Planet Interactions Research Articles

On the migration-induced resonances in a system of two planets with masses in the Earth mass range

We investigate orbital resonances expected to arise when a system of two planets, with masses in the range 1–4 M⊕, undergoes convergent migration while embedded in a section of gaseous disc where the flow is laminar. We consider surface densities corresponding to 0.5–4 times that expected for a minimum mass solar nebula at 5.2 au. For the above mass range, the planets undergo type I migration. Using hydrodynamic simulations, we find that, when the configuration is such that convergent migration occurs, the planets can become locked in a first-order commensurability for which the period ratio is (p+ 1)/p with p being an integer and migrate together maintaining it for many orbits. Slow convergent migration results in commensurabilities with small p such as 1 or 2. Instead, when the convergent migration is relatively rapid as tends to occur for disparate masses, higher p commensurabilities are realized such as 4:3, 5:4, 7:6 and 8:7. However, in these cases the dynamics is found to have a stochastic character with some commensurabilities showing long-term instability with the consequence that several can be visited during the course of a simulation. Furthermore, the successful attainment of commensurabilities is also a sensitive function of initial conditions. When the convergent migration is slower, such as occurs in the equal-mass case, lower p commensurabilities such as 3:2 are obtained, which show much greater stability. Resonant capture leads to a rise in eccentricities that can be predicted using a simple analytic model that assumes the resonance is isolated, constructed in this paper. We find that, once the commensurability has been established, the system with an 8:7 commensurability is fully consistent with this prediction. We find that very similar behaviour is found when the systems are modelled using an N-body code with simple prescriptions for the disc–planet interaction. Comparisons with the hydrodynamic simulations indicate reasonably good agreement with predictions for these prescriptions obtained using the existing semi-analytic theories of type I migration. We have run our hydrodynamic simulations for up to 103–104 orbits of the inner planet. Longer times could only be followed in the simpler N-body approach. Using that, we found that, on the one hand, an 8:7 resonance established in a hydrodynamic simulation could be maintained for more than 105 orbits. On the other hand, other similar cases show instability leading to another resonance and ultimately a close scattering. There is already one known example of a system with nearly equal masses in the range of several Earth masses, namely the two pulsar planets in PSR B1257+12, which are intriguingly, in view of the results obtained here, close to a 3:2 commensurability. This will be considered in a future publication. Future detection of other systems with masses in the Earth mass range that display orbital commensurabilities will give useful information on the role and nature of orbital migration in planet formation.

The heliosphere and the interaction of the terrestrial planets with the solar wind

The heliosphere and the interaction of the terrestrial planets with the solar wind

Global numerical simulations of differentially rotating disks with free eccentricity

We study the nonlinear evolution of global m = 1 modes with low pattern speeds in a differentially rotating non magnetic disk, by means of two and three dimensional numerical simulations. The modes make disk streamlines eccentric with maximum eccentricity in the range 0.13−0.24. We found that long lived patterns corresponding to eccentricities ∼0.1 lasted for the duration of the simulations of ∼64 orbits at the disk outer boundary. They had slow retrograde precession with period ∼10 orbits at the outer boundary, in good agreement with that found from linear normal mode analysis. As expected from linear stability analysis, which leads one to expect a parametric instability associated with the non circular streamlines, we also found that three dimensional simulations showed the growth of a local instability to producing vertical motions on a local scale that eventually results in small amplitude turbulence with root mean square vertical velocity typically ∼0.03cs, cs being the sound speed. This turbulence together with much more effective shocks produced by the interaction of the eccentric disk with the inner boundary were responsible for damping the disk eccentricity. Our first estimate of the damping time associated with turbulence for eccentricities ∼0.1, corresponded to that associated with a turbulent viscosity acting to circularize the eccentric streamlines with α parameter ∼10 −3 . For parameters appropriate to protostellar disks at 5 AU this corresponds to a decay time ∼10 5−6 y. Thus although the indication is that disks with eccentric streamlines are long lived, there is associated turbulence even in the non magnetic case leading to estimated decay times that may be significant when the possibility of the growth of orbital eccentricity for extrasolar planets is considered through disk planet interaction on sufficiently long timescales.

The interaction of the stellar wind with an extrasolar planet—3D hybrid and drift-kinetic simulations

To be able to simulate the interaction of extrasolar planets with the stellar wind, a number of planetary parameters are required. Some of these (like planetary mass and radius) can be obtained directly from observational data. Other properties are not known very precisely. For example, up to now, there is no observation providing information on the strength of planetary magnetic moments. However, there is good reason to expect only very small magnetic moments for planets in very close orbits around their stars (like HD 209458 b and OGLE-TR-56 b). Thus, as a first step towards a more complete treatment, it seems reasonable to treat the interaction of the stellar wind with an unmagnetized planet. Calculations were performed for a nonconducting as well as for a weakly conducting planet. The interaction with the stellar wind and the resulting induced magnetosphere was simulated using a three dimensional hybrid code as well as in the drift-kinetic approximation. The effect of a interplanetary magnetic field oriented perpendicular to the incoming stellar wind was included. In the case of a weakly conducting body an asymmetrical Mach cone is formed, whereas for a nonconducting body no Mach cone is observed. These investigations will serve as the first step in the search for particular effects occurring at extrasolar planets, which could possibly lead to observable effects, e.g. radio emission. The results are also relevant for plasma structures near weakly conducting, unmagnetized bodies like the Earth's moon.

Geliosfera i vzaimodeistvie planet zemnoi gruppy s solnechnym vetrom

Geliosfera i vzaimodeistvie planet zemnoi gruppy s solnechnym vetrom

Jumping Jupiters in Binary Star Systems

We investigate the outcomes of the dynamical interaction of Jupiter-mass planets orbiting the central star in a binary system. These systems are unstable and lead to the hyperbolic ejection of one or more planets, while the surviving bodies are inserted in inner eccentric orbits. The gravitational perturbations of the companion star, set at an intermediate distance (50 AU) and typically on an eccentric orbit, influence both the development of instability and the outcome of the subsequent chaotic evolution. We compute the statistical properties of the resulting planetary systems when they reach a stable configuration. The binary eccentricity and the number of initial planets (two or three) are strong predictors of the final configuration of the planetary system. Cases of apsidal resonance between two final planets, Kozai resonance between a single surviving planet and the companion star, and retrograde orbits with respect to the binary orbit are naturally produced.

A comparison of magnetohydrodynamic instabilities at the Martian ionopause

It is known from Pioneer Venus measurements that at the Venusian ionopause wave-like structures develop, which can detach in the form of ionospheric plasma clouds. This phenomenon is assumed to occur due to the Kelvin–Helmholtz instability, which can appear in large regions of the Venusian ionopause. Recent studies of Mars Global Surveyor measurements indicate that wave–like structures and plasma clouds also detach from the Martian ionopause. Therefore, these features seem to be common for the solar wind interaction of non-magnetized planets. We study the conditions at the Martian ionopause with respect to the occurrence of several MHD instabilities. The conditions in the magnetosheath are modeled by a semi-analytical MHD simulation that includes mass loading. The ionospheric parameter needed for the model calculations are taken from a global hybrid model. The stability of the Martian ionopause against the Kelvin–Helmholtz, the Rayleigh–Taylor, and the interchange instability is analyzed. Further, we suggest that including the Hall term in the description of the Kelvin–Helmholtz instability gives a current in the planetary boundary layer resulting in a shear flow compared with the ionospheric plasma, which can lead to an unstable boundary layer near the subsolar point. Since the interchange instability depends on the curvature of the magnetic field lines, we additionally study the influence of the strong curvature of the Martian ionopause due to the localized, remnant, crustal magnetism appearing mainly in the southern hemisphere of Mars.

Timing regulation in insect reproduction

Summary The interaction of planets, the earth, the moon, and the sun, causes rhythmic changes in the environment. Many aspects of the lives of insects have to coincide with these changes in order for them to survive and prosper. For insects to anticipate the rhythmic changes of the environment, an endogenous clock (or clocks) is employed in almost all known species. The canonical circadian clock is the best known system used in temporal regulation of insect daily life processes. It is employed in the regulation of reproductive physiology and behavior of insects, providing a timing mechanism that ensures sexual receptivity, mate finding, sperm release, and ovulation, for instance, occurring at appropriate times, thereby maximizing fitness. Several insect species exhibit life cycles longer than 1 year, such as the 17-year cicada or the two-year life cycle of some mayflies. However, it is more common in insects to have reproductive cycles shorter than 1 year. The requirement for a circannual clock, therefore, is less common in insects compared to the circadian clock. To outline the underlying mechanism and the role of the circadian clock in an overt rhythm which is much longer than 24 h is a challenging subject. However, a photoperiodic system with the ability to respond to the duration of periods of light and dark can certainly drive a seasonal rhythm in temperate regions where the photoperiodic change is significant. It is possible that insects use a photoperiod to gate the reproductive season, but they may also depend on the circadian clock to control the time of mate-pairing within the breeding season. Several species have been investigated to reveal the driving force behind the timing of reproduction, including our own research on the German cockroach. Although molecular expression of the circadian clock has been advancing rapidly, the functional elements of the putative circannual clock remain unexplored. The linkage between overt behavioral rhythms with internal physiological conditions and the molecular mechanism of clock expression certainly can unveil the mystery of timing regulation in insect reproduction.

The interaction of giant planets with a disc with MHD turbulence - IV. Migration rates of embedded protoplanets

We present the results of global cylindrical disc simulations and local shearing box simulations of protoplanets interacting with a disc undergoing magnetohydrodynamic (MHD) turbulence. The specific emphasis of this paper is to examine and quantify the magnitude of the torque exerted by the disc on the embedded protoplanets as a function of the protoplanet mass, and thus to make a first study of the induced orbital migration of protoplanets resulting from their interaction with magnetic, turbulent discs. This issue is of crucial importance in understanding the formation of gas giant planets through the so-called core instability model, and the subsequent orbital evolution post-formation prior to the dispersal of the protostellar disc. Current estimates of the migration time of protoplanetary cores in the 3–30 Earth mass range in standard disc models are τmig≃ 104–105 yr, which is much shorter than the estimated gas accretion time-scale of Jupiter-type planets. The global simulations were carried out for a disc with constant aspect ratio H/r= 0.07 and protoplanet masses of Mp= 3, 10, 30 Earth masses, and 3 Jupiter masses. The local shearing box simulations were carried out for values of the dimensionless parameter (Mp/M*)/(H/R)3= 0.1, 0.3, 1.0 and 2.0, with M*, R and H being the central mass, the orbital radius and the local disc semithickness, respectively. These allow both embedded and gap-forming protoplanets for which the disc response is non-linear to be investigated. In all cases the instantaneous net torque experienced by a protoplanet showed strong fluctuations on an orbital time-scale, and in the low-mass embedded cases oscillated between negative and positive values. Consequently, in contrast to the laminar disc type I migration scenario, orbital migration would occur as a random walk. Running time averages for embedded protoplanets over typical run times of 20–25 orbital periods indicated that the averaged torques from the inner and outer disc took on values characteristic of type I migration. However, large fluctuations occurring on longer than orbital time-scales remained, preventing convergence of the average torque to well-defined values or even to a well-defined sign for these lower mass cases. Fluctuations became relatively smaller for larger masses indicating better convergence properties, to the extent that in the 30-M⊕ simulation consistently inward, albeit noisy, migration was indicated. Both the global and local simulations showed this trend with increasing protoplanet mass, which is due to its perturbation on the disc increasing to become comparable to and then dominate the turbulence in its neighbourhood. The turbulence then becomes unable to produce very large long-term fluctuations in the torques acting on the protoplanet. Eventually gap formation occurs and there is a transition to the usual type II migration at a rate determined by the angular momentum transport in the distant parts of the disc. The existence of significant fluctuations occurring in the turbulent discs on long time-scales is an important unexplored issue for the lower mass embedded protoplanets, which are unable to modify the turbulence in their neighbourhood, and which have been studied here. If significant fluctuations occur on the longest disc evolutionary time-scales, convergence of torque running averages for practical purposes will not occur and the migration behaviour of low-mass protoplanets considered as an ensemble will be very different from predictions of type I migration theory for laminar discs. The fact that noise levels were relatively smaller in the local simulations may indicate the presence of long-term global fluctuations, but the issue remains an important one for future investigation.

Bow Shock and Upstream Phenomena at Mars

Mars Global Surveyor is the sixth spacecraft to return measurements of the Martian bow shock. The earlier missions were Mariner 4 (1964), Mars 2 and 3 (1972), Mars 5 (1975) and Phobos 2 (1989) (see reviews by Gringauz, 1981; Slavin and Holzer, 1982; Russell, 1985; Vaisberg, 1992a,b; Zakharov, 1992). Previous investigations of planetary bow shocks have established that their position, shape and jump conditions are functions of the upstream flow parameters and the nature of the solar wind — planet interaction (Spreiter and Stahara, 1980; Slavin et al., 1983; Russell, 1985). At Mars, however, the exact nature of the solar wind interaction was elusive due to the lack of low altitude plasma and magnetic field measurements (e.g., Axford, 1991). In fact our knowledge of the nature of the interaction of Mars with the solar wind was incomplete until the arrival of MGS and the acquisition of close-in magnetic field data (Acuna et al., 1998). As detailed by a series of review papers in this monograph, the Mars Global Surveyor (MGS) mission has now shown that the Mars environment is very complex with strong, highly structured crustal magnetic remnants in the southern hemisphere, while the northern hemisphere experiences the direct impingement of solar wind plasma. This review paper first presents a survey of the observations on the Martian bow shock and the upstream phenomena in the light of results from all the missions to date. It also discusses the kinetic properties of the Martian bow shock compared to the predictions of simulations studies. Then it examines the current status of understanding of these phenomena, including the possible sources of upstream low-frequency waves and the interpretations of localized disturbances in the upstream solar wind around Mars. Finally, it briefly discusses the open issues and questions that require further study.

The interaction of giant planets with a disc with MHD turbulence - III. Flow morphology and conditions for gap formation in local and global simulations

We present the results of both global cylindrical disc simulations and local shearing box simulations of protoplanets interacting with a disc undergoing magnetohydrodynamic (MHD) turbulence with zero net flux magnetic fields. We investigate the nature of the disc response and conditions for gap formation. This issue is an important one for determining the type and nature of the migration of the protoplanet, with the presence of a deep gap being believed to enable slower migration. For both types of simulation we find a common pattern of behaviour for which the main parameter determining the nature of the response is 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 local disc semi-thickness respectively. We find that as M p R 3 /(M*H 3 ) is increased to ∼ 0.1 the presence of the protoplanet is first indicated by the appearance of the well-known trailing wake which, although it may appear to be erratic on account of the turbulence, appears to be well-defined. Once M p R 3 /(M*H 3 ) exceeds a number around unity, a gap starts to develop inside which the magnetic energy density tends to be concentrated in the high-density wakes. This condition for gap formation can be understood from simple dimensional considerations of the conditions for non-linearity and the balance of angular momentum transport due to Maxwell and Reynolds' stresses with that due to tidal torques applied to the parameters of our simulations. An important result is that the basic flow morphology in the vicinity of the protoplanet is very similar in both the local and global simulations. This indicates that, regardless of potentially unwanted effects arising from the periodic boundary conditions, local shearing box simulations, which are computationally less demanding, capture much of the physics of disc-planet interactions. Thus they may provide a useful tool for studying the local interaction between forming protoplanets and turbulent, protostellar discs.

Simulations of gaseous disc-embedded planet interaction

We present three-dimensional self-gravitating smoothed-particle hydrodynamics (SPH) simulations of an isothermal gaseous disc interacting with an embedded planet. Discs of varying stability are simulated with planets ranging from 10 Earth-masses to 2 Jupiter-masses. The SPH technique provides the large dynamic range needed to accurately capture the large scale behavior of the disc as well as the small scale interaction of the planet with surrounding material. Most runs used 10^5 gas particles, giving us the spatial resolution required to observe the formation of planets. We find four regions in parameter space: low-mass planets undergo Type I migration; higher-mass planets can form a gap; the gravitational instability mode of planet formation in marginally stable discs can be triggered by embedded planets; discs that are completely unstable can fragment to form many planets. The disc stability is the most important factor in determing which interaction a system will exhibit. For the stable disc cases, our migration and accretion time-scales are shorter and scale differently than previously suggested.

Growth and interaction of planets

We discuss theories of origin and evolution of the newly discovered extrasolar planetary systems. As these systems failed to fulfill prior expectations concerning their orbital structure, we are challenged to extend and/or revise many preexisting theories. Important extensions include migration of bodies in disks and planetary eccentricity pumping by planet-planet interaction and primordial disk-planet interaction. Progress in observational techniques will allow us to find which of these two types of interaction is responsible for the observed variety of orbits and masses of planets. New insights into the formation of giant planets in our system can be obtained by asking why Jupiter and Saturn are not larger, closer to the sun and/or do not follow noticeably elliptic orbits.

On the tidal interaction of massive extrasolar planets on highly eccentric orbits

In this paper we develop a theory of disturbances induced by the stellar tidal field in a fully convective slowly rotating planet orbiting on a highly eccentric orbit around a central star. In this case it is appropriate to treat the tidal influence as a succession of impulsive tidal interactions occurring at periastron passage. For a fully convective planet mainly the l = 2 fundamental mode of oscillation is excited. We show that there are two contributions to the mode energy and angular momentum gain due to impulsive tidal interaction: (i) 'the quasi-static' contribution, which requires dissipative processes operating in the planet, and (ii) the dynamical contribution associated with excitation of modes of oscillation. These contributions are obtained self-consistently from a single set of the governing equations. We calculate a critical 'equilibrium' value of angular velocity of the planet Ω crit determined by the condition that action of the dynamic tides does not alter the angular velocity at this rotation rate. We show that this can be much larger than the corresponding rate associated with quasi-static tides and that at this angular velocity, the rate of energy exchange is minimized. We also investigate the conditions for the stochastic increase in oscillation energy that may occur if many periastron passages are considered and dissipation is not important. We provide a simple criterion for this instability to occur. Finally, we make some simple estimates of the time-scale of evolution of the orbital semimajor axis and circularization of the initially eccentric orbit due to tides, using a realistic model of the planet and its cooling history, for orbits with periods after circularization typical of those observed for extrasolar planets P obs ≥ 3 d. Quasi-static tides are found to be ineffective for semimajor axes ≥0.1 au. On the other hand, dynamic tides could have produced a very large decrease of the semimajor axis of a planet with mass of the order of the Jupiter mass M J and final period P obs ∼ 1-4.5 d on a time-scale ≥ a few Gyrs. In this case the original semimajor axis may be as large as ≥10 2 au. For planets with masses ≥5M J dynamic tides excited in the star appear to be more important than the tides excited in the planet. They may also, in principle, result in orbital evolution in a time less than or comparable to the lifetime of the planetary systems. Finally, we point out that there are several issues in the context of the scenario of the circularization of the orbit solely due to dynamic tides that remain to be resolved. Their possible resolution is discussed.

The Effect of Neptune's Accretion on Pluto and the Plutinos

The peculiar relationship of Pluto to Neptune, its resonances and high eccentricity and inclination, have led to the theory that the relationship arose from the migration of the outer planets, particularly the outward migration of Neptune, during the early solar system. In support of this scenario is the fact that the formation of Neptune at its current location would have been complicated by long dynamical times and low densities in the solar nebula. Here we address the following questions: Though the formation of Neptune at its current location seems unfavorable, are there dynamical obstacles to the capture of Pluto and the Plutinos under this scenario? Or are there features of the Neptune-Pluto system that would allow us to preclude this possibility of Neptune forming near its current orbit? Levison & Stern have examined the effect of the purely gravitational interactions of the giant planets on Pluto and concluded that the most important dynamical aspects of the Neptune-Pluto system could be reproduced. The exception was the amplitude of the 3 : 2 resonant argument, which was found to be too large in their model. We performed simulations of the outer solar system that included a slowly accreting Neptune and found that the efficiency of capture of dynamically cold particles into the 3 : 2 resonance was increased by a factor of 3, and that the resonant argument was substantially decreased. However, further dissipation is still required to match all aspects of the Plutino population and to produce truly Pluto-like orbits. Given that cold initial conditions did not reproduce the observations completely, simulations of initially dynamically hot particles near the 3 : 2 resonance with Neptune were also examined. These results, though resulting from seemingly ad hoc starting conditions, are reported as they produce remarkably good matches with both the Plutino population and Pluto’s own orbit, including all three of its known resonances. These simulations reveal that Pluto could have arisen from an initially low-e (� 0) but high-i (� 25 � ) orbit, both a clue to its origin and an illustration of the difficulty in understanding Pluto’s current orbital configuration.

Statistical properties of exoplanets

Interesting emerging observational properties of the period-mass distribution of extra-solar planets are discussed. New recent detections confirm the already emphasized lack of massive planets () on short-period orbits ( days). Furthermore, we point out i) a shortage of planets in the 10–100 day period range as well as ii) a lack of light planets () on orbits with periods larger than ~100 days. The latter feature is shown not to be due to small-number statistics with Monte-Carlo simulations. These observational period-related characteristics are discussed in the context of the migration process of exoplanets. They are found to be in agreement with recent simulations of planet interactions with viscous disks. The observed valley at a few tens of days in the period distribution is interpreted as a transition region between two categories of planets that suffered different migration scenarios. The lack of light planets on longer-period orbits and the corresponding intriguing sharp limit in mass is tentatively explained by the runaway migration process recently studied by Masset & Papaloizou (2003). The observed properties also have implications for the observation strategies of the on-going surveys and of future higher-precision searches.

Disc–Planet Interactions: Migration and Resonances in Extrasolar Planetary Systems

We consider orbital resonances in multiplanet systems. These are expected to arise during or just after formation in a gaseous disc. Disc–planet interaction naturally produces orbital migration and circularization through the action of tidal torques which in turn may lead to an orbital resonance. The mass and angular momentum content of the disc is likely to be comparable to that in the planets so that it is essential to fully incorporate the disc in the analysis.

Resonance in PSR B1257+12 Planetary System

In this paper we present a new method that can be used for analysis of times of arrival (TOAs) of pulsar pulses. It is especially designed to detect quasi-periodic variations of TOAs. We apply our method to timing observations of PSR B1257+12 and demonstrate that by using it, it is possible to detect not only first harmonics of a periodic variations, but also the presence of a resonance effect. The resonance effect detected independently of its physical origin can appear only when there is a nonlinear interaction between two periodic modes. The explanation of TOAs variations as an effect of the existence of planets was, until now, the only known and well-justified explanation. In this context, the existence of the resonance frequency in TOAs is the most significant signature of the gravitational interaction of planets.

Interactions between Planets and Small Bodies: Introduction

The exploration of our Solar System is rapidly growing in importance as a scientific discipline. During the last decades, great progress has been achieved as the result of space missions to planets and small bodies and improved remote-sensing methods, as well as due to refined techniques of laboratory measurements and a rapid progress in theoretical studies, involving the development of various astrophysical and geophysical evolutionary models, based in particular on the approach of comparative planetology. In the crossroads of astronomy and geophysics, recent years have seen a growing understanding of the importance of impact phenomena throughout the history of the Solar System and, therefore, the necessity to get more insight into the problem of interactions of planets and small bodies. This importance is clearly manifested by the observed cratering records of planetary surfaces and such dramatic events as the explosions of the comet P/Shoemaker-Levy 9 fragments in Jupiter’s atmosphere in 1994, that of the Tunguska object over Siberia in 1908, and the Chicxulub event dating back to the end of the Cretaceous.

Mapping the Effects of Distant Perturbations on Particle–Planet Interactions

Monte-Carlo codes generally treat planestesimal–planet encounters using the two-body scattering approximation, which can be inaccurate when relative velocities are low; however, Monte-Carlo codes using the two-body approximation frequently produce results consistent with more accurate codes using numerical integration. To better understand why this breakdown occurs at low velocities, and to test a hypothesis from Greenberget al. (1988,Icarus75, 1–29) that may explain the unexpected accuracy of Monte-Carlo codes, we numerically integrate test body trajectories using a unique set of orbital elements defined by the geometry of the two-body approximation. This new coordinate system is ideal for examining the effects of distant planetary perturbations on particle trajectories all the way to encounter with the planet.Our results show that the failure of the two-body approximation is caused by distant planetary perturbations modifying the approach geometry of the test bodies; behavior at encounter follows two-body scattering even at very low relative velocities. By testing particle swarms encountering a planet, we found that some test bodies, whose approach orbits were shifted by distant planetary perturbations, were then replaced by similarly shifted nearby test bodies. The “particle replacement” mechanism explains why Monte-Carlo codes frequently yield outcome results comparable to numerical integration results. Moreover, we found that the relative velocity of a test body at encounter is not the critical parameter in determining the “breakdown” of two-body scattering outcome statistics; instead, we found that the semimajor axis of the test body relative to the size of the planet's Hill sphere (or the synodic period of the test body when mass is included) is much more diagnostic. Thus, our results verify that Monte-Carlo models can yield statistically accurate results, even if individual particles do not behave as assumed in those codes.