Possible long-lived asteroid belts in the inner Solar System (original) (raw)
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Hypothetical planet in the asteroid belt: An undergraduate exercise in celestial mechanics
AIP Conference Proceedings
We report results from numerical integrations of Solar System orbit for 100 mega years (Myr) from current epoch to the future with the integrator package EVORB. The model of the Solar System for asteroid belt region that previous researchers have proposed is reviewed in this paper. The previous results were very promising, up to more than 70% of the simulated cases have succeeded in removing terrestrial planets reside in the asteroid belt, region between the orbit of planet Mars and Jupiter. This is very close to the currently observed fact that massive objects no longer inhabit the asteroid belt region. We, therefore, simulated the presence of a massive object with a mass of the planet Mars at the current location of the dwarf planet Ceres (model02). The simulation results are compared with a Solar System model that excludes the presence of a hypothetical planet in the asteroid belt (model01). We found no significant difference in the orbital evolution of the Earth. At least for the next 100 Myr, Earth will remain a habitable planet. This computational laboratory activity provided an excellent case study for our upper-level undergraduate students in celestial mechanics course as their research project.
Existence of Asteroids in the Inner Solar System
Symposium - International Astronomical Union, 2004
Ensembles of in-plane and inclined orbits in the vicinity of the Lagrange points of the terrestrial planets are integrated for up to 100 million years. Mercurian Trojans probably do not exist, although there is evidence for long-lived, corotating horseshoe orbits with small inclinations. Both Venus and the Earth are much more promising, as they possess rich families of stable tadpole and horseshoe orbits. Our survey of in-plane test particles near the Martian Lagrange points shows no survivors after 60 million years. Low inclination test particles do not persist, as their inclinations are quickly increased until the effects of a secular resonance with Jupiter cause de-stabilisation. Numerical integrations of inclined test particles for timespans of 25 million years show stable zones for inclinations between 14°and 40°. Both Martian Trojans 5261 Eureka and 1998 VF31 lie deep within the stable zones, which suggests they may be of primordial origin.
Asteroids in the inner Solar system - I. Existence
Monthly Notices of the Royal Astronomical Society, 2002
Ensembles of in-plane and inclined orbits in the vicinity of the Lagrange points of the terrestrial planets are integrated for up to 100 Myr. The integrations incorporate the gravitational effects of the Sun and the eight planets (Pluto is neglected). Mercury is the least promising planet, as it is unable to retain tadpole orbits over 100-Myr timescales. Mercurian Trojans probably do not exist, although there is evidence for long-lived, corotating horseshoe orbits with small inclinations. Both Venus and the Earth are much more promising, as they possess rich families of stable tadpole and horseshoe orbits. Our survey of Trojans in the orbital plane of Venus is undertaken for 25 Myr. Some 40 per cent of the survivors are on tadpole orbits. For the Earth, the integrations are pursued for 50 Myr. The stable zones in the orbital plane are larger for the Earth than for Venus, but fewer of the survivors (,20 per cent) are tadpoles. Both Venus and the Earth also have regions in which inclined test particles can endure near the Lagrange points. For Venus, only test particles close to the orbital plane i & 168 are stable. For the Earth, there are two bands of stability, one at low inclinations i & 168 and one at moderate inclinations 248 & i & 348X The inclined test particles that evade close encounters are primarily moving on tadpole orbits. Two Martian Trojans (5261 Eureka and 1998 VF31) have been discovered over the last decade and both have orbits moderately inclined to the ecliptic (208 X 3 and 318 X 3 respectively). Our survey of in-plane test particles near the Martian Lagrange points shows no survivors after 60 Myr. Low-inclination test particles do not persist, as their inclinations are quickly increased until the effects of a secular resonance with Jupiter cause destabilization. Numerical integrations of inclined test particles for time-spans of 25 Myr show stable zones for inclinations between 148 and 408. However, there is a strong linear resonance with Jupiter that destabilizes a narrow band of inclinations at ,298. Both 5261 Eureka and 1998 VF31 lie deep within the stable zones, which suggests that they may be of primordial origin.
Long-term integrations and stability of planetary orbits in our Solar system
Monthly Notices of the Royal Astronomical Society, 2002
We present the results of very long-term numerical integrations of planetary orbital motions over 10 9-yr time-spans including all nine planets. A quick inspection of our numerical data shows that the planetary motion, at least in our simple dynamical model, seems to be quite stable even over this very long time-span. A closer look at the lowest-frequency oscillations using a low-pass filter shows us the potentially diffusive character of terrestrial planetary motion, especially that of Mercury. The behaviour of the eccentricity of Mercury in our integrations is qualitatively similar to the results from Jacques Laskar's secular perturbation theory (e.g. e max ∼ 0.35 over ∼±4 Gyr). However, there are no apparent secular increases of eccentricity or inclination in any orbital elements of the planets, which may be revealed by still longerterm numerical integrations. We have also performed a couple of trial integrations including motions of the outer five planets over the duration of ±5 × 10 10 yr. The result indicates that the three major resonances in the Neptune-Pluto system have been maintained over the 10 11-yr time-span.
Solar System dynamics, beyond the two-body-problem approach
AIP Conference Proceedings, 2006
When one thinks of the solar system, he has usually in mind the picture based on the solution of the two-body problem approximation presented by Newton, namely the ordered clockwork motion of planets on fixed, non-intersecting orbits around the Sun. However, already by the end of the 18th century this picture was proven to be wrong. As discussed by Laplace and Lagrange (for a modern approach see or [2]), the interaction between the various planets leads to secular changes in their orbits, which nevertheless were believed to be corrections of higher order to the Keplerian elliptical motion.
Icarus, 2005
We present a kinetic model of a disk of solid particles, orbiting a primary and experiencing inelastic collisions. In distinction to other collisional models that use a 2D (mass-semimajor axis) binning and perform a separate analysis of the velocity (eccentricity, inclination) evolution, we choose mass and orbital elements as independent variables of a phase space. The distribution function in this space contains full information on the combined mass, spatial, and velocity distributions of particles. A general kinetic equation for the distribution function is derived, valid for any set of orbital elements and for any collisional outcome, specified by a single kernel function. The first implementation of the model utilizes a 3D phase space (mass-semimajor axis-eccentricity) and involves averages over the inclination and all angular elements. We assume collisions to be destructive, simulate them with available material-and size-dependent scaling laws, and include collisional damping. A closed set of kinetic equations for a mass-semimajor axis-eccentricity distribution is written and transformation rules to usual mass and spatial distributions of the disk material are obtained. The kinetic "core" of our approach is generic. It is possible to add inclination as an additional phase space variable, to include cratering collisions and agglomeration, dynamical friction and viscous stirring, gravity of large perturbers, drag forces, and other effects into the model. As a specific application, we address the collisional evolution of the classical population in the Edgeworth-Kuiper belt (EKB). We run the model for different initial disk's masses and radial profiles and different impact strengths of objects. Our results for the size distribution, collisional timescales, and mass loss are in agreement with previous studies. In particular, collisional evolution is found to be most substantial in the inner part of the EKB, where the separation size between the survivors over EKB's age and fragments of earlier collisions lies between a few and several tens of km. The size distribution in the EKB is not a single Dohnanyi-type power law, reflecting the size dependence of the critical specific energy in both strength and gravity regimes. The net mass loss rate of an evolved disk is nearly constant and is dominated by disruption of larger objects. Finally, assuming an initially uniform distribution of orbital eccentricities, we show that an evolved disk contains more objects in orbits with intermediate eccentricities than in nearly circular or more eccentric orbits. This property holds for objects of any size and is explained in terms of collisional probabilities. The effect should modulate the eccentricity distribution shaped by dynamical mechanisms, such as resonances and truncation of perihelia by Neptune.
Successive Refinements in Long-Term Integrations of Planetary Orbits
Astrophysical Journal, 2003
We report on accurate, long-term numerical simulations of the orbits of the major planets in our solar system. The equations of motion are directly integrated by a Störmer multi-step scheme, which is optimized to reduce round-off errors. The physical models are successively refined to include corrections due to general relativity and the finite size of the lunar orbit. In one case, the Earth-Moon system is resolved as two separate bodies and the results are compared to those based on analytically averaging the lunar orbit. Through this comparison, a better analytical model is obtained. The computed orbits are in good agreement with those of previous studies for the past five million years but not for earlier times. The inner planets exhibit chaotic behavior with a Lyapunov time of exponential separation of nearby orbits equal to about 4 million years.
Structure of possible long-lived asteroid belts
Monthly Notices of the Royal Astronomical Society, 2002
High resolution simulations are used to map out the detailed structure of two long-lived stable belts of asteroid orbits in the inner Solar system. The Vulcanoid belt extends from 0.09 to 0.20 astronomical units (au), though with a gaps at 0.15 and 0.18 au corresponding to de-stabilising mean motion resonances with Mercury and Venus. As collisional evolution proceeds slower at larger heliocentric distances, kilometre-sized or larger Vulcanoids are most likely to be found in the region between 0.16 and 0.18 au. The optimum location in which to search for Vulcanoids is at geocentric ecliptic longitudes 9 • ≤ | g | ≤ 10 • and latitudes |β g | < 1 •. Dynamically speaking, the Earth-Mars belt between 1.08-1.28 au is an extremely stable repository for asteroids on nearly circular orbits. It is interrupted at 1.21 au due to the 3:4 commensurability with the Earth, while secular resonances with Saturn are troublesome beyond 1.17 au. These detailed maps of the fine structure of the belts can be used to plan search methodologies. Strategies for detecting members of the belts are discussed, including the use of infrared wide-field imaging with VISTA, and forthcoming European Space Agency satellite missions like GAIA and BepiColombo.
Order and chaos in the asteroid belt
Celestial Mechanics, the first branch of Astronomy where mathematical modelling managed to interpret and predict celestial phenomena, has a record of impressive achievements. In less than three hundred years, starting from Newton's law of gravitation and his three laws of dynamics, Celestial Mechanics managed finally to accurately describe, in practice, the trajectories of all planets and minor bodies of our solar system. The theory, behind all this, starts from the solution of the two-body problem and its particular properties.
The Primordial Excitation and Clearing of the Asteroid Belt
Icarus, 2001
In this paper, we use N -body integrations to study the effect that planetary embryos spread between ∼0.5 and 4 AU would have on primordial asteroids. The most promising model for the formation of the terrestrial planets assumes the presence of such embryos at the time of formation of Jupiter. At the end of their runaway growth phase, the embryos are on quasi-circular orbits, with masses comparable to that of the Moon or Mars. Due to gravitational interactions among them, and with the growing Jupiter, their orbits begin to cross each other, and they collide, forming bigger bodies. A general outcome of this model is that a few planets form in a stable configuration in the terrestrial planet region, while the asteroid belt is cleared of embryos. Due to combined gravitational perturbations from Jupiter and the embryos, the primordial asteroids are dynamically excited. Most of the asteroids are ejected from the system in a very short time, the dynamical lifetime being on the order of 1 My. A few asteroids (less than 1%) survive, mostly in the region 2.8-3.3 AU, and their eccentricity and inclination distribution qualitatively resembles the observed one. The surviving asteroids have undergone changes in semimajor axis of several tenths of an AU, which could explain the observed radial mixing of asteroid taxonomic types. When the distribution of massive embryos is truncated at 3 AU, we obtain too many asteroids in the outer part of the belt, especially too many Hildas. This suggests that the formation of Jupiter did not prohibit the formation of large embryos in the outer belt and Jupiter did not accrete them while it was still growing.