Restricted Two-body Problem Research Articles

Revisiting the Dynamics of Two-Body Problem in the Framework of the Continued Fraction Potential

In this analytical study, a novel solving method for determining the precise coordinates of a mass point in orbit around a significantly more massive primary body, operating within the confines of the restricted two-body problem (R2BP), has been introduced. Such an approach entails the utilization of a continued fraction potential diverging from the conventional potential function used in Kepler’s formulation of the R2BP. Furthermore, a system of equations of motion has been successfully explored to identify an analytical means of representing the solution in polar coordinates. An analytical approach for obtaining the function t = t(r), incorporating an elliptic integral, is developed. Additionally, by establishing the inverse function r = r(t), further solutions can be extrapolated through quasi-periodic cycles. Consequently, the previously elusive restricted two-body problem (R2BP) with a continued fraction potential stands fully and analytically solved.

Dark matter and motion of bodies in space

This paper investigates the degree of influence of the gravitational field of dark matter on the laws of motion of bodies in a medium in a restricted two-body problem, when a test body (planet, asteroid, artificial satellite of a star, in particular, the Sun, etc.) has its own rotation, i. e. own angular momentum impulse. The study was carried out within the framework of the post-Newtonian approximation of the general theory of relativity. In accordance with the latest experimental data, hypotheses about the average densities of dark matter ρD.M. and visible matter ρvis. in planetary systems are accepted. In particular, in the Solar system the following is accepted: ρD.M ≈ 2,8 · 10–19 g · cm–3, ρvis ≈ 3 · 10–20 g · cm–3 and ρΣ = ρvis + ρD.M ≈ 3,1 · 10–19 g · cm–3. In the post-Newtonian approximation of the general theory of relativity, the equation for the trajectory of a rotating test body with respect to ρΣ is derived, and working formulas are obtained that give the laws of secular changes in the direction of the vector of the proper angular momentum impulse of the test body and the modulus of this vector. It is shown that accounting ρD.M changes the magnitude of the periastron shift. For example, in the Solar System when taking into account ρvis, all the planets except Pluto have a directly shifted perihelion in the post-Newtonian approximation of the general theory of relativity. When taking into account ρΣ the planets from Mercury to Saturn included, they have a direct shift of perihelion, and Uranus, Neptune, Pluto have the reverse (against the planets in orbit). There is also a secular change in the eccentricity of the orbit. The formula is derived that can be used to calculate the secular deviation of the translational motion of a rotating body from motion in a plane. Accounting ρΣ enhances deviation. It is emphasized that all the noted effects for planetary systems in the vicinity of neutron stars, radio pulsars and other dense objects can be many orders of magnitude greater than in the solar system.

Гравитационные маневры при осуществлении марсианской пилотируемой экспедиции с электроракетной двигательной установкой

The paper focuses on the analysis of energy-ballistic efficiency of gravity-assist maneuvers in the implementation of the Martian manned expedition in the period 2049–2050. The purpose of this analysis was to identify the opportunities for improving the energy-ballistic performance indicators of the Martian manned expedition through gravity-assist maneuvers around the Earth and Venus. The methodical approach to calculating the main energy-ballistic indicators of the Martian manned expedition is based on dividing the flight trajectory into sections. To determine the main characteristics of these sections, the statement corresponding to the restricted two-body problem was used. The heliocentric trajectory sections were optimized using the Pontryagin maximum principle. The families of solutions with a gravity-assist maneuver near Venus were obtained, differing in the direction of the flyby of Venus and the height of the flight orbit pericenter. The research shows the existence of extremals close in characteristics, which are with the fly-by orbit pericenter altitude restriction and without it. A comparison was made in terms of the duration of the expedition and the initial mass with solutions without a gravity-assist maneuver.

Некоторые модели оптимальных траекторий сближения космического аппарата с астероидом

We consider the optimal trajectories in the rendezvous problem calculated by calculus of variations techniques. We formulate the problem of translational motion of the spacecraft taking the asteroid rotation into consideration. The motion is supposed to occur outside of a sphere of influence of any planet, the gravity of the asteroid is defined like the one in the restricted two-body problem. We represent the equations of controlled motion in central gravity field and construct the calculus of variations problem, given the control constraints. The problem is solved applying Valentine’s method. We solve the Euler–Lagrange equations generated from the functional related to the fuel consumption and show the solutions in the special case of neglecting the asteroid gravity, both taking and not taking the Coriolis force into consideration. The optimal control in these solutions is occurred to be twofold: continuous (either smooth and piecewise smooth one) and piecewise continuous one, having one or more points of jump discontinuities. In case of these discontinuities the Weierstrass–Erdmann condition is applied. The optimality is validated by the Legendre condition. We note the way the control function and the trajectory change while varying some of the parameters of the problem: the initial speed, the flight duration, the control limit, the angular velocity of the asteroid. The descripted method allows one to evaluate some parameters of the flight which may be used in the rendezvous mission planning.

Closed-form Solutions for Optimal Orbital Transfers Around Oblate Planets

Optimal spacecraft orbit control has been the subject of extensive research, which resulted in solutions for optimal orbit transfers. A common orbital maneuver problem is the fuel-optimal impulsive transfer between coplanar circular orbits. Three such well-known transfers are the Hohmann transfer, which is an optimal bi-impulsive transfer, the bi-elliptic tri-impulsive transfer, and the bi-parabolic transfer. These solutions were developed based on the Keplerian restricted two-body problem. However, the omission of perturbations results in deviated target orbits and leads to maneuvers that are not actually fuel-optimal. In this paper, the well-known Hohmann, bi-elliptic, and bi-parabolic transfers are modified to accommodate the J2 zonal harmonic, and new closed-form solutions for the optimal maneuvers are presented. An improvement in maneuver precision is obtained by using an analytical model based on closed-form solutions of motion in the equatorial plane under the effect of J2. The performance improvement is validated using high-fidelity simulations, which include a myriad of orbital perturbations.

A first-order secular theory for the post-Newtonian two-body problem with spin – I. The restricted case

We revisit the relativistic restricted two-body problem with spin employing a perturbation scheme based on Lie series. Starting from a post-Newtonian expansion of the field equations, we develop a first-order secular theory that reproduces well-known relativistic effects such as the precession of the pericentre and the Lense-Thirring and geodetic effects. Additionally, our theory takes into full account the complex interplay between the various relativistic effects, and provides a new explicit solution of the averaged equations of motion in terms of elliptic functions. Our analysis reveals the presence of particular configurations for which non-periodical behaviour can arise. The application of our results to real astrodynamical systems (such as Mercury-like and pulsar planets) highlights the contribution of relativistic effects to the long-term evolution of the spin and orbit of the secondary body.

Two-Body Problem with Drag and High Tangential Speeds

This paper considers the restricted two-body problem with atmospheric drag. A simple formula is presented that approximates the atmospheric density from raw data, replaces previous models, and is amenable to closed-form solution of the orbit equation for high tangential speeds. A procedure for subdividing an altitude interval and calculating the parameters of the formula over each subinterval leads to highly improved accuracy in the solutions and compares favorably with numerical integration. To validate our model we compare it with the trajectory and flight time of a satellite in an exponential atmosphere starting from a near-circular orbit at 7120 km from the Earth's center.

Two-Body Problem With High Tangential Speeds and Quadratic Drag

This paper considers the restricted two-body problem with atmospheric drag, and its results apply to arcs of orbits in which the tangential speed is much greater than the radial speed. In principle this treatment applies to certain arcs of any near-Keplerian orbit but its usefulness is primarily restricted to near-circular orbits. The atmospheric density is approximated by a model that enables us to derive closed-form solutions for the radial distance of a satellite in orbit that are more accurate than other closed-form solutions found in previous works. However, these new formulas for the orbit do not retain the similarity to those of the two-body problem without drag as much as the previous ones, and require the determination of two additional parameters. Nevertheless, the increase in accuracy is significant.

The restricted two-body problem in constant curvature spaces

We perform the bifurcation analysis of the Kepler problem on $S^3$ and $L^3$. An analogue of the Delaunay variables is introduced. We investigate the motion of a point mass in the field of the Newtonian center moving along a geodesic on $S^2$ and $L^2$ (the restricted two-body problem). When the curvature is small, the pericenter shift is computed using the perturbation theory. We also present the results of the numerical analysis based on the analogy with the motion of rigid body.

Closed-Form Solutions for Near-Circular Arcs with Quadratic Drag

This paper considers the restricted two-body problem in the presence of drag that varies with a power of the magnitude of the velocity. In general, the orbit equation for this problem is an integrodifferential equation. For extreme cases in which the motion is mostly tangential and the drag varies with the square of the magnitude of the velocity, a new transformation of the orbit equation results in an ordinary differential equation. For a model atmosphere in which the density varies inversely as the square of the distance from the center of attraction, we provide closed-form solutions to this differential equation. This extends previous work in which the density of the model atmosphere varies inversely with the distance from the center of attraction.

On the formation flying of satellite constellations

From the viewpoint of orbit dynamics, the formation flying of the satellites of a constellation and the accompanying flying in satellite-satellite tracking are affected by the following dynamical mechanisms: (1) The main characteristics of the variations of disturbed orbits determine the space configuration of the satellites. (2) When the distances among the satellites are small, conditional periodic motion near the collinear equilibrium points in the degenerate restricted three-body problem (actually, the restricted two-body problem) can also constrain, within a certain time interval, the satellite configuration. In this paper, we will clarify the principles of these two dynamical mechanisms and the corresponding satellite configurations, and we will verify their applicable range by simulative calculations, so providing a theoretical and practical basis for orbit control in the formation flying of satellite constellations.

Nonintegrability of a restricted two-body problem for an elastic-interaction potential on a sphere

Nonintegrability of a restricted two-body problem for an elastic-interaction potential on a sphere

Finsler generalization of the hypothesis of Riemannian geodesics for the motion of test objects. II

In a continuation of a study begun in an earlier paper [1], the Lagrangian density is constructed for the gravitational field in the Finsler approach on the basis of the Einstein Lagrangian density. A restricted two-body problem is discussed in the Finsler approach.