Resonance and Capture of Jupiter Comets

A number of Jupiter family comets such as Otermaand Gehrels 3make a rapid transition from heliocentric orbits outside the orbit of Jupiter to heliocentric orbits inside the orbit of Jupiter and vice versa. During this transition, the comet can be captured temporarily by Jupiter for one to several orbits around Jupiter. The interior heliocentric orbit is typically close to the 3:2 resonance while the exterior heliocentric orbit is near the 2:3 resonance. An important feature of the dynamics of these comets is that during the transition, the orbit passes close to the libration points L ₁and L ₂, two of the equilibrium points for the restricted three-body problem for the Sun-Jupiter system. Studying the libration point invariant manifold structures for L ₁and L ₂is a starting point for understanding the capture and resonance transition of these comets. For example, the recently discovered heteroclinic connection between pairs of unstable periodic orbits (one around the L ₁and the other around L ₂) implies a complicated dynamics for comets in a certain energy range. Furthermore, the stable and unstable invariant manifold tubes associated to libration point periodic orbits, of which the heteroclinic connections are a part, are phase space conduits transporting material to and from Jupiter and between the interior and exterior of Jupiter's orbit.     

Numerical results to the Sitnikov-problem

We present numerical results of the so-called Sitnikov-problem, a special case of the three-dimensional elliptic restricted three-body problem. Here the two primaries have equal masses and the third body moves perpendicular to the plane of the primaries' orbit through their barycenter. The circular problem is integrable through elliptic integrals; the elliptic case offers a surprisingly great variety of motions which are until now not very well known. Very interesting work was done by J. Moser in connection with the original Sitnikov-paper itself, but the results are only valid for special types of orbits. As the perturbation approach needs to have small parameters in the system we took in our experiments as initial conditions for the work moderate eccentricities for the primaries' orbit (0.33e _primaries 0.66) and also a range of initial conditions for the distance of the 3 ^rd body (= the planet) from very close to the primaries orbital plane of motion up to distance 2 times the semi-major axes of their orbit. To visualize the complexity of motions we present some special orbits and show also the development of Poincaré surfaces of section with the eccentricity as a parameter. Finally a table shows the structure of phase space for these moderately chosen eccentricities.     

Chaos in the solar system

We have accumulated thousands of orbits of test particles in the Solar System from the asteroid belt to beyond the orbit of Neptune. We find that the time for an orbit to make a close encounter with a perturbing planet, T _c ,is a function of the Lyapunov time, T _ty .The relation is log (T _c /T _o )= a + b log (T _ly T _o )where T _o is a fiducial period which we have taken as the period of the principal perturber or the period of the asteroid. There are exceptions to this rule interior to the 2/3 resonance with Jupiter. There, at least in the restricted problem, for sufficiently small Jupiter mass, orbits may have a positive Lyapunov exponent and still be blocked from having a close approach to Jupiter by a zero velocity curve. Of more serious concern is whether the relation holds for purely secular resonances, and if it does, how to choose T _o .This is the case of interest for the planets in the solar system.     

A procedure of selection of meteors from major streams for determination of mean orbits

A procedure of selection of meteoroids from major streams is suggested and applied to the IAU Lund photographic database modified by a check for internal consistency among orbital elements (3411 orbits). Limits for choice of stream members were defined by break points on the plots of the cumulative numberN _C vs. the Southworth-HawkinsD discriminant. For the break points were considered the points from which the dependenceN _C vs.D changes to a quasi-linear one, and with the increasingD, N _C changes only moderately. Except for the Taurids which desire a separate analysis, theN _C vs.D diagrams are presented for the following major meteoroid streams: Quadrantids, Lyrids, Aquarids, Capricornids, N and S Aquarids, Perseids, Orionids, Leonids and Geminids. The mean orbits, velocities and radiants of the streams are derived and compared with the osculating orbits of their parent bodies. The limitingD _B was found to be a function of the number of the stream membersN _CB. Omitting the exceptionally concentrated Geminids, the relation is in the formD _B = 0.058 *ln(N _CB) – 0.04.     

Unified iterative methods in orbit determination

The classical problem of Keplerian orbit determination from only three measurements of time and angular coordinates (t _i, _i, _i) has been solved here numerically in two different ways, using Newton's method for non-linear equations in both cases. The first method (Perov, 1989) is based on KS variables, whereas the second emphasizes the fundamental part played by the unified Lambert's equation and the related formulae in that kind of applications. These two methods have been compared and put into practice in various numerical tests based on real asteroid orbits and ficitious Keplerian asteroid, comet and artificial satellite orbits in order to try the stability of these methods for peculiar orbits.     

The Phase Space Structure Around L4 in the Restricted Three-Body Problem

The phase space structure around L ₄ in the restricted three-body problem is investigated. The connection between the long period family emanating from L ₄ and the very complex structure of the stability region is shown by using the method of Poincarés surface of section. The zero initial velocity stability region around L ₄ is determined by using a method based on the calculation of finite-time Lyapunov characteristic numbers. It is shown that the boundary of the stability region in the configuration space is formed by orbits suffering slow chaotic diffusion.     

A closed-form solution to the minimum $${\Delta V_{\rm tot}^2}$$ Lambert’s problem

A closed form solution to the minimum DV_tot²{\Delta V_{\rm tot}^2} Lambert problem between two assigned positions in two distinct orbits is presented. Motivation comes from the need of computing optimal orbit transfer matrices to solve re-configuration problems of satellite constellations and the complexity associated in facing these problems with the minimization of DV_tot{\Delta V_{\rm tot}}. Extensive numerical tests show that the difference in fuel consumption between the solutions obtained by minimizing DV_tot²{\Delta V_{\rm tot}^2} and DV_tot{\Delta V_{\rm tot}} does not exceed 17%. The DV_tot²{\Delta V_{\rm tot}^2} solution can be adopted as starting point to find the minimum DV_tot{\Delta V_{\rm tot}}. The solving equation for minimum DV_tot²{\Delta V_{\rm tot}^2} Lambert problem is a quartic polynomial in term of the angular momentum modulus of the optimal transfer orbit. The root selection is discussed and the singular case, occurring when the initial and final radii are parallel, is analytically solved. A numerical example for the general case (orbit transfer “pork-chop” between two non-coplanar elliptical orbits) and two examples for the singular case (Hohmann and GTO transfers) are provided.     

The arrangement in mean elements space of the periodic orbits close to that of the Moon

In a simplified model of the Earth-Moon-Sun system based on the restricted circular 3-dimensional 3-body problem, it is possible to find numerically a set of 8 periodic orbits whose time evolutions closely resemble that of the Moon's orbit. These orbits have a period of 223 synodic months (i.e. the period of the Saros cycle known for more than two millennia as a means of predicting eclipses), and are characterized by a secular rotation of the argument of perigee . Periodic orbits of longer durations exhibiting this last feature are very abundant in Earth-Moon-Sun dynamical models. Their arrangement in the space of the mean orbital elements- for various values of the lunar mean motion is presented.     

Newtonian and relativistic periodic orbits around two fixed black holes

We compare families of simple periodic orbits of test particles in the Newtonian and relativistic problems of two fixed centers (black holes). The Newtonian problem is integrable, while the relativistic problem is highly non-integrable.The orbits are calculated on the meridian plane through the fixed centersM ₁ (atz=+1) andM ₂ (atz=–1) for energies smaller than the escape energyE=1. We use prolate spheroidal coordinates (, , =const) and also the variables =cosh and =–cos . The orbits are inside a curve of zero velocity (CZV). The Newtonian orbits are also limited by an ellipse and a hyperbola, or by two eillipses. There are 3 main types of periodic orbits (1) elliptic type (around both centers), (2) hyperbolic-type, and (3) resonant-type.The elliptic type orbits are stable in the Newtonian case and both stable and unstable in the relativistic case. From the stable orbits bifurcate double period orbits both symmetric and asymmetric with respect to thez-axis. There are also higher order bifurcations. The hyperbolic-type orbits are unstable. The Newtonian resonant orbits are defined by the ratiot _µ/t =n/m of oscillations along and during one period, and they are all marginally unstable. The corresponding relativistic orbits are stable, or unstable. The main families are figure eight orbits aroundM ₁, or aroundM ₂ (3/1 orbits); gamma, or inverse gamma orbits (4/2); higher resonant families 5/1,7/1,...,8/2,12/2,...;, more complicated orbits, like 5/3, and bifurcations from the above orbits. Satellite orbits aroundM ₁, orM ₂, and their bifurcations (e.g. double period) exist in the relativistic case but not in the Newtonian case. The characteristics of the various families are quite different in the Newtonian and the relativistic cases. The sizes of the orbits and their stabilities are also quite different in general. In the Appendix we study the various types of straight line orbits and prove that some subcases introduced by Charlier (1902) are impossible.     

Metrics in Keplerian orbits quotient spaces

Quotient spaces of Keplerian orbits are important instruments for the modelling of orbit samples of celestial bodies on a large time span. We suppose that variations of the orbital eccentricities, inclinations and semi-major axes remain sufficiently small, while arbitrary perturbations are allowed for the arguments of pericentres or longitudes of the nodes, or both. The distance between orbits or their images in quotient spaces serves as a numerical criterion for such problems of Celestial Mechanics as search for common origin of meteoroid streams, comets, and asteroids, asteroid families identification, and others. In this paper, we consider quotient sets of the non-rectilinear Keplerian orbits space \(\mathbb H\). Their elements are identified irrespective of the values of pericentre arguments or node longitudes. We prove that distance functions on the quotient sets, introduced in Kholshevnikov et al. (Mon Not R Astron Soc 462:2275–2283, 2016), satisfy metric space axioms and discuss theoretical and practical importance of this result. Isometric embeddings of the quotient spaces into \(\mathbb R^n\), and a space of compact subsets of \(\mathbb H\) with Hausdorff metric are constructed. The Euclidean representations of the orbits spaces find its applications in a problem of orbit averaging and computational algorithms specific to Euclidean space. We also explore completions of \(\mathbb H\) and its quotient spaces with respect to corresponding metrics and establish a relation between elements of the extended spaces and rectilinear trajectories. Distance between an orbit and subsets of elliptic and hyperbolic orbits is calculated. This quantity provides an upper bound for the metric value in a problem of close orbits identification. Finally the invariance of the equivalence relations in \(\mathbb H\) under coordinates change is discussed.     

The structure of chaos in a potential without escapes

We study the structure of chaos in a simple Hamiltonian system that does no have an escape energy. This system has 5 main periodic orbits that are represented on the surface of section by the points (1)O(0,0), (2)C ₁,C ₂(±y _c, 0), (3)B ₁,B ₂(O,±1) and (4) the boundary . The periodic orbits (1) and (4) have infinite transitions from stability (S) to instability (U) and vice-versa; the transition values of are given by simple approximate formulae. At every transitionS U a set of 4 asymptotic curves is formed atO. For larger the size and the oscillations of these curves grow until they destroy the closed invariant curves that surroundO, and they intersect the asymptotic curves of the orbitsC ₁,C ₂ at infinite heteroclinic points. At every transitionU S these asymptotic curves are duplicated and they start at two unstable invariant points bifurcating fromO. At the transition itself the asymptotic curves fromO are tangent to each other. The areas of the lobes fromO increase with ; these lobes increase even afterO becomes stable again. The asymptotic curves of the unstable periodic orbits follow certain rules. Whenever there are heteroclinic points the asymptotic curves of one unstable orbit approach the asymptotic curves of another unstable orbit in a definite way. Finally we study the tangencies and the spirals formed by the asymptotic curves of the orbitsB ₁,B ₂. We find indications that the number of spiral rotations tends to infinity as . Therefore new tangencies between the asymptotic curves appear for arbitrarily large . As a consequence there are infinite new families of stable periodic orbits that appear for arbitrarily large .     

Resonant motion in the restricted three body problem

The resonant structure of the restricted three body problem for the Sun- Jupiter asteroid system in the plane is studied, both for a circular and an elliptic orbit of Jupiter. Three typical resonances are studied, the 2 : 1, 3 : 1 and 4 : 1 mean motion resonance of the asteroid with Jupiter. The structure of the phase space is topologically different in these cases. These are typical for all other resonances in the asteroid problem. In each case we start with the unperturbed two-body system Sun-asteroid and we study the continuation of the periodic orbits when the perturbation due to a circular orbit of Jupiter is introduced. Families of periodic orbits of the first and of the second kind are presented. The structure of the phase space on a surface of section is also given. Next, we study the families of periodic orbits of the asteroid in the elliptic restricted problem with the eccentricity of Jupiter as a parameter. These orbits bifurcate from the families of the circular problem. Finally, we compare the above families of periodic orbits with the corresponding families of fixed points of the averaged problem. Different averaged Hamiltonians are considered in each resonance and the range of validity of each model is discussed.     

THE PHASE SPACE STRUCTURE OF THE EXTENDED SITNIKOV-PROBLEM

In this article we treat the 'Extended Sitnikov Problem' where three bodies of equal masses stay always in the Sitnikov configuration. One of the bodies is confined to a motion perpendicular to the instantaneous plane of motion of the two other bodies (called the primaries), which are always equally far away from the barycenter of the system (and from the third body). In contrary to the Sitnikov Problem with one mass less body the primaries are not moving on Keplerian orbits. After a qualitative analysis of possible motions in the 'Extended Sitnikov Problem' we explore the structure of phase space with the aid of properly chosen surfaces of section. It turns out that for very small energies H the motion is possible only in small region of phase space and only thin layers of chaos appear in this region of mostly regular motion. We have chosen the plane ( ) as surface of section, where r is the distance between the primaries; we plot the respective points when the three bodies are 'aligned'. The fixed point which corresponds to the 1 : 2 resonant orbit between the primaries' period and the period of motion of the third mass is in the middle of the region of motion. For low energies this fixed point is stable, then for an increased value of the energy splits into an unstable and two stable fixed points. The unstable fixed point splits again for larger energies into a stable and two unstable ones. For energies close toH = 0 the stable center splits one more time into an unstable and two stable ones. With increasing energy more and more of the phase space is filled with chaotic orbits with very long intermediate time intervals in between two crossings of the surface of section. We also checked the rotation numbers for some specific orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stable Orbits of Planets of a Binary Star System in the Three-Dimensional Restricted Problem

The present research was motivated by the recent discovery of planets around binary stars. Our initial intention was thus to investigate the 3-dimensional nearly circular periodic orbits of the circular restricted problem of three bodies; more precisely Stromgren's class L, (direct) and class m, (retrograde). We started by extending several of Hénon's vertical critical orbits of these 2 classes to three dimensions, looking especially for orbits which are near circular and have stable characteristic exponents.We discovered early on that the periodic orbits with the above two qualifications are fairly rare and we decided thus to undertake a systematic exploration, limiting ourselves to symmetric periodic orbits. However, we examined all 16 possible symmetry cases, trying 10000 sets of initial values for periodicity in each case, thus 160000 integrations, all with z _o or _o equal to 0.1 This gave us a preliminary collection of 171 periodic orbits, all fairly near the xy-plane, thus with rather low inclinations. Next, we integrated a second similar set of 160000 cases with z _o or _o equal to 0.5, in order to get a better representation of the large inclinations. This time, we found 167 periodic orbits, but it was later discovered that at least 152 of them belong to the same families as the first set with 0.1Our paper quickly describes the definition of the problem, with special emphasis on the symmetry properties, especially for the case of masses with equal primaries. We also allow a section to describe our approach to stability and characteristic exponents, following our paper on this subject, (Broucke, 1969). Then we describe our numerical results, as much as space permits in the present paper.We found basically only about a dozen families with sizeable segments of simple stable periodic orbits. Some of them are around one of the two stars only but we do not describe them here because of a lack of space. We extended about 170 periodic orbits to families of up to 500 members, (by steps of 0.005 in the parameter), although, in many cases, we do not know the real end of the families. We also give an overview of the different types of periodic orbits that are most often encountered. We describe some of the rather strange orbits, (some of which are actually stable).     

Three-dimensional periodic motion of three finite masses around collinear equilibrium configurations

Three-dimensional periodic motions of three bodies are shown to exist in the infinitesimal neighbourhood of their collinear equilibrium configurations. These configurations and some characteristic quantities of the emanating three-dimensional periodic orbits are given for many values of the two mass parameters, =m ₂/(m ₁+m ₂) andm ₃, of the general three-body problem, under the assumption that the straight line containing the bodies at equilibrium rotates with unit angular velocity. The analysis of the small periodic orbits near the equilibrium configurations is carried out to second-order terms in the small quantities describing the deviation from plane motion but the analytical solution obtained for the horizontal components of the state vector is valid to third-order terms in those quantities. The families of three-dimensional periodic orbits emanating from two of the collinear equilibrium configurations are continued numerically to large orbits. These families are found to terminate at large vertical-critical orbits of the familym of retrograde periodic orbits ofm ₃ around the primariesm ₁ andm ₂. The series of these termination orbits, formed when the value ofm ₃ varies, are also given. The three-dimensional orbits are computed form ₃=0.1.     

Implications of the Galilean satellites ice envelope explosions III. The origin of the Trojans and of some comets

Secondary explosions of the primary ice fragments ejected in the explosion of the electrolyzed massive ice envelopes of the Galilean satellites are capable of imparting velocities of up to ~5km s^–1 to the secondary fragments. As a result, the secondary fragments can enter the orbits of the irregular satellites (Agafonova and Drobyshevski, 1984b) and the Trojan libration orbits. In the latter case a perturbation velocity of V 0.3–2 km s^–1 is sufficient.The primary fragments ejected by the gravitational perturbations due to the Galilean satellites sunward from Jupiter's sphere of action move faster relative to the Sun than Jupiter does and therefore reach their first aphelion ahead of Jupiter in the neighborhood of L ₄. At the same time the fragments propelled from Jupiter's sphere of action beyond the planet's orbit approach it again in their perihelia behind Jupiter in the region of L ₅. The concentration of the fragments and, hence, the frequency of their collisions and explosions at L ₄ turn out to be much greater than those at L ₅. As a result, the number of the secondary fragments of diameter 15 km captured into libration orbits ahead of Jupiter can be as high as many hundreds and should exceed by more than a factor 3.5 that captured behind Jupiter.Since the icy mix of the fragments contains hydrocarbons and particulate material (silicates and the like), after ice sublimation from the surface layers the Trojans should reveal type C and RD spectra typical for Jupiter's irregular satellites, comet nuclei and other distant ice bodies of similar origin. Among the Trojans there cannot be rocky or metallic objects which are known to exist in the main asteroid belt.It is shown that a velocity perturbation of 150–200 m s^–1 resulting from a purely mechanical impact of two bodies may be sufficient to move collision fragments from the orbits of the Trojans to horseshoe-shaped trajectories with a subsequent transfer to the cometary orbits of Jupiter's family.     

Symmetric and Asymmetric Librations in Planetary and Satellite Systems at the 2/1 Resonance

Families of asymmetric periodic orbits at the 2/1 resonance are computed for different mass ratios. The existence of the asymmetric families depends on the ratio of the planetary (or satellite) masses. As models we used the Io-Europa system of the satellites of Jupiter for the case m₁>m₂, the system HD82943 for the new masses, for the case m₁=m₂ and the same system HD82943 for the values of the masses m₁<m₂ given in previous work. In the case m₁≥ m₂ there is a family of asymmetric orbits that bifurcates from a family of symmetric periodic orbits, but there exist also an asymmetric family that is independent of the symmetric families. In the case m₁<m₂ all the asymmetric families are independent from the symmetric families. In many cases the asymmetry, as measured by and by the mean anomaly M of the outer planet when the inner planet is at perihelion, is very large. The stability of these asymmetric families has been studied and it is found that there exist large regions in phase space where we have stable asymmetric librations. It is also shown that the asymmetry is a stabilizing factor. A shift from asymmetry to symmetry, other elements being the same, may destabilize the system.     

The ‘Halo’ family of 3-dimensional periodic orbits in the Earth-Moon restricted 3-body problem

The Halo orbits originating in the vicinities of both,L ₁ andL ₂ grow larger, but shorter in period, as they shift towards the Moon. There is in each case a narrow band of stable orbits roughly half-way to the Moon. Nearer to the Moon, the orbits are fairly well-approximated by an almost rectilinear analysis. TheL ₂ family shrinks in size as it approaches the Moon, becoming stable again shortly before penetrating the lunar surface. TheL ₁-family becomes longer and thinner as it approaches the Moon, with a second narrow band of stable orbits with perilune, however, below the lunar surface.