Coplanar Circular Orbits Research Articles

Warm start for optimal transfer between close circular orbits with first generation E-sail

The Electric Solar Wind Sail (E-sail) is a propellantless propulsion system for deep space navigation that exploits the dynamic pressure of the solar wind to generate thrust using a web of long, conducting tethers. The heliocentric trajectory of an E-sail-based spacecraft in a classic transfer between two Keplerian orbits is usually analyzed, in an optimal framework, by minimizing the total flight time. This paper discusses an analytical, approximate procedure for solving the two-point boundary value problem associated with the trajectory optimization process in a heliocentric, two-dimensional scenario involving a first-generation E-sail. The procedure is applied in a typical transfer between two coplanar circular orbits whose (assigned) radii are sufficiently close to each other. Paralleling the approach proposed in the recent literature, the mathematical model presented in this paper discusses a set of analytical equations that give an accurate approximation of both the minimum flight time and the (unknowns) initial adjoint variables. The paper also analyzes a set of mission applications, which involve interplanetary transfers to some near-Earth asteroids whose orbits have a very small value of both eccentricity and inclination.

JWST/NIRCam Imaging of Young Stellar Objects. II. Deep Constraints on Giant Planets and a Planet Candidate Outside of the Spiral Disk Around SAO 206462

We present JWST/NIRCam F187N, F200W, F405N, and F410M direct imaging data of the disk surrounding SAO 206462. Previous images show a very structured disk, with a pair of spiral arms thought to be launched by one or more external perturbers. The spiral features are visible in three of the four filters, with the nondetection in F410M due to the large detector saturation radius. We detect with a signal-to-noise ratio of 4.4 a companion candidate that, if on a coplanar circular orbit, would orbit SAO 206462 at a separation of ∼300 au, 2.25σ away from the predicted separation for the driver of the eastern spiral. No other companion candidates were detected. At the location predicted by simulations of both spirals generated by a single massive companion, the NIRCam data exclude objects more massive than ∼2.2 M J assuming the BEX evolutionary models. In terms of temperatures, the data are sensitive to objects with T eff ∼ 650–850 K, when assuming planets emit like blackbodies (R p between 1 and 3R J). From these results, we conclude that if the spirals are driven by gas giants, these must be either cold or embedded in circumplanetary material. In addition, the NIRCam data provide tight constraints on ongoing accretion processes. In the low extinction scenario we are sensitive to mass accretion rates of the order Ṁ∼10−9MJ yr−1. Thanks to the longer wavelengths used to search for emission lines, we reach unprecedented sensitivities to processes with Ṁ∼10−7MJ yr−1 even toward highly extincted environments (A V ≈ 50 mag).

Statistical Study of Uncontrolled Geostationary Satellites near an Unstable Equilibrium Point

The growth of the population of space debris in the geostationary ring and the resulting threat to active satellites require insight into the dynamics of uncontrolled objects in the region. A Monte Carlo simulation analyzed the sensitivity to initial conditions of the long-term evolution of geostationary spacecraft near an unstable point of the geopotential, where irregular behavior (e.g., transitions between long libration and continuous circulation) occurs. A statistical analysis unveiled sudden transitions from order to disorder, interspersed with intervals of smooth evolution. There is a periodicity of approximately half a century in the episodes of disorder, suggesting a connection with the precession of the orbital plane, due to Earth’s oblateness and lunisolar perturbations. The third-degree harmonics of the geopotential also play a vital role. They introduce an asymmetry between the unstable equilibrium points, enabling the long libration mode. The unpredictability occurs just in a small fraction of the precession cycle, when the inclination is close to zero. A simplified model including only gravity harmonics up to degree 3 and the Earth and moon in circular coplanar orbits is capable of reproducing most features of the high-fidelity simulation.

Solar Sail Orbit Raising with Electro-Optically Controlled Diffractive Film

The aim of this paper is to analyze the transfer performance of a spacecraft whose primary propulsion system is a diffractive solar sail with active, switchable panels. The spacecraft uses a propellantless thruster that converts the solar radiation pressure into propulsive acceleration by taking advantage of the diffractive property of an electro-optically controlled (binary) metamaterial. The proposed analysis considers a heliocentric mission scenario where the spacecraft is required to perform a two-dimensional transfer between two concentric and coplanar circular orbits. The sail attitude is assumed to be Sun-facing, that is, with its sail nominal plane perpendicular to the incoming sunlight. This is possible since, unlike a more conventional solar sail concept that uses metalized highly reflective thin films to reflect the photons, a diffractive sail is theoretically able to generate a component of the thrust vector along the sail nominal plane also in a Sun-facing configuration. The electro-optically controlled sail film is used to change the in-plane component of the thrust vector to accomplish the transfer by minimizing the total flight time without changing the sail attitude with respect to an orbital reference frame. This work extends the mathematical model recently proposed by the authors by including the potential offered by an active control of the diffractive sail film. The paper also thoroughly analyzes the diffractive sail-based spacecraft performance in a set of classical circle-to-circle heliocentric trajectories that model transfers from Earth to Mars, Venus and Jupiter.

Motion and Radiation for Charged Particles Revolving Round Centre of Force of Attraction

Nuclear and non-nuclear models of hydrogen atom are proposed, each with N equal to 42 or 43 orbits of revolution. Aberration of electric field is invoked to describe radiation from nuclear model with particle of electronic charge -e and mass nm of electronic mass m, revolving in nth orbit, with angular momentum nL, at radius nr1 from a nucleus of charge +Ne and mass (Nm/2)(1 + N), where n = 1, 2, 3...N, and r1 is radius of first orbit. The revolving particles have total charge -Ne and same total mass as the nucleus. The non-nuclear model has two particles of same mass nm and charges +e and -e, revolving diametrically in nth orbit, at radius nr1 from a centre, with particles of total mass Nm(1 + N). Distinct masses nm gives discrete coplanar circular orbits. A particle dislodged from a circular orbit, revolves in an elliptic path with emission of radiation, at the frequency of revolution, in fine structure, before reverting to a stable circular orbit. The nuclear atom is identified with solid or liquid state of hydrogen and non-nuclear atom with gaseous state

Meta-reinforcement learning for adaptive spacecraft guidance during finite-thrust rendezvous missions

In this paper, a meta-reinforcement learning approach is investigated to design an adaptive guidance algorithm capable of carrying out multiple rendezvous space missions. Specifically, both a standard fully-connected network and a recurrent neural network are trained by proximal policy optimization on a wide distribution of finite-thrust rendezvous transfers between circular co-planar orbits. The recurrent network is also provided with the control and reward at the previous simulation step, thus allowing it to build, thanks to its history-dependent state, an internal representation of the considered task distribution. The ultimate goal is to generate a model which could adapt to unseen tasks and produce a nearly-optimal guidance law along any transfer leg of a multi-target mission. As a first step towards the solution of a complete multi-target problem, a sensitivity analysis on the single rendezvous leg is carried out in this paper, by varying the radius either of the initial or the final orbit, the transfer time, and the initial phasing between the chaser and the target. Numerical results show that the recurrent-network-based meta-reinforcement learning approach is able to better reconstruct the optimal control in almost all the analyzed scenarios, and, at the same time, to meet, with greater accuracy, the terminal rendezvous condition, even when considering problem instances that fall outside the original training domain.

Nonlinear stability analysis of a collinear libration point in the planar circular restricted photogravitational three-body problem

We investigate the problem on stability of collinear libration point in the photogravitational three-body problem. That is, we study the stability of stationary motion of an infinitesimal small body (particle) in gravitational field of two bodies (primaries) acting on the particle with radiation pressure repulsive forces. The primaries move in circular coplanar orbits about their barycenter and the particle lies in the plane of primaries motion. By using the method of normal forms and applying theorems of the Kolmogorov-Arnold-Moser theory we perform nonlinear stability study of collinear point, which located on the line connecting the primaries. Rigorous conclusions on stability in the sense of Lyapunov have been obtained both for non-resonant case and for cases of third and fourth order resonances. These results can be interesting for spaceflight and astronomical applications such as studies of dynamics of artificial satellites and observations of cosmic dust.

Electron Mass Predicted From Substructure Stability in Electrodynamical Model

Modern physics has characterized spacetime, the interactions between particles, but not the nature of the particles themselves. Previous models of the electron have not specified its substance nor justified its cohesion. Here we present a relativistic electrodynamical model of the electron at rest, founded on natural interpretations of observables. Essentially intertwined positively and negatively charged subparticles revolve at light velocity in coplanar circular orbits, forming some coherent envelopeand nucleus, possibly responsible for its wavelike and corpuscular behaviors, respectively. We show that the model can provide interpretations of fundamental constants, satisfy the Virial theorem, and exhibit cohesion and stability without invoking Poincare stresses. Remarkably, the stability condition allows predicting electron mass, regarded as being a manifestation of its total (kinetic and potential) electromagnetic cohesion energy, and muon mass, directly from the substructure. Our study illustrates the possibility of constructing causal and objectively realist models of particles beneath the Compton scale. Finally, wave-corpuscle duality and the relation to quantum mechanics are discussed in the light of our electron model.

Orbital Stability of Exomoons and Submoons with Applications to Kepler 1625b-I

Abstract An intriguing question in the context of dynamics arises: could a moon possess a moon itself? Such a configuration does not exist in the solar system, although this may be possible in theory. Kollmeier & Raymond determined the critical size of a satellite necessary to host a long-lived subsatellite, or submoon. However, the orbital constraints for these submoons to exist are still undetermined. Domingos et al. indicated that moons are stable out to a fraction of the host planet's Hill radius R H,p, which in turn depend on the eccentricity of its host’s orbit. Motivated by this, we simulate systems of exomoons and submoons for 105 planetary orbits, while considering many initial orbital phases to obtain the critical semimajor axis in terms of R H,p or the host satellite’s Hill radius R H,sat, respectively. We find that, assuming circular coplanar orbits, the stability limit for an exomoon is 0.40 R H,p and for a submoon is 0.33 R H,sat. Additionally, we discuss the observational feasibility of detecting these subsatellites through photometric, radial velocity, or direct imaging observations using the Neptune-sized exomoon candidate Kepler 1625b-I and identify how stability can shape the identification of future candidates.

Optimal Constant-Thrust Radius Change in Circular Orbit

The problem of minimum-time, constant-thrust orbital transfer between coplanar circular orbits is revisited using a relative motion approach in curvilinear coordinates for the dynamics and an indirect optimization method. This is a continuation of a previous work where the authors studied the orbit rephasing problem. A linearization of state and costate equations leads to approximate analytical relations characterizing the evolution of the system dynamics and the optimal thrust profile as a function of a fundamental nondimensional parameter, , which characterizes the maneuver duration. This approach allows one to study the range from very short to multirevolution maneuvers using a unified framework. The optimal solution is seen to undergo a structural change as increases, moving from a short-maneuver to a long-maneuver regime and passing through a transition zone. Approximate expressions are obtained that can predict the maneuver duration with reasonable accuracy when sufficiently far from the transition zone. The full nonlinear problem characterized by an additional nondimensional parameter is studied numerically, showing that the effect of nonlinearities can be accommodated by adopting a specific intermediate orbit as a reference. Examples of applications to Earth orbit, interplanetary missions, and orbit control around small bodies are presented.

Hohmann transfer via constrained optimization

In the first part of this paper, inspired by the geometric method of Jean-Pierre Marec, we consider the two-impulse Hohmann transfer problem between two coplanar circular orbits as a constrained nonlinear programming problem. By using the Kuhn-Tucker theorem, we analytically prove the global optimality of the Hohmann transfer. Two sets of feasible solutions are found, one of which corresponding to the Hohmann transfer is the global minimum, and the other is a local minimum. In the second part, we formulate the Hohmann transfer problem as two-point and multi-point boundary-value problems by using the calculus of variations. With the help of the Matlab solver bvp4c, two numerical examples are solved successfully, which verifies that the Hohmann transfer is indeed the solution of these boundary-value problems. Via static and dynamic constrained optimization, the solution to the orbit transfer problem proposed by W. Hohmann ninety-two years ago and its global optimality are re-discovered.

Resonance capture and dynamics of three-planet systems

We present a series of dynamical maps for fictitious three-planet systems in initially circular coplanar orbits. These maps have unveiled a rich resonant structure involving two or three planets, as well as indicating possible migration routes from secular to double resonances or pure three-planet commensurabilities. These structures are then compared to the present-day orbital architecture of observed resonant chains. In a second part of the paper, we describe N-body simulations of type-I migration. Depending on the orbital decay time-scale, we show that three-planet systems may be trapped in different combinations of independent commensurabilities: (i) double resonances, (ii) intersection between a two-planet and a first-order three-planet resonances, and (iii) simultaneous libration in two first-order three-planet resonances. These latter outcomes are found for slow migrations, while double resonances are almost always the final outcome in high-density discs. Finally, we discuss an application to the TRAPPIST-1 system. We find that, for low migration rates and planetary masses of the order of the estimated values, most three-planet sub-systems are able to reach the observed double resonances after following evolutionary routes defined by pure three-planet resonances. The final orbital configuration shows resonance offsets comparable with present-day values without the need of tidal dissipation. For the 8/5 resonance proposed to dominate the dynamics of the two inner planets, we find little evidence of its dynamical significance; instead, we propose that this relation between mean motions could be a consequence of the interaction between a pure three-planet resonance and a two-planet commensurability between planets c and d.

Concurrent rendezvous control of underactuated spacecraft

The concurrent control of spacecraft equipped with one-axis unilateral thruster and three-axis attitude actuator is considered in this paper. The proposed control law utilizes attitude control channels along with the single thrust force concurrently, for three-dimensional trajectory tracking and rendezvous with a target object. The concurrent controller also achieves orbital transfer to low Earth orbits with long range separation. To demonstrate the orbit transfer capabilities of the concurrent controller, a smooth elliptical orbit transfer trajectory for co-planar circular orbits is designed. The velocity change and energy consumption of the designed orbit transfer trajectory is observed to be equivalent to that of Hohmann transfer.

A Shape-Based Method for Continuous Low-Thrust Trajectory Design between Circular Coplanar Orbits

The shape-based method can provide suitable initial guesses for trajectory optimization, which are useful for quickly converging a more accurate trajectory. Combined with the optimal control theory, an optimized shape-based method using the finite Fourier series is proposed in this paper. Taking the flight time-fixed case and the time-free case into account, respectively, the optimized shape-based method, which considers the first-order optimal necessary conditions, can guarantee that not only an orbit designed during the preliminary phase is optimal, but also the thrust direction is not constrained to be tangential. Besides, the traditional shape-based method using the finite Fourier series, in which the thrust direction is constrained to be tangential, is developed for the time-free case in this paper. The Earth-Mars case and the LEO-GEO case are used to verify the optimized shape-based method’s feasibility for time-fixed and time-free continuous low-thrust trajectory design between circular coplanar orbits, respectively. The optimized shaped-based method can design a lower cost trajectory.

Impulsive Ground-Track Adjustment for Assigned Final Orbit

This paper provides analytical approximate solutions to the impulsive coplanar ground-track adjustment problem for an assigned final orbit (e.g., repeated ground-track orbit) whose ground track passes over a user-specified Earth site. A three-impulse method associated with bielliptical transfer and four-impulse methods based on the optimal slewing of the apse line are proposed for coplanar circular and noncoaxial elliptical orbits, respectively. All impulses are required to be tangential. A semi-analytical method is proposed to obtain the longitude difference for the assigned ground-site latitude. Then, approximate solutions are obtained by solving a quartic equation in a high-fidelity model. Numerical examples with both circular and elliptical orbits are provided to validate this approach for one-day repeated ground-track final orbits.

Variable-time-domain neighboring optimal guidance applied to space trajectories

This research describes and applies the recently introduced, general-purpose variable-time-domain neighboring optimal guidance scheme, which is capable of driving a space vehicle in the proximity of a specified nominal, optimal path. This goal is achieved by minimizing the second differential of the objective function (related to fuel consumption) along the perturbed trajectory. This minimization principle leads to deriving all the corrective maneuvers, in the context of a closed-loop guidance scheme. Several time-varying gain matrices, referring to the nominal trajectory, are defined, computed offline, and stored in the onboard computer. Original analytical developments, based on optimal control theory and adoption of a variable time domain, constitute the theoretical foundation for three relevant features that characterize the guidance algorithm used in this work: (i) a new, efficient law for the real-time update of the time of flight (usually referred to as time-to-go), (ii) a new, effective termination criterion, and (iii) a new formulation of the sweep method. As a first application, this paper considers minimum-time lunar ascent and descent paths, under the flat Moon approximation. The second application is represented by the minimum-time, continuous-thrust orbit transfer between two coplanar circular orbits. In both cases, the nominal trajectories are two-dimensional, while the corresponding perturbed paths are three-dimensional. Specifically, perturbations arising from the imperfect knowledge of the propulsive parameters and from errors in the initial conditions are included in the dynamical simulations. Extensive Monte Carlo tests are performed, and definitely prove the effectiveness and accuracy of the variable-time-domain neighboring optimal guidance algorithm.

Global optimization of fuel consumption in rendezvous scenarios by the method of interval analysis

To reduce the optimal but large Δv of the fixed-short-time two impulse Lambert rendezvous between two spacecrafts along two coplanar circular orbits, the three-impulse Lambert rendezvous optimized via the optimization algorithm-interval analysis (IA) is proposed in this paper. The purpose of optimization is to minimize the velocity increment of the fixed-short-time three-impulse Lambert rendezvous. The optimization algorithm IA is given for solving the rendezvous optimization problem with multiple uncertain variables, and strong nonlinearity and nonconvexity. Numerical examples of the time-open, coplanar-circular-orbit, multiple-revolution Lambert rendezvous with a parking time optimized via the optimization algorithm IA are firstly undertaken to validate the feasibility of the optimization algorithm IA by comparing the optimization results with those of the globally optimal Hohmann transfer. The results indicate that the globally optimal parameters of the time-open coplanar-circular-orbit multiple-revolution Lambert rendezvous can be obtained by the optimization algorithm IA, and the initial separation angle of two spacecrafts with different orbit radius can be adjusted to obtain the globally optimal and small Δv by distributing an optimal parking time. After that, for the fixed-short-time two-impulse Lambert rendezvous problem without sufficient time to adjust the separation angle by distributing a parking time like the open-time Lambert rendezvous problem, three-impulse Lambert rendezvous involving multiple optimization variables is given and the variables are optimized by the optimization algorithm IA to obtain an optimal and small Δv. Numerical simulation indicates that the optimal and small Δv of the fixed short time, three-impulse Lambert rendezvous can be obtained using the optimization algorithm IA.

Fast solar sail rendezvous mission to near Earth asteroids

The concept of fast solar sail rendezvous missions to near Earth asteroids is presented by considering the hyperbolic launch excess velocity as a design parameter. After introducing an initial constraint on the hyperbolic excess velocity, a time optimal control framework is derived and solved by using an indirect method. The coplanar circular orbit rendezvous scenario is investigated first to evaluate the variational trend of the transfer time with respect to different hyperbolic excess velocities and solar sail characteristic accelerations. The influence of the asteroid orbital inclination and eccentricity on the transfer time is studied in a parametric way. The optimal direction and magnitude of the hyperbolic excess velocity are identified via numerical simulations. The found results for coplanar circular scenarios are compared in terms of fuel consumption to the corresponding bi-impulsive transfer of the same flight time, but without using a solar sail. The fuel consumption tradeoff between the required hyperbolic excess velocity and the achievable flight time is discussed. The required total launch mass for a particular solar sail is derived in analytical form. A practical mission application is proposed to rendezvous with the asteroid 99942 Apophis by using a solar sail in combination with the provided hyperbolic excess velocity.

Tour of Jupiter Galilean moons: Winning solution of GTOC6

The paper presents the trajectory designed by the Italian joint team Politecnico di Torino & Sapienza Università di Roma (Team5), winner of the 6th edition of the Global Trajectory Optimization Competition (GTOC6). In the short time available in these competitions, Team5 resorted to basic knowledge, simple tools and a powerful indirect optimization procedure. The mission concerns a 4-year tour of the Jupiter Galilean moons. The paper explains the strategy that was preliminarily devised and eventually implemented by looking for a viable trajectory. The first phase is a capture that moves the spacecraft from the arrival hyperbola to a low-energy orbit around Jupiter. Six series of flybys follow; in each one the spacecraft orbits Jupiter in resonance with a single moon; criteria to construct efficient chains of resonant flybys are presented. Transfer legs move the spacecraft from resonance with a moon to another one; precise phasing of the relevant moons is required; mission opportunities in a 11-year launch window are found by assuming ballistic trajectories and coplanar circular orbits for the Jovian satellites. The actual trajectory is found by using an indirect technique.

Trajectory Control During an Aeroassisted Maneuver Between Coplanar Circular Orbits

This paper presents the simulation results of an aeroassisted maneuver around the Earth, between coplanar circular orbits, from a geostationary orbit to a low orbit. The simulator developed considers a reference trajectory and a trajectory perturbed by external disturbances combined with non-idealities of sensors and actuators. It is able to operate in closed loop, controlling the trajectory (drag-free control) at each instant of time using a Proportional-Integral-Derivative (PID) controller and propulsive jets. We adopted a spacecraft with a cubic body composed of two rectangular plates arranged perpendicular to the velocity vector of the vehicle. Propulsive jets are applied at the apogee of the transfer orbit in order to keep the perigee altitude and control the rate of heat transfer suffered by the vehicle during atmospheric passage. A PID controller is used to correct the deviation in the state vector and in the keplerian elements. The U.S. Standard Atmosphere is adopted as the atmospheric model. The results have shown that the aeroassisted transfer presents a smaller fuel consumption when compared to a Hohmann transfer or a bi-elliptic transfer.