Numerical investigation of motions of resonant asteroids in the three-dimensional space

Motions of asteroids in mean motion resonances with Jupiter are studied in three-dimensional space. Orbital changes of fictitious asteroids in the Kirkwood gaps are calculated by numerical integrations for 10⁵ – 10⁶ years. The main results are as follows: (1) There are various motions of resonant asteroids, and some of them are very complicated and chaotic and others are regular. (2) The eccentricity of some asteroids becomes very large, and the variation of the inclination is large while the eccentricity is large. (3) In the 3:1 resonance, there is a long periodic change in the variation of the inclination, when (7 : ) is a simple ratio (7: longitude of perihelion, : longitude of node). (4) In the 7:3 resonance, the variation of the inclination of some resonant asteroids is so large that prograde motion becomes retrograde. Some asteroids in the 7:3 resonance can collide with the Sun as well as with the inner planets.     

A symplectic mapping model for the 3/4 exterior

We present a symplectic mapping model to study the evolution of a small body at the 3/4 exterior resonance with Neptune, for planar and for three dimensional motion. The mapping is based on the averaged Hamiltonian close to this resonance and is constructed in such a way that the topology of its phase space is similar to that of the Poincaré map of the elliptic restricted three-body problem. Using this model we study the evolution of a small object near the 3/4 resonance. Both chaotic and regular motions are found, and it is shown that the initial phase of the object plays an important role on the appearance of chaos. In the planar case, objects that are phase-protected from close encounters with Neptune have regular orbits even at eccentricities up to 0.44. On the other hand objects that are not phase protected show chaotic behaviour even at low eccentricities. The introduction of the inclination to our model affects the stable areas around the 3/4 mean motion resonance, which now become thinner and thinner and finally at is=10° the whole resonant region becomes chaotic. This may justify the absence of a large population of objects at this resonance.     

Asteroid motion near the 3 : 1 resonance

A mapping model is constructed to describe asteroid motion near the 3 : 1 mean motion resonance with Jupiter, in the plane. The topology of the phase space of this mapping coincides with that of the real system, which is considered to be the elliptic restricted three body problem with the Sun and Jupiter as primaries. This model is valid for all values of the eccentricity. This is achieved by the introduction of a correcting term to the averaged Hamiltonian which is valid for small values of the ecentricity.We start with a two dimensional mapping which represents the circular restricted three body problem. This provides the basic framework for the complete model, but cannot explain the generation of a gap in the distribution of the asteroids at this resonance. The next approximation is a four dimensional mapping, corresponding to the elliptic restricted problem. It is found that chaotic regions exist near the 3 : 1 resonance, due to the interaction between the two degrees of freedom, for initial conditions close to a critical curve of the circular model. As a consequence of the chaotic motion, the eccentricity of the asteroid jumps to high values and close encounters with Mars and even Earth may occur, thus generating a gap. It is found that the generation of chaos depends also on the phase (i.e. the angles andv) and as a consequence, there exist islands of ordered motion inside the sea of chaotic motion near the 3 : 1 resonance. Thus, the model of the elliptic restricted three body problem cannot explain completely the generation of a gap, although the density in the distribution of the asteroids will be much less than far from the resonance. Finally, we take into account the effect of the gravitational attraction of Saturn on Jupiter's orbit, and in particular the variation of the eccentricity and the argument of perihelion. This generates a mixing of the phases and as a consequence the whole phase space near the 3 : 1 resonance becomes chaotic. This chaotic zone is in good agreement with the observations.     

Two-body problem with slowly decreasing mass

Summary Evolution of the orbital elements of a two-body system with slowly decreasing mass according to Jeans' mode is described by a non-linear, non-autonomous system of differential equations.In general the system contains one stationary solution (e=1,f=), for which an instability criterion is derived. For example the stationary solution is unstable for all Jeans-Eddington functionsm _n (t) with 1n3 which characterize the loss of mass. Furthermore, it is possible to describe the quantitative behaviour ofE+,e anda for arbitrarym(t) in a large number of cases. In the case of the Jeans-Eddington functions we find that the amplitude of the oscillations ine is monotone decreasing with time ifn>3 and it is monotone increasing with time ifn<3.By comparing these analytical results with the numerical calculations of Hadjidemetriou we explain the rapid rotation of the line of apsides which occurs if the initial value ofe is nearly-circular.

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, , . (e=1,f=), . , -m _n (t), , 1n3. , E+,e a m(t) . - , , n>3, , n<3. , , - e.

     

A statistical theory of resonance motion in the Sun-Jupiter system

We consider the application of the statistical method of phase mixing to the approximate Poincaré solution to resonant motion. The two Poincaré integrals of the motion for the restricted problem of three bodies are introduced to first order in the eccentricity. The theory of the phase mixing of an initialad hoc distribution of particles is then developed for this dynamical system, and the absence of significant evolution of the system far from resonance is verified.A selection of results is given for the 21, 31, and 52 resonances, which show in general a peak on the low side of exact resonance and a gap on the high side. The amplitudes of both the peak and the gap decrease, and their relative separation increases as the resonance order increases, or as the initial distribution is shifted to higher eccentricities. Comparison with large numbers of numerically integrated orbits gives good agreement with the model, at least for small eccentricities. However, the model is unable to exhibit the clean gaps shown by the real asteroid belt. Hence, a purely statistical model of the Kirkwood gaps is ruled out, and we must search for an additional mechanism. Some speculation on possible additional mechanisms is offered.     

The observed limb effect in Fraunhofer lines,II

In this paper the absolute limb effect for a number of Fraunhofer lines observed at the McMath Solar Observatory is given. Results, uncorrected for scattered light, are given for the following lines: Fei 37334.9, 3735.3, 5123.7, 5250.2, 5434.5, 6678.0, and 8886.6. Additional lines observed are five lines of CN 3876.3–3880.0, two lines of Cai 6161.3 and 6162.2, one line of Nai, 6160.7, and one CN line (7957.0) of the red system.Operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under cooperative agreement with the National Science Foundation.     

The existence of a chaotic region due to the overlap of secular resonances ν5 and ν6

A nonlinear theory of secular resonances is developed. Both terms corresponding to secular resonances ₅ and ₆ are taken into account in the Hamiltonian. The simple overlap criterion is applied and the condition for the overlap of these resonances is found. It is shown that in given approximation the value p = (1 - e²)^1/2(1 - cosI) is an integral of motion, where the mean eccentricity e and mean inclination I are obtained by eliminating short-period perturbations as well as the nonresonant terms from the planets. The overlap criterion yields a critical value of parameter p depending on the semi-major axis a of the asteroid. For p greater than the critical value, resonance overlap occurs and chaotic motion has to be expected. A mapping is presented for fast calculation of the trajectories. The results are illustrated by level curves in surfaces of section method.     

On the coupling of lunisolar resonances for Earth satellite orbits

The resonance C1 occurs when the longitude of the perigee measured from the equinox becomes a slow angle in the doubly averaged equations of motion. This resonance is one of the critical inclination family with I 46°. For prograde Earth satellite orbits, up to five critical points can be identified. Only simple pitchfork bifurcations occur for the single resonance C1. A two degrees of freedom system is studied to check how a coupling of two lunisolar resonances affects the results furnished by the analysis of an isolated resonance case. In the system with two critical angles (g+h and h,+2 , seven types of critical points have been identified. The critical points arise and change their stability through 11 bifurcations. If the initial conditions are selected close to the critical points, the system becomes chaotic as shown in Poincaré maps.     

On the motion of a satellite in resonance with its rotating planet

The influence of resonance perturbations due to the gravitational field of an oblate planet on its satellite whose motion is commensurable with rotation of the planet has been investigated. It has been shown that in special case of the critical inclination or circular orbit the Lagrange equations can be integrated for all resonance terms simultaneously. The method is applied to the investigation of the motion of the 12-hour communication and navigation satellites of the Molniya and Navstar type. The computations has been performed by the use of four models of the geopotential.     

The three principal secular resonances ν5, ν6, and ν16 in the asteroidal belt

We review theoretical and numerical results obtained for secular resonant motion in the asteroidal belt. William's theory (1969) yields the locations of the principal secular resonances ₅, ₆, and ₁₆ in the asteroidal belt. Theories by Nakai and Kinoshita (1985) and by Yoshikawa (1987) allow us to model the basic features of orbital evolution at the secular resonances ₁₆ and ₆, respectively. No theory is available for the secular resonance v₅. Numerical experiments by Froeschlé and Scholl yield quantitative and new qualitative results for orbital evolutions at the three principal secular resonances ₅, ₆, and ₁₆. These experiments indicate possible chaotic motion due to overlapping resonances. A secular resonance may overlap with another secular resonance or with a mean motion resonance. The role of the secular resonances as possible sources of meteorites is discussed.     

The ‘critical inclination’: Another look

The occurrence and uniqueness of the critical inclination in satellite theory is discussed.An infinite set of canonical transformations in Hill variables are shown to exist whereby the first order secular part of the disturbing function can be changed into an alternative form. As a result of such a transformation the critical inclination can become (a)any other real or complex inclination or a function independent of the satellite's orbital inclination and (b) a function dependent on the semi-major axis, eccentricity and inclination of the satellite's orbit. It is also shown that all transformations of types (a) and (b) are only valid for short intervals of time of the order of a few satellite revolutions. Furthermore if such transformations are modified so that they become valid for greater intervals of time, then the resulting solutions in all cases containno singular divisor other than the critical inclination.It is concluded that the singularity at the critical inclination is unique and that it represents an actual physical resonance rather than something resulting from the method of solution or the type of variable used in the analysis. This conclusion is supported by numerical evidence which shows that a satellite's perigee height does suffer resonant changes when the orbital inclination is equal to the critical inclination of 63.°4.     

Ambiguity in Determining the Diffusion Coefficient from Shock-Associated Energetic Particle Increases

A numerical solution to the Fisk and Lee (1980) equations for the particle intensity upstream of a corotating interplanetary shock is considered for the November 1991 event observed at Ulysses. A numerically derived parallel diffusion coefficient is available for this region (Quenby et al., 1993), based upon in-situ magnetometer data. Fitting the transport equations solution to the upstream energetic particle distribution function, employing a radial diffusion coefficient = ₀ r, where r and are, respectively, radial distance from the Sun and particle velocity, and with ₀ fixed from the magnetometer derived coefficient yielded a range of statistically acceptable values of (, ). These ran from (0.5, 0.0) to (1.8, 1.6) along a thin strip of — space, hence demonstrating the improbability that the velocity and radial dependence of the particle diffusion can be fixed from such particle and magnetic field data alone.     

Discovery of a New Class of Predicted Resonance Objects beyond Jupiter

In the regions of mean diurnal motions between the orbits of Jupiter and Saturn, predicted earlier by the authors, five asteroids have been discovered that move in 1:2 and 2:3 Lindblad orbital resonances with Jupiter (external orbital commensurability) and in 2:1 resonance with Saturn (internal version of commensurability). In addition to this, in the precalculated stable resonance zones between the giant planets Saturn and Uranus, three objects have been found that possess third-order (2:5) orbital commensurability with Saturn; nine objects have been discovered between the orbits of Uranus and Neptune, whose mean motions are in 1:3 and 1:4 orbital resonances with Saturn, and more than 200 libration-stable objects, linked by lower-order orbital resonances with Neptune and Uranus have been found in the Kuiper belt.     

Resonances and close approaches. I. The Titan-Hyperion case

The orbits of Titan and Hyperion represent an interesting case of orbital resonance of order one (ratio of periods 3/4), which can be studied within a reasonable accuracy by means of the planar restricted three-body problem. The behaviour of this resonance has been investigated by numerical integrations, of which we show the results in terms of the Poincaré mapping in the plane of the coordinates = [(2L – 2G)] cos ( _H – t)and = –,[(2L – 2G)] sin ( _H –t)keeping a constant value of the Jacobi integral throughout all integrations. We find the numerical invariant curves corresponding to low and high eccentricity resonance locking (which seem stable, at least during the limited time span of our experiments) and show that the observed libration of Hyperion's pericenter about the conjunction lies inside the stable high eccentricity region. If initial conditions are chosen outside the stable zones, we have no more stable librations, but a chaotic behaviour causing successive close approaches to Titan.We discuss these results both from the point of view of the mathematical theory of invariant curves, and with the aim of understanding the origin of the resonance locking in this case. The tidal evolution theory cannot be rigorously tested by such experiments (because of the dissipative terms which change the Jacobi constant); however, we note that the time scale of chaotic evolution is by many orders of magnitude smaller than the tidal dissipation time scale, so that the chaotic regions of the phase space cannot be crossed by a slow and smooth evolution. Therefore, our results seem to favour the hypothesis that Hyperion was formed via accumulation of the planetesimals originally inside a stable island of libration, while Titan was depleting by collisions or ejections the zones where the bodies could not escape the chaotic behaviour.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

The limits of possible values of inclination of Mercury's equator

The method of estimation of the limits, containing the equator inclination of a celestial body, had been developed. In this method it is necessary to know the orbital elements and the mass of a celestial body. Another condition is that the axial rotation of a body should be in the resonance with its orbital motion. It has been found that the equator inclinations should have the values between 1 .7 and 2 .6 for Mercury and between 1 .0 and 1 .8 for the Moon. It also has been found that largest harmonics in Mercury's physical libration are the harmonics sin( – 3g), cos( – 3g), sin g and sin 2.     

Stability of the triangular libration points of the circular restricted problem in the presence of resonances

The stability of triangular libration points, when the bigger primary is a source of radiation and the smaller primary is an oblate spheroid. has been investigated in the resonance cases ₁ = 2₂ and ₁ = 3₂. The motion is unstable for all the values of parameters q and A when ₁ = 2₂ and the motion is unstable and stable depending upon the values of the parameters q and A when ₁ = 3₂. Here q is the radiation parameter and A is the oblateness parameter.     

Numerical experiments in the 3/1 andv 6 overlapping resonance region

Numerical experiments of fictitious small bodies with initial eccentricities e=0.1 have been performed in the overlapping region of the 3/1 mean motion resonance and of thev ₆ secular resonance 2.48a2.52AU for different values of the initial inclination 16°i20°. An analysis for thev ₆ secular resonance shows that the topology is different from the one found outside the overlapping region: the critical argument for thev ₆ resonance in the overlapping region rotates in opposite direction as compared to the purev ₆ region. In the 3/1 resonance region the secular resonancev ₅ is dominant, and some secondary secular resonances asv ₆ –v ₁₆ andv ₅ +v ₆ are present.     

Angular momenta in the solar system

A statistical study of the orbital parameters of comets, asteroids and meteor streams shows that the vectors representing their angular momenta per mass unit (or the average angular momentum for meteor streams) are not arbitrarily distributed in the space: They are clustered around determinated values of angles . This synthesizes the eccentricities and inclinations of the orbital planes in a unique parameter adequated for the statistical purposes of the present work being defined by cos = cos (arc sin e) cos i.The discreteness of the obtained distribution N() and its relation with the components of the angular momenta per mass unit is analysed having this distribution common features for objects of different nature and located in different places in the solar potential well. Some hypotheses concerning to these effects are discussed.     

Resonant structure of the outer asteroid belt

An analysis of ordered and chaotic regions of motion in the outer asteroid belt has shown that once the eccentricity of Jupiter is introduced the chaotic regions of the circular model are quite easily depleted. This suggests that also objects in neighbouring regions must be strongly perturbed. Therefore it is not surprising that many outer belt asteroids have been reported in the literature as resonant or anyway dynamically protected. By using the planar elliptic restricted 3-body model we have investigated the motion of outer belt asteroids which had not been suspected to librate. We find 3 cases of libration and 11 cases of e, coupling that can be explained within the theory of secular resonances. It is thus established that in the outer belt only resonant and dynamically protected asteroids can have lifetimes of the same order as the age of the Solar System.     

Dust particle and solar radiation

Influence of the solar radiation (electromagnetic and corpuscular - solar wind) on the motion of the interplanetary dust particle is investigated. The ratio time of inspiralling toward the Sun: time of inspiralling neglecting the change of mass of the particle is presented as a function of initial eccentricities.