Rectilinear Orbit Research Articles

Relative motion guidance for near-rectilinear lunar orbits with path constraints via actor-critic reinforcement learning

This paper presents a feedback guidance algorithm for proximity operation in cislunar environment based on actor-critic reinforcement learning. The algorithm is lightweight, closed-loop, and capable of taking path constraints into account. The method relies on reinforcement learning to make the well known Zero-Effort-Miss/Zero-Effort-Velocity guidance state dependent and allow for path constraints to be directly embedded. The algorithm is tested in the circular restricted three-body problem (CRTBP) framework for Near Rectilinear Orbits (NRO) in the Earth-Moon system. It shows promising results in terminal guidance error and satisfies path constraints in constraint scenarios comprising spherical constraints and keep-out-spheres with approach corridors. Furthermore, this approach indicates that reinforcement learning can be effectively used to solve constrained relative spacecraft guidance problems in complex environments and thus can be effective for autonomous relative motion operations in the Earth-Moon dynamical environment.

Solar sail H-reversal trajectory: A review of its advances and applications

The set of the orbital angular-momentum reversal, or H-reversal, sailcraft trajectory was born as a type of unconventional precursor interstellar mission trajectory by using highperformance solar sails. Starting from an outline of the H-reversal sail trajectory, this paper mainly focuses on the 2D reversal-mode solution to the general solar-photon sail motion equations. The feasible region for H-reversal trajectories in fixed sail attitude angles is illustrated. Some interesting applications of the H-reversal trajectory are presented in detail to show its advantages. As a special case, a precursor interstellar probe can be delivered with a constant sail orientation in the H-reversal trajectory to be compared with the direct-motion sail flyby of the Sun. Of importance are the heliocentric periodic orbits in double H-reversal modes, obtained via both fixed and time-varying sail attitude angles. Two more applications involving H-reversal trajectories are discussed in terms of asteroid deflection and transfer trajectory to rectilinear orbits. Finally, some items of the mathematics behind the 3D motion-reversal trajectories are summarized.

The family of planar periodic orbits generated by the equal-mass four-body Schubart interplay orbit

We locate members of a one-parameter family of equal-mass four-body periodic orbits in the plane. The family begins and ends with the rectilinear four-body equal-mass Schubart interplay orbit and passes through a double choreography orbit. The first-order stability of these orbits is computed. Some members of this symmetric family are stable to symmetric perturbations; however, they are unstable when all perturbations are allowed.

Flexural-gravity wave resistances due to a moving point source on 2-D infinite floating beam

The flexural-gravity wave responses due to a load steadily moving or suddenly accelerated along a rectilinear orbit are analytically studied within the framework of the linear potential theory. A thin viscoelastic plate model is used for a very large floating structure. The initially quiescent fluid in the ocean is assumed to be homogenous, incompressible, and inviscid, and the disturbed motion be irrotational. A moving line source on the plate surface is considered as a moving point in the two-dimensional coordinates. Under the assumptions of small-amplitude wave motion and small plate deflection, a linear fluid-plate coupling model is established. The integral solutions for the surface deflections and the wave resistances are analytically obtained by the Fourier transform method. To study the dynamic characteristics of the flexural-gravity wave response, the asymptotic representations of the wave resistances are derived by the residue theorem and the methods of stationary phase. It shows that the steady wave resistance is zero when the speed of moving load is less than the minimal phase speed. The wave resistances due to the accelerate motion consist of two parts, namely the steady and transient wave responses. Eventually the transient wave resistance declines toward zero and the wave resistance approaches the steady component as the time goes to the infinity. Furthermore, the effect of the strain relaxation time for this viscoelastic plate is studied and it exhibits more influence for a high-speed motion.

Equinoctial elements for Vinti theory: Generalizations to an oblate spheroidal geometry

Vinti theory constructs orbits on an oblate spheroidal geometry, naturally encoding the gravitational potential of an oblate spheroid in the coordinates. Classical techniques use spherical geometry. Recent work applied Vinti theory to the relative motion problem by way of a linear dynamical model, which is nonsingular in the oblate spheroidal element space. But as with classical spherical elements, the linear mapping between classical spheroidal elements and inertial rectangular coordinates becomes singular for small eccentricities and/or inclinations in the sense of linear dependence of columns in the Jacobian. To mitigate these practical issues, the standard (spherical) equinoctial elements are chosen to inform in a natural way their generalization to a new nonsingular element set: the oblate spheroidal equinoctial orbital elements. The spherical equinoctial elements can be considered a special case of the spheroidal equinoctial elements in the same way that spherical coordinates can be considered a special case of oblate spheroidal coordinates. The new element set is defined and algorithms for converting between spheroidal equinoctial elements and inertial coordinates are derived. Similarities and differences between spheroidal and spherical equinoctial elements are emphasized for clarity, both in terms of the form of equations and geometrical interpretation. The transformations are valid away from the nearly rectilinear orbit regime and are exact except near the poles. When near the poles, the transformations match the accuracy of the approximate analytical solution, which has been developed to the third order in J2 in the literature. As a result, the singularity on the poles is completely eliminated for the first time.

Quantum mechanics of rectilinear orbits of the Coulomb-Kepler problem

Based on coherent states of the reversed harmonic oscillator, a wave function is proposed with the following three properties: (1) The mean values of position, r0, and velocity, v0, are parallel to the x axis; (2) The wave function is extended by a curve parameter w ≥ 0 such that, as a function of w, the mean values rw and vw describe a rectilinear orbit along the x axis with the mean initial values r0 and v0; (3) The mean energy is constant with respect to w by the leading order of the perturbation parameter ∈. For comparison, the classical one-dimensional equation of motion is integrated. Exact analytical equivalence between classical and the ∈ = 0 quantum limit is found, both for positive and negative mean energy. The curve parameter w is in 1-1 correspondence with time t ≥ 0.

Optimal periodic relative orbit and rectilinear relative orbits with eccentric reference orbits

The problem of two-body linearized periodic relative orbits with eccentric reference orbits is studied in this paper. The periodic relative orbit in the target-orbital coordinate system can be used in fly-around and formation-flying orbit design. Based on the closed-form solutions to the Tschauner–Hempel equations, the initial condition for periodic relative orbits is obtained. Then the minimum-fuel periodic-orbit condition with a single impulse is analytically derived for given initial position and velocity vectors. When considering the initial coasting time, the impulse position of the global minimum-fuel periodic orbit is proved to be near to the perigee of the target and can be obtained by numerical optimization algorithms. Moreover, the condition for a special periodic orbit, i.e., the rectilinear relative orbit in the target-orbital frame, is obtained. Numerical simulations are used to demonstrate the efficacy of the method, and show the geometry of the periodic relative orbit and the rectilinear relative orbit.

THE 2-D SEXTIC HAMILTONIAN OSCILLATOR

The 2-D sextic oscillator is studied as a family of axial symmetric parametric integrable Hamiltonian systems, presenting a bifurcation analysis of the different flows. It includes the "elliptic core" model in 1-D nonlinear oscillators, recently proposed in the literature. We make use of the energy-momentum mapping, which will give us the fundamental fibration of the four-dimensional phase space. Special attention is given to the singular values of the energy-momentum mapping connected with rectilinear and circular orbits. They are related to the saddle-center and pitchfork scenarios with the associated homoclinic and heteroclinic trajectories. We also study how the geometry of the phase space evolves during the transition from the one-dimensional to the two-dimensional model. Within an elliptic function approach, the solutions are given using Legendre elliptic integrals of the first and third kind and the corresponding Jacobi elliptic functions.

Optimal solar sail transfer to linear trajectories

The aim of this paper is to characterize a mission toward a heliocentric linear trajectory (that is, a rectilinear orbit) and to investigate the performance of a solar-sail-based spacecraft to accomplish the transfer phase using an ideal or an optical force model. A similar problem has been recently discussed under simplified assumptions, but new results are now provided with a thorough analysis that uses a three dimensional transfer orbit or a piecewise constant steering law. The paper investigates a complex mission scenario, where the solar sail returns back to Earth after a scientific probe is released along a target linear trajectory.

Solar Sail Capabilities to Reach Elliptic Rectilinear Orbits

A N ELLIPTIC rectilinear orbit (ERO) is a particular conic orbit in which the semimajor axis takes a finite value and the eccentricity is unity [1]. Froma geometrical point of view anERO is a line segment connecting both orbital foci: one endpoint of the segment coincides with the primary focus (the sun in a heliocentric system), while the other endpoint is the orbit apocenter. A spacecraft (or a celestial body) that tracks a heliocentric ERO experiences a rectilinear motion toward the sunwith a purely radial velocity, that is, directed along the spacecraft-sun direction. The velocity magnitude is zero in correspondence of the orbit apocenter and takes its maximum value at the primary focus. Apart from the theoretical mathematical interest of such orbits, the practical importance of a rectilinear trajectory toward the sun involves different possible scientific missions, such as those concerning the test of the equivalence principle, the analysis of the interstellar dust, or the study of the radial variation of the solar wind. In addition, an ERO could be useful for a detailed analysis of the sun’s gravitational harmonics [2,3]. Themain obstacle against the use of anERO for scientificmissions is mainly due to the substantial amount of propellant required to reach these orbits. To get a rough estimate, consider the minimum V necessary to transfer a spacecraft from a heliocentric circular orbit of radius r0 to a coplanar ERO with an aphelion radius ra > r0. Assuming a bi-impulsive maneuver and a Hohmann-like elliptic transfer orbit with semimajor axis r0 ra =2, the total velocity variation necessary to complete the transfer is

Beam reception and data transmission performance analysis of optical taper used in a mobile wireless optical communication

Mobile wireless optical communication system was demonstrated experimentally by using a special solid coupling optical taper. The performance of special solid coupling optical taper and the data transmission performance with 240 Mbit/s were investigated on the rectilinear orbit between the mobile terminal and the fixed terminal. The experimental results shown that the scheme used in the mobile wireless optical communication system is reliable and quite simple, which could be widely used in future.

The 1/1 resonance in extrasolar planetary systems

We study orbits of planetary systems with two planets, for planar motion, at the 1/1 resonance. This means that the semimajor axes of the two planets are almost equal, but the eccentricities and the position of each planet on its orbit, at a certain epoch, take different values. We consider the general case of different planetary masses and, as a special case, we consider equal planetary masses. We start with the exact resonance, which we define as the 1/1 resonant periodic motion, in a rotating frame, and study the topology of the phase space and the long term evolution of the system in the vicinity of the exact resonance, by rotating the orbit of the outer planet, which implies that the resonance and the eccentricities are not affected, but the symmetry is destroyed. There exist, for each mass ratio of the planets, two families of symmetric periodic orbits, which differ in phase only. One is stable and the other is unstable. In the stable family the planetary orbits are in antialignment and in the unstable family the planetary orbits are in alignment. Along the stable resonant family there is a smooth transition from planetary orbits of the two planets, revolving around the Sun in eccentric orbits, to a close binary of the two planets, whose center of mass revolves around the Sun. Along the unstable family we start with a collinear Euler–Moulton central configuration solution and end to a planetary system where one planet has a circular orbit and the other a Keplerian rectilinear orbit, with unit eccentricity. It is conjectured that due to a migration process it could be possible to start with a 1/1 resonant periodic orbit of the planetary type and end up to a satellite-type orbit, or vice versa, moving along the stable family of periodic orbits.

Stationary orbits of comets perturbed by Galactic tides

Using the first-order normalized equations describing the heliocentric cometary motion perturbed by the Galactic tides, we identify ‘stationary solutions’ with constant values of the eccentricity, inclination, argument of perihelion and longitude of the ascending node in the reference frame rotating with the Galaxy. The families found involve circular orbits, orbits in the Galactic equatorial plane, rectilinear orbits normal to the equatorial plane, elliptic orbits symmetric with respect to the direction to the Galactic Centre or to its perpendicular, and asymmetrically oriented elliptic orbits. Linear stability of the stationary solution is studied analytically and confirmed by numerical experiments. Most, but not all, of the unstable solutions prove chaotic with the Lyapunov times at least 100 Myr.

Head-on Collision of Disk–Sphere Galaxies

Numerical simulations are performed to study the tidal effects of non-merging rapid head-on collision between a disk galaxy and a spherical galaxy. The disk consists of three components – a disk, a bulge and a halo – and the spherical galaxy is a Plummer model. The galaxies have the same dimensions with different mass ratios viz., 2, 1 and 0.5. They move in a rectilinear orbit with a relative velocity of 1000 km s−1.

A Geometrical Interpretation of the Deflection of Almost Rectilinear Orbits in a Central Field

We give a geometrical interpretation for the deflection at the origin of rectilinear orbits in a central field. This interpretation is based on the correspondence between the plane orbits of a conservative force field and the geodesics of a certain surface. If the field is ‘hard’ near the origin, the surface is tangent to a cone. By considering the development of this cone, we obtain the deflection. We study also almost rectilinear orbits.

'On Solving Kepler's Equation For Nearly Rectilinear Hyperbolic Orbits'

We deal here with efficient starting points for Kepler's equation in a special case of nearly rectilinear hyperbolic orbits, that is these ones with the eccentricities e ≫ 1. These orbits appear in stellar dynamics when considering encounters of stars. We test efficiency of these starters for the method for successive approximation (MSA) in its two often applied variants, that is the Newton's method with the quadratic convergence (NM) and in the fixed point method (FPM). Moreover, we determine a dynamical domain of Kepler's equation for this motion.

Two-Body Problems with Drag or Thrust: Qualitative Results

We consider two‐body problems in which the drag is proportional to the velocity divided by the square of the distance and whose radial and tangential components have distinct coefficients. For all parameters, we study the flow of the system obtained by suitable coordinate and time transformations and draw conclusions about the qualitative behavior of solutions. In each case, we examine the existence of collision–ejection, collision–escape, capture–collision, capture–escape, and oscillatory rectilinear orbits, study the motion near collision, and show that if periodic orbits exist they must be limit cycles.

Wing galaxies - A formation mechanism of the clumpy irregular galaxy Markarian 297

In order to contribute to an understanding of collision-induced starburst activities, the authors present a detailed case study on the starburst galaxy Markarian 297 (= NGC 6052 = Arp 209; hereafter Mrk 297). This galaxy is classified as a clumpy irregular galaxy (CIG) according to its morphological properties (cf. Heidmann, 1987). Two major clumps and many small clumps are observed in the entire region of Mrk 297 (Hecquet, Coupinot, and Maucherat 1987). The overall morphology of Mrk 297 is highly chaotic and thus it seems difficult to determine possible orbits of galaxy-galaxy collision. However, the authors have serendipitously found a possible orbit during a course of numerical simulations for a radial-penetration collision between galaxies. The radial-penetration collision means that an intruder penetrates a target galaxy radially passing by its nucleus. This kind of collision is known to explain a formation mechanism of ripples around disk galaxies (Wallin and Struck-Marcell 1988). Here, the authors show that the radial-penetration collision between galaxies successfully explains both overall morphological and kinematical properties of Mrk 297. The authors made two kinds of numerical simulations for Mrk 297. One is N-body (1x10(exp 4) particles) simulations in which effects of self gravity of the stellar disk are taken into account. These simulations are used to study detailed morphological feature of Mrk 297. The response of gas clouds are also investigated in order to estimate star formation rates in such collisions. The other is test-particle simulations, which are utilized to obtain a rough picture of Mrk 297 and to analyze the velocity field of Mrk 297. The techniques of the numerical simulations are the same as those in Noguchi (1988) and Noguchi and Ishibashi (1986). In the present model, an intruding galaxy with the same mass of a target galaxy moves on a rectilinear orbit which passes the center of the target.

Symmetric rectilinear periodic orbits of three bodies

In this paper the existence of families of symmetric periodic orbits in the rectilinear three body problem with the middle mass much larger than the masses on the outside is rigorously established. A number of these families are continued numerically and their stability properties as orbits of the planar general problem of three bodies are studied.

The rotation number of bounded orbits in a central field

After reviewing known results on bounded motions in a central field and on the angle between two consecutive apocenters, we investigate the limit of the orbits as the pericenter tends to the origin. A ‘deflection’ of the almost rectilinear orbits is found which depends on the shape of the potential at the origin. We then study the rotation numbers of all, bounded motions in a field homogeneous inr. Finally on a few examples we show how to represent the set of bounded motions in an inhomogeneous field.