A finite element method for computational full two-body problem: I. The mutual potential and derivatives over bilinear tetrahedron elements Abstract A finite element method (FEM) for computing the gravitational interactions between two arbitrarily shaped celestial bodies is proposed. Expressions for the gravitational potential, attraction and torques are derived in terms of the finite element mesh division and mass density distribution. This method is implemented to a parallel-simulation package on a local cluster. Benchmarking tests are performed to confirm the convergence properties and to measure the computational costs. For a representative application, we construct the FEM model of the binary Near-Earth asteroid 65803 Didymos and simulate the coupled spin–orbit motion of its two components. The results show our method propagates the binary motion precisely, which is significantly dependent on the primary’s internal structure. In this numeric example, we show the finite element method is capable of modeling complex geometry and dissimilar material properties, which is useful to address questions in predicting the evolution of actual binary asteroids.

On the solution to every Lambert problem Abstract Lambert’s problem is the two-point boundary value problem for Keplerian dynamics. The parameter and solution space is surveyed for both the zero- and multiple-revolution problems, including a detailed look at the stress cases that typically plague Lambert solvers. The problem domain, independent of formulation, is shown to be rectangular for each revolution case, making the elusive initial guess and the solution itself amenable for interpolation. Biquintic splines are implemented to achieve continuous derivatives and quick evaluation. Resulting functions may be used directly as low-fidelity solutions or used with a single update iteration without safeguards. A concise, improved vercosine formulation of the Lambert problem is presented, including new singularity-free and precision-saving equations. The interpolation scheme is applied for up to 100 revolutions. The domain considered includes all practically conceivable flight times, and every possible geometry except a small region near the only physical singularity of the problem: the equal terminal vector case. The solutions are archived and benchmarked for accuracy, memory footprint, and speed. For typical scenarios, users can expect \(\sim 6\) or more digits of velocity vector accuracy using an interpolated solution without iteration. Using a single, unguarded iteration leads to solutions with near machine precision accuracy over the full domain, including the most extreme scenarios. Depending on desired resolution, coefficient files vary in size from \(\sim 3\) to 65 MB for each revolution case. Evaluation runtimes vary from \(\sim 2\) to 5 times faster than the industry benchmark Gooding algorithm. The coefficient files and driver routines are provided online. While the method is currently demonstrated on the vercosine formulation, the 2D interpolation scheme stands to benefit all Lambert problem formulations.

Charged dust close to outer mean-motion resonances in the heliosphere Abstract We investigate the dynamics of charged dust close to outer mean-motion resonances with planet Jupiter. The importance of the interplanetary magnetic field on the orbital evolution of dust is clearly demonstrated. New dynamical phenomena are found that do not exist in the classical problem of uncharged dust. We find changes in the orientation of the orbital planes of dust particles, an increased amount of chaotic orbital motions, sudden ’jumps’ in the resonant argument, and a decrease in time of temporary capture due to the Lorentz force. Variations in the orbital planes of dust grain orbits are found to be related to the angle between the orbital angular momentum and magnetic axes of the heliospheric field and the rotation rate of the Sun. These variations are bound using a simplified model derived from the full dynamical problem using the first-order averaging theory. It is found that the interplanetary magnetic field does not affect the capture process, that is still dominated by the other non-gravitational forces. Our study is based on a dynamical model in the framework of the inclined circular restricted three-body problem. Additional forces include solar radiation pressure, solar wind drag, the Poynting–Robertson effect, and the influence of a Parker spiral-type interplanetary magnetic field model. The analytical estimates are derived on the basis of Gauss’ form of planetary equations of motion. Numerical results are obtained by simulations of dust grain orbits together with the system of variational equations. Chaotic regions in phase space are revealed by means of fast Lyapunov chaos indicators.

A novel analytic continuation power series solution for the perturbed two-body problem Abstract Inspired by the original developments of recursive power series by means of Lagrange invariants for the classical two-body problem, a new analytic continuation algorithm is presented and studied. The method utilizes kinematic transformation scalar variables differentiated to arbitrary order to generate the required power series coefficients. The present formulation is extended to accommodate the spherical harmonics gravity potential model. The scalar variable transformation essentially eliminates any divisions in the analytic continuation and introduces a set of variables that are closed with respect to differentiation, allowing for arbitrary-order time derivatives to be computed recursively. Leibniz product rule is used to produce the needed arbitrary-order expansion variables. With arbitrary-order time derivatives available, Taylor series-based analytic continuation is applied to propagate the position and velocity vectors for the nonlinear two-body problem. This foundational method has been extended to also demonstrate an effective variable step size control for the Taylor series expansion. The analytic power series approach is demonstrated using trajectory calculations for the main problem in satellite orbit mechanics including high-order spherical harmonics gravity perturbation terms. Numerical results are presented to demonstrate the high accuracy and computational efficiency of the produced solutions. It is shown that the present method is highly accurate for all types of studied orbits achieving 12–16 digits of accuracy (the extent of double precision). While this double-precision accuracy exceeds typical engineering accuracy, the results address the precision versus computational cost issue and also implicitly demonstrate the process to optimize efficiency for any desired accuracy. We comment on the shortcomings of existing power series-based general numerical solver to highlight the benefits of the present algorithm, directly tailored for solving astrodynamics problems. Such efficient low-cost algorithms are highly needed in long-term propagation of cataloged RSOs for space situational awareness applications. The present analytic continuation algorithm is very simple to implement and efficiently provides highly accurate results for orbit propagation problems. The methodology is also extendable to consider a wide variety of perturbations, such as third body, atmospheric drag and solar radiation pressure.

The evolution of the Line of Variations at close encounters: an analytic approach Abstract We study the post-encounter evolution of fictitious small bodies belonging to the so-called Line of Variations (LoV) in the framework of the analytic theory of close encounters. We show the consequences of the encounter on the local minimum of the distance between the orbit of the planet and that of the small body and get a global picture of the way in which the planetocentric velocity vector is affected by the encounter. The analytical results are compared with those of numerical integrations of the restricted three-body problem.

Central configurations in planar n -body problem with equal masses for $$n=5,6,7$$ n = 5 , 6 , 7 Abstract We give a computer-assisted proof of the full listing of central configuration for n-body problem for Newtonian potential on the plane for \(n=5,6,7\) with equal masses. We show all these central configurations have a reflective symmetry with respect to some line. For \(n=8,9,10\) , we establish the existence of central configurations without any reflectional symmetry.

Analytic orbit theory with any arbitrary spherical harmonic as the dominant perturbation Abstract In the gravitational potential of Earth, the oblateness term is the dominant perturbation, with its coefficient at least three orders of magnitude greater than that of any other zonal or tesseral spherical harmonic. Therefore, analytic orbit theories (or satellite theories) are developed using the Keplerian Hamiltonian as the unperturbed solution, oblateness term as the first-order and the remaining spherical harmonics as the second-order perturbations. These orbit theories are generally constructed by applying multiple near-identity canonical transformations to the perturbed Hamiltonian in conjunction with averaging to obtain a secular Hamiltonian, from which the short-period and long-period terms are removed. If the oblateness term is the only first-order perturbation, then the long-period terms appear only in the second- or higher-order terms of the single-averaged Hamiltonian, from which the short-period terms are removed. These second-order long-period terms are separated from the Hamiltonian by the first-order generating function using a second canonical transformation. This results in a secular Hamiltonian dependent only on the momenta. However, in the case of other gravitational bodies with more deformed shapes compared to Earth such as moons and asteroids, the oblateness coefficient may have the same order of magnitude as some of the higher spherical harmonic coefficients. If these higher harmonics are treated as the first-order perturbation along with the oblateness term, then the long-period terms appear in the first-order single-averaged Hamiltonian. These first-order long-period terms cannot be separated using the generating function in the conventional way. This problem occurs because the zeroth-order Hamiltonian, i.e., the Keplerian part, is degenerate in the angular momentum. In this paper, a new approach to the long-period transformation is proposed to resolve this issue and obtain a fully analytic orbit theory when for the perturbing gravitational body, any arbitrary zonal or tesseral harmonic is the dominant perturbation. The proposed theory is closed form in the eccentricity as well. It is applied to predict the motion of artificial satellites for the two test cases: a lunar orbiter and a satellite of 433 Eros asteroid. The prediction accuracy is validated against the numerical propagation using a force model with \(6\times 6\) gravity field.

Structure of the centre manifold of the $$L_1,L_2$$ L 1 , L 2 collinear libration points in the restricted three-body problem Abstract We present a global analysis of the centre manifold of the collinear points in the circular restricted three-body problem. The phase-space structure is provided by a family of resonant 2-DOF Hamiltonian normal forms. The near 1:1 commensurability leads to the construction of a detuned Birkhoff–Gustavson normal form. The bifurcation sequences of the main orbit families are investigated by a geometric theory based on the reduction of the symmetries of the normal form, invariant under spatial mirror symmetries and time reversion. This global picture applies to any values of the mass parameter.

Phase space description of the dynamics due to the coupled effect of the planetary oblateness and the solar radiation pressure perturbations Abstract The aim of this work is to provide an analytical model to characterize the equilibrium points and the phase space associated with the singly averaged dynamics caused by the planetary oblateness coupled with the solar radiation pressure perturbations. A two-dimensional differential system is derived by considering the classical theory, supported by the existence of an integral of motion comprising semi-major axis, eccentricity and inclination. Under the single resonance hypothesis, the analytical expressions for the equilibrium points in the eccentricity-resonant angle space are provided, together with the corresponding linear stability. The Hamiltonian formulation is also given. The model is applied considering, as example, the Earth as major oblate body, and a simple tool to visualize the structure of the phase space is presented. Finally, some considerations on the possible use and development of the proposed model are drawn.

A new radial, natural, higher-order intermediary of the main problem four decades after the elimination of the parallax Abstract Simplifications in dealing with the equation of the center when the short-period effects are removed from a zonal Hamiltonian are commonly attributed to the elimination of parallactic terms. But this interpretation is incorrect, and the simplifications rather stem from the removal of concomitant long-period terms, an outcome that can also be achieved without need of eliminating the parallax. To show that, a Lie transforms simplification is invented that augments the exponents of the inverse of the radius and still achieves analogous simplifications in handling the equation of the center to those provided by the classical elimination of the parallax simplification. The particular case in which the new transformation does not modify the exponents of the parallactic terms of the original problem leads to a new intermediary of the main problem that, while keeping higher-order effects of \(J_2\) , is formally analogous to Cid’s first-order radial intermediary.