 10 within the protosolar disk. Our result is that, at the stage of disk evolution considered here, gas drag force affects the radial migration of planetesimals by a few orders of magnitude more than gravitational interaction.Paper presented at the Conference on Planetary Systems: Formation, Evolution, and Detection held 7–10 December, 1992 at CalTech, Pasadena, California, U.S.A. 相似文献
8.
We have performed N-body simulations on the stage of protoplanet formation from planetesimals, taking into account so-called “type-I migration,” and damping of orbital eccentricities and inclinations, as a result of tidal interaction with a gas disk without gap formation. One of the most serious problems in formation of terrestrial planets and jovian planet cores is that the migration time scale predicted by the linear theory is shorter than the disk lifetime (106-107 years). In this paper, we investigate retardation of type-I migration of a protoplanet due to a torque from a planetesimal disk in which a gap is opened up by the protoplanet, and torques from other protoplanets which are formed in inner and outer regions. In the first series of runs, we carried out N-body simulations of the planetesimal disk, which ranges from 0.9 to 1.1 AU, with a protoplanet seed in order to clarify how much retardation can be induced by the planetesimal disk and how long such retardation can last. We simulated six cases with different migration speeds. We found that in all of our simulations, a clear gap is not maintained for more than 105 years in the planetesimal disk. For very fast migration, a gap cannot be created in the planetesimal disk. For migration slower than some critical speed, a gap does form. However, because of the growth of the surrounding planetesimals, gravitational perturbation of the planetesimals eventually becomes so strong that the planetesimals diffuse into the vicinity of the protoplanets, resulting in destruction of the gap. After the gap is destroyed, close encounters with the planetesimals rather accelerate the protoplanet migration. In this way, the migration cannot be retarded by the torque from the planetesimal disk, regardless of the migration speed. In the second series of runs, we simulated accretion of planetesimals in wide range of semimajor axis, 0.5 to 2-5 AU, starting with equal mass planetesimals without a protoplanet seed. Since formation of comparable-mass multiple protoplanets (“oligarchic growth”) is expected, the interactions with other protoplanets have a potential to alter the migration speed. However, inner protoplanets migrate before outer ones are formed, so that the migration and the accretion process of a runaway protoplanet are not affected by the other protoplanets placed inner and outer regions of its orbit. From the results of these two series of simulations, we conclude that the existence of planetesimals and multiple protoplanets do not affect type-I migration and therefore the migration shall proceed as the linear theory has suggested. 相似文献
9.
10.
As planetary embryos grow, gravitational stirring of planetesimals by embryos strongly enhances random velocities of planetesimals and makes collisions between planetesimals destructive. The resulting fragments are ground down by successive collisions. Eventually the smallest fragments are removed by the inward drift due to gas drag. Therefore, the collisional disruption depletes the planetesimal disk and inhibits embryo growth. We provide analytical formulae for the final masses of planetary embryos, taking into account planetesimal depletion due to collisional disruption. Furthermore, we perform the statistical simulations for embryo growth (which excellently reproduce results of direct N-body simulations if disruption is neglected). These analytical formulae are consistent with the outcome of our statistical simulations. Our results indicate that the final embryo mass at several AU in the minimum-mass solar nebula can reach about ∼0.1 Earth mass within 107 years. This brings another difficulty in formation of gas giant planets, which requires cores with ∼10 Earth masses for gas accretion. However, if the nebular disk is 10 times more massive than the minimum-mass solar nebula and the initial planetesimal size is larger than 100 km, as suggested by some models of planetesimal formation, the final embryo mass reaches about 10 Earth masses at 3-4 AU. The enhancement of embryos’ collisional cross sections by their atmosphere could further increase their final mass to form gas giant planets at 5-10 AU in the Solar System. 相似文献
11.
P. Thebault 《Celestial Mechanics and Dynamical Astronomy》2011,111(1-2):29-49
HD 196885 Ab is the most ??extreme?? planet-in-a-binary discovered to date, whose orbit places it at the limit for orbital stability. The presence of a planet in such a highly perturbed region poses a clear challenge to planet-formation scenarios. We investigate this issue by focusing on the planet-formation stage that is arguably the most sensitive to binary perturbations: the mutual accretion of kilometre-sized planetesimals. To this effect we numerically estimate the impact velocities dv amongst a population of circumprimary planetesimals. We find that most of the circumprimary disc is strongly hostile to planetesimal accretion, especially the region around 2.6 AU (the planet??s location) where binary perturbations induce planetesimal-shattering dv of more than 1 kms?1. Possible solutions to the paradox of having a planet in such accretion-hostile regions are (1) that initial planetesimals were very big, at least 250 km (2) that the binary had an initial orbit at least twice the present one, and was later compacted due to early stellar encounters (3) that planetesimals did not grow by mutual impacts but by sweeping of dust (the ??snowball?? growth mode identified by Xie et al., in Astrophys J 724:1153, 2010b), or (4) that HD 196885 Ab was formed not by core-accretion but by the concurrent disc instability mechanism. All of these 4 scenarios remain however highly conjectural. 相似文献
12.
《Icarus》1987,69(1):51-69
A method is described for computing the probability distributions of the new orbital elements (a, e, i, q, q′) of a minor body which is subject ot close encounters with a planet. By including the frequency of such encounters the rate at which one class of orbit is transposed into a new class (e.g., Mars-crossing asteroids changed into Apollos) can be estimated. By applying this technique to the cases of Hidalgo and Chiron its uses are illustrated, and its limitations due to the two, two-body approximation utilized are pointed out. 相似文献
13.
We have made numerical experiments of the collisional and gravitational interaction of a planetesimal swarm in the early Solar System. In particular we study the dynamical evolution of an initial population of kilometer-size planetesimals subject to collisions (accretion, rebound, cratering, and catastrophic fragmentation). This study is based on a Monte-Carlo statistical method and provides the mass and velocity distributions of the planetesimal swarm as a function of time as well as their distribution in heliocentric distance. Several experiments have been performed and three of them are presented here. They simulate the accretional growth of numerous planetesimals in the absence (or presence) of gaseous drag, with (or without) one larger embryo among them, and with (or without) a size gradient. The results show that (i) for a population of planetesimals submitted to a negative gradient in size as the heliocentric distance increases, the outer planetesimals spiral toward the Sun faster than inner ones, leading after some time to an accumulation of bodies inside the cloud which allows the formation of an embryo; (ii) the growth of one embryo among a population of planetesimals is accelerated by the presence of gas and is warranted as long as its feeding zone is fed by the inward flow of planetesimals due to gas drag. These results offer some complementary new insights in the understanding of the accretional formation of 4–5 terrestrial planets instead of the numerous Moon-size planets generally found in numerical experiments. 相似文献
14.
We investigate classical planetesimal accretion in a binary star system of separation ab?50 AU by numerical simulations, with particular focus on the region at a distance of 1 AU from the primary. The planetesimals orbit the primary, are perturbed by the companion and are in addition subjected to a gas drag force. We concentrate on the problem of relative velocities Δv among planetesimals of different sizes. For various stellar mass ratios and binary orbital parameters we determine regions where Δv exceed planetesimal escape velocities vesc (thus preventing runaway accretion) or even the threshold velocity vero for which erosion dominates accretion. Gaseous friction has two crucial effects on the velocity distribution: it damps secular perturbations by forcing periastron alignment of orbits, but at the same time the size-dependence of this orbital alignment induces a significant Δv increase between bodies of different sizes. This differential phasing effect proves very efficient and almost always increases Δv to values preventing runaway accretion, except in a narrow eb?0 domain. The erosion threshold Δv>vero is reached in a wide (ab,eb) space for small <10-km planetesimals, but in a much more limited region for bigger ?50-km objects. In the intermediate vesc<Δv<vero domain, a possible growth mode would be the type II runaway growth identified by Kortenkamp et al. [Kortenkamp, S., Wetherill, G., Inaba, S., 2001. Science 293, 1127-1129]. 相似文献
15.
Zoë M. LeinhardtDerek C. Richardson 《Icarus》2002,159(2):306-313
We present results from direct N-body simulations of collisions between gravitational aggregates of varying size as part of a study to parameterize planetesimal growth in the Solar System. We find that as the ratio of projectile to target mass departs from unity, the impact angle has less effect on the outcome. At the same time, the probability of planetesimal growth increases. Conversely, for a fixed impact energy, collisions between impactors with mass ratio near unity are more dispersive than those with impactor mass ratio far from unity. We derive an expression for the accretion probability as a function of mass ratio. For an average mass ratio of 1:5, we find an accretion probability of ∼60% over all impact parameters. We also compute the critical specific dispersal energy Q*D as a function of projectile size. Extrapolating to a projectile size of 1 m with a 1-km target, we find Q*D=103−104 J kg−1, in agreement with several other collision models that use fundamentally different techniques. Our model assumes that the components of each gravitational aggregate are identical and indestructible over the range of sampled impact speeds. In future work we hope to incorporate a simple fracture model to extend the range of applicable speeds and we plan to implement our results in a large-scale planetesimal evolution code. 相似文献
16.
Tidal disruption is a potentially important process for the accumulation of the planets from planetesimals. The fact that stable equilibria do not exist for circular orbits inside the Roche limit has often been hypothesized to mean that any object that passes within the Roche limit is totally disrupted. We have disproven this hypothesis by solving the dynamic problem of the tidal disruption of a dissipative planetestimal during a close encounter with a protoplanet. The solution consists of a numerical integration of the three-dimensional, nonlinear equations of motion, including an approximate treatment of viscous dissipation in the solid regions of the planetesimal. The numerical methods have been extensively tested on a series of one-, two- (Jeans), and three-(Roche) dimensional test problems involving the equilibrium of a body subjected to tidal forces. The results may be scaled to planetesimals of arbitrary size, providing that the scaled equation of state applied. The calculations show that a strongly dissipative planetesimal which passes by the Earth on a parabolic orbit with a perigee within the Roche limit (≈3REarth) is not tidally disrupted (even for grazing incidence), and loses no more than a few percent of its mass. This result applies to bodies of radius R which have a kinematic viscosity ν ? 1012(R/1000km)2 cm2sec?1. Less dissipative planetesimals (ν ≈ 1013(R/1000 km)2 cm2sec?1) may lose up to about 20% of their mass. There are two coupled reasons why this result differs from previous hypotheses: (1) in a dynamic encounter, there is insufficient time to disrupt the planetesimal, and (2) even in circular orbit, the small velocities in the solid region imply that many orbital periods are necessary to completely disrupt the planetesimal. Hence solid and partially molten planetesimals will not experience substantial tidal disruption; completely molten bodies may be sufficiently inviscid to undergo tidal disruption. 相似文献
17.
Conventional planet formation models via coagulation of planetesimals require timescales in the range of several 10 or even 100 Myr in the outer regions of a protoplanetary disk. But according to observational data, the lifetime of a protoplanetary disk is limited to about 6 Myr. Therefore the existence of Uranus and Neptune poses a problem. Planet formation via gravitational instability may be a solution for this discrepancy. We present a parameter study of the possibility of gravitationally triggered disk instability. Using a restricted N‐body model which allows for a survey of an extended parameter space, we show that a passing dwarf star with a mass between 0.1 and 1 M⊙ can probably induce gravitational instabilities in the pre‐planetary solar disk for prograde passages with minimum separations below 80‐170 AU. Inclined and retrograde encounters lead to similar results but require slightly closer passages. Such encounter distances are quite likely in young moderately massive star clusters. The induced gravitational instabilities may lead to enhanced planetesimal formation in the outer regions of the protoplanetary disk, and could therefore be relevant for the formation of Uranus and Neptune. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
18.
We follow the contraction and evolution of a typical Jupiter-mass clump created by the disk instability mechanism, and compute the rate of planetesimal capture during this evolution. We show that such a clump has a slow contraction phase lasting ∼3×105 yr. By following the trajectories of planetesimals as they pass through the envelope of the protoplanet, we compute the cross-section for planetesimal capture at all stages of the protoplanet's evolution. We show that the protoplanet can capture a large fraction of the solid material in its feeding zone, which will lead to an enrichment of the protoplanet in heavy elements. The exact amount of this enrichment depends upon, but is not very sensitive to the size and random speed of the planetesimals. 相似文献
19.
Sergei I. Ipatov 《Meteoritics & planetary science》2023,58(6):752-774
The estimates of the delivery of icy planetesimals from the feeding zone of Proxima Centauri c (with mass equal to 7mE, mE is the mass of the Earth) to inner planets b and d were made. They included the studies of the total mass of planetesimals in the feeding zone of planet c and the probabilities of collisions of such planetesimals with inner planets. This total mass could be about 10–15mE. It was estimated based on studies of the ratio of the mass of planetesimals ejected into hyperbolic orbits to the mass of planetesimals collided with forming planet c. At integration of the motion of planetesimals, the gravitational influence of planets c and b and the star was taken into account. In most series of calculations, planetesimals collided with planets were excluded from integrations. Based on estimates of the mass of planetesimals ejected into hyperbolic orbits, it was concluded that during the growth of the mass of planet c the semi-major axis of its orbit could decrease by at least a factor of 1.5. Depending on possible gravitational scattering due to mutual encounters of planetesimals, the total mass of material delivered by planetesimals from the feeding zone of planet c to planet b was estimated to be between 0.002mE and 0.015mE. Probably, the amount of water delivered to Proxima Centauri b exceeded the mass of water in Earth's oceans. The amount of material delivered to planet d could be a little less than that delivered to planet b. 相似文献
20.
An analytical theory is developed for the velocity evolution of nonaccreting planetesimal populations, based on the Boltzmann and Fokker-Planck equations. Adapting Shkarofsky's calculation of plasma viscosities, the rate of increase in random velocities due to gravitational encounters between planetesimals of equal mass is found to be one-third to one-half Safronov's result. Comparison with Wetherill's numerical experiments suggests that the Fokker-Planck equation underestimates the effectiveness of encounters and that Safronov's value is approximately correct. For populations of nonuniform sizes, the Fokker-Planck equation indicates an efficient redistribution of energy from the largest bodies to the smaller ones. By conserving angular momentum, the rate of radial spreading of orbits is also derived. 相似文献