Analytical and numerical construction of vertical periodic orbits about triangular libration points based on polynomial expansion relations among directions

In this work, the single-mode motions around the collinear and triangular libration points in the circular restricted three-body problem are studied. To describe these motions, we adopt an invariant manifold approach, which states that a suitable pair of independent variables are taken as modal coordinates and the remaining state variables are expressed as polynomial series of them. Based on the invariant manifold approach, the general procedure on constructing polynomial expansions up to a certain order is outlined. Taking the Earth–Moon system as the example dynamical model, we construct the polynomial expansions up to the tenth order for the single-mode motions around collinear libration points, and up to order eight and six for the planar and vertical-periodic motions around triangular libration point, respectively. The application of the polynomial expansions constructed lies in that they can be used to determine the initial states for the single-mode motions around equilibrium points. To check the validity, the accuracy of initial states determined by the polynomial expansions is evaluated.

The construction of periodic orbits close to triangular libration points for the three-body problem in a resistive medium

Zero, minimum and maximum relative radial acceleration for planar formation flight dynamics near triangular libration points in the Earth–Moon system

Formation flying missions to the libration points are becoming common avenues of research. While nearly all of the literature explores the dynamics, navigation, and control of formations near the Sun -Earth/Moon L2, little attention has been paid to the triangular libration points. This paper explores the dynamics of relative motion near the Earth -Moon L4 point within the circular restricted three -body problem . T his research is intended to provide a baseline for and serve as a precursor to more detailed investigation of formation design, control, and navigation at the triangular libration points.

Zero drift regions and control strategies to keep satellite in formation around triangular libration point in the restricted Sun–Earth–Moon scenario

In the real solar system, due to various perturbations, the triangular libration points of the earth-moon system are unstable. However, there are quasi-periodic orbits around them. These orbits show mild instability. Due to the instability, orbit control is necessary for the spacecrafts around these points. In the paper, solar sails were taken to fulfill the control. Numerical simulations were made in the earth-moon system. The results showed that taking the surface of the solar sail as the control parameter can achieve better results. Triangular Libration points in the earth-moon system have fixed configurations with the earth and the moon. This geometrical property is ideal for some space missions. For example, relay stations can be put around them. Spacecrafts can also be sent to these points for deep space observations. Along with the observations on earth, better observation data can be obtained. Besides this, these points are ideal potential candidates for space VLBI observations. In the restricted three-body problem of the earth-moon system, the mass ratio μ is smaller than Routh’s critical value μ1=0.03852…. They are linearly stable in the restricted three-body problem 1) . However, in the real solar system, various perturbations (mainly the gravitational perturbations from the sun) change their dynamical properties and these points are no longer stable anymore. Nevertheless, quasi-periodic orbits staying around them can be found. These orbits show mild instability. Due to their instability, orbit control is necessary for the spacecrafts moving on these orbits. Solar sails can produce long-lasting thrust and do not require any fuels. It’s an ideal choice for space missions. They were taken to fulfill the orbit control in this paper. The magnitude and direction of the thrust produced by the solar sail are coupled, and the force is always directed away from the sun. This is one shortcoming of the solar sails. Due to this property, instead of the tight control strategy, we took a loose one. This strategy needs a pre-determined nominal orbit. A quasi-periodic orbit around the triangular libration points was chosen as the nominal orbit. Two sets of different control variables were used. One set is the surface of the solar sail, and the other one is the normal direction angles of the solar sail. Numerical simulations were done in the earth-moon system. The results showed that taking the surface of the solar sail as the control variable is better. During the lifetime of the spacecraft, it may be shadowed from the sun by the earth or the moon. The solar sail is of no use when the spacecraft is in the shadow. In order to avoid the shadows, the initial epoch of the nominal orbit is carefully chosen. Since the inclination angle between the moon’s and the sun’s orbit planes is small, the z amplitude motion of the nominal orbit is chosen to be large to avoid the shadows too. For a short time mission, these steps can avoid the shadows. For a long time mission, the spacecraft will encounter the shadows anyway. At the end of the paper, a discussion of this case was made. For the shadows in the paper, cylindrical model was considered.

Periodic and quasi-periodic orbit design based on the center manifold theory

Periodic orbit is one of the most important issues among all kinds of large-scale motion types, and has been regarded as the breakthrough of three-body problem by Poincare. Specifically, to orbital dynamics around small bodies, there are at least two reasons for us to focus on the periodic motion, which is related to orbital evolution of real small body system, and is crucial for mission design of approaching the target body. Differing from equilibrium points, to search periodic orbits in the phase space is still an art to date, especially for the highly dimensional systems like Eq. 2.23. The orbital motion equation near a small body Eq. 2.22 has similar formulation with the equation of CRTBP, which suggests that it might include periodic motion of the same level of abundance as the latter. This chapter starts with a specific asteroid. Section 4.2 proposes an algorithm to search large-scale periodic orbits around irregular bodies, which is then applied to find out the periodic orbital families of the target small body. Section 4.3 surveys the stabilities of these orbits, and Sect. 4.4 further describes the topologies of different orbital families, based on which a classification method is proposed to track the topological evolution. Section 4.5 discusses the general motion forms about the periodic orbits, which discriminates different orbital patterns according to the linearized map on the section.

The present paper deals with the stationary solution of the planar restricted three body problem when both the primaries are heterogeneous oblate spheroid with three layers of different densities and sources of radiation as well. It derives equations of motion and examines the existence and the linear stability of libration points. It is seen that there are five libration points, two triangular points and three collinear ones. It is further observed that the collinear points are unstable, while the triangular points are stable for the mass parameter 0 ≤ μ < μcrit. Here μcrit (the critical mass parameter) depends upon the combined effects of radiation and that due to the three different layers in the primaries. It is further seen that long or short periodic elliptical orbits emanate from the triangular points in the aforementioned range of μ.

In this paper we have considered the restricted three body problem (R3BP) in which the more massive primary is triaxial, the less massive primary and infinitesimal body are oblate spheroids, and are encompassed by a belt of homogenous material points. Analytically and numerically, we have studied the effects of triaxiality of the more massive primary, oblateness of both the less massive primary and infinitesimal body and the gravitational potential generated by the belt on the location of the triangular libration points L 4,5 and their linear stability. L 4,5 do not form equilateral triangles with the primaries in the presence of all or any of the aforementioned perturbations. Due to triaxiality of the more massive primary and oblateness of the infinitesimal body the triangular libration points are seen to move away from the line linking the primaries, whereas they shift closer to the line owing to the oblateness of the less massive primary and the potential from the belt. The range 0 < μ < μ c of stability of the triangular points is reduced in the presence of any of the perturbations, except when considering the potential from the belt the range increases, where μ c is the critical mass ratio. The oblateness of a test particle (of infinitesimal mass) shifts the location of its libration positions away from the primaries and reduces its range of stability.

In the restricted three-body problem, motion about the triangular Lagrange libration points is stable, under certain conditions, in the sense that if a particle is “slightly” disturbed it will still remain “close” to the original libration point. In the case of the earth-moon system, the sun exerts a strong perturbing influence and the equilibrium points no longer exist. To investigate stability in this case, a very restricted four-body problem is formulated and “stable” trajectories are found. Their existence is shown to be markedly influenced by the initial configuration of the three large bodies. The stable initial state conditions are found from a first-order variational solution and later refined by numerical integration of the equations of motion. Real world trajectories are shown to exhibit qualitatively the same configuration effects.

We study the classical and quantum correspondence for a two-dimensional Sinai billiard system. By using the Stationary state expansion method and Gutzwiller's periodic orbit theory, we analyze the quantum length spectrum obtained through the Fourier transformation of the quantum density of state for the Sinai billiard system, and by comparing the peak position with the length of the classical periodic orbit we find their excellent correspondence. We observe that some quantum states are localized near some short period orbits, forming the quantum scarred states or superscarred states. In this paper we also investigate the nearest-neighbor spacing distribution of levels for both concentric and nonconcentric Sinai billiard systems, and find that the concentric Sinai billiard system is nearintegrable, and for the nonconcentric Sinai billiard system with =3/8 its nearest-neighbor spacing distribution of levels transits from nearintegrable to the Wigner distribution as the distance between the two centers increases.

The relationship between the hyperbolic automorphisms of the unit 2-torus and their quantum counterparts is studied. An exact and explicit representation of the quantum density of states in terms of the periodic orbits of the classical motion is derived. This is then used to obtain exact formulae for certain statistics of the distribution of the eigenvalues of the quantum propagator. It is shown that these statistics are strongly dependent upon the arithmetical nature of the dimensionless Planck's constant. The authors investigate their semiclassical form and, using results concerning the distribution of the long period orbits, find that they do not correspond to those of any of the universality classes previously studied. Finally, formulae are derived which relate properties of the quantum eigenfunctions to the periodic orbits.

In this research work, the effects of perturbing and oblateness on the generalised photogravitational restricted three body problem (RTBP) are examined when the primary are regarded as radiation sources and oblate spheroid. The perturbing forces were included in the equations of motion that we developed for the system. The triangular libration points were derived in terms of the perturbing parameters using linear approximations, and their influence was seen. Applying Murray's criteria, we determined that the triangular libration points remained unstable due to the nature of the characteristics equation matching to the variational equations of motion of the system generated.

Preliminary designs of low-thrust transfer trajectories are developed in the Earth-moon three-body problem with variable specific impulse engines and fixed engine power. The solution for a complete time history of the thrust magnitude and direction is initially approached as a calculus of variations problem to locally maximize the final spacecraft mass. The problem is then solved directly by sequential quadratic programming, using either single or multiple shooting. The coasting phase along the transfer exploits invariant manifolds and, when possible, considers locations along the entire manifold surface for insertion. Such an approach allows for a nearly propellant-free final coasting phase along an arc selected from a family of known trajectories that contract to the periodic libration point orbit. This investigation includes transfer trajectories from an Earth parking orbit to some sample libration point trajectories, including L 1 halo orbits, L 1 and L 2 vertical orbits, and L 2 butterfly orbits. Given the availability of variable specific impulse engines in the future, this study indicates that fuel-efficient transfer trajectories could be used in future lunar missions, such as south pole communications satellite architectures.