Two‐dimension potentials which generate spatial families of orbits

We consider the following case of the 3D inverse problem of dynamics: Given a spatial two‐parametric family of curves f (x, y, z) = c₁, g (x, y, z) = c₂, find possibly existing two‐dimension potentials under whose action the curves of the family are trajectories for a unit mass particle. First we establish the conditions which must be fulfilled by the family so that potentials of the form w (y, z) give rise to the curves of the family, and we present some applications. Then we examine briefly the existence of potentials depending on (x, z), respectively (x, y), which are compatible with the given family (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of the three-dimensional inverse problem

The three dimensional inverse problem for a material point of unit mass, moving in an autonomous conservative field, is solved. Given a two-parametric family of space curvesf(x, y, z)=c ₁,g(x, y, z)=c ₂, it is shown that, in general, no potentialU=U(x, y, z) exists which can give rise to this family. However, if the given functionsf(x, y, z) andg(x, y, z) satisfy certain conditions, the corresponding potentialU(x, y, z), as well as the total energyE=E(f, g) are determined uniquely, apart from a multiplicative and an additive constant.     

Generalization of Szebehely's equation

Szebehely's partial differential equation for the force functionU=U(x,y) which gives rise to a given family of planar orbitsf(x,y)=Constant is generalized to account for velocity-dependent potentials V^*=V^*(x,y, ). The new partial differential equation is quasi-linear and of the first order. An example is given and a comparison is made of the two equations.     

Families of planar orbits generated by homogeneous potentials

The second order partial differential equation which relates the potentialV(x,y) to a family of planar orbitsf(x,y)=c generated by this potential is applied for the case of homogeneousV(x,y) of any degreem. It is shown that, if the functionf(x,y) is also homogeneous, there exists, for eachm, a monoparametric set of homogeneous potentials which are the solutions of an ordinary, linear differential equation of the second order. Iff(x,y) is not homogeneous, in general, there is not a homogeneous potential which can create the given family; only if =f _y /f _x satisfies two conditions, a homogeneous potential does exist and can be determined uniquely, apart from a multiplicative constant. Examples are offered for all cases.     

Special Families of Orbits in the Direct Problem of Dynamics

The direct problem of dynamics in two dimensions is modeled by a nonlinear second-order partial differential equation, which is therefore difficult to be solved. The task may be made easier by adding some constraints on the unknown function = f _y /f _x , where f(x, y) = c is the monoparametric family of orbits traced in the xy Cartesian plane by a material point of unit mass, under the action of a given potential V(x, y). If the function is supposed to verify a linear first-order partial differential equation, for potentials V satisfying a differential condition, can be found as a common solution of certain polynomial equations.The various situations which can appear are discussed and are then illustrated by some examples, for which the energy on the members of the family, as well as the region where the motion takes place, are determined. One example is dedicated to a Hénon—Heiles type potential, while another one gives rise to families of isothermal curves (a special case of orthogonal families). The connection between the inverse/direct problem of dynamics and the possibility of detecting integrability of a given potential is briefly discussed.This revised version was published online in October 2005 with corrections to the Cover Date.     

Continuity and stability of families of figure eight orbits with finite angular momentum

Numerical solutions are presented for a family of three dimensional periodic orbits with three equal masses which connects the classical circular orbit of Lagrange with the figure eight orbit discovered by C. Moore [Moore, C.: Phys. Rev. Lett. 70, 3675–3679 (1993); Chenciner, A., Montgomery, R.: Ann. Math. 152, 881–901 (2000)]. Each member of this family is an orbit with finite angular momentum that is periodic in a frame which rotates with frequency Ω around the horizontal symmetry axis of the figure eight orbit. Numerical solutions for figure eight shaped orbits with finite angular momentum were first reported in [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and mathematical proofs for the existence of such orbits were given in [Marchal, C.: Celest. Mech. Dyn. Astron. 78, 279–298 (2001)], and more recently in [Chenciner, A. et al.: Nonlinearity 18, 1407–1424 (2005)] where also some numerical solutions have been presented. Numerical evidence is given here that the family of such orbits is a continuous function of the rotation frequency Ω which varies between Ω = 0, for the planar figure eight orbit with intrinsic frequency ω, and Ω = ω for the circular Lagrange orbit. Similar numerical solutions are also found for n > 3 equal masses, where n is an odd integer, and an illustration is given for n = 21. Finite angular momentum orbits were also obtained numerically for rotations along the two other symmetry axis of the figure eight orbit [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and some new results are given here. A preliminary non-linear stability analysis of these orbits is given numerically, and some examples are given of nearby stable orbits which bifurcate from these families.     

Inverse problem with two-parametric families of planar orbits

As a possible extension of recent work we study the following version of the inverse problem in dynamics: Given a two-parametric familyf(x, y, b)=c of plane curves, find an autonomous dynamical system for which these curves are orbits.We derive a new linear partial differential equation of the first order for the force componentsX(x, y) andY(x, y) corresponding to the given family. With the aid of this equation we find that, depending on the given functionf, the problem may or may not have a solution. Based on given criteria, we present a full classification of the various cases which may arise.     

The Inverse Problem for Collinear Central Configurations

We consider the problem: given a collinear configuration of n bodies, find the masses which make it central. We prove that for n ≤ 6, each configuration determines a one-parameter family of masses (after normalization of the total mass). The parameter is the center of mass when n is even and the square of the angular velocity of the corresponding circular periodic orbit when n is odd. The result is expected to be true for any n. This revised version was published online in July 2006 with corrections to the Cover Date.     

Families of orbits in planar anisotropic fields

The aim of the planar inverse problem of dynamics is: given a monoparametric family of curves f(x, y) = c, find the potential V (x, y) under whose action a material point of unit mass can describe the curves of the family. In this study we look for V in the class of the anisotropic potentials V(x, y) = v(a²x² + y²), (a=constant). These potentials have been used lately in the search of connections between classical, quantum, and relativistic mechanics. We establish a general condition which must be satisfied by all the families produced by an anisotropic potential. We treat special cases regarding the families (e. g. families traced isoenergetically) and we present certain pertinent examples of compatible pairs of families of curves and anisotropic potentials. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the determination of potential function

Szebehely's renowned equation given in 1974, allowing for potential determination from a given orbit or family of orbits, is proved to be equivalent with an equation deduced in 1963 by Drǎmbǎ. This basic equation in the inverse problem of dynamics, for which the denomination of Drǎmbǎ –Szebehely equation is proposed, is generalized for the motion in the n-dimensional Euclidean space. A method for the determination of the potential function from motion equations is extended to this space.     

Nonintegrability Detected from Geometrically Similar Orbits

H. Yoshida’s criterion of nonintegrability is restated in the light of the inverse problem of dynamics. It is shown that, given a monoparametric family of geometrically similar orbits, one may be able to assert nonintegrability of all or some homogeneous potentials of integer degreem (≠ 0, ± 2)which can produce this family. This revised version was published online in July 2006 with corrections to the Cover Date.     

Volume and mass distribution in selected asteroid families

The main focus of this paper is calculation of the diameters of asteroids belonging to five families (Vesta, Eos, Eunomia, Koronis, and Themis). To do that, we used the HCM algorithm applied for a data set containing 292,003 numbered asteroids, and a numerical procedure for choosing the crucial parameter of the HCM, called “the cutting velocity” v_cut. It was established with a precision as high as 1 m s^?1. Thereafter, we used the WISE (Wide‐field Infrared Survey Explorer) catalog to set a range of albedo for the largest members of each family considered. The albedo data were supported by the data concerning color classification (SDSS MOC4). The asteroids with albedo out of this range were classified as interlopers and were therefore disqualified as family members. Sizes were calculated for the asteroids with albedo within the acceptable range. For the other asteroids (those chosen by means of the HCM, but with albedo not listed in the WISE), the value of albedo of the largest member of the family was adopted. Results are given in a set of figures showing the families on the planes (a, e), (a, i), (e, i). Diameters and volumes of the asteroids that are the individual members of a family were calculated on the basis of their known or assumed albedo and on their absolute magnitude. Volumes of the parent bodies of the families were found on the basis of the cumulative volume distribution of these families. We also studied the secular resonances of the family members. We have shown that the locations of members of the considered asteroid families are related to the lines of secular resonances z₁, z₂, and z₃ with Saturn.     

Numerical explorations of the rectilinear problem of three bodies

We present some results of a numerical exploration of the rectilinear problem of three bodies, with the two outer masses equal. The equations of motion are first given in relative coordinates and in regularized variables, removing both binary collision singularities in a single coordinate transformation. Among our most important results are seven periodic solutions and three symmetric triple collision solutions. Two of these periodic solutions have been continued into families, the outer massm ₃ being the family parameter. One of these families exists for all masses while the second family is a branch of the first at a second-kind critical orbit. This last family ends in a triple collision orbit.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.     

Bifurcation points and intersections of families of periodic orbits in the three-dimensional restricted three-body problem

Intersections of families of three-dimensional periodic orbits which define bifurcation points are studied. The existence conditions for bifurcation points are discussed and an algorithm for the numerical continuation of such points is developed. Two sequences of bifurcation points are given concerning the family of periodic orbits which starts and terminates at the triangular equilibrium pointsL ₄,L ₅. On these sequences two trifurcation points are identified forµ = 0.124214 andµ = 0.399335. The caseµ = 0.5 is studied in particular and it is found that the space families originating at the equilibrium pointsL ₂,L ₃,L ₄,L ₅ terminate on the same planar orbitm _1v of the familym.     

The manifolds of families of 3D periodic orbits associated to Sitnikov motions in the restricted three-body problem

This paper deals with the Sitnikov family of straight-line motions of the circular restricted three-body problem, viewed as generator of families of three-dimensional periodic orbits. We study the linear stability of the family, determine several new critical orbits at which families of three dimensional periodic orbits of the same or double period bifurcate and present an extensive numerical exploration of the bifurcating families. In the case of the same period bifurcations, 44 families are determined. All these families are computed for equal as well as for nearly equal primaries (μ = 0.5, μ = 0.4995). Some of the bifurcating families are determined for all values of the mass parameter μ for which they exist. Examples of families of three dimensional periodic orbits bifurcating from the Sitnikov family at double period bifurcations are also given. These are the only families of three-dimensional periodic orbits presented in the paper which do not terminate with coplanar orbits and some of them contain stable parts. By contrast, all families bifurcating at single-period bifurcations consist entirely of unstable orbits and terminate with coplanar orbits.     

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.     

Compatible potentials and orbits in synodic systems

For a given family of orbits f(x,y) = c ^* which can be traced by a material point of unit in an inertial frame it is known that all potentials V(x,y) giving rise to this family satisfy a homogeneous, linear in V(x,y), second order partial differential equation (Bozis,1984). The present paper offers an analogous equation in a synodic system Oxy, rotating with angular velocity . The new equation, which relates the synodic potential function (x,y), = –V(x, y) + % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSqaaSqaai% aaigdaaeaacaaIYaaaaaaa!3780!\[\tfrac{1}{2}\]²(x ² + y ²) to the given family f(x,y) = c ^*, is again of the second order in (x,y) but nonlinear.As an application, some simple compatible pairs of functions (x,y) and f(x, y) are found, for appropriate values of , by adequately determining coefficients both in and f.     

On the adelphic potentials compatible with a set of planar orbits

Given a planar potentialB=B(x, y), compatible with a monoparametric family of planar orbitsf(x, y)=c, we face the problem of producing potentialsA=A(x, y), adelphic toB(x, y), i.e. nontrivial potentials which have in common withB(x, y) the given set of orbits. We establish a linear, second order partial differential equation for a functionP(x, y) and we prove that, to any definite positive solution of this equation, there corresponds a potentialA(x, y) adelphic toB(x, y).     

Effects of resonances on the stability of retrograde satellites

The stability evolution of family f of the planar circular restricted three-body problem in the Earth–Moon case is explored numerically using the Poincaré surface of section. It is shown that third order resonances are the main cause of the reduction of the stability region of retrograde satellites. Several branches of family f are also computed and these are seen by the configuration of their family characteristics to roughly determine the stability region. Previous results on smaller mass ratios of primaries are thus extended to the Earth–Moon system.     

On the Family of the Binary Asteroid 423 Diotima

We identified the family of the binary asteroid 423 Diotima consisting of 411 members in the phase space of orbital elements—semimajor axes a (or mean motions n), eccentricities e, and inclinations i—by using an electronic version of the ephemerides of minor planets EMP-2003 containing osculating orbital elements for 34992 asteroids of the main belt. The 9/4 resonance with Jupiter clearly separates the family of 423 Diotima from the family of Eos, which, according to EMP for 2003, contains 1204 asteroids.