Conservative two-body problem with variable masses

We consider the conservative two-body problem with a constant total mass, but with variable individual masses. The problem is shown to be completely integrable for any mass variation law. The Keplerian motion known for the classical two-body problem with constant masses remains valid for the relative motion of the bodies. The absolute motions of the bodies depend on the center-of-mass motion. Hitherto unknown quadratures that depend on the mass variation law were derived for the integrals of motion of the center of mass. We consider some of the laws that are of interest in studying the motion of close binary stars with mass transfer.     

Non-Continuation of Integrals of the Rotating Two-body Problem in Hill's Lunar Problem

We consider Hill's lunar problem as a perturbation of the integrable two-body problem. For this we avoid the usual normalization in which the angular velocity of the rotating frame of reference is put equal to unity and consider as the perturbation parameter. We first express the Hamiltonian H of Hill's lunar problem in the Delaunay variables. More precisely we deduce the expressions of H along the orbits of the two-body problem. Afterwards with the help of the conserved quantities of the planar two-body problem (energy, angular momentum and Laplace–Runge–Lenz vector) we prove that Hill's lunar problem does not possess a second integral of motion, independent of H, in the sense that there exist no analytic continuation of integrals, which are linear functions of in the rotating two-body problem. In connection with the proof of this main result we give a further restrictive statement to the nonintegrability of Hill's lunar problem.     

Complete closed-form solutions of the Stark problem

Perturbed two-body problems play a special role in Celestial Mechanics as they capture the dominant dynamics for a broad range of natural and artificial satellites. In this paper, we investigate the classic Stark problem, corresponding to motion in a Newtonian gravitational field subjected to an additional uniform force of constant magnitude and direction. For both the two-dimensional and three-dimensional cases, the integrals of motion are determined, and the resulting quadratures are analytically integrated. A complete list of exact, closed-form solutions is deduced in terms of elliptic functions. It is found that all expressions rely on only seven fundamental solution forms. Particular attention is given to ensure that the expressions are well-behaved for very small perturbations. A comprehensive study of the phase space is also made using a boundary diagram to describe the domains of the general types of possible motion. Numerical examples are presented to validate the solutions.     

The uniform,regular differential equations of the KS transformed perturbed two-body problem

The Newtonian differential equations of motion for the two-body problem can be transformed into four, linear, harmonic oscillator equations by simultaneously applying the regularizing time transformation dt/ds=r and the Kustaanheimo-Stiefel (KS) coordinate transformation. The time transformation changes the independent variable from time to a new variables, and the KS transformation transforms the position and velocity vectors from Cartesian space into a four-dimensional space. This paper presents the derivation of uniform, regular equations for the perturbed twobody problem in the four-dimensional space. The variation of parameters technique is used to develop expressions for the derivatives of ten elements (which are constants in the unperturbed motion) for the general case that includes both perturbations which can arise from a potential and perturbations which cannot be derived from a potential. These element differential equations are slightly modified by introducing two additional elements for the time to further improve long term stability of numerical integration.Originally presented at the AAS/AIAA Astrodynamics Specialists Conference, Vail, Colorado, July 1973     

Analytical solutions for J 2-perturbed unbounded equatorial orbits

While solutions for bounded orbits about oblate spheroidal planets have been presented before, similar solutions for unbounded motion are scarce. This paper develops solutions for unbounded motion in the equatorial plane of an oblate spheroidal planet, while taking into account only the J ₂ harmonic in the gravitational potential. Two cases are distinguished: A pseudo-parabolic motion, obtained for zero total specific energy, and a pseudo-hyperbolic motion, characterized by positive total specific energy. The solutions to the equations of motion are expressed using elliptic integrals. The pseudo-parabolic motion unveils a new orbit, termed herein the fish orbit, which has not been observed thus far in the perturbed two-body problem. The pseudo-hyperbolic solutions show that significant differences exist between the Keplerian flyby and the flyby performed under the the J ₂ zonal harmonic. Numerical simulations are used to quantify these differences.     

The post-Newtonian relative orbit of the two-body system. Gravitational radiation

The average loss of energy over one period of the elliptical motion of the two-body system is given, within the quadrupole approximation, by using the relative motion in the post-Newtonian centre of the mass frame. More explicit formulae are derived for the elliptical orbits and detailed results are presented for the circular orbits assuming small orbital velocities compared to the velocity of light. On the other hand, using the defined Lagrangian we give the integrals of motion.     

On the Milankovitch orbital elements for perturbed Keplerian motion

We consider sets of natural vectorial orbital elements of the Milankovitch type for perturbed Keplerian motion. These elements are closely related to the two vectorial first integrals of the unperturbed two-body problem; namely, the angular momentum vector and the Laplace–Runge–Lenz vector. After a detailed historical discussion of the origin and development of such elements, nonsingular equations for the time variations of these sets of elements under perturbations are established, both in Lagrangian and Gaussian form. After averaging, a compact, elegant, and symmetrical form of secular Milankovitch-like equations is obtained, which reminds of the structure of canonical systems of equations in Hamiltonian mechanics. As an application of this vectorial formulation, we analyze the motion of an object orbiting about a planet (idealized as a point mass moving in a heliocentric elliptical orbit) and subject to solar radiation pressure acceleration (obeying an inverse-square law). We show that the corresponding secular problem is integrable and we give an explicit closed-form solution.     

Circular binary star and an interstellar interloper: The analytical solution

We consider an interstellar interloper moving at a relatively large distance from a circular binary star. We use the analytical method of separating rapid and slow subsystems, the rapid subsystem being the binary and the slow subsystem being the interstellar interloper. We show that due to the higher than geometrical symmetry of the problem, in addition to the conservation of the energy and the projection of the angular momentum on the axis of the rotation of the binary, the square of the angular momentum is also conserved. In the course of the time evolution, the vector of the angular momentum rotates about that axis at the constant angle to the axis. After obtaining this general counterintuitive result, we focus at the case where the interstellar interloper is coplanar with the binary. We provide an explicit equation of the motion of the interloper. Then we calculate analytically the angle of deflection of the interloper from the straight line. We analyze the difference in the angle of deflection between this three-body problem and the corresponding two-body problem: we show that this difference remains almost constant (a negative constant) at the range of the eccentricities of the interloper trajectory relatively close to unity and linearly increases (by the absolute value, remaining negative) with the eccentricity as the latter becomes much greater than unity.     

Exact solution of a triaxial gyrostat with one rotor

The problem of the attitude dynamics of a triaxial gyrostat under no external torques and one constant internal rotor, is a three degrees-of-freedom system, although thanks to the existence of integrals of motion it can be reduced to only one degree-of-freedom problem. We introduce coordinates to represent the orbits of constant angular momentum as a flow on a sphere. This representation shows that the problem is equivalent to a quadratic Hamiltonian depending on two parameters. We find the exact solution of the orbits in terms of elliptic functions. By making use of properties of elliptic functions we find the solution at each region of the parametric partition from the solution of one region. We also prove that heteroclinic orbits are planar curves.     

The Lagrange and Two-Center Problems in the Lobachevsky Space

The generalization of the two-center problem and the Lagrange problem (a mass point motion under the action of attracting center field and the analog of a constant homogeneous field) to the case of a constant curvature space, in the three-dimensional space of Lobachevsky (³), is investigated in this paper. The integrability of these problems is proved. The bifurcation set in the plane of integrals of motion is constructed and the classification of the domains of possible motion is carried out. An analog of a constant homogeneous field is obtained in the Lobachevsky space.     

The effects of integrals on the totality of solutions of dynamical systems

Regions of possible motions are established for dynamical systems possessing time-independent Hamiltonians or for systems which are reducible to that form by means of integrals of the motion using only extended point transformations. The method is applied to the problem of three bodies in a plane and surfaces of zero velocity are found. The governing parameters are the energy, angular momentum and the masses of the participating bodies. The analytical and geometrical properties of these surfaces provide qualitative results for given constants of the motion.     

Perturbation theory in rectangular coordinates

For treating the perturbed two-body problem in rectangular coordinates a new method is developed. The method is based on the reduction of the variational equations of the two-body problem with arbitrary elements to the Jordan system. The equations of perturbed motion rewritten in the quasi-Jordan form are subjected to a transformation excluding fast variables and leading to a system governing the long term evolution of motion. The method may be easily extended to the problem of the heliocentric motion of the major planets. For performing this method on computer it is suitable to use facilities of Poissonian and Keplerian processors.     

Construction of a general planetary theory of the first order

All the necessary formulae for constructing a general solution for the motion of a planet, in rectangular coordinates, at the first order of the disturbing masses, in purely literal form in eccentricities and inclinations, are given. The authors present the transformation formulae in the two-body problem which give the correspondence between the constants of integration introduced in the theory and the classical keplerian elements. The practical elaboration of the algorithm and some partial results for the couple of planets Jupiter and Saturn are described.     

About the first integrals of the generalized problem of translatory-rotary motion of rigid bodies

The existence of ten first integrals for the classical problem of the motion of a system of material points, mutually attracting according to Newtonian law, is well known.The existence of the analogous ten first integrals for the more complicated problem of the motion of a system of absolutely rigid bodies, whose elementary particles mutually attract according to the Newtonian law, was established by the author (Duboshin, 1958, 1963, 1968).In his later papers (Duboshin, 1969, 1970), the problem of the motion of a system of material points, attracting each other according to a more general law, was considered and, in particular, it was shown under what conditions the ten first integrals, analogous to the classical integrals, may exist for this problem.In the present paper, the generalized problem of translatory-rotatory motion of rigid bodies, whose elementary particles acting upon each other according to arbitrary laws of forces along the straight line joining them, is discussed.The author has shown that the first integrals for this general problem, analogous to the integrals of the problem of the translatory-rotatory motion of rigid bodies, whose elementary particles acting according to the Newtonian law, exist under certain well known conditions.That is, it has been established that if the third axiom of dynamics (action = reaction) is satisfied, then the integrals of the motion of centre of inertia and the integrals of the moment of momentum exist for this generalized problem.If the third axiom is not satisfied, then the above mentioned integrals do not exist.The third axiom is a necessary but not a sufficient condition for the existence of the tenth integral-the energy integral. The tenth integral always exists if the elementary particles of the bodies acting with a force, depend only on the mutual distances between them. In this case the force function exists for the problem and the energy integral can be expressed in a well known form.The tenth integral may exist for some more general case, without expressing the principle of conservation of energy, but permitting calculation of the kinetic energy, if the configuration of a system is given.The problem, in which the elementary particles acting according to the generalized Veber's law (Tisserand, 1896) has been cited as an example of this more general case.     

Integration of the full two-body problem by using generalized inertia integrals

A new algorithm to integrate the full two-body problem based on generalized inertia integrals is given. The computation speed is comparable to the fastest algorithm available till now which is based on spherical harmonics.     

Classification of Motions for Generalization of the two Centers Problem on a Sphere

The generalization of a test particle motion in a central field of two immovable point-like centers to the case of a constant curvature space, on a three-dimensional sphere, is investigated in the paper. The bifurcation set in the plane of integrals of motion is constructed and the classification of the domains of possible motion is carried out on a two-dimensional sphere. The regularization of the Kepler’s problem on a two-dimensional sphere is carried out. This revised version was published online in July 2006 with corrections to the Cover Date.     

Regularization of Motion Equations with L-Transformation and Numerical Integration of the Regular Equations

The sets of L-matrices of the second, fourth and eighth orders are constructed axiomatically. The defining relations are taken from the regularization of motion equations for Keplerian problem. In particular, the Levi-Civita matrix and KS-matrix are L-matrices of second and fourth order, respectively. A theorem on the ranks of L-transformations of different orders is proved. The notion of L-similarity transformation is introduced, certain sets of L-matrices are constructed, and their classification is given. An application of fourth order L-matrices for N-body problem regularization is given. A method of correction for regular coordinates in the Runge–Kutta–Fehlberg integration method for regular motion equations of a perturbed two-body problem is suggested. Comparison is given for the results of numerical integration in the problem of defining the orbit of a satellite, with and without the above correction method. The comparison is carried out with respect to the number of calls to the subroutine evaluating the perturbational accelerations vector. The results of integration using the correction turn out to be in a favorable position.     

Orbit determination with the two-body integrals

We investigate a method to compute a finite set of preliminary orbits for solar system bodies using the first integrals of the Kepler problem. This method is thought for the applications to the modern sets of astrometric observations, where often the information contained in the observations allows only to compute, by interpolation, two angular positions of the observed body and their time derivatives at a given epoch; we call this set of data attributable. Given two attributables of the same body at two different epochs we can use the energy and angular momentum integrals of the two-body problem to write a system of polynomial equations for the topocentric distance and the radial velocity at the two epochs. We define two different algorithms for the computation of the solutions, based on different ways to perform elimination of variables and obtain a univariate polynomial. Moreover we use the redundancy of the data to test the hypothesis that two attributables belong to the same body (linkage problem). It is also possible to compute a covariance matrix, describing the uncertainty of the preliminary orbits which results from the observation error statistics. The performance of this method has been investigated by using a large set of simulated observations of the Pan-STARRS project.     

New formulation of the two body problem using a continued fractional potential

In the present work, the two body problem using a potential of a continued fractions procedure is reformulated. The equations of motion for two bodies moving under their mutual gravity is constructed. The integrals of motion, angular momentum integral, center of mass integral, total mechanical energy integral are obtained. New orbit equation is obtained. Some special cases are followed directly. Some graphical illustrations are shown. The only included constant of the continued fraction procedure is adjusted so as to represent the so called J ₂ perturbation term of the Earth’s potential.