Synthetic Proper Elements for Outer Main Belt Asteroids

For the orbits with low to moderate inclination and eccentricity, in the asteroid main belt, the analytically computed proper elements are accurate to a level very close to the best result achievable by any analytical theory. This fundamental limitation results from the infinite web of resonances and because of the occurrence of chaotic motions. Still, there are some regions of the belt in which these proper elements are of degraded accuracy, thus preventing a reliable definition of asteroid families and detailed studies of the dynamical structure. We have used a different method to compute asteroid proper elements, following the approach introduced in the LONGSTOP project to describe the secular dynamics of the major outer planets. By applying purely numerical techniques, we produced so-called synthetic proper elements for a catalog of 10,256 asteroids with osculating semimajor axes between 2.5 and 4.0AU.The procedure consisted of simultaneous integration of asteroid and planetary orbits for 2Myr, with online filtering of the short-periodic perturbations. The output of the integration was spectrally resolved, and the principal harmonics (proper values) extracted from the time series. For each asteroid we have also tested the accuracy and stability in time of the proper elements, and estimated the maximum Lyapunov Characteristic Exponent to monitor the chaotic behaviors. This provided information on the reliability of the data for each orbit, in particular allowing to select 1,852 cases for an extended integration (10Myr) of the orbits showing instability. The results indicate that for more than half of the cases the proper elements have a time stability improved by more than a factor 3 with respect to the elements computed by the previous analytical theory. But of course there are also unstable cases for which the proper elements are less accurate and reliable, the extreme examples being 23 orbits exhibiting hyperbolic escape from the solar system. This form of escape from the asteroid belt could be responsible for a significant mass loss over the age of the solar system.     

Palomar-Leiden minor planets: Proper elements,frequency distributions,belt boundaries,and family memberships

Proper elements have been calculated for 1227 higher accuracy orbits from the Palomar-Leiden Survey of faint minor planets. Tabulations are given for the special orbits: Earth and deep Mars crossers, Trojans, Hildas, and one 2:1 librator. The frequency distributions of the proper semimajor axis, eccentricity, sine inclination, longitudes of perihelion and node plus their rates, and the closest distances of approach to Mars and Jupiter are displayed as histograms and discussed. The distribution of the closest approach distance to Mars drops off sharply near zero while that for Jupiter vanishes near 1.1 AU. Mars and Jupiter have apparently caused these boundaries and the asteroid belt must have been larger early in the history of the solar system. 3.5% of the sample can impact Mars. Most of these potential impactors are shallow crossers which require occasional fortuitous alignments of the secular terms to intersect Mars' orbit so that long lifetimes result and moderate populations remain. As these fortuitous alignments occur with near simultaneity for many, but not all, asteroids the shallow crossers in the observed size range will episodically bombard Mars. Proper elements have been used to recognize families and 49% of the sample of minor planets fall into these families. The proper elements and family assignments are tabulated.     

Meteorites from the asteroid 6 Hebe

We have numerically integrated the orbits of 18 fictitious fragments ejected from the asteroid 6 Hebe, an S-type object about 200km across which is located very close to theg=g ₆ (orv ₆) secular resonance at a semimajor axis of 2.425AU and a (proper) inclination of 15° .0. A realistic ejection velocity distribution, with most fragments escaping at relative speeds of a few hundredsm/s, has been assumed. In four cases we have found that the resonance pumps up the orbital eccentricity of the fragments to values >0.6, which result into Earth-crossing, within a time span of 1Myr; subsequent close encounters with the Earth cause strongly chaotic orbital evolution. The closest Earth and Mars encounters recorded in our integration occur at miss distances of a few thousandths ofAU, implying collision lifetimes <10⁹ yr. Some other fragments affected by the secular resonance become Mars-crossers but not Earth-crossers over the integration time span. Two bodies are injected into the 3 : 1 mean motion resonance with Jupiter, and also display macroscopically chaotic behaviour leading to Earth-crossing. 6 Hebe is the first asteroid for which a realistic collisional/dynamical evolutionroute to generate meteorites has been fully demonstrated. It may be the parent body of one of the ordinary chondrite classes.     

The dynamics of Neptune Trojan – I. The inclined orbits

The stability of Trojan type orbits around Neptune is studied. As the first part of our investigation, we present in this paper a global view of the stability of Trojans on inclined orbits. Using the frequency analysis method based on the fast Fourier transform technique, we construct high-resolution dynamical maps on the plane of initial semimajor axis a ₀ versus inclination i ₀. These maps show three most stable regions, with i ₀ in the range of  (0°, 12°), (22°, 36°)  and  (51°, 59°),  respectively, where the Trojans are most probably expected to be found. The similarity between the maps for the leading and trailing triangular Lagrange points L ₄ and L ₅ confirms the dynamical symmetry between these two points. By computing the power spectrum and the proper frequencies of the Trojan motion, we figure out the mechanisms that trigger chaos in the motion. The Kozai resonance found at high inclination varies the eccentricity and inclination of orbits, while the  ν₈  secular resonance around   i ₀∼ 44°  pumps up the eccentricity. Both mechanisms lead to eccentric orbits and encounters with Uranus that introduce strong perturbation and drive the objects away from the Trojan like orbits. This explains the clearance of Trojan at high inclination  (>60°)  and an unstable gap around  44°  on the dynamical map. An empirical theory is derived from the numerical results, with which the main secular resonances are located on the initial plane of  ( a ₀, i ₀)  . The fine structures in the dynamical maps can be explained by these secular resonances.     

Long term stability of Earth Trojans

We explore the long-term stability of Earth Trojans by using a chaos indicator, the Frequency Map Analysis. We find that there is an extended stability region at low eccentricity and for inclinations lower than about $50^{\circ }$ even if the most stable orbits are found at $i \le 40^{\circ }$ . This region is not limited in libration amplitude, contrary to what found for Trojan orbits around outer planets. We also investigate how the stability properties are affected by the tidal force of the Earth–Moon system and by the Yarkovsky force. The tidal field of the Earth–Moon system reduces the stability of the Earth Trojans at high inclinations while the Yarkovsky force, at least for bodies larger than 10 m in diameter, does not seem to strongly influence the long-term stability. Earth Trojan orbits with the lowest diffusion rate survive on timescales of the order of $10^9$  years but their evolution is chaotic. Their behaviour is similar to that of Mars Trojans even if Earth Trojans appear to have shorter lifetimes.     

On the Stability of High Inclined L4 and L5Trojans

The orbits of real asteroids around the Lagrangian points L₄ and L ₅of Jupiter with large inclinations (i > 20°) were integrated for 50 Myrs. We investigated the stability with the aid of the Lyapunov characteristic exponents (LCE) but tested also two other methods: on one hand we integrated four neighbouring orbits for each asteroid and computed the maximum distance in every group, on the other hand we checked the variation of the Delaunay element H of the asteroid. In a second simulation – for a grid of initial eccentricity versus initial inclination – we examined the stability of the orbits around both Lagrangian points for 20° < i < 55° and 0.0 < e < 0.20. For the initial semimajor axes we have chosen the one ofJupiter(a = 5.202 AU). We determined the stability with the aid of the LCEs and also the maximum eccentricity of the orbits during the whole integration time. The region around L₄ turned out to be unstable for large inclinations and eccentricities (i > 55° and e > 0.12). The stable region shrinks for orbits around L₅: we found that they become unstable already for i > 45° and e > 0.10. We interpret it as a first hint why we observe more Trojans around the leading Lagrangian point. The results confirm the stability behaviour of the real Trojans which we computed in the first part of the paper.     

Chaotic Diffusion And Effective Stability of Jupiter Trojans

It has recently been shown that Jupiter Trojans may exhibit chaotic behavior, a fact that has put in question their presumed long term stability. Previous numerical results suggest a slow dispersion of the Trojan swarms, but the extent of the ‘effective’ stability region in orbital elements space is still an open problem. In this paper, we tackle this problem by means of extensive numerical integrations. First, a set of 3,200 fictitious objects and 667 numbered Trojans is integrated for 4 Myrs and their Lyapunov time, T_L, is estimated. The ones following chaotic orbits are then integrated for 1 Gyr, or until they escape from the Trojan region. The results of these experiments are presented in the form of maps of T_Land the escape time, T_E, in the space of proper elements. An effective stability region for 1 Gyr is defined on these maps, in which chaotic orbits also exist. The distribution of the numbered Trojans follows closely the T_E=1 Gyr level curve, with 86% of the bodies lying inside and 14% outside the stability region. This result is confirmed by a 4.5 Gyr integration of the 246 chaotic numbered Trojans, which showed that 17% of the numbered Trojans are unstable over the age of the solar system. We show that the size distributions of the stable and unstable populations are nearly identical. Thus, the existence of unstable bodies should not be the result of a size-dependent transport mechanism but, rather, the result of chaotic diffusion. Finally, in the large chaotic region that surrounds the stability zone, a statistical correlation between T_LandT_E is found.     

Sun-synchronous orbits near critical inclination

The long period dynamics of Sun-synchronous orbits near the critical inclination 116.6° are investigated. It is known that, at the critical inclination, the average perigee location is unchanged by Earth oblateness. For certain values of semimajor axis and eccentricity, orbit plane precession caused by Earth oblateness is synchronous with the mean orbital motion of the apparent Sun (a Sun-synchronism). Sun-synchronous orbits have been used extensively in meteorological and remote sensing satellite missions. Gravitational perturbations arising from an aspherical Earth, the Moon, and the Sun cause long period fluctuations in the mean argument of perigee, eccentricity, inclination, and ascending node. Double resonance occurs because slow oscillations in the perigee and Sun-referenced ascending node are coupled through the solar gravity gradient. It is shown that the total number and infinitesimal stability of equilibrium solutions can change abruptly over the Sun-synchronous range of semimajor axis values (1.54 to 1.70 Earth radii). The effect of direct solar radiation pressure upon certain stable equilibria is investigated.     

A survey of near-mean-motion resonances between Venus and Earth

It is known since the seminal study of Laskar (1989) that the inner planetary system is chaotic with respect to its orbits and even escapes are not impossible, although in time scales of billions of years. The aim of this investigation is to locate the orbits of Venus and Earth in phase space, respectively, to see how close their orbits are to chaotic motion which would lead to unstable orbits for the inner planets on much shorter time scales. Therefore, we did numerical experiments in different dynamical models with different initial conditions—on one hand the couple Venus–Earth was set close to different mean motion resonances (MMR), and on the other hand Venus’ orbital eccentricity (or inclination) was set to values as large as e = 0.36 (i = 40°). The couple Venus–Earth is almost exactly in the 13:8 mean motion resonance. The stronger acting 8:5 MMR inside, and the 5:3 MMR outside the 13:8 resonance are within a small shift in the Earth’s semimajor axis (only 1.5 percent). Especially Mercury is strongly affected by relatively small changes in initial eccentricity and/or inclination of Venus, and even escapes for the innermost planet are possible which may happen quite rapidly.     

An analysis of the region of the Phocaea dynamical family

In this work, I conduct a preliminary analysis of the Phocaea family region. I obtain families and clumps in the space of proper elements and proper frequencies, study the taxonomy of the asteroids for which this information is available, analyse the albedo and absolute magnitude distribution of objects in the area, obtain a preliminary estimate of the possible family age, study the cumulative size distribution and collision probabilities of asteroids in the region, the rotation rate distribution and obtain dynamical map of averaged elements and Lyapunov times for grids of objects in the area.
Among my results, I identified the first clump visible only in the frequency domain, the (6246) Komurotoru clump, obtained a higher limit for the possible age of the Phocaea family of 2.2 Byr, identified a class of Phocaea members on Mars-crossing orbits characterized by high Lyapunov times and showed that an apparently stable region on time-scales of 20 Myr near the  ν₆  secular resonance is chaotic, possibly because of the overlapping of secular resonances in the region. The Phocaea dynamical group seems to be a real S-type collisional family, formed up to 2.2 Byr ago, whose members with a large semimajor axis have been dynamically eroded by the interaction with the local web of mean-motion and secular resonances. Studying the long-term stability of orbits in the chaotic regions and the stability of family and clumps identified in this work remain challenges for future works.     

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.     

High-precision repeat-groundtrack orbit design and maintenance for Earth observation missions

The focus of this paper is the design and station keeping of repeat-groundtrack orbits for Sun-synchronous satellites. A method to compute the semimajor axis of the orbit is presented together with a station-keeping strategy to compensate for the perturbation due to the atmospheric drag. The results show that the nodal period converges gradually with the increase of the order used in the zonal perturbations up to \(J_{15}\). A differential correction algorithm is performed to obtain the nominal semimajor axis of the reference orbit from the inputs of the desired nodal period, eccentricity, inclination and argument of perigee. To keep the satellite in the proximity of the repeat-groundtrack condition, a practical orbit maintenance strategy is proposed in the presence of errors in the orbital measurements and control, as well as in the estimation of the semimajor axis decay rate. The performance of the maintenance strategy is assessed via the Monte Carlo simulation and the validation in a high fidelity model. Numerical simulations substantiate the validity of proposed mean-elements-based orbit maintenance strategy for repeat-groundtrack orbits.     

Averaged model to study long-term dynamics of a probe about Mercury

This paper provides a method for finding initial conditions of frozen orbits for a probe around Mercury. Frozen orbits are those whose orbital elements remain constant on average. Thus, at the same point in each orbit, the satellite always passes at the same altitude. This is very interesting for scientific missions that require close inspection of any celestial body. The orbital dynamics of an artificial satellite about Mercury is governed by the potential attraction of the main body. Besides the Keplerian attraction, we consider the inhomogeneities of the potential of the central body. We include secondary terms of Mercury gravity field from \(J_2\) up to \(J_6\), and the tesseral harmonics \(\overline{C}_{22}\) that is of the same magnitude than zonal \(J_2\). In the case of science missions about Mercury, it is also important to consider third-body perturbation (Sun). Circular restricted three body problem can not be applied to Mercury–Sun system due to its non-negligible orbital eccentricity. Besides the harmonics coefficients of Mercury’s gravitational potential, and the Sun gravitational perturbation, our average model also includes Solar acceleration pressure. This simplified model captures the majority of the dynamics of low and high orbits about Mercury. In order to capture the dominant characteristics of the dynamics, short-period terms of the system are removed applying a double-averaging technique. This algorithm is a two-fold process which firstly averages over the period of the satellite, and secondly averages with respect to the period of the third body. This simplified Hamiltonian model is introduced in the Lagrange Planetary equations. Thus, frozen orbits are characterized by a surface depending on three variables: the orbital semimajor axis, eccentricity and inclination. We find frozen orbits for an average altitude of 400 and 1000 km, which are the predicted values for the BepiColombo mission. Finally, the paper delves into the orbital stability of frozen orbits and the temporal evolution of the eccentricity of these orbits.     

Chaotic zone boundary for low free eccentricity particles near an eccentric planet

We consider particles with low free or proper eccentricity that are orbiting near planets on eccentric orbits. Through collisionless particle integration, we numerically find the location of the boundary of the chaotic zone in the planet's corotation region. We find that the distance in semimajor axis between the planet and boundary depends on the planet mass to the 2/7 power and is independent of the planet eccentricity, at least for planet eccentricities below 0.3. Our integrations reveal a similarity between the dynamics of particles at zero eccentricity near a planet in a circular orbit and with zero free eccentricity particles near an eccentric planet. The 2/7th law has been previously explained by estimating the semimajor at which the first-order mean motion resonances are large enough to overlap. Orbital dynamics near an eccentric planet could differ due to first-order corotation resonances that have strength proportional to the planet's eccentricity. However, we find that the corotation resonance width at low free eccentricity is small; also the first-order resonance width at zero free eccentricity is the same as that for a zero-eccentricity particle near a planet in a circular orbit. This accounts for insensitivity of the chaotic zone width to planet eccentricity. Particles at zero free eccentricity near an eccentric planet have similar dynamics to those at zero eccentricity near a planet in a circular orbit.     

The positions of secular resonance surfaces

The surfaces for the three strongest secular resonances have been located as a function of proper semimajor axis, eccentricity, and inclination for semimajor axes between 1.25 and 3.5 AU. The results are presented graphically. The ν₅ resonance only occurs at high inclinations (?23°). The ν₆ resonance passes through both the main belt and Mars-crossing space. The ν₁₆ resonance starts near the inner edge of the belt and, at low inclinations at least, folds around a portion of the Mars-crossing space until it runs nearly parallel with the Earth-crossing boundary.     

Explicit evolution relations with orbital elements for eccentric,inclined, elliptic and hyperbolic restricted few-body problems

Planetary, stellar and galactic physics often rely on the general restricted gravitational $N$ -body problem to model the motion of a small-mass object under the influence of much more massive objects. Here, I formulate the general restricted problem entirely and specifically in terms of the commonly used orbital elements of semimajor axis, eccentricity, inclination, longitude of ascending node, argument of pericentre, and true anomaly, without any assumptions about their magnitudes. I derive the equations of motion in the general, unaveraged case, as well as specific cases, with respect to both a bodycentric and barycentric origin. I then reduce the equations to three-body systems, and present compact singly- and doubly-averaged expressions which can be readily applied to systems of interest. This method recovers classic Lidov–Kozai and Laplace–Lagrange theory in the test particle limit to any order, but with fewer assumptions, and reveals a complete analytic solution for the averaged planetary pericentre precession in coplanar circular circumbinary systems to at least the first three nonzero orders in semimajor axis ratio. Finally, I show how the unaveraged equations may be used to express resonant angle evolution in an explicit manner that is not subject to expansions of eccentricity and inclination about small nor any other values.     

Analytical short-term orbit predictions withJ 3 andJ 4 in terms of KS elements

Analytical theory for short-term orbit motion of satellite orbits with Earth's zonal harmonicsJ ₃ andJ ₄ is developed in terms of KS elements. Due to symmetry in KS element equations, only two of the nine equations are integrated analytically. The series expansions include terms of third power in the eccentricity. Numerical studies with two test cases reveal that orbital elements obtained from the analytical expressions match quite well with numerically integrated values during a revolution. Typically for an orbit with perigee height, eccentricity and inclination of 421.9 km, 0.17524 and 30 degrees, respectively, maximum differences of 27 and 25 cm in semimajor axis computation are noted withJ ₃ andJ ₄ term during a revolution. For application purposes, the analytical solutions can be used for accurate onboard computation of state vector in navigation and guidance packages.     

Balanced earth satellite orbits

An analytic model for third-body perturbations and for the second zonal harmonic of the central body's gravitational field is presented. A simplified version of this model applied to the Earth-Moon-Sun system indicates the existence of high-altitude and highly-inclined orbits with their apsides in the equator plane, for which the apsidal as well as the nodal motion ceases. For special positions of the node, secular changes of eccentricity and inclination disappear too (balanced orbits). For an ascending node at vernal equinox, the inclination of balanced orbits is 94.56°, for a node at autumnal equinox 85.44°, independent of the eccentricity of the orbit. For a node perpendicular to the equinox, there exist circular balanced orbits at 90° inclination. By slightly adjusting the initial inclination as suggested by the simplified model, orbits can be found — calculated by the full model or by different methods — that show only minor variations in eccentricity, inclination, argument of perigee, and longitude of the ascending node for 10⁵ revolutions and more. Orbits near the unstable equilibria at 94.56° and 85.44° inclination show very long periodic librations and oscillations between retrogade and prograde motion.Retired from IBM Vienna Software Development Laboratory.     

Stochasticity of planetary orbits in double star systems. II

Cosmogonical theories as well as recent observations allow us to expect the existence of numerous exo-planets, including in binaries. Then arises the dynamical problem of stability for planetary orbits in double star systems. Modern computations have shown that many such stable orbits do exist, among which we consider orbits around one component of the binary (called S-type orbits). Within the framework of the elliptic plane restricted three-body problem, the phase space of initial conditions for fictitious S-type planetary orbits is systematically explored, and limits for stability had been previously established for four nearby binaries which components are nearly of solar type. Among stable orbits, found up to distance of their sun of the order of half the binarys periastron distance, nearly-circular ones exist for the three binaries (among the four) having a not too high orbital eccentricity. In the first part of the present paper, we compare these previous results with orbits around a 16 Cyg B-like binarys component with varied eccentricities, and we confirm the existence of stable nearly-circular S-type planetary orbits but for very high binarys eccentricity. It is well-known that chaos may destroy this stability after a very long time (several millions years or more). In a first paper, we had shown that a stable planetary orbit, although chaotic, could keep its stability for more than a billion years (confined chaos). Then, in the second part of the present paper, we investigate the chaotic behaviour of two sets of planetary orbits among the stable ones found around 16 Cyg B-like components in the first part, one set of strongly stable orbits and the other near the limit of stability. Our results show that the stability of the first set is not destroyed when the binarys eccentricity increases even to very high values (0.95), but that the stability of the second set is destroyed as soon as the eccentricity reaches the value 0.8.