Periodic orbits of the planar collision restricted 3-body problem

Using the continuation method we prove that the circular and the elliptic symmetric periodic orbits of the planar rotating Kepler problem can be continued into periodic orbits of the planar collision restricted 3–body problem. Additionally, we also continue to this restricted problem the so called “comets orbits”.

Periodic orbits near a heteroclinic loop formed by one dimensional orbit and a 2-dimensional manifold: application to the charged collinear 3-body problem

The paper is devoted to the study of a type of differential systems which appear usually in the study of some Hamiltonian systems with 2 degrees of freedom. We prove the existence of infinitely many periodic orbits on each negative energy level. All these periodic orbits pass near the total collision. Finally we apply these results to study the existence of periodic orbits in the charged collinear 3–body problem.

Families of periodic orbits for the spatial isosceles 3-body problem

We study the families of periodic orbits of the spatial isosceles 3-body problem (for small enough values of the mass lying on the symmetry axis) coming via the analytic continuation method from periodic orbits of the circular Sitnikov problem. Using the first integral of the angular momentum, we reduce the dimension of the phase space of the problem by two units. Since periodic orbits of the reduced isosceles problem generate invariant two-dimensional tori of the nonreduced problem, the analytic continuation of periodic orbits of the (reduced) circular Sitnikov problem at this level becomes the continuation of invariant two-dimensional tori from the circular Sitnikov problem to the nonreduced isosceles problem, each one filled with periodic or quasi-periodic orbits. These tori are not KAM tori but just isotropic, since we are dealing with a three-degrees-of-freedom system. The continuation of periodic orbits is done in two different ways, the first going directly from the reduced circular Sitnikov problem to the reduced isosceles problem, and the second one using two steps: first we continue the periodic orbits from the reduced circular Sitnikov problem to the reduced elliptic Sitnikov problem, and then we continue those periodic orbits of the reduced elliptic Sitnikov problem to the reduced isosceles problem. The continuation in one or two steps produces different results. This work is merely analytic and uses the variational equations in order to apply Poincar´e’s continuation method.

Bifurcation of relative equilibria of the (1+3)-body problem

We study the relative equilibria of the limit case of the pla- nar Newtonian 4{body problem when three masses tend to zero, the so-called (1 + 3){body problem. Depending on the values of the in- nitesimal masses the number of relative equilibria varies from ten to fourteen. Always six of these relative equilibria are convex and the oth- ers are concave. Each convex relative equilibrium of the (1 + 3){body problem can be continued to a unique family of relative equilibria of the general 4{body problem when three of the masses are su ciently small and every convex relative equilibrium for these masses belongs to one of these six families.

New classes of spatial central configurations for the n+3-body problem

In this paper, we show the existence of new families of spatial central configurations for the n + 3-body problem, n >= 3. We study spatial central configurations where n bodies are at the vertices of a regular n-gon T and the other three bodies are symmetrically located on the straight line that is perpendicular to the plane that contains T and passes through the center of T. The results have simple and analytic proofs. (c) 2010 Elsevier Ltd. All rights reserved.

The number of planar central configurations for the 4-body problem is finite when 3 mass positions are fixed

In the n{body problem a central con guration is formed when the position vector of each particle with respect to the center of mass is a common scalar multiple of its acceleration vector. Lindstrom showed for n = 3 and for n > 4 that if n ? 1 masses are located at xed points in the plane, then there are only a nite number of ways to position the remaining nth mass in such a way that they de ne a central con guration. Lindstrom leaves open the case n = 4. In this paper we prove the case n = 4 using as variables the mutual distances between the particles.

Families of periodic horseshoe orbits in the restricted three-body problem

We compute families of symmetric periodic horseshoe orbits in the restricted three-body problem. Both the planar and three-dimensional cases are considered and several families are found.We describe how these families are organized as well as the behavior along and among the families of parameters such as the Jacobi constant or the eccentricity. We also determine the stability properties of individual orbits along the families. Interestingly, we find stable horseshoe-shaped orbit up to the quite high inclination of 17◦

Caustics of the restricted three-body problem

Trajectories of the planar, circular, restricted three-body problem are given in the configuration space through the caustics associated to the invariant tori of quasi-periodic orbits. It is shown that the caustics of trajectories librating in any particular resonance display some features associated to that resonance. This method can be considered complementary to the Poincare surface of section method, because it provides information not accessible by the other method.

A study of the errors of the averaged models in the restricted three-body problem in a short time scale

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Asymptotic solution for the two-body problem with constant tangencial acceleration

An analytical solution of the two body problem perturbed by a constant tangential acceleration is derived with the aid of perturbation theory. The solution, which is valid for circular and elliptic orbits with generic eccentricity, describes the instantaneous time variation of all orbital elements. A comparison with high-accuracy numerical results shows that the analytical method can be effectively applied to multiple-revolution low-thrust orbit transfer around planets and in interplanetary space with negligible error.

A new set of integrals of motion to propagate the perturbed two-body problem

A formulation of the perturbed two-body problem that relies on a new set of orbital elements is presented. The proposed method represents a generalization of the special perturbation method published by Peláez et al. (Celest Mech Dyn Astron 97(2):131?150,2007) for the case of a perturbing force that is partially or totally derivable from a potential. We accomplish this result by employing a generalized Sundman time transformation in the framework of the projective decomposition, which is a known approach for transforming the two-body problem into a set of linear and regular differential equations of motion. Numerical tests, carried out with examples extensively used in the literature, show the remarkable improvement of the performance of the new method for different kinds of perturbations and eccentricities. In particular, one notable result is that the quadratic dependence of the position error on the time-like argument exhibited by Peláez?s method for near-circular motion under the J2 perturbation is transformed into linear.Moreover, themethod reveals to be competitive with two very popular elementmethods derived from theKustaanheimo-Stiefel and Sperling-Burdet regularizations.

EDROMO: An accurate propagator for elliptical orbits in the perturbed two-body problem

EDROMO is a special perturbation method for the propagation of elliptical orbits in the perturbed two-body problem. The state vector consists of a time-element and seven spatial elements, and the independent variable is a generalized eccentric anomaly introduced through a Sundman time transformation. The key role in the derivation of the method is played by an intermediate reference frame which enjoys the property of remaining fixed in space as long as perturbations are absent. Three elements of EDROMO characterize the dynamics in the orbital frame and its orientation with respect to the intermediate frame, and the Euler parameters associated to the intermediate frame represent the other four spatial elements. The performance of EDromo has been analyzed by considering some typical problems in astrodynamics. In almost all our tests the method is the best among other popular formulations based on elements.

Asymptotic Solution for the Two Body Problem with Radial Perturbing Acceleration

In this article, an approximate analytical solution for the two body problem perturbed by a radial, low acceleration is obtained, using a regularized formulation of the orbital motion and the method of multiple scales. The results reveal that the physics of the problem evolve in two fundamental scales of the true anomaly. The first one drives the oscillations of the orbital parameters along each orbit. The second one is responsible of the long-term variations in the amplitude and mean values of these oscillations. A good agreement is found with high precision numerical solutions.

Destruction of invariant curves in the restricted circular planar three body problem using comparison of action

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