A general canonical transformation increasing the number of variables with application to the two-body problem

In this paper, we present a canonical transformation that extends the change of coordinates of Cartesian type into the associate homogeneous coordinates, and provides a redundant set of eight canonical variables to describe the orbital motion of a particle. The transformed problem has two additional integrals, since the transformation increases the number of variables. Using these variables and a time proportional to the true anomaly, the Kepler problem can be reduced to a 4-dimensional oscillator, whose frequency can be selected to be either the magnitude of the angular momentum or unity, depending on a suitable scaling.Perturbed problems are represented by perturbed harmonic oscillators, whatever the type of the orbit is, and in the special case of central force fields, the resulting equations can be linearized exactly.     

On the Szebehely-bond equation generalized Sundman's transformation for the perturbed two-body problem

In this paper, starting with the Szebehely and Bond (1983) equation, we rediscuss the regularization and linearization of the perturbed planar two-body problem.We study the generalization of the Sundman's (1912) transformation proposed by Szebehely and Bond and obtain the radial and transverse perturbations (represented by powers of the radial distance r), which can be linearized with these transformations. In this way we generalize some previous results of Belen'kii (1981a, b) and Szebehely and Bond (1983).We also consider another generalization of Sundman's transformation, introduced by Cidet al. (1983), in the case when the radial and transverse perturbations are presented by polynomials in the reciprocal of the distance. As a consequence we give a partial answer to a problem suggested by Szebehely and Bond (1983).     

A complex exponential solution to the unified two-body problem

A complex exponential solution has been derived which unifies the elliptic and hyperbolic trajectories into a single set of equations and provides an exact, analytical solution to the unperturbed, Keplerian two-body problem. The formulation eliminates singularities associated with the elliptic and hyperbolic trajectories that arise from these orbits. Using this complex exponential solution formulation, a variation of parameters formulation for the perturbed two-body problem has been derived. In this paper, we present the analytical formulation of the complex exponential solution, numerical simulations, a comparison with classical solution methods, and highlight the benefits of this approach compared with the classical developments. Previously presented as AAS 07-136 at the 17th AAS/AIAA Spaceflight Mechanics Meeting Sedona, Arizona, AAS 08-206 and AAS 08-230 at the 18th AAS/AIAA Spaceflight Mechanics Meeting Galveston, Texas.     

A note on velocity-related series expansions in the two-body problem

The present note describes a few important series expansions in the two-body problem. They are related to the magnitudeV of the velocity vector and they are important for the treatment of atmospheric drag with the method of general perturbations. These series have been obtained with computerized Poisson series Manipulations. The results are given to order seven in the eccentricity, for both the Mean Anomaly and the True Anomaly.     

The eleventh motion constant of the two-body problem

The two-body problem is a twelfth-order time-invariant dynamic system, and therefore has eleven mutually-independent time-independent integrals, here referred to as motion constants. Some of these motion constants are related to the ten mutually-independent algebraic integrals of the n-body problem, whereas some are particular to the two-body problem. The problem can be decomposed into mass-center and relative-motion subsystems, each being sixth-order and each having five mutually-independent motion constants. This paper presents solutions for the eleventh motion constant, which relates the behavior of the two subsystems. The complete set of mutually-independent motion constants describes the shape of the state-space trajectories. The use of the eleventh motion constant is demonstrated in computing a solution to a two-point boundary-value problem.     

A note on expansions of functions of velocity in the two-body problem

Fourier expansions of functions of velocity in the two-body problem are obtained in terms of both the true anomaly and the mean anomaly.     

Some nonlinear effects in the relativistic two-body problem

The test-particle motion in the centrally symmetric gravitational field can be described by the equation in the form appropriate for a nonlinear oscillator — the nonlinear terms being due to the nonrelativistic effects. This enables us to apply to this equation the well-known asymptotic methods of the theory of nonlinear oscillations. Typical nonlinear oscillation phenomena arising from the action of external forces are shown to take place. The form of equations and the main results remain valid in the problem of two bodies of comparable mass in the post-Newtonian approximation.     

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.     

A noncanonical analytic solution to theJ 2 perturbed two-body problem

The motion of a satellite subject to an inverse-square gravitational force of attraction and a perturbation due to the Earth's oblateness as theJ ₂ term is analyzed, and a uniform, analytic solution correct to first-order inJ ₂, is obtained using a noncanonical approach. The basis for the solution is the transformation and uncoupling of the differential equations for the model. The resulting solution is expressed in terms of elementary functions of the independent variable (the ‘true anomaly’), and is of a compact and simple form. Numerical results are comparable to existing solutions.     

Counting relative equilibrium configurations of the full two-body problem

Consider a system of two rigid, massive bodies interacting according to their mutual gravitational attraction. In a relative equilibrium motion, the bodies rotate rigidly and uniformly about a fixed axis in \({\mathbb {R}}^3\). This is possible only for special positions and orientations of the bodies. After fixing the angular momentum, these relative equilibrium configurations can be characterized as critical points of a smooth function on configuration space. The goal of this paper is to use Morse theory and Lusternik–Schnirelmann category theory to give lower bounds for the number of critical points when the angular momentum is sufficiently large. In addition, the exact number of critical points and their Morse indices are found in the limit as the angular momentum tends to infinity.     

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 $J_{2}$ perturbation is transformed into linear. Moreover, the method reveals to be competitive with two very popular element methods derived from the Kustaanheimo-Stiefel and Sperling-Burdet regularizations.     

Extended two-body problem for rotating rigid bodies

Given the interest in future space missions devoted to the exploration of key moons in the solar system and that may involve libration point orbits, an efficient design strategy for transfers between moons is introduced that leverages the dynamics in these multi-body systems. The moon-to-moon analytical transfer (MMAT) method is introduced, comprised of a general methodology for transfer design between the vicinities of the moons in any given system within the context of the circular restricted three-body problem, useful regardless of the orbital planes in which the moons reside. A simplified model enables analytical constraints to efficiently determine the feasibility of a transfer between two different moons moving in the vicinity of a common planet. In particular, connections between the periodic orbits of such two different moons are achieved. The strategy is applicable for any type of direct transfers that satisfy the analytical constraints. Case studies are presented for the Jovian and Uranian systems. The transition of the transfers into higher-fidelity ephemeris models confirms the validity of the MMAT method as a fast tool to provide possible transfer options between two consecutive moons.

     

Bounded motion in a generalized two-body problem

In this paper we prove the existence of ring-type bounded motion in an isolated system consisting of a massive point particle and a homogeneous cube. We study the case of planar motion where the particle moves in a symmetry plane of the cube and we use a rotating frame of reference with its center at the mass center of the cube and its axes coinciding with the symmetry axes of the cube. We prove that, for negative values of the total energy and properly chosen values of the total angular momentum, the relative distance of the bodies has an upper and a lower bound-i.e., the regions of possible motion lie inside an annulus around the cube (motion inside a ring or an island).     

The almost constant-speed two-body problem with resistance

The almost constant-speed motion of a mass acted upon by a Newtonian attraction and a resisting force is treated. The equation of orbit is derived for a specific type of resistance which covers the familiar case of Danby's drag(=r ^–2) whilst the vector invariants are obtained by direct operation on the vector form of the equation of motion.     

On Broucke's velocity-related series expansions in the two-body problem

This short article supplements a recent paper by Dr R. Broucke on velocity-related series expansions in the two-body problem. The derivations of the Fourier and Legendre expansions of the functionsF(v), \(\sqrt {F(\upsilon )} \) and \(\sqrt {{1 \mathord{\left/ {\vphantom {1 {F(\upsilon )}}} \right. \kern-0em} {F(\upsilon )}}} \) are given, where $$F(\upsilon ) = (1 - e^2 )/(1 + 2e\cos \upsilon + e^2 ), e< 1$$ In the two-body problem,v is identified with the true anomaly,e the eccentricity andF(v) equals (an/V)². Some interesting relations involving Legendre polynomials are also noted.     

Semi-analytic solution of a two-body problem with drag

An approximate semi-analytic solution of a two-body problem with drag is presented. The solution describesnon-lifting orbital motion in a central, inverse-square gravitational field. Drag deceleration is a non-linear function of velocity relative to a rotating atmosphere due to dynamic pressure and velocity-dependent drag coefficient. Neglected are aerodynamic lift, gravitational perturbations of the inverse-square field, and kinematic accelerations due to coordinate frame rotation at earth angular rate. With these simplifications, it is shown that (i) orbital motion occurs in an earth-fixed invariable plane defined by the radius and relative velocity vectors, and (ii) the simplified equations of motion are autonomous and independent of central angle measured in the invariable plane. Consequently, reduction of the differential equations from sixth to second-order is possible. Solutions for the radial and circumferential components of relative velocity are reduced to quadratures with respect to radial distance. Since the independent variable is radial distance, the solutions are singular at zero radial velocity (e. g., for circular orbits). General atmospheric density and drag coefficient models may be used to evaluate the velocity quadratures. The central angle and time variables are recovered from two additional quadratures involving the velocity quadratures. Theoretical results are compared with numerical simulation results.Presently affiliated with AVCO Systems Division, Wilmington, MA 01887, U.S.A.     

Analytical treatment of the two-body problem with slowly varying mass

The present work is concerned with the two-body problem with varying mass in case of isotropic mass loss from both components of the binary systems. The law of mass variation used gives rise to a perturbed Keplerian problem depending on two small parameters. The problem is treated analytically in the Hamiltonian frame-work and the equations of motion are integrated using the Lie series developed and applied, separately by Delva (1984) and Hanslmeier (1984). A second order theory of the two bodies eject mass is constructed, returning the terms of the rate of change of mass up to second order in the small parameters of the problem.     

Transformations of the perturbed two-body problem to unperturbed harmonic oscillators

Transformations are given which change the perturbed planar problem of two bodies into unperturbed and undamped harmonic oscillators with constant coefficients. The orginally singular, nonlinear and Lyapunov unstable equations become in this way regular, linear, and the stable solution may be written down immediately in terms of the new variables. Transformations of the independent and dependent variables are treated separately as well as jointly. Using arbitrary and special functions for the transformations allows a systematic discussion of previously introduced and new anomalies. For the unperturbed two-body problem the theorem is proved according to which if the transformations are power-functions of the radial variable, then only the eccentric and the true anomalies with the corresponding transformations of the radial variable will result in harmonic oscillators. Important practical applications are to increase autonomous operations in space, since by replacing lengthy numerical integrations by transformations, computer requirements are significantly reduced.