Trojan orbits in secular resonances

A near equality between the nodal rates of suitably defined Trojan orbits and Jupiter represents an important type of a secular resonance. This case is realized by the model Sun-Jupiter-Saturn-Trojan, referred to the invariable plane. A second theoretical example is based on the elliptic three-body problem Sun-Jupiter-Trojan, where the vanishing nodal rate of a special Trojan orbit and the vanishing rate of Jupiter's longitude of perihelion define a secular resonance.We investigate the perturbations in the asteroidal inclinations and the nodes and consider the possibility of a libration.     

Long term evolution of quasi-circular Trojan orbits

Trojan asteroids undergo very large perturbations because of their resonance with Jupiter. Fortunately the secular evolution of quasi circular orbits remains simple—if we neglect the small short period perturbations. That study is done in the approximation of the three dimensional circular restricted three-body problem, with a small mass ratio μ—that is about 0.001 in the Sun Jupiter case. The Trojan asteroids can be defined as celestial bodies that have a “mean longitude”, M + ω + Ω, always different from that of Jupiter. In the vicinity of any circular Trojan orbit exists a set of “quasi-circular orbits” with the following properties: (A) Orbits of that set remain in that set with an eccentricity that remains of the order of the mass ratio μ. (B) The relative variations of the semi-major axis and the inclination remain of the order of ${\sqrt{\mu}}$ . (C) There exist corresponding “quasi integrals” the main terms of which have long-term relative variations of the order of μ only. For instance the product c(1 – cos i) where c is the modulus of the angular momentum and i the inclination. (D) The large perturbations affect essentially the difference “mean longitude of the Trojan asteroid minus mean longitude of Jupiter”. That difference can have very large perturbations that are characteristics of the “horseshoes orbit”. For small inclinations it is well known that this difference has two stable points near ±60° (Lagange equilibrium points L₄ and L₅) and an unstable point at 180° (L₃). The stable longitude differences are function of the inclination and reach 180° for an inclination of 145°41′. Beyond that inclination only one equilibrium remains: a stable difference at 180°.     

The analysis of periodic orbits generated by Lagrangian solutions of the restricted three-body problem with non-spherical primaries

The present paper deals with the periodic orbits generated by Lagrangian solutions of the restricted three-body problem when both the primaries are oblate bodies. We have illustrated the periodic orbits for different values of μ, h, σ₁ and σ₂ (h is energy constant, μ is mass ratio of the two primaries, σ₁ and σ₂ are oblateness factors). These orbits have been determined by giving displacements along the tangent and normal to the mobile coordinates as defined by (Karimov and Sokolsky, 1989). We have applied the predictor-corrector algorithm to construct the periodic orbits in an attempt to unveil the effect of oblateness of the primaries by taking the fixed values of parameters μ, h, σ₁ and σ₂.     

Periodic orbits and bifurcations in the Sitnikov four-body problem when all primaries are oblate

We study the motions of an infinitesimal mass in the Sitnikov four-body problem in which three equal oblate spheroids (called primaries) symmetrical in all respect, are placed at the vertices of an equilateral triangle. These primaries are moving in circular orbits around their common center of mass. The fourth infinitesimal mass is moving along a line perpendicular to the plane of motion of the primaries and passing through the center of mass of the primaries. A relation between the oblateness-parameter ‘A’ and the increased sides ‘ε’ of the equilateral triangle during the motion is established. We confine our attention to one particular value of oblateness-parameter A=0.003. Only one stability region and 12 critical periodic orbits are found from which new three-dimensional families of symmetric periodic orbits bifurcate. 3-D families of symmetric periodic orbits, bifurcating from the 12 corresponding critical periodic orbits are determined. For A=0.005, observation shows that the stability region is wider than for A=0.003.     

On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries

The existence of new equilibrium points is established in the restricted three-body problem with equal prolate primaries. These are located on the Z-axis above and below the inner Eulerian equilibrium point L ₁ and give rise to a new type of straight-line periodic oscillations, different from the well known Sitnikov motions. Using the stability properties of these oscillations, bifurcation points are found at which new types of families of 3D periodic orbits branch out of the Z-axis consisting of orbits located entirely above or below the orbital plane of the primaries. Several of the bifurcating families are continued numerically and typical member orbits are illustrated.     

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.     

A variable mass of the primaries and librations around the triangular points

The delimitations of the librational motion around the Lagrangian triangular pointsL ₄,L ₅ are investigated within the framework of the restricted circular three body problem according to Brown's and Thüring's theory. The isotropic mass variation of the primaries does not exceed the order of the small primary and the derivatives of the masses with respect to the time are negligible second order quantities. The amplitude of the maximum elongations with respect to the small primary remains unchanged. The expression for the maximum variation of the distance of the particle from the large mass has the same form as in the classical problem with constant masses.     

Dynamics of close binary systems with mass exchange

The effects of non-isotropic ejection of mass from either component of a binary system on the orbital elements are studied, for the case of a small initial eccentricity of the relative orbit, when all the ejected mass falls on the other component. The problem is transformed to an equivalent two-body problem with isotropic variation of mass, plus a perturbing force which is a function of the intial conditions of ejection of the particles and their final, positions and velocities when they fall on the surface of the other star. The variation of the orbital elements are derived. It is shown that, to first-order terms in the eccentricity, the secular change of the semimajor axis is equal to the one corresponding to the case of zero initial eccentricity. On the contrary, the secular change of the eccentricity is smaller and it depends on the variations of mass ejection due to the finite eccentricity.     

Periodic orbits near ℒ4 for mass ratios near the critical mass ratio of routh

The Hamiltonian for orbits near ₄ and mass ratios near ₁ is brought into a normal form. A theorem shows that two coefficients in this expansion predict the behavior of the periodic orbits.This research is partially supported by Office of Naval Research Grant N00014-67-A-0113-0019.     

Periodic three body orbits in the case of small third mass

Near a symmetric periodic orbit of the plane circular or elliptic restricted probelm, the conditions for a symmetric periodic orbit of the plane general three body problem are reduced, under a natural nondegeneracy condition, to the vanishing of a single real valued function. The implicit function theorem and Hörmander's generalized Morse's lemma are then used to analyze the set of zeros of this function.     

On the restricted three body problem with oblate primaries

We present a study of the Lagrangian triangular equilibria in the planar restricted three body problem where the primaries are oblate homogeneous spheroids steadily rotating around their axis of symmetry and whose equatorial planes coincide throughout their motion.     

Extension of the rotating dipole model with oblateness of both primaries

A rotating mass dipole can be used to understand the dynamical behaviors around elongated asteroids as well as binary asteroids. In this paper an improved dipole model with oblateness in both primaries is investigated. The dynamical equations of a particle around the improved model are first derived by introducing the oblateness coefficients. The characteristic equations of equilibrium points are obtained, resulting in the emergence of new equilibria in the equatorial plane and the plane xoz depending on the shape of the spheroid. Numerical simulations are performed to illustrate the distribution of these equilibrium points. Significant influence from the oblateness of the primaries on the topological structure is also analyzed via zero-velocity curves.     

The three-dimensional equilibria of magnetic-binary systems with oblate primaries

In this paper the three-dimensional equilibria in a Magnetic-Binary system with oblate primaries are studied. It is also examined how the primaries oblateness affect the equilibria configuration of the spherical case.     

Formulation of the restricted three-body problem with two-center problem orbits as intermediate orbits

The restricted three-body Hamiltonian is partitioned into a two-center type principal part and its accompanying perturbational part. The mathematical analysis, involving the Jacobian elliptic functions, is adapted for the case of figure-eight orbits winding around the two given mass points. For many such orbits the elliptic function modulusk is small and can serve as a small parameter.Fourier expansions in terms of a parameter related tot are obtained for the intermediate orbit functions which provide representations in terms of elementary functions.     

Theory of the Trojan asteroids

In the previously published Parts I and II of the paper, the author has constructed a formal long-periodic solution for the case of 11 resonance in the restricted problem of three bodies to 0(m ^3/2), wherem is the small mass parameter of the system. The time-dependencet(, ,m), where is the mean synodic longitude and is related to the Jacobi constant, has been expressed by ahyperelliptic integral. It is shown here that with the approximationm=0 in the integrand, the functiont(, , 0) can be expanded in a series involving standardelliptic functions. Then the problem of inversion can be formally solved, yielding the function (t, , 0).Similarly, the normalized period (,m) of the motion can be approximated by theHagihara hyperelliptic integral (, 0), corresponding tom=0. This integral is also expanded into elliptic functions. Asymptotic forms for (, 0) are derived for 0 and for 1, corresponding to the extreme members of thetadpole branch of the family of orbits.     

Accurate Mars Express orbits to improve the determination of the mass and ephemeris of the Martian moons

The determination of the ephemeris of the Martian moons has benefited from observations of their plane-of-sky positions derived from images taken by cameras onboard spacecraft orbiting Mars. Images obtained by the Super Resolution Camera (SRC) onboard Mars Express (MEX) have been used to derive moon positions relative to Mars on the basis of a fit of a complete dynamical model of their motion around Mars. Since, these positions are computed from the relative position of the spacecraft when the images are taken, those positions need to be known as accurately as possible. An accurate MEX orbit is obtained by fitting two years of tracking data of the Mars Express Radio Science (MaRS) experiment onboard MEX. The average accuracy of the orbits has been estimated to be around 20–25 m. From these orbits, we have re-derived the positions of Phobos and Deimos at the epoch of the SRC observations and compared them with the positions derived by using the MEX orbits provided by the ESOC navigation team. After fit of the orbital model of Phobos and Deimos, the gain in precision in the Phobos position is roughly 30 m, corresponding to the estimated gain of accuracy of the MEX orbits. A new solution of the GM of the Martian moons has also been obtained from the accurate MEX orbits, which is consistent with previous solutions and, for Phobos, is more precise than the solution from the Mars Global Surveyor (MGS) and Mars Odyssey (ODY) tracking data. It will be further improved with data from MEX-Phobos closer encounters (at a distance less than 300 km). This study also demonstrates the advantage of combining observations of the moon positions from a spacecraft and from the Earth to assess the real accuracy of the spacecraft orbit. In turn, the natural satellite ephemerides can be improved and participate to a better knowledge of the origin and evolution of the Martian moons.     

Circular restricted three-body problem when both the primaries are heterogeneous spheroid of three layers and infinitesimal body varies its mass

The circular restricted three-body problem, where two primaries are taken as heterogeneous oblate spheroid with three layers of different densities and infinitesimal body varies its mass according to the Jeans law, has been studied. The system of equations of motion have been evaluated by using the Jeans law and hence the Jacobi integral has been determined. With the help of system of equations of motion, we have plotted the equilibrium points in different planes (in-plane and out-of planes), zero velocity curves, regions of possible motion, surfaces (zero-velocity surfaces with projections and Poincaré surfaces of section) and the basins of convergence with the variation of mass parameter. Finally, we have examined the stability of the equilibrium points with the help of Meshcherskii space–time inverse transformation of the above said model and revealed that all the equilibrium points are unstable.