Orbital dynamics and equilibrium points around an asteroid with gravitational orbit–attitude coupling perturbation

The strongly perturbed dynamical environment near asteroids has been a great challenge for the mission design. Besides the non-spherical gravity, solar radiation pressure, and solar tide, the orbital motion actually suffers from another perturbation caused by the gravitational orbit–attitude coupling of the spacecraft. This gravitational orbit–attitude coupling perturbation (GOACP) has its origin in the fact that the gravity acting on a non-spherical extended body, the real case of the spacecraft, is actually different from that acting on a point mass, the approximation of the spacecraft in the orbital dynamics. We intend to take into account GOACP besides the non-spherical gravity to improve the previous close-proximity orbital dynamics. GOACP depends on the spacecraft attitude, which is assumed to be controlled ideally with respect to the asteroid in this study. Then, we focus on the orbital motion perturbed by the non-spherical gravity and GOACP with the given attitude. This new orbital model can be called the attitude-restricted orbital dynamics, where restricted means that the orbital motion is studied as a restricted problem at a given attitude. In the present paper, equilibrium points of the attitude-restricted orbital dynamics in the second degree and order gravity field of a uniformly rotating asteroid are investigated. Two kinds of equilibria are obtained: on and off the asteroid equatorial principal axis. These equilibria are different from and more diverse than those in the classical orbital dynamics without GOACP. In the case of a large spacecraft, the off-axis equilibrium points can exist at an arbitrary longitude in the equatorial plane. These results are useful for close-proximity operations, such as the asteroid body-fixed hovering.     

Orbits and manifolds near the equilibrium points around a rotating asteroid

We study the orbits and manifolds near the equilibrium points of a rotating asteroid. The linearised equations of motion relative to the equilibrium points in the gravitational field of a rotating asteroid, the characteristic equation and the stable conditions of the equilibrium points are derived and discussed. First, a new metric is presented to link the orbit and the geodesic of the smooth manifold. Then, using the eigenvalues of the characteristic equation, the equilibrium points are classified into 8 cases. A theorem is presented and proved to describe the structure of the submanifold as well as the stable and unstable behaviours of a massless test particle near the equilibrium points. The linearly stable, the non-resonant unstable, and the resonant equilibrium points are discussed. There are three families of periodic orbits and four families of quasi-periodic orbits near the linearly stable equilibrium point. For the non-resonant unstable equilibrium points, there are four relevant cases; for the periodic orbit and the quasi-periodic orbit, the structures of the submanifold and the subspace near the equilibrium points are studied for each case. For the resonant equilibrium points, the dimension of the resonant manifold is greater than 4, and we find at least one family of periodic orbits near the resonant equilibrium points. As an application of the theory developed here, we study relevant orbits for the asteroids 216 Kleopatra, 1620 Geographos, 4769 Castalia and 6489 Golevka.     

Equilibrium points and Lyapunov families in the circular restricted three-body problem with an oblate primary and a synchronous rotating dipole secondary: Application to Luhman-16 binary system

We numerically study a version of the synchronous circular restricted three-body problem, where an infinitesimal mass body is moving under the Newtonian gravitational forces of two massive bodies. The primary body is an oblate spheroid while the secondary is an elongated asteroid of a combination of two equal masses forming a rotating dipole which is synchronous to the rotation of the primaries of the classic circular restricted three-body problem. In this paper, we systematically examine the existence, positions, and linear stability of the equilibrium points for various combinations of the model's parameters. We observe that the perturbing forces have significant effects on the positions and stability of the equilibrium points as well as the regions where the motion of the particle is allowed. The allowed regions of motion as determined by the zero-velocity surface and the corresponding isoenergetic curves as well as the positions of the equilibrium points are given. Finally, we numerically study the binary system Luhman-16 by computing the positions of the equilibria and their stability as well as the allowed regions of motion of the particle. The corresponding families of periodic orbits emanating from the collinear equilibrium points are computed along with their stability properties.     

Equilibrium points and associated periodic orbits in the gravity of binary asteroid systems: (66391) 1999 KW4 as an example

The motion of a massless particle in the gravity of a binary asteroid system, referred as the restricted full three-body problem (RF3BP), is fundamental, not only for the evolution of the binary system, but also for the design of relevant space missions. In this paper, equilibrium points and associated periodic orbit families in the gravity of a binary system are investigated, with the binary (66391) 1999 KW4 as an example. The polyhedron shape model is used to describe irregular shapes and corresponding gravity fields of the primary and secondary of (66391) 1999 KW4, which is more accurate than the ellipsoid shape model in previous studies and provides a high-fidelity representation of the gravitational environment. Both of the synchronous and non-synchronous states of the binary system are considered. For the synchronous binary system, the equilibrium points and their stability are determined, and periodic orbit families emanating from each equilibrium point are generated by using the shooting (multiple shooting) method and the homotopy method, where the homotopy function connects the circular restricted three-body problem and RF3BP. In the non-synchronous binary system, trajectories of equivalent equilibrium points are calculated, and the associated periodic orbits are obtained by using the homotopy method, where the homotopy function connects the synchronous and non-synchronous systems. Although only the binary (66391) 1999 KW4 is considered, our methods will also be well applicable to other binary systems with polyhedron shape data. Our results on equilibrium points and associated periodic orbits provide general insights into the dynamical environment and orbital behaviors in proximity of small binary asteroids and enable the trajectory design and mission operations in future binary system explorations.     

Families of periodic orbits in Hill’s problem with solar radiation pressure: application to Hayabusa 2

This work studies periodic solutions applicable, as an extended phase, to the JAXA asteroid rendezvous mission Hayabusa 2 when it is close to target asteroid 1999 JU3. The motion of a spacecraft close to a small asteroid can be approximated with the equations of Hill’s problem modified to account for the strong solar radiation pressure. The identification of families of periodic solutions in such systems is just starting and the field is largely unexplored. We find several periodic orbits using a grid search, then apply numerical continuation and bifurcation theory to a subset of these to explore the changes in the orbit families when the orbital energy is varied. This analysis gives information on their stability and bifurcations. We then compare the various families on the basis of the restrictions and requirements of the specific mission considered, such as the pointing of the solar panels and instruments. We also use information about their resilience against parameter errors and their ground tracks to identify one particularly promising type of solution.     

Attitude stability of a spacecraft on a stationary orbit around an asteroid subjected to gravity gradient torque

Attitude stability of spacecraft subjected to the gravity gradient torque in a central gravity field has been one of the most fundamental problems in space engineering since the beginning of the space age. Over the last two decades, the interest in asteroid missions for scientific exploration and near-Earth object hazard mitigation is increasing. In this paper, the problem of attitude stability is generalized to a rigid spacecraft on a stationary orbit around a uniformly-rotating asteroid. This generalized problem is studied via the linearized equations of motion, in which the harmonic coefficients $C_{20}$ and $C_{22}$ of the gravity field of the asteroid are considered. The necessary conditions of stability of this conservative system are investigated in detail with respect to three important parameters of the asteroid, which include the harmonic coefficients $C_{20}$ and $C_{22}$ , as well as the ratio of the mean radius to the radius of the stationary orbit. We find that, due to the significantly non-spherical shape and the rapid rotation of the asteroid, the attitude stability domain is modified significantly in comparison with the classical stability domain predicted by the Beletskii–DeBra–Delp method on a circular orbit in a central gravity field. Especially, when the spacecraft is located on the intermediate-moment principal axis of the asteroid, the stability domain can be totally different from the classical stability domain. Our results are useful for the design of attitude control system in the future asteroid missions.     

Stability of Surface Motion on a Rotating Ellipsoid

The dynamical environment on the surface of a rotating, massive ellipsoid is studied, with applications to surface motion on an asteroid. The analysis is performed using a combination of classical dynamics and geometrical analysis. Due to the small sizes of most asteroids, their shapes tend to differ from the classical spheroids found for the planets. The tri-axial ellipsoid model provides a non-trivial approximation of the gravitational potential of an asteroid and is amenable to analytical computation. Using this model, we study some properties of motion on the surface of an asteroid. We find all the equilibrium points on the surface of a rotating ellipsoid and we show that the stability of these points is intimately tied to the conditions for a Jacobi or MacLaurin ellipsoid of equilibria. Using geometrical analysis we can define global constraints on motion as a function of shape, rotation rate, and density, we find that some asteroids should have accumulation of material at their ends, while others should have accumulation of surface material at their poles. This study has implications for motion of a rover on an asteroid, and for the distribution of natural material on asteroids, and for a spacecraft hovering over an asteroid.     

Parametric excitation induced by solar pressure torque on the roll-yaw attitude motion of a gravity-gradient stabilized spacecraft

The parametric excitation of a gravity gradient stabilized spacecraft induced by the periodic solar pressure torque is discussed. The solar pressure torque in the linearized equations of motion appears as linear terms with periodic coefficients. The attitude stability is analyzed numerically through the calculation of the Floquet multiplier. The perturbation method is also applied to identify the instability condition analytically. It is made clear that the periodic solar pressure torque can destabilize the coupled roll and yaw attitude motion of the spacecraft. It is also shown that the conditions of parametric resonance are included in the gravity gradient stability condition. Nonlinear simulations are also carried out to verify the effect of the parametric resonance. The numerical simulation using actual parameters shows that the spacecraft inevitably experiences a large amplitude attitude motion due to the periodic solar pressure torque even if the gravity gradient stability condition is satisfied.     

Instabilities and bifurcations of the families of collision periodic orbits in the restricted three-body problem

By using Birkhoff's regularizing transformation, we study the evolution of some of the infinite j-k type families of collision periodic orbits with respect to the mass ratio μ as well as their stability and dynamical structure, in the planar restricted three-body problem. The μ-C characteristic curves of these families extend to the left of the μ-C diagram, to smaller values of μ and most of them go downwards, although some of them end by spiralling around the constant point S* (μ=0.47549, C=3) of the Bozis diagram (1970). Thus we know now the continuation of the families which go through collision periodic orbits of the Sun-Jupiter and Earth-Moon systems. We found new μ-C and x-C characteristic curves. Along each μ-C characteristic curve changes of stability to instability and vice versa and successive very small stable and very large unstable segments appear. Thus we found different types of bifurcations of families of collision periodic orbits. We found cases of infinite period doubling Feigenbaum bifurcations as well as bifurcations of new families of symmetric and non-symmetric collision periodic orbits of the same period. In general, all the families of collision periodic orbits are strongly unstable. Also, we found new x-C characteristic curves of j-type classes of symmetric periodic orbits generated from collision periodic orbits, for some given values of μ. As C varies along the μ-C or the x-C spiral characteristics, which approach their focal-terminating-point, infinite loops, one inside the other, surrounding the triangular points L₄ and L₅ are formed in their orbits. So, each terminating point corresponds to a collision asymptotic symmetric periodic orbit for the case of the μ-C curve or a non-collision asymptotic symmetric periodic orbit for the case of the x-C curve, that spiral into the points L₄ and L₅, with infinite period. All these are changes in the topology of the phase space and so in the dynamical properties of the restricted three-body problem.     

Equilibria and orbits around asteroid using the polyhedral model

In the current study, we use the polyhedral model to compute the potential of the asteroid. There are five equilibrium points in the gravitational field of the asteroid 283 Emma. We concluded that the zero-velocity surfaces and the equilibrium points change with the suppositive variation of the rotational speed of the asteroid. It is found that if the rotational speed equals a half as it is in present, the number of equilibrium points is also five. However, if the rotational speed equals twice as it is in present, there are only three equilibrium points left. Four different periodic orbits are calculated using the hierarchical grid searching method. We calculated characteristic multipliers of periodic orbits to investigate the stability of these periodic orbits. The orbit near the primary's equatorial plane is more likely to be stable when the separation/ primary-radius is a large number.     

On the families of periodic orbits which bifurcate from the circular Sitnikov motions

In this paper we deal with the circular Sitnikov problem as a subsystem of the three-dimensional circular restricted three-body problem. It has a first analytical part where by using elliptic functions we give the analytical expressions for the solutions of the circular Sitnikov problem and for the period function of its family of periodic orbits. We also analyze the qualitative and quantitative behavior of the period function. In the second numerical part, we study the linear stability of the family of periodic orbits of the Sitnikov problem, and of the families of periodic orbits of the three-dimensional circular restricted three-body problem which bifurcate from them; and we follow these bifurcated families until they end in families of periodic orbits of the planar circular restricted three-body problem. We compare our results with the previous ones of other authors on this problem. Finally, the characteristic curves of some bifurcated families obtained for the mass parameter close to 1/2 are also described.     

Computation of the Liapunov Orbits in the Photogravitational RTBP with Oblateness

We study the periodic motion around the collinear equilibrium points of the restricted three-body problem when the primary is a source of radiation and the secondary is an oblate spheroid. In particular, the Liapunov families of two and three dimensional periodic orbits are computed. In order to gain the appropriate initial conditions a third-fourth order Lindstedt-Poincaré local analysis is used. The stability of these families is also computed.     

Numerical treatment of non-integrable dynamical systems

A systematic and detailed discussion of the gravitational spring-pendulum problem is given for the first time. A procedure is developed for the numerical treatment of non-integrable dynamical systems which possess certain properties in common with the gravitational problem. The technique is important because, in contrast to previous studies, it discloses completely the structure of two-dimensional periodic motion by examining the stability of the one-dimensional periodic motion. Through the parameters of this stability, points have been predicted from which the one-dimensional motion bifurcates into two-dimensional motion. Consequently, families of two-dimensional periodic solutions emanated from these points are studied. These families constitute the generators of the mesh of all the families of periodic solutions of the problem.     

Origin and continuation of 3/2, 5/2, 3/1, 4/1 and 5/1 resonant periodic orbits in the circular and elliptic restricted three-body problem

We consider a planetary system consisting of two primaries, namely a star and a giant planet, and a massless secondary, say a terrestrial planet or an asteroid, which moves under their gravitational attraction. We study the dynamics of this system in the framework of the circular and elliptic restricted three-body problem, when the motion of the giant planet describes circular and elliptic orbits, respectively. Originating from the circular family, families of symmetric periodic orbits in the 3/2, 5/2, 3/1, 4/1 and 5/1 mean-motion resonances are continued in the circular and the elliptic problems. New bifurcation points from the circular to the elliptic problem are found for each of the above resonances, and thus, new families continued from these points are herein presented. Stable segments of periodic orbits were found at high eccentricity values of the already known families considered as whole unstable previously. Moreover, new isolated (not continued from bifurcation points) families are computed in the elliptic restricted problem. The majority of the new families mainly consists of stable periodic orbits at high eccentricities. The families of the 5/1 resonance are investigated for the first time in the restricted three-body problems. We highlight the effect of stable periodic orbits on the formation of stable regions in their vicinity and unveil the boundaries of such domains in phase space by computing maps of dynamical stability. The long-term stable evolution of the terrestrial planets or asteroids is dependent on the existence of regular domains in their dynamical neighbourhood in phase space, which could host them for long-time spans. This study, besides other celestial architectures that can be efficiently modelled by the circular and elliptic restricted problems, is particularly appropriate for the discovery of terrestrial companions among the single-giant planet systems discovered so far.     

Critical cases of 3-dimensional systems

We study the evolution of families of periodic orbits of simple 3-dimensional models representing the central parts of deformed galaxies. In some cases the evolution is non-unique, i.e. if we follow a closed path in the parameter space we do not return with the same periodic orbit. This happens when the path surrounds a critical point. We found that critical points are generated at particular collisions of bifurcations in limiting cases when the 3-D system is separated into a 2-D system and an independent oscillation along the third axis. The regions of stability and instability of some families of periodic orbits change in remarkable ways near the various collisions of bifurcations and around the critical points.     

Bifurcating families around collinear libration points

The planar and the vertical Lyapunov families are two basic periodic families around the collinear libration points. The stability curves of these two families are given first, and then periodic families bifurcating from them are explored in detail. Several properties of these bifurcating families are found. This study follows a series of the authors’ publications on periodic families around the libration points in the restricted three-body problem.     

The Structure of Periodic Solutions in the Restricted Problem of the Three Bodies II

In this work we consider four families of plane periodic orbits direct around the Sun which approach Jupiter but they are sufficiently far from it so as to be considered as predominantly two body orbits of the Sun-asteroid system. We study their horizontal and vertical stabilities and we give the exact orbits of bifurcations of these families with three-dimensional families of the same multiplicity or twice the multiplicity of the above families of plane symmetric periodic orbits. Moreover, we give the first segments of the three dimensional families of symmetric periodic orbits which emanate from these plane bifurcations and we study their stability relating it with the stability of the plane bifurcations.     

On critical values concerning the evolution of the long period families

In a previous paper, we proposed another special critical value concerning the evolution of the long period family around the equilateral equilibrium points, besides the two values given by Henrard. Are there any other special critical values? After studying the stability curves of the long period family carefully, we gave a negative answer. During the study, we found an interesting family of periodic orbits which we called the homo family. We studied the evolution of this family following the increase of μ. With these findings, we were able to explain the origin of the four branches of periodic families emanating from L4 and the stability results of the equilateral equilibrium points.