Roger A. Broucke

Solution of the Elliptic Rendezvous Problem with the Time as Independent Variable

An explicit solution is given for the linearized motion of a chaser in a close neighborhood of a target in an elliptic orbit. The solution is a direct generalization of the Clohessy-Wiltshire equations that are widely used for circular orbits. In other words, when the eccentricity is set equal to zero in the new formulas, the well-known Clohessy-Wiltshire formulas are obtained. The solution is completely explicit in the time. As a starting point, a closed-form solution is found of the de Vries equations of 1963. These are the linearized equations of elliptic motion in a rotating coordinate system, rotating with a variable angular velocity. This solution is shown to be obtained simply by taking the partial derivatives, with respect to the orbit elements, of the two-body solution in polar coordinates. When four classical elements are used, four linearly independent solutions of the de Vries equations are obtained. However, the classical orbit elements turn out to be singular for circular orbits. The singularity is removed by taking appropriate linear combinations of the four solutions. This gives a 4 by 4 fundamental solution matrix R that is nonsingular and reduces to the Clohessy-Wiltshire solution matrix when the eccentricity is set equal to zero.

Long-Term Third-Body Effects via Double Averaging

Westudy thelong-term effects of a third body on a satelliteof negligiblemass. Wehavein mind to study thelunisolar effects on an Earth satellite but many other applications can be imagined. We begin with the representation of the disturbing function in an ine nite series in Legendre polynomials but we truncate it at the second-degree terms. Our approach consists of a double analytic averaging: with the period of the satellite and with the period of the third body. We concentrate on using the two important integrals (energy and angular momentum ) to discuss and classify properties of the perturbed orbits of the satellite. For inclinations below 39 deg, the perigee of the orbits always circulates. There are circular and elliptic orbits but the eccentricity does not vary much. For the inclinations above 39 deg, the circular orbits are unstable and the eccentricities can increase rapidly but we have the appearance of new stableorbits: two ellipticfrozen orbits with constant eccentricity and e xed perigee location, at either 90 or 270 deg.

Numerical integration of periodic orbits in the main problem of artificial satellite theory

We describe a collection of results obtained by numerical integration of orbits in the main problem of artificial satellite theory (theJ2 problem). The periodic orbits have been classified according to their stability and the Poincaré surfaces of section computed for different values ofJ2 andH (whereH is thez-component of angular momentum). The problem was scaled down to a fixed value (−1/2) of the energy constant. It is found that the pseudo-circular periodic solution plays a fundamental role. They are the equivalent of the Poincaré first-kind solutions in the three-body problem. The integration of the variational equations shows that these pseudo-circular solutions are stable, except in a very narrow band near the critical inclincation. This results in a sequence of bifurcations near the critical inclination, refining therefore some known results on the critical inclination, for instance by Izsak (1963), Jupp (1975, 1980) and Cushman (1983). We also verify that the double pitchfork bifurcation around the critical inclination exists for large values ofJ2, as large as |J2|=0.2. Other secondary (higher-order) bifurcations are also described. The equations of motion were integrated in rotating meridian coordinates.

On Szebehely's equation for the potential of a prescribed family of orbits

In the present note we first give a simple proof of the Dainelli formulas for the force field generating a given family of orbits. We also show that the Szebehely partial differential equation for the potential can be derived from the Dainelli formulas if the energy integral is assumed. The Szebehely equation can be solved directly with the method of characteristics or indirectly with the Joukovsky formulas. Several examples are briefly described in the article. In particular we find some rather general potential functions corresponding to circular motion.

Transfer orbits in restricted problem

This paper studies transfer orbits in the planar restricted three-body problem. In particular, we are searching for orbits that can be used in two situations: 1) to transfer a spacecraft from one body back to the same body (known in the literature as Henons problem) and 2) to transfer a spacecraft from one body to the respective Lagrangian points 1,4 and L§. To avoid numerical problems during close approaches, the global Lemaitre regularization is used. Under this model, Henons problem becomes a Lambert three-body problem. After the simulations, we found orbits to transfer a spacecraft between any two points in the group formed by Earth and the Lagrangian points Z/j, £4, LS (in the Earth-sun system) with near-zero A V (near 10 ~ in canonical units). We also found several orbits that can be used to make a tour to the Lagrangian points for reconnaissance purposes with near-zero AF for the entire tour. The method employed was to solve the two-point boundary value problem for each transfer using the results available from the two-body version of this problem as a first guess.

Transfer orbits in the Earth-moon system using a regularized model

In a continuation of previous research where the problem was studied for the Earth-sun system, we search for transfer orbits from one body back to the same body (known in the literature as Henons problem) in the Earth-moon system. In particular, we are searching for orbits that can be used in three situations: 1) to transfer a spacecraft from the moon back to the moon (passing close to the Lagrangian point L

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

in most of the cases); 2) to transfer a spacecraft from the moon to the respective Lagrangian points L

Area-preserving mappings and deterministic chaos for nearly parabolic motions

, L^ and LS; and 3) to transfer a spacecraft to an orbit that passes close to the moon and to the Earth several times, with the goal of building a transportation system between these two celestial bodies. The model used for the dynamics is the planar and circular restricted three-body problem. The global Lemaltre regularization is used to avoid numerical problems during close approaches. An interesting result that was obtained in this research is a family of transfer orbits from the moon back to the moon that requires an impulse with magnitude lower than the escape velocity from the

Effects of atmospheric drag in swing-by trajectory

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 Stromgrens class L, (direct) and class m, (retrograde). We started by extending several of Hénons 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 10 000 sets of initial values for periodicity in each case, thus 160 000 integrations, all with zo or żo 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 160 000 cases with zo or żo 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).

Anatomy of the Constant Radial Thrust Problem

The present work investigates a mechanism of capturing processes in the restricted three-body problem. The work has been done in a set of variables which is close to Delaunays elements but which allows for the transition from elliptic to hyperbolic orbits. The small denominator difficulty in the perturbation theory is overcome by embedding the small denominator in an analytic function through a suitable analytic continuation. The results indicate that motions in nearly parabolic orbits can become chaotic even though the model is deterministic. The theoretical results are compared with numerical results, showing an agreement of about one percent. Some possible applications to cometary orbits are also given.