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1.
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   

2.
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   

3.
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   

4.
A new method for the development of the disturbing function of the three-body problem is outlined in this paper. A special process is devised to get the distance between two planetsP 1 andP 2 in terms of their heliocentric distances. It is then shown that the differential equations of relative motion of this problem can be brought in an homogeneous set of differential equations.  相似文献   

5.
Green's Theorem is developed for the spherically-symmetric steady-state cosmic-ray equation of transport in interplanetary space. By means of it the momentum distribution functionF o(r,p), (r=heliocentric distance,p=momentum) can be determined in a regionr arrbwhen a source is specified throughout the region and the momentum spectrum is specified on the boundaries atr a andr b . Evaluation requires a knowledge of the Green's function which corresponds to the solution for monoenergetic particles released at heliocentric radiusr o , Examples of Green's functions are given for the caser a =0,r b = and derived for the cases of finiter a andr b . The diffusion coefficient is assumed of the form = o(p)r b . The treatment systematizes the development of all analytic solutions for steady-state solar and galactic cosmic-ray propagation and previous solutions form a subset of the present solutions.  相似文献   

6.
In Sections 1–6, we determine an approximate analytical model for the density and temperature distribution in the protoplanetary could. The rotation of the planets is discussed in Section 7 and we conclude that it cannot be determined from simple energy conservation laws.The velocity of the gas of the protoplanetary cloud is found to be smaller by about 5×103 cm s–1 in comparison to the Keplerian circular velocity. If the radius of the planetesimals is smaller than a certain limitr 1, they move together with the gas. Their vertical and horizontal motion for this case is studied in Sections 8 and 9.As the planetesimals grow by accretion their radius becomes larger thanr 1 and they move in Keplerian orbits. As long as their radius is betweenr 1 and a certain limitr 2 their gravitational interaction is negligible. In Section 10, we study the accretion for this case.In Section 11, we determine the change of the relative velocities due to close gravitational encounters. The principal equations governing the late stages of accretion are deduced in Section 12, In Section 13 there are obtained approximate analytical solutions.The effect of gas drag and of collisions is studied in Sections 14 and 15, respectively. Numerical results and conclusions concerning the last and principal stage of accretion are drawn in Section 16.  相似文献   

7.
A number of Jupiter family comets such as Otermaand Gehrels 3make a rapid transition from heliocentric orbits outside the orbit of Jupiter to heliocentric orbits inside the orbit of Jupiter and vice versa. During this transition, the comet can be captured temporarily by Jupiter for one to several orbits around Jupiter. The interior heliocentric orbit is typically close to the 3:2 resonance while the exterior heliocentric orbit is near the 2:3 resonance. An important feature of the dynamics of these comets is that during the transition, the orbit passes close to the libration points L 1and L 2, two of the equilibrium points for the restricted three-body problem for the Sun-Jupiter system. Studying the libration point invariant manifold structures for L 1and L 2is a starting point for understanding the capture and resonance transition of these comets. For example, the recently discovered heteroclinic connection between pairs of unstable periodic orbits (one around the L 1and the other around L 2) implies a complicated dynamics for comets in a certain energy range. Furthermore, the stable and unstable invariant manifold tubes associated to libration point periodic orbits, of which the heteroclinic connections are a part, are phase space conduits transporting material to and from Jupiter and between the interior and exterior of Jupiter's orbit.  相似文献   

8.
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   

9.
We investigated the collision rate (i.e., the growth rate) of a migrating protoplanet with planetesimals. The collision rate strongly depends on the orbital elements of planetesimals (e.g., their eccentricities and inclinations). Thus we calculated the orbital evolutions of 2000 planetesimals in the vicinity of the migrating protoplanet and obtained the collision rate by counting the number of collisions with the protoplanet. For slow migration, the protoplanet makes a gap around its orbit in the planetesimals disk. On the other hand, for rapid migration, the protoplanet cannot shepherd planetesimals and keeps catching planetesimals. The obtained collision rate becomes larger with an increase in the migration speed. The comparison of the obtained collision rates with that of the previous work with no migration shows that the rapid migration of a protoplanet can enhance the collision rate by more than the factor 10. Using the obtained collision rate, we examined the growth of a migrating protoplanet. Our results suggest that, due to the enhancement of the collision rate, planets can be formed before they fall to the sun.  相似文献   

10.
This study presents a method of obtaining asymptotic approximations for motions near a Lagrange point in the planar, elliptic, restricted three-body problem by using a von Zeipel-type method. The calculations are carried out for a second-order escape solution in the proximity of the equilateral Lagrange point, L 4, where the primaries' orbital eccentricity is taken as the small parameter .  相似文献   

11.
The correlations angular momentaL to massesM are studied for different types of spectroscopic binaries. The functionsL=AM b have the coefficientb with the values expected from a Keplerian mechanics, but the valuesA(q, T), A(q, a), A(q, v), associated tob=5/3, 3/2, and 2, respectively, are given (statistically speaking) by multiples or submultiples of discrete values of: the mass ratiosq, the semi-major axesa, periodsT, and velocitiesv of the reduced mass. This indicates the existence of a discrete unit of actionL=(1/2)×potential energy xperiod. Postulates about equivalent states of angular momenta for different orbital parameters are introduced, being this coherent with the analysis of the up-to-date data. Among other examples of the application of such equivalence postulates, we haveL(M) (W-type of the WUMa systems)L(M) (main group of the Algol binaries). The quantum units of action seen here are equivalent to those seen in the solar system in one of our previous works. From comparisons with galaxies and single stars, it is evidence that there is not an unique universal functionL=AM b, when the fine structure of the relation is analysed: each type of object has its own coefficients,A, b. It sems to be that there are an upper and a lower limit for all the possible functions. The upper limit isL=A gM5/3, withA g1 associated to periodsT Hubble time, and the lower limit isL=GM 2/c, with 1. The existence of the upper limit can be investigated with studies of pairs of galaxies, and the lower limit can be tested with analysis of single G, K, M stars. The quantical hypothesis introduced here can be checked definitely, when available larger samples of data with low errors, with similar quality as the selected list of almost 80 eclipsing binaries (mainly detached systems) analysed here.  相似文献   

12.
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, Icarus137, 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, Icarus106, 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.  相似文献   

13.
An expansion of the force function ofn-body dynamical systems, where the equations of motion are expressed in the Jacobian coordinate system, is shown to give rise naturally to a set of (n–1) (n–2) dimensionless parameters ki li {i = 2,...,n;k = 2,...,i – 1 (i 3);l =i + 1,...,n (i n – 1)}, representative of the size of the disturbances on the Keplerian orbits of the various bodies. The expansion is particularized to the casen=3 which involves the consideration of only two parameters 23 and 32. Further, the work of Szebehely and Zare (1977) is reviewed briefly with reference to a sufficient condition for the stability of corotational coplanar three-body systems, in which two of the bodies form a binary system. This condition is sufficient in the sense that it precludes any possibility of an exchange of bodies, i.e. Hill type stability, however, it is not a necessary condition. These two approaches are then combined to yield regions of stability or instability in terms of the parameters 23 and 32 for any system of given masses and orbital characteristics (neglecting eccentricities and inclinations) with the following result: that there is a readily applicable rule to assess the likelihood of stability or instability of any given triple system in terms of 23 and 32.Treating a system ofn bodies as a set of disturbed three-body systems we use existing data from the solar system, known triple systems and numerical experiments in the many-body problem to plot a large number of triple systems in the 23, 32 plane and show the results agree well with the 23, 32 analysis above (eccentricities and inclinations as appropriate to most real systems being negligible). We further deal briefly with the extension of the criteria to many-body systems wheren>4, and discuss several interesting cases of dynamical systems.  相似文献   

14.
Ravit Helled  Attay Kovetz 《Icarus》2006,185(1):64-71
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   

15.
We study and review disk protoplanet interactions using local shearing box simulations. These suffer the disadvantage of having potential artefacts arising from periodic boundary conditions but the advantage, when compared to global simulations, of being able to capture much of the dynamics close to the protoplanet at high resolution for low computational cost. Cases with and without self sustained MHD turbulence are considered. The conditions for gap formation and the transition from type I migration are investigated and found to depend on whether the single parameter M p R 3/(M* H 3), with M p, M*, R, and H being the protoplanet mass, the central mass, the orbital radius and the disk semi-thickness, respectively, exceeds a number of order unity. We also investigate the coorbital torques experienced by a moving protoplanet in an inviscid disk. This is done by demonstrating the equivalence of the problem for a moving protoplanet to one where the protoplanet is in a fixed orbit which the disk material flows through radially as a result of the action of an appropriate external torque. For sustainable coorbital torques to be realized a quasi steady state must be realized in which the planet migrates through the disk without accreting significant mass. In that case, although there is sensitivity to computational parameters, in agreement with earlier work by Masset and Papaloizou [2003, ApJ, 588, 494] based on global simulations, the coorbital torques are proportional to the migration speed and result in a positive feedback on the migration, enhancing it and potentially leading to a runaway. This could lead to fast migration for protoplanets in the Saturn mass range in massive disks and may be relevant to the mass period correlation for extrasolar planets which gives a preponderance of sub Jovian masses at short orbital periods.  相似文献   

16.
A comparison of the lunar frontside gravity field with topography indicates that low-density ( 2.9 g cm–3) types of rock form a surface layer or crust of variable thickness: 40-60 km beneath terrae; 20-40 km beneath non-mascon maria; 0-20 km beneath mascon maria. The observed offset between lunar centers of mass and figure is consistent with farside crustal thicknesses of 40-50 km, similar to frontside terra thicknesses.The Moon is asymmetric in crustal thickness, and also in the distribution of maria and gamma radioactivity. Early bombardment of the Moon by planetesimals, in both heliocentric and geocentric orbits, is examined as a possible cause of the asymmetries. The presence of a massive companion (Earth) causes a spin-orbit coupled Moon to be bombarded non-uniformly. The most pronounced local concentration of impacts would have occurred on the west limb of the Moon, when it orbited close to the Earth, if low-eccentricity heliocentric planetesimals were still abundant in the solar system at that time.A very intense bombardment of this type could have redistributed crustal material on the Moon, thinning the west limb crust appreciably. This would have caused a change in position of the principal axes of inertia, and a reorientation of the spin-orbit coupled Moon such that the thinnest portion of its crust turned toward one of the poles. Erupting lavas would have preferentially flooded such a thin-crusted, low-lying area. This would have caused another readjustment of principal moments, and a reorientation of the Moon such that the mare areas tipped toward the equator. The north-south and nearside-farside asymmetries of mare distribution on the present Moon can be understood in terms of such a history.Paper dedicated to Prof. Harold C. Urey on the occasion of his 80th birthday on 29 April 1973.  相似文献   

17.
The equation of transfer for interlocked multiplets has been solved exactly by the method used by Busbridge and Stibbs (1954) for exponential form of the Planck functionB v (T)=b 0+b 1 e –.  相似文献   

18.
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   

19.
Assuming that an unknown mechanism (e.g., gas turbulence) removes most of the subnebula gas disk in a timescale shorter than that for satellite formation, we develop a model for the formation of regular (and possibly at least some of the irregular) satellites around giant planets in a gas-poor environment. In this model, which follows along the lines of the work of Safronov et al. [1986. Satellites. Univ. of Arizona Press, Tucson, pp. 89-116], heliocentric planetesimals collide within the planet's Hill sphere and generate a circumplanetary disk of prograde and retrograde satellitesimals extending as far out as ∼RH/2. At first, the net angular momentum of this proto-satellite swarm is small, and collisions among satellitesimals leads to loss of mass from the outer disk, and delivers mass to the inner disk (where regular satellites form) in a timescale ?105 years. This mass loss may be offset by continued collisional capture of sufficiently small <1 km interlopers resulting from the disruption of planetesimals in the feeding zone of the giant planet. As the planet's feeding zone is cleared in a timescale ?105 years, enough angular momentum may be delivered to the proto-satellite swarm to account for the angular momentum of the regular satellites of Jupiter and Saturn. This feeding timescale is also roughly consistent with the independent constraint that the Galilean satellites formed in a timescale of 105-106 years, which may be long enough to accommodate Callisto's partially differentiated state [Anderson et al., 1998. Science 280, 1573; Anderson et al., 2001. Icarus 153, 157-161]. In turn, this formation timescale can be used to provide plausible constraints on the surface density of solids in the satellitesimal disk (excluding satellite embryos for satellitesimals of size ∼1 km), which yields a total disk mass smaller than the mass of the regular satellites, and means that the satellites must form in several ∼10 collisional cycles. However, much more work will need to be conducted concerning the collisional evolution both of the circumplanetary satellitesimals and of the heliocentric planetesimals following giant planet formation before one can assess the significance of this agreement. Furthermore, for enough mass to be delivered to form the regular satellites in the required timescale one may need to rely on (unproven) mechanisms to replenish the feeding zone of the giant planet. We compare this model to the solids-enhanced minimum mass (SEMM) model of Mosqueira and Estrada [2003a. Icarus 163, 198-231; 2003b. Icarus 163, 232-255], and discuss its main consequences for Cassini observations of the saturnian satellite system.  相似文献   

20.
This paper studies the properties of self-gravity of accretion discs around supermassive black holes. With integration on the thin disc configuration, this paper has calculated the radial and vertical components of self-gravity of accretion discs. The discussion mainly concentrates on the radial component, and the results are briefly as follows: for accretion discs around supermassive black holes (M10 8–1010 M). At the distance where (R/R g)105–104, the radial component of self-gravity dominates over the central one where the dynamical structure of the accretion discs completely differs from that of Keplerian disc. A turbulence driven by radial self-gravity instability as a kind of energy source is proposed. This paper has two criteria for the comparison of magnitude between the self-gravity of accretion discs and the gravity of the central black hole, from which an analytic estimation for the outer radius of the accretion discs has been derived. The results of this paper may be used to explain the accretion discs of quasars and AGNs.This research was supported by the National Natural Science Foundation of China.  相似文献   

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