Shengping Gong

Analysis of the potential field and equilibrium points of irregular-shaped minor celestial bodies

The equilibrium points of the gravitational potential field of minor celestial bodies, including asteroids, comets, and irregular satellites of planets, are studied. In order to understand better the orbital dynamics of massless particles moving near celestial minor bodies and their internal structure, both internal and external equilibrium points of the potential field of the body are analyzed. In this paper, the location and stability of the equilibrium points of 23 minor celestial bodies are presented. In addition, the contour plots of the gravitational effective potential of these minor bodies are used to point out the differences between them. Furthermore, stability and topological classifications of equilibrium points are discussed, which clearly illustrate the topological structure near the equilibrium points and help to have an insight into the orbital dynamics around the irregular-shaped minor celestial bodies. The results obtained here show that there is at least one equilibrium point in the potential field of a minor celestial body, and the number of equilibrium points could be one, five, seven, and nine, which are all odd integers. It is found that for some irregular-shaped celestial bodies, there are more than four equilibrium points outside the bodies while for some others there are no external equilibrium points. If a celestial body has one equilibrium point inside the body, this one is more likely linearly stable.

Coupled Control of Reflectivity Modulated Solar Sail for GeoSail Formation Flying

The use of the electrochromic device has been proposed for the attitude control of spacecraft for decades because its optical characteristics can be altered by electrical excitation. The technology was recently employed and verified successfully on Interplanetary Kite Craft Accelerated by Radiation of the Sun, the world’s first interplanetary sailcraft. In addition to its ability to control the attitude of a solar sail, the reflectivity control device can also be used directly for the orbit control of the sail. In this study, a new approach for joint active orbit and attitude control of a sail using the reflectivity modulation technology is proposed. The force and torque models of a reflectivity modulated spinning disk sail experienced by solar radiation pressure are discussed, and two reflectivity modulation modes are provided and compared in terms of control performance. One mode, which was used in the Interplanetary Kite Craft Accelerated by Radiation of the Sun mission, is investigated for the active ...

Solar Sail Formation Flying Around Displaced Solar Orbits

R ECENTLY, attention has been focused on solar sail missions, such as the new artificial Lagrange points created by solar sails to be used to provide early warning of solar plasma storms, before they reach Earth [1,2]. There are several prior references with regard to such orbits in the literature. As early as 1929, Oberth mentioned in his study that solar radiation pressure would displace a reflector in an Earth polar orbit in the anti-sun direction, so that the orbit plane did not contain the center-of-mass of the Earth [3]. Later, in 1977, Austin et al. [4] noted that propulsive thrust can be used to displace the orbit of an artificial body, but only small displacements were considered for spacecraft proximity operations, and no analysis of the problem was provided. Similarly, Nock suggested a displaced orbit above Saturn’s rings for in situ observation, however, again no analysis was given [3]. In 1981, Forward [5] considered a displaced solar sail north or south of the geostationary ring. However, because he did not use an active control, subsequent analysis has criticized thiswork and claimed that such orbits were impossible. More recently, McInnes and Simmons have done work in which large families of displaced orbits were found by considering the dynamics of a solar sail in a rotating frame [6], and the dynamics, stability, and control of different families of displaced orbits were investigated in detail [7,8]. Based on McInnes’ and Simmons’ work [6], Molostov and Shvartsburg considered a more realistic solar sail model with nonperfect reflectivity and discussed the effect of finite absorption of the sail on the displaced orbits [9,10]. However, studies of the relative motion of solar sails are rare in the literature. The original idea of formation flying around a displaced orbit considered in this note comes from the concept of combining a displaced orbit with formation flying to achieve greater resolution than a single sail for science missions. This note outlines the characteristics of the relative motion around a displaced solar orbit and proposes some possible control strategies. Because the relative distance between the sails is very small compared with the distance from the sun to the sails, the relative equation of motion is linearized in the vicinity of a displaced solar orbit. Based on the linearized equation, two types of formations, seminatural and controlled formations, are discussed. The seminatural formations are performed with only sail attitude variations, but configurations of the relative orbits strongly depend on the orbit of the leader sail. Therefore, more complex controllers are adopted to build more sophisticated formations to meet special demands on the relative orbit configurations.

Analysis of Displaced Solar Sail Orbits with Passive Control

T HEuse of solar radiation pressure wasfirst proposed by a Soviet pioneer of astronautics, Tsiolkovski, and the technology was greatly developed by NASA for a proposed comet Halley rendezvous mission in the 1970s [1,2]. Recently, many space applications of solar sails are proposed because solar sails enable some special missions which would be impossible for any conventional space propulsion. Such missions include displaced solar orbits, geocentric halo orbits, Mercury sun-synchronous polar orbit, artificial Lagrange points, and so on. Leipold and Wagner investigated the Mercury sun-synchronous polar orbit using solar sail propulsion to explore the inner solar system [3]. West investigated the new artificial Lagrange points created by solar sails to provide early warning of solar plasma storms before they reach the Earth [4]. McInnes and Simmons have done much work on the dynamics and control of solar sails on different exotic trajectories [5,6]. The stability of solar sails on displaced solar orbitswith passive control is investigated in [7], and the results show that the sails are stable if the sail pitch angle is fixed with respect to a rotating frame. The passive stability can be realized by designing the configuration of the sail, which is investigated in [8]. Passive control is a good option for the solar sail because its large and complex structure may introduce some difficulties for active control. In this Note, the global stability of the solar sail with passive control is investigated by considering the dynamics in an inertial frame. It is found that the sail is stable with any initial values, and the sail will oscillate in the vicinity of a nominal orbit that is uniquely determined by the angular momentum of the sail. The amplitudes of the oscillations are determined by the initial values of the radius and angular velocity.

Solar Sail Body-Fixed Hovering over Elongated Asteroids

Solar sail spacecraft are proposed to accomplish body-fixed hovering missions over elongated asteroids. The body-fixed hovering flight is to maintain a fixed position relative to the surface of the spinning asteroid. A solar sail without fuel consumption can greatly expand the range of hovering locations in a variable lightness number for an extended period. A rotating mass dipole is used to produce the gravitational field created by an elongated asteroid. Dynamic equations are obtained for the approximate model in terms of the specified hovering conditions. Feasible hovering regions over elongated asteroids are presented and analyzed via numerical simulations. A parametric study is made to investigate the influence of solar latitude angles and the lightness number on the feasible hovering region. The hovering orbits around the realistic asteroid 951 Gaspra are performed to evaluate the effectiveness of the method in this paper.

Solar Sail Three-Body Transfer Trajectory Design

This paper discusses design methods of solar sail transfer trajectories in Hill’s restricted three-body problem. Basedonthepremisethatthesailattitudeiskept fixedwithrespecttothesunlight,theconceptsofthegeneralenergy constant and solar sail invariant manifolds are proposed. The invariant manifolds are used to design transfer trajectories fromEarthorbits toequilibriumpointsandtrajectories betweenequilibriumpoints.Firstly, thegeneral energy constant is employed to evaluate the energy requirement analytically for transfer trajectories from Earth orbits to equilibrium points. It is compared with a direct transfer using a rocket propellant as power source, and eventually the conclusion is reached that a velocity increment of hundreds of meters per second could be saved. For no-impulse transfer trajectories, active control and passive flying stage are incorporated to achieve transfer trajectories. The design problem is converted into a parameter optimization problem, and different objective functionsare optimizedforcomparisons andanalysis.Theresultsofcomparisonsshowthatthetotal transfertimeis insensitive to different objective functions. Finally, using the symmetry of the dynamical system, the invariant manifolds are patched to develop symmetrical transfer trajectories between symmetrical equilibrium points. Simulations are given for each case to validate the design method.

Solar sail time-optimal interplanetary transfer trajectory design ∗

The fuel consumptionassociated with some interplanetarytransfer trajecto- ries using chemical propulsionis not affordable.A solar sail is a method of propulsion that doesnotconsumefuel. Transfertime is oneof the mostpressing problemsof solar sail transfer trajectory design. This paper investigates the time-optimal interplanetary transfer trajectories to a circular orbit of given inclination and radius. The optimal control law is derived from the principle of maximization. An indirect method is used to solve the optimal control problem by selecting values for the initial adjoint vari- ables, which are normalized within a unit sphere. The conditions for the existence of the time-optimal transfer are dependent on the lightness number of the sail and the inclination and radius of the target orbit. A numerical method is used to obtain the boundary values for the time-optimal transfer trajectories. For the cases where no time-optimal transfer trajectories exist, first-order necessary conditions of the optimal control are proposed to obtain feasible solutions. The results show that the transfer time decreases as the minimum distance from the Sun decreases during the transfer duration. For a solar sail with a small lightness number, the transfer time may be evaluated analytically for a three-phase transfer trajectory. The analytical results are comparedwith previousresults and the associated numericalresults. The transfer time of the numerical result here is smaller than the transfer time from previous results and is larger than the analytical result.

New applications of the H-reversal trajectory using solar sails

Advanced solar sailing has been an increasingly attractive propulsion system for highly non-Keplerian orbits. Three new applications of the orbital angular momentum reversal (H-reversal) trajectories using solar sails are presented: space observation, heliocentric orbit transfer and collision orbits with asteroids. A theoretical proof for the existence of double H-reversal trajectories (referred to as ‘H2RTs’) is given, and the characteristics of the H2RTs are introduced before a discussion of the mission applications. A new family of H2RTs was obtained using a 3D dynamic model of the two-body frame. In a time-optimal control model, the minimum period H2RTs both inside and outside the ecliptic plane were examined using an ideal solar sail. Due to the quasi-heliostationary property at its two symmetrical aphelia, the H2RTs were deemed suitable for space observation. For the second application, the heliocentric transfer orbit was able to function as the time-optimal H-reversal trajectory, since its perihelion velocity is a circular or elliptic velocity. Such a transfer orbit can place the sailcraft into a clockwise orbit in the ecliptic plane, with a high inclination or displacement above or below the Sun. The third application of the H-reversal trajectory was simulated impacting an asteroid passing near Earth in a head-on collision. The collision point can be designed through selecting different perihelia or different launch windows. Sample orbits of each application were presented through numerical simulation. The results can serve as a reference for theoretical research and engineering design.

Trajectory optimization and applications using high performance solar sails

The high performance solar sail can enable fast missions to the outer solar system and produce exotic non-Keplerian orbits. As there is no fuel consumption, mission trajectories for solar sail spacecraft are typically optimized with respect to flight time. Several investigations focused on interstellar probe missions have been made, including optimal methods and new objective functions. Two modes of interstellar mission trajectories, namely “direct flyby” and “angular momentum reversal trajectory”, are compared and discussed. As a foundation, a 3D non-dimensional dynamic model for an ideal plane solar sail is introduced as well as an optimal control framework. A newly found periodic double angular momentum reversal trajectory is presented, and some properties and potential applications of this kind of inverse orbits are illustrated. The method how to achieve the minimum periodic inverse orbit is also briefly elucidated.

Bifurcation of equilibrium points in the potential field of asteroid 101955 Bennu

The stability and topological structure of equilibrium points in the potential field of the asteroid 101955 Bennu have been investigated with a variable density and rotation period. A dimensionless quantity is introduced for the nondimensionalization of the equations of motion, and this quantity can indicate the effect of both the rotation period and bulk density of the asteroid. Using the polyhedral model of the asteroid Bennu, the number and position of the equilibrium points are calculated and illustrated by a contour plot of the gravitational effective potential field. The topological structure and the stability of the equilibrium points are also investigated using the linearized method. The results show that there are nine equilibrium points in the potential field of the asteroid Bennu, eight in the exterior of the body and one in the interior of the body. Moreover, bifurcation will occur with a variation of the density and rotation period. Different equilibrium points will encounter each other and mix together. Thus, the number of equilibrium points will change. The stability and topological structure of the equilibrium points will also change because of the variation of the density and rotation period of the asteroid. When considering the error of the density of Bennu, the range of the dimensionless quantity covers the critical values that will lead to bifurcation. This means that the stability of the equilibrium points is uncertain, making the dynamical environment of Bennu much more complicated. These bifurcations can help better understand the dynamic environment of an irregular-shaped asteroid.