Andre Deprit

A Manifold of Periodic Orbits

Abstract : In the restricted problem of three bodies, when the mass ratio is such that the characteristic exponents at L sub 4 are equal in pair, the triangular equilibrium is a point of ramification in the analytical manifold of periodic orbits emanating from L sub 4: the branch L sub 4 superscript s of short period orbits can be continued through L sub 4 by the branch L sub 4 superscript l of long period orbits, and this real analytical continuation is unique. The branch L sub 4 superscript l ends with an orbit traveled twice which is an element of L sub 4 superscript s. The branch L sub 4 superscript s meets its mirror image L sub 5 superscript s of short period orbits around L sub 5 on a symmetric orbit which is also an element of the branch L sub 3 of periodic solutions emanating from the collinear equilibrium L sub 3. Around L sub 4, the branches L sub 4 superscript l and L sub 4 superscript s are connected by bridges B(pL,qS) of periodic orbits which start from a long period orbit traveled p times and end with a short period one traveled q times. We have completely explored the bridges B(2L,3S), B(3L,4S) and B(4L,5S). (Author)

Trojan orbits: I. d'Alembert Series at L4

Abstract : The analytical continuation of the family of long period orbits at L sub 4 represents the normal Cartesian coordinates of the asteroid as dAlembert series in function of an orbital parameter. By means of a program that enables a computer to calculate symbolically in the real graded algebra of dAlembert series, the family of Trojan orbits is computed numerically up to the fourteenth order. Within the range of amplitudes of real Trojan planets, the dAlembert series are in very close agreement with the numerical results of Prof. E. Rabe. Outside this range, the poor convergence of the Fourier series is found to be imputed, not to a 13:1 commensurability between the Trojans libration period and Jupiters orbital period, but to the slow convergence of the dAlembert series all along the family of long period librations. (Author)

THE TROJAN MANIFOLD - SURVEY AND CONJECTURES

Recent results concerning the families of periodic orbits emanating from the triangular equilibrium L 4 are interpreted in an attempt to establish the evolution of these manifolds as the mass ratio varies from Routh’s critical value down to its value in the system sun-jupiter.

TROJAN ORBITS. II. BIRKHOFF'S NORMALIZATION,

Abstract In the Restricted Problem of Three Bodies for the system Sun-Jupiter, all terms of short and long period up to degree 13 are eliminated from the Hamiltonian expanded in power series in the neighborhood of an equilateral center of libration. The normalizing canonical transformation expresses the Cartesian phase variables as double Fourier series in two angle coordinates whose coefficients are power series in the square roots of two action momenta. In the normalized part of the transformed Hamiltonian, the angle coordinates are ignorable. It is shown here how such a Birkhoffs normalization can be implemented automatically on a computer. As a first application, the characteristic exponents along the singular families of long and short period orbits originating at the equilibrium are developed as power series of the orbital parameter.

A note concerning the collinear libration centers

Abstract The synodical barycentric abscissas of the collinear libration centers L1 and L2 are expanded in power series of the cubic root of the mass ratio up to the seventh order. A proof given by Wintner concerning an essential property of the characteristic exponents at the collinear Lagrangian points is shown to be wrong and is corrected. Through numerical computation for different values of the mass ratio, it is found that the absolute value of both characteristic exponents at L1 and L3 is a strictly increasing function of the mass ratio on the interval (0, 1/2), while, at L2, it is a strictly decreasing function.

Regularisation du probleme restreint plan des trois corps par representations conformes

Abstract For the general three-body problem, canonical coordinates were introduced recently, that regularize the binary collisions by means of a conformal mapping. It is shown here how this regularization applies to the plane restricted three-body problem, and while this new set of coordinates is compared to Thieles and Birkhoffs conformal mappings, it is advocated that, for numerical computations on a fast electronic machine, parabolic coordinates offers advantages in accuracy and speed that are not matched by others.

NATURAL ORBITS OF THE FIRST KIND IN THE RESTRICTED PROBLEM

Abstract Lindstedts method is applied to expand analytically the natural families of periodic orbits of the first kind in the restricted problem of three bodies. The mass ratio is kept as a literal variable; the series proceed in the powers of Hills ratio of periods. They cover all four cases at once, namely, the direct or retrograde orbits for either inferior planets or for satellites. When the mass ratio is given a numerical value, the expansions contain fewer terms and thus can be carried up to degree 17 within a 32K core of an IBM 7094. For the system Sun-Jupiter, the initial conditions provided by the series have been corrected to yield the beginnings of all four natural families, and the characteristic exponents have been computed. Comparison between the values computed out of the series and their improvement by numerical integrations and successive variational corrections show that, within an accuracy of one part in ten thousand, the series represent moderately well even the orbit of J VIII; the retrograde planetary orbits are fairly well covered up to 90% of the distance from Sun-Jupiter, whereas the direct planetary orbits are covered only up to 60% of that distance.