Orbital evolution of asteroids with the effects of mutual gravitational attraction

The effects of the mutual gravitational attraction between asteroids were analyzed by two N-body calculations, in which N=4,516 (the Sun, the nine planets, and 4,506 asteroids). In one calculation the gravity of the asteroids was taken into account, and in the other it was ignored. These calculations were carried out for a time period of about 100 years. The largest difference in the positions of the asteroids between these two calculations is about 10^?3 AU. For the orbital elements of the semimajor axis, the eccentricity, and the inclination, the largest differences were 9 × 10^?6 AU, 4 × 10^?6, and 5 × 10^?4 degrees, respectively. It was found that the distribution of the differences of the semimajor axis between the two calculations is quite similar to the Cauchy distribution.     

Earth albedo and the orbit of Lageos

The long-period perturbations in the orbit of Lageos satellite due to the earth's albedo have been found using a new analytical formalism. The earth is assumed to be a sphere whose surface diffusely reflects sunlight according to Lambert's law. Specular reflection is not considered. The formalism is based on spherical harmonics; it produces equations which hold regardless of whether the terminator is seen by the satellite or not. Specializing in the case of a realistic zonal albedo shows that Lageos' orbital semimajor axis changes periodically by only about a centimeter and the eccentricity by two parts in 10⁵. The longitude of the node increases secularly by about 6×10^–4 arc sec yr^–1. The effect considered here can explain neither the secular decay of 1.1 mm day^–1 in the semimajor axis nor the observed along-track variations in acceleration of order 2×10^–12 ms^–2.     

Implications from the motion of outer belt asteroids

A principal feature of the asteroidal distribution is the rapid truncation of its population outward from ～3.4 AU. This paper presents further evidence, based on the motion and distribution of certain minor planets with large semimajor axes, that this truncation cannot be strictly the result of gravitational perturbations of the major planets even acting over times of ～10⁹ years. The motion of other outer asteroids sets a probable upper limit of 0.081 on Jupiter's eccentricity.     

Asteroid sizes and albedos

The radiometric method of determining diameters of asteroids is reviewed, and a synthesis of radiometric and polarimetric measurements of the diameters and geometric albedos of a total of 187 asteroids is presented. All asteroids with diameters greater than 250 km are identified, and statistical studies can be carried out of the size distributions of different albedo classes down to 80-km diameter for the entire main asteroid belt (2.0–3.5 AU). The distribution of albedos is strongly bimodal, with mean albedos for the C and S groups of 0.035 and 0.15, respectively. The C asteroids outnumber the S at all sizes and all values of semi-major axis, increasing from a little over half the population inside 2.5 AU to more than 95% beyond 3.0 AU; for all objects with D > 70 km, the ratio C/(C+S) is 0.88 ± 0.04. More than half of all asteroids in this size range have a > 3.0 AU. The M asteroids constitute about 5% of the population for a < 3.0 AU, but no members of of this class have been identified in the outer belt. There are no significant differences between the distributions of C, S, and M asteroids for the largest asteroids (D > 200 km) and for those of intermediate size (200–270 km). The total mass in the belt, down to 70-km size, but excluding Ceres, is about 2 × 10²⁴ g. Evidence is presented that several large asteroids rotate in a prograde sense, and that a real difference existsbetween the bulk densities of Ceres and Vesta.     

A Specific Nongravitational Effect in the Asteroid Belt

The evolution of orbits of asteroids found in the IRAS and WISE albedo databases was calculated numerically from 2005 to 2016. It follows from the analysis of the obtained results that a certain nongravitational effect (NGE) currently affects the motion of a considerable fraction of main-belt asteroids with diameters up to 40 km. This conclusion agrees with the available data regarding the axial rotation of asteroids. The NGE manifests itself in an increase in the semimajor axes of orbits of low-albedo asteroids relative to the semimajor axes of orbits of high-albedo bodies. The NGE-induced rate of elongation of semimajor axes of asteroids with albedos р_v < 0.1 may be as high as (2–8) × 10^–8 AU/year. Errors in orbital elements of asteroids (unrelated to the accuracy of observational data used to determine these orbital elements) were found in one of the MPC catalogues for 2003 in the process of estimation of the accuracy of calculations.     

On the transport of bodies within and from the asteroid belt

Abstract— This paper explores two processes, sweeping secular resonance (Ward, 1981) and gas drag (Lecar and Franklin, 1997), at work during the dispersal of the solar nebula. we have two aims not previously considered for the two mechanisms: (1) to explain the likely depletion, by a factor of 1000 or so, of the rocky material in the inner belt (2.0 < a < 3.2 AU); (2) to introduce a means for providing—or contributing to—the dispersion in semimajor axis of the various asteroidal taxonomic classes. We suggest that large asteroids with birthplaces separated by an astronomical unit or more can be finally deposited, owing to drag, at the same semimajor axis. For example, we find that bodies with radii up to 100 km can be transferred by gas drag from the outer belt (a > 3.3 AU) well into the inner one, and that an object already in the inner belt as large or even larger than Vesta (r = 250 km)—thought to be the parent body of many meteorites—can be inwardly displaced by as much as an astronomical unit if the nebula dispersal times lie close to 10⁵ years. For such times, a large fraction of the inner belt's primordial mass can be ejected, with most of it passing into the inner solar system.     

The frequency and intensity of comet showers from the Oort cloud

We simulate the Oort comet cloud to study the rate and properties of new comets and the intensity and frequency of comet showers. An ensemble of ∼10⁶ comets is perturbed at random times by a population of main sequence stars and white dwarfs that is described by the Bahcall-Soneira Galaxy model. A cloning procedure allows us to model a large ensemble of comets efficiently, without wasting computer time following a large number of low eccentricity orbits. For comets at semimajor axis a = 20,000 AU, about every 100 myr a star with mass in the range 1M_⊙−2M_⊙ passes within ∼10,000 AU of the Sun and triggers a shower that enhances the flux of new comets by more than a factor of 10. The time-integrated flux is dominated by the showers for comets with semimajor axes less than ∼30,000 AU. For semimajor axes greater than ∼30,000 AU the comet loss rate is roughly constant and strong showers do not occur. In some of our simulations, comets are also perturbed by the Galactic tidal field. The inclusion of tidal effects increases the loss rate of comets with semimajor axes between 10,000 and 20,000 AU by about a factor of 4. Thus the Galactic tide, rather than individual stellar perturbations, is the dominant mechanism which drives the evolution of the Oort cloud.     

Power spectra of the interplanetary magnetic field

Power spectra based on Pioneer 6 interplanetary magnetic field data in early 1966 exhibit a frequency dependence of f ^–2 in the range 2.8 × 10^–4 to 1.6 × 10^–2 cps for periods of both quiet and disturbed field conditions. Both the shape and power levels of these spectra are found to be due to the presence of directional discontinuities in the microstructure (< 0.01 AU) of the interplanetary magnetic field. Power spectra at lower frequencies, in the range of 2.3 × 10^–6 to 1.4 × 10^–4 cps, reflect the field macrostructure (> 0.1 AU) and exhibit a frequency dependence roughly between f ^–1 and f ^–3/2. The results are related to theories of galactic cosmic-ray modulation and are found to be consistent with recent observations of the modulation.     

Origin and sustainability of the population of asteroids captured in the exterior resonance 1:2 with Mars

At present, approximately 1500 asteroids are known to evolve inside or sticked to the exterior 1:2 resonance with Mars at a ? 2.418 AU, being (142) Polana the largest member of this group. The effect of the forced secular modes superposed to the resonance gives rise to a complex dynamical evolution. Chaotic diffusion, collisions, close encounters with massive asteroids and mainly orbital migration due to the Yarkovsky effect generate continuous captures to and losses from the resonance, with a fraction of asteroids remaining captured over long time scales and generating a concentration in the semimajor axis distribution that exceeds by 20% the population of background asteroids. The Yarkovsky effect induces different dynamics according to the asteroid size, producing an excess of small asteroids inside the resonance. The evolution in the resonance generates a signature on the orbits, mainly in eccentricity, that depends on the time the asteroid remains captured inside the resonance and on the magnitude of the Yarkovsky effect. The greater the asteroids, the larger the time they remain captured in the resonance, allowing greater diffusion in eccentricity and inclination. The resonance generates a discontinuity and mixing in the space of proper elements producing misidentification of dynamical family members, mainly for Vesta and Nysa-Polana families. The half-life of resonant asteroids large enough for not being affected by the Yarkovsky effect is about 1 Gyr. From the point of view of taxonomic classes, the resonant population does not differ from the background population and the excess of small asteroids is confirmed.     

Interplanetary propagation of relativistic solar protons

Particle fluxes and pitch angle distributions of relativistic solar protons at Earth's orbit have been determined by Monte Carlo calculations. The analysis covers two hours after the release of the particles from the Sun and total of 8 × 10⁶ particle trajectories were simulated. The pitch angle scattering was assumed to be isotropic and the scattering mean free path was varied from 0.1 to 4 AU.The intensity-time profiles after a delta-like injection from the Sun show that the interplanetary propagation is clearly non-diffusive at scattering mean-free paths above 0.5 AU. All pitch angle distributions have a steady minimum at 90 °, and they become similar about 20 min after the arrival of first particles.As an application, the solar injection profile and the interplanetary scattering mean-free path of particles that gave rise to the GLE on 7 May, 1978 were determined. In contrast to the values of 3–5 AU published by other authors, the average scattering mean-free path was found to be about 1 AU.     

Some requirements of a colliding comet source of gamma ray bursts

Colliding comets in the Solar System may be an important source of gamma ray bursts. The spherical gamma ray comet cloud required by the results of the Venera Satellites (Mazets and Golenetskii, 1987) and the BATSE detector on the Compton Satellite (Meeganet al., 1992a, b) is neither the Oort Cloud nor the Kuiper Belt. To satisfy observations ofN(>P _max) vsP _max for the maximum gamma ray fluxes,P _max > 10^–5 erg cm^–2 s^–1 (about 30 bursts yr^–1), the comet density,n, should increase asn a ¹ from about 40 to 100 AU wherea is the comet heliocentric distance. The turnover above 100 AU requiresn a ^–1/2 to 200 AU to fit the Venera results andn a ^1/4 to 400 AU to fit the BATSE data. Then the masses of comets in the 3 regions are from: 40–100 AU, about 9 earth masses,m _E; 100–200 AU about 25m _E; and 100–400 AU, about 900m _E. The flux of 10^–5 erg cm^–2 s^–1 corresponds to a luminosity at 100 AU of 3 × 10²⁶ erg s^–1. Two colliding spherical comets at a distance of 100 AU, each with nucleus of radiusR of 5 km, density of 0.5 g cm^–3 and Keplerian velocity 3 km s^–1 have a combined kinetic energy of 3 × 10²⁸ erg, a factor of about 100 greater than required by the burst maximum fluxes that last for one second. Betatron acceleration in the compressed magnetic fields between the colliding comets could accelerate electrons to energies sufficient to produce the observed high energy gamma rays. Many of the additional observed features of gamma ray bursts can be explained by the solar comet collision source.     

The Hypothesis of Additional Acceleration in the Motion of Comet Harrington–Abell from Jupiter

By processing 494 observations of Comet Harrington–Abell, we obtained a unified system of elements that includes its turn around the Sun during which it closely approached Jupiter to a minimum distance of 0.037 AU in 1974. A study of the cometary orbit before and after the approach showed that, probably, at the approach of the comet to Jupiter, apart from the well-known gravitational perturbations, its motion was affected by an additional force. An improvement of the cometary orbit by assuming that an additional acceleration inversely proportional to the square of the distance to Jupiter exists in its motion yielded the following values: (4.57 ± 0.42) × 10^–10 and (–7.20 ± 0.42) × 10^–10 AU day^–2 for the radial and transversal acceleration components, respectively. As a plausible explanation of the changes in the cometary orbit, we additionally considered a model based on the hypothesis of partial disintegration of the cometary nucleus. The parameter that characterizes the instant displacement of the center of inertia along the jovicentric radius vector was estimated to be –1.83 ± 0.75 km. Based on a unified numerical theory of cometary motion, we determined the nongravitational parameters using Marsden's model for two periods: A ₁ = (11.68 ± 1.74) × 10^–10 AU day^–2, A ₂ = (0.53 ± 0.0357) × 10^–10 AU day^–2 for 1975–1999 and A ₁ = (5.92 ± 5.86) × 10^–10 AU day^–2, A ₂ = (0.08 ± 0.028) × 10^–10 AU day^–2 for 1955–1969, under the assumption that the nongravitational acceleration changed at the approach of the comet to Jupiter.     

Evolution of comet orbits under the perturbing influence of the giant planets and nearby stars

The orbital evolution of 500 hypothetical comets during 10⁹ years is studied numerically. It is assumed that the birthplace of such comets was the region of Uranus and Neptune from where they were deflected into very elongated orbits by perturbations of these planets. Then, we adopted the following initial orbital elements: perihelion distances between 20 and 30 AU, inclinations to the ecliptic plane smaller than 20°, and semimajor axes from 5 × 10³ to 5 × 10⁴ AU. Gravitational perturbations by the four giant planets and by hypothetical stars passing at distances from the Sun smaller than 5 × 10⁵ AU are considered. During the simulation, somewhat more than 50% of the comets were lost from the solar system due to planetary or stellar perturbations. The survivors were removed from the planetary region and left as members of what is generally known as the cometary cloud. At the end of the studied period, the semimajor axes of the surviving comets tend to be concentrated in the interval 2 × 10⁴ < a < 3 × 10⁴ AU. The orbital planes of the comets with initial $\text{a ≧ 3 × 10}{}^4\text{AU}$ acquired a complete randomization while the others still maintain a slight predominance of direct orbits. In addition, comet orbits with final a < 6 × 10⁴AU preserve high eccentricities with an average value greater than 0.8 Most “new” comets from the sample entering the region interior to Jupiter's orbit had already registered earlier passages through the planetary region. By scaling up the rate of paritions of hypothetical new comets with the observed one, the number of members of the cometary cloud is estimated to be about 7 × 10¹⁰ and the conclusion is drawn that Uranus and Neptune had to remove a number of comets ten times greater.     

Tidal and rotational distortions in figures of Pluto and Charon

The supposition is, the tidal and rotational distortions should be fully responsible for the Pluto's and Charon's figure parameters. The mean polar and equatorial flattenings have been estimated about 10^–3, the second sectorial Stokes parameters about 9 × 10^–5, the differences between equatorial principal moments of inertia about 6 × 10³⁰ kg m² (Pluto) and 2 × 10²⁹ kg m² (Charon).     

Significant high number commensurabilities in the main lunar problem II: The occurrence of Saros-like near periodicities

The effect of changes in the Moon's semimajor axis and the Earth's orbital eccentricity on the occurrence of Saros-like cycles is examined. The Earth-Moon-Sun dynamical system exhibits such cycles for only 25 to 30% of the time interval between –5×10⁷ to +5×10⁷ years. Not only has the present Saros the smallest period during this time, but it also has one of the longest durations and the period closest to an integral number of anomalistic years, thus making it one of the most efficient Saros-like cycles for reversing solar perturbations in the main lunar problem. During the lifetime of a Saros-like cycle, variations of the Earth's orbital eccentricity cause frequent disappearances and reappearances of the cycle.     

Interplanetary acceleration of energetic particles at 1 and 5 AU

Energetic particle (0.1 to 100 MeV protons) acceleration is studied by using high resolution interplanetary magnetic field and plasma measurements at 1 AU (HEOS-2) and at 5 AU (Pioneer 10). Energy changes of a particle population are followed by computing test particle trajectories and the energy changes through the particle interaction with the time varying magnetic field. The results show that considerable particle acceleration takes place throughout the interplanetary medium, both in the corotating interaction regions (CIR) (5 AU), and in quiet regions (1 AU). Although shocks may contribute to acceleration we suggest statistical acceleration within the CIRs is sufficient to explain most energetic particle observations (e.g., McDonaldet al., 1975; Barnes and Simpson, 1976).The first and second order statistical acceleration coefficients which include transit time damping and Alfvén resonance interactions, are found to be well represented byD _T 8.5×10^–6 T ^0.5 MeV s^–1 andD _TT 4×10^–6 T ^1.5 MeV² s^–1 at 5 AU.By comparison, Fisk's estimates (1976), based on quasi-linear theory for transit-time damping, gaveD _TT 5×10^–7 T MeV² s^–1 at 1 AU.     

The formation and origin of the IRAS zodiacal dust bands as a consequence of single collisions between asteroids

The zodiacal dust bands discovered by IRAS can be explained as products of single collisions between asteroids. Debris from such a collision is distributed about the plane of the ecliptic as particles experience differential precession of their ascending nodes due to dispersion of their semimajor axes. For each collision, two bands, one on each side of the ecliptic, are formed on time scales of 10⁵ to 10⁶ years. The band pairs observed by IRAS are most likely the result of collisions between asteroids ∼15 km in diameter that occured within the last several million years. Further analysis of the IRAS sky survey data and of any future, more sensitive surveys should reveal additional, fainter band pairs. Our model suggests that asteroid collisions are sufficient to account for the bulk of the observed zodiacal thermal emission.     

Orbital stability of systems of closely-spaced planets,II: configurations with coorbital planets

We numerically investigate the stability of systems of 1 \({{\rm M}_{\oplus}}\) planets orbiting a solar-mass star. The systems studied have either 2 or 42 planets per occupied semimajor axis, for a total of 6, 10, 126, or 210 planets, and the planets were started on coplanar, circular orbits with the semimajor axes of the innermost planets at 1 AU. For systems with two planets per occupied orbit, the longitudinal initial locations of planets on a given orbit were separated by either 60° (Trojan planets) or 180°. With 42 planets per semimajor axis, initial longitudes were uniformly spaced. The ratio of the semimajor axes of consecutive coorbital groups in each system was approximately uniform. The instability time for a system was taken to be the first time at which the orbits of two planets with different initial orbital distances crossed. Simulations spanned virtual times of up to 1 × 10⁸, 5 × 10⁵, and 2 × 10⁵ years for the 6- and 10-planet, 126-planet, and 210-planet systems, respectively. Our results show that, for a given class of system (e.g., five pairs of Trojan planets orbiting in the same direction), the relationship between orbit crossing times and planetary spacing is well fit by the functional form log(t _c /t ₀) = b β + c, where t _c is the crossing time, t ₀ = 1 year, β is the separation in initial orbital semimajor axis (in terms of the mutual Hill radii of the planets), and b and c are fitting constants. The same functional form was observed in the previous studies of single planets on nested orbits (Smith and Lissauer 2009). Pairs of Trojan planets are more stable than pairs initially separated by 180°. Systems with retrograde planets (i.e., some planets orbiting in the opposite sense from others) can be packed substantially more closely than can systems with all planets orbiting in the same sense. To have the same characteristic lifetime, systems with 2 or 42 planets per orbit typically need to have about 1.5 or 2 times the orbital separation as orbits occupied by single planets, respectively.     

Additional criteria for identifying the asteroid families and confirmation of the effect of spatial separation of family members according to their albedos

The D(a) distribution of asteroid sizes by their semimajor axes and the N(p) distribution of the number of asteroids by their albedo values for individual families are used to isolate the asteroid families more clearly. The families identified by Masiero et al. (2013) are analyzed with the use of these distributions, and correctly and incorrectly isolated families are found. A reduction in the mean albedo with increasing semimajor axis is observed for almost all correctly identified families that are not truncated by resonances. This reduction is statistically significant for the majority of these families. Not a single family exhibits a statistically significant increase in albedo. This confirms our previous conclusions that a nongravitational effect acting in the asteroid belt results in the spatial separation of asteroids with different albedos.     

Where are the Uranus Trojans?

The area of stable motion for fictitious Trojan asteroids around Uranus’ equilateral equilibrium points is investigated with respect to the inclination of the asteroid’s orbit to determine the size of the regions and their shape. For this task we used the results of extensive numerical integrations of orbits for a grid of initial conditions around the points L ₄ and L ₅, and analyzed the stability of the individual orbits. Our basic dynamical model was the Outer Solar System (Jupiter, Saturn, Uranus and Neptune). We integrated the equations of motion of fictitious Trojans in the vicinity of the stable equilibrium points for selected orbits up to the age of the Solar system of 5 × 10⁹ years. One experiment has been undertaken for cuts through the Lagrange points for fixed values of the inclinations, while the semimajor axes were varied. The extension of the stable region with respect to the initial semimajor axis lies between 19.05 ≤ a ≤ 19.3 AU but depends on the initial inclination. In another run the inclination of the asteroids’ orbit was varied in the range 0° < i < 60° and the semimajor axes were fixed. It turned out that only four ‘windows’ of stable orbits survive: these are the orbits for the initial inclinations 0° < i < 7°, 9° < i < 13°, 31° < i < 36° and 38° < i < 50°. We postulate the existence of at least some Trojans around the Uranus Lagrange points for the stability window at small and also high inclinations.