Collinear Equilibrium Research Articles

A note on the dynamics around the Lagrange collinear points of the Earth–Moon system in a complete Solar System model

In this paper we study the dynamics of a massless particle around the L1,2 libration points of the Earth–Moon system in a full Solar System gravitational model. The study is based on the analysis of the quasi-periodic solutions around the two collinear equilibrium points. For the analysis and computation of the quasi-periodic orbits, a new iterative algorithm is introduced which is a combination of a multiple shooting method with a refined Fourier analysis of the orbits computed with the multiple shooting. Using as initial seeds for the algorithm the libration point orbits of Circular Restricted Three Body Problem, determined by Lindstedt-Poincare methods, the procedure is able to refine them in the Solar System force-field model for large time-spans, that cover most of the relevant Sun–Earth–Moon periods.

Equilibrium configurations of the tethered three-body formation system and their nonlinear dynamics

This paper considers nonlinear dynamics of tethered three-body formation system with their centre of mass staying on a circular orbit around the Earth, and applies the theory of space manifold dynamics to deal with the nonlinear dynamical behaviors of the equilibrium configurations of the system. Compared with the classical circular restricted three body system, sixteen equilibrium configurations are obtained globally from the geometry of pseudo-potential energy surface, four of which were omitted in the previous research. The periodic Lyapunov orbits and their invariant manifolds near the hyperbolic equilibria are presented, and an iteration procedure for identifying Lyapunov orbit is proposed based on the differential correction algorithm. The non-transversal intersections between invariant manifolds are addressed to generate homoclinic and heteroclinic trajectories between the Lyapunov orbits. (3,3)-and (2,1)-heteroclinic trajectories from the neighborhood of one collinear equilibrium to that of another one, and (3,6)- and (2,1)-homoclinic trajectories from and to the neighborhood of the same equilibrium, are obtained based on the Poincare mapping technique.

Motion in the generalized restricted three-body problem

This paper investigates the motion of an infinitesimal body in the generalized restricted three-body problem. It is generalized in the sense that both primaries are radiating, oblate bodies, together with the effect of gravitational potential from a belt. It derives equations of the motion, locates positions of the equilibrium points and examines their linear stability. It has been found that, in addition to the usual five equilibrium points, there appear two new collinear points L n1, L n2 due to the potential from the belt, and in the presence of all these perturbations, the equilibrium points L 1, L 3 come nearer to the primaries; while L 2, L 4, L 5, L n1 move towards the less massive primary and L n2 moves away from it. The collinear equilibrium points remain unstable, while the triangular points are stable for 0<μ<μ c and unstable for \(\mu_{c} \le\mu\le\frac{1}{2}\), where μ c is the critical mass ratio influenced by the oblateness and radiation of the primaries and potential from the belt, all of which have destabilizing tendency. A practical application of this model could be the study of the motion of a dust particle near the oblate, radiating binary stars systems surrounded by a belt.

Passive Control Feasibility of Collinear Equilibrium Points with Solar Balloons

A = solar balloon’s surface area, m E = Young’s modulus of the skin, Pa h = auxiliary variable, see Eq. (13) i, ĵ, k = unit vectors of rotating frame K = second-order tensor, see Eq. (5) k = gain ‘ = sun–planet distance (with ‘ ≜ 1 AU), astronomical unit m = mass, kg O = center of mass p = internal pressure, Pa R = solar balloon’s radius, m r = absolute position vector, r krk T = spacecraft equilibrium temperature, K T O; x; y; z = rotating reference frame u = vector, see Eq. (6) W = thermal power flux,W=m W = solar constant, 1350 W=m 2 = coefficient of absorptivity = lightness number = variation = coefficient of emissivity = angular coordinate, deg = planet’s dimensionless mass = skin’s Poisson ratio = dimensionless x coordinate = relative position vector, k k = skin’s tensile stress, Pa ~ = Stefan–Boltzmann constant = coefficient of linear expansion, K 1 ! = angular velocity, rad=s

Robe’s problem: its extension to 2+2 bodies

In the problem of 2+2 bodies in the Robe’s setup, one of the primaries of mass \(m^{*}_{1}\) is a rigid spherical shell filled with a homogeneous incompressible fluid of density ρ1. The second primary is a mass point m2 outside the shell. The third and the fourth bodies (of mass m3 and m4 respectively) are small solid spheres of density ρ3 and ρ4 respectively inside the shell, with the assumption that the mass and the radius of third and fourth body are infinitesimal. We assume m2 is describing a circle around \(m^{*}_{1}\). The masses m3 and m4 mutually attract each other, do not influence the motion of \(m^{*}_{1}\) and m2 but are influenced by them. We also assume masses m3 and m4 are moving in the plane of motion of mass m2. In the paper, the equations of motion, equilibrium solutions, linear stability of m3 and m4 are analyzed. There are four collinear equilibrium solutions for the given system. The collinear equilibrium solutions are unstable for all values of the mass parameters μ,μ3,μ4. There exist an infinite number of non collinear equilibrium solutions each for m3 and m4, lying on circles of radii λ,λ′ respectively (if the densities of m3 and m4 are different) and the centre at the second primary. These solutions are also unstable for all values of the parameters μ,μ3,μ4, φ, φ′. Such a model may be useful to study the motion of submarines due to the attraction of earth and moon.

Collinear Equilibrium Solutions of Four-body Problem

We discuss the equilibrium solutions of four different types of collinear four-body problems having two pairs of equal masses. Two of these four-body models are symmetric about the center-of-mass while the other two are non-symmetric. We define two mass ratios as μ1 = m1/MT and μ2 = m2/MT, where m1 and m2 are the two unequal masses and MT is the total mass of the system. We discuss the existence of continuous family of equilibrium solutions for all the four types of four-body problems.

EQUILIBRIUM POINTS AND THEIR STABILITY IN THE RESTRICTED FOUR-BODY PROBLEM

We study numerically the problem of four bodies, three of which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, according to the solution of Lagrange where they are always at the vertices of an equilateral triangle, while the fourth is infinitesimal. The fourth body does not affect the motion of the three bodies (primaries). The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves as well as the positions of the equilibrium points are given. The existence and the number of collinear and noncollinear equilibrium points of the problem depend on the mass parameters of the primaries. For three unequal masses, collinear equilibrium solutions do not exist. Critical masses associated with the existence and the number of equilibrium points, are given. The stability of the relative equilibrium solutions in all cases is also studied. The regions of the basins of attraction for the equilibrium points of the present dynamical model for some values of the mass parameters are illustrated.

Location of collinear equilibrium points in the generalised photogravitational elliptic restricted three body problem

We have discussed the location of collinear equilibrium points in the generalised photogravitational restricted three body problem. The problem is generalised in the sense that both primaries are oblate spheroid. They are source of radiation as well. We have found the solution for the location of collinear point L1. We found that location of collinear point L1 is affected by eccentricity, oblateness and radiation factor terms. The same method may be applied for location of collinear points L2and L3.

Permissible three dimensional areas of motion surround planar equilibrium points in the four dipole problem

We have extended Stormer’s problem considering four magnetic dipoles in motion. In this environment, because of the enormous three dimensional instability, the permissible areas of motion of a charged particle moving under the infl uence of the electromagnetic field of the system, consist layers of small thickness similar to natural phenomena we observe in the rings of Saturn. We give the planar equilibrium points for two sets of values of the magnetic moments and for every one of the no collinear equilibrium points, we present the corresponding surfaces, which determine the three dimensional area which surrounds it, within the motion is permitted. We also give the height of the lowest and highest points of the corresponding trapping space regions according to the third dimension; it is clear that these points are zero velocity curves with zero arc length.

Nonlinear Charge Control for a Collinear Fixed-Shape Three-Craft Equilibrium

The collinear equilibrium three-craft-formation charge feedback control problem is investigated. Given a charged-equilibrium collinear configuration of spacecraft flying in deep space, a nonlinear charge feedback control algorithm is developed to stabilize the formation to the desired shape and size. The Coulomb forces are assumed to be acting along the line-of-sight directions between the bodies and thus do not provide general vehicle position controllability. A study of the charged collinear three-craft formation shows that there exists an infinite number of equilibrium charge solutions for any collinear configuration. Given a real value of one spacecraft charge, the required equilibrium charges of the remaining two vehicles are solved analytically. A Lyapunov-based nonlinear control algorithm is developed to stabilize the configuration to the equilibrium state using only the spacecraft charges as the control states. Real charge solutions are ensured by using the one-dimensional solution null-space. Numerical simulations illustrate the new charge feedback control performance. In contrast to earlier efforts, collinear configurations are stabilized even in the presence of very large initial position errors.

Equilibrium points of the restricted three-body problem with equal prolate and radiating primaries, and their stability

Paper presents a complete discussion of the existence, location and stability of the equilibrium points of the coplanar restricted three-body problem with equal prolate and radiating primaries. Depending on the values of the radiation and negative oblateness parameters, two or four additional collinear equilibrium points exist, in addition to the three Eulerian points of the classical case, making up a total of up to seven collinear points. Four of these additional points, as well as the classical central equilibrium point located at the origin, are stable for certain ranges of the parameters. Also, depending on the values of the parameters, up to six additional non-collinear equilibrium points exist, in addition to the triangular Lagrangian points of the classical case. Two of these additional points are located symmetrically above and below the origin and are stable, while the other four are located symmetrically in the four quadrants and are unstable.

Stability of Collinear Points in the Generalized Photogravitational Robes Restricted Three-Body Problem

In studying the effects of radiation and oblateness of the primaries on the stability of collinear equilibrium points in the Robes restricted three-body problem we observed the variations of the density parameter k with the mass parameter μ for constant radiation and oblateness factors on the location and stability of the collin-ear points L1, L2and L3. It is also discovered that the collinear points are unstable for k > 0 and stable for k < 0.

Keeping a Spacecraft on a Vertical Circular Collinear Lagrange Point Orbit

A propulsively managed stationkeeping strategy for a spacecraft orbiting a collinear equilibrium point in a three-body system is introduced. A nominal circular solution with nonuniform speed, which is derived from the Jacobi integral equation, employing elliptic integral theory, is used in a plane perpendicular to the line joining the two primaries. Thrust control inputs, which are found to be nonlinear functions of time, are used to negate the instability ofthe nominal orbit. Analytical relationships for control requirements are developed. Orbit parameters are analyzed and chosen so that the cost of maintaining the nominal orbit is minimized. Resulting thrust requirements are assessed for feasibility with respect to existing propulsion capabilities. Significant freedoms are found to exist for tailoring thrust demands. If proven feasible, the thrust managed orbit could find applications in communications, in situ space measurements, observation platforms, forecasting-warning systems, and loitering. An in situ magnetosphere sampling mission based on a lunar synchronized geopolar orbit is proposed and analyzed using the new results.

Periodic and quasi-periodic motions of a solar sail close to SL 1 in the Earth–Sun system

Solar sails are a proposed form of spacecraft propulsion using large membrane mirrors to propel a satellite taking advantage of the solar radiation pressure. To model the dynamics of a solar sail we have considered the Earth–Sun Restricted Three Body Problem including the Solar radiation pressure (RTBPS). This model has a 2D surface of equilibrium points parametrised by the two angles that define the sail orientation. In this paper we study the non-linear dynamics close to an equilibrium point, with special interest in the bounded motion. We focus on the region of equilibria close to SL 1, a collinear equilibrium point that lies between the Earth and the Sun when the sail is perpendicular to the Sun–sail direction. For different fixed sail orientations we find families of planar, vertical and Halo-type orbits. We have also computed the centre manifold around different equilibria and used it to describe the quasi-periodic motion around them. We also show how the geometry of the phase space varies with the sail orientation. These kind of studies can be very useful for future mission applications.

Collinear equilibrium points of Hill’s problem with radiation and oblateness and their fractal basins of attraction

We discuss the existence, location, and stability of the collinear equilibrium points of a generalized Hill problem with radiation of the primary (the Sun) and oblateness of the secondary (the planet), and present some remarkable fractals created as basins of attraction of Newton’s method applied for their computation in several cases of the parameters.

Linear stability of collinear equilibrium points around an asteroid as a two-connected-mass: Application to fast rotating Asteroid 2000EB 14

This note discusses the stability of collinear equilibrium points around a rotating system composed of two masses rigidly connected by a massless rod in the case, where the centripetal force outweighs the gravitational force. It is found that a stable region appears at L 1 when the ratio of gravitational to centripetal acceleration is less than 0.125, and that there is always no stable area at L 2 and L 3 ; the result is applied to the fast rotating Asteroid 2000EB 14.

Numerical continuation of families of homoclinic connections of periodic orbits in the RTBP

The goal of this paper is the numerical computation and continuation of homoclinic connections of the Lyapunov families of periodic orbits (p.o.) associated with the collinear equilibrium points, L1, L2 and L3, of the planar circular restricted three-body problem (RTBP). We describe the method used that allows us to follow individual families of homoclinic connections by numerical continuation of a system of (nonlinear) equations that has as unknowns the initial condition of the p.o., the linear approximation of its stable and unstable manifolds and a point in a given Poincaré section in which the unstable and stable manifolds match. For the L3 case, some comments are made on the geometry of the manifold tubes and the possibility of obtaining trajectories with prescribed itineraries.

Asymptotic orbits in the (N+1)-body ring problem

In this paper we study the asymptotic solutions of the (N+1)-body ring planar problem, N of which are finite and ν=N−1 are moving in circular orbits around their center of masses, while the Nth+1 body is infinitesimal. ν of the primaries have equal masses m and the Nth most-massive primary, with m 0=β m, is located at the origin of the system. We found the invariant unstable and stable manifolds around hyperbolic Lyapunov periodic orbits, which emanate from the collinear equilibrium points L 1 and L 2. We construct numerically, from the intersection points of the appropriate Poincare cuts, homoclinic symmetric asymptotic orbits around these Lyapunov periodic orbits. There are families of symmetric simple-periodic orbits which contain as terminal points asymptotic orbits which intersect the x-axis perpendicularly and tend asymptotically to equilibrium points of the problem spiraling into (and out of) these points. All these families, for a fixed value of the mass parameter β=2, are found and presented. The eighteen (more geometrically simple) families and the corresponding eighteen terminating homo- and heteroclinic symmetric asymptotic orbits are illustrated. The stability of these families is computed and also presented.

Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP

We consider the planar restricted three-body problem and the collinear equilibrium point L 3, as an example of a center × saddle equilibrium point in a Hamiltonian with two degrees of freedom. We explore numerically the existence of symmetric and non-symmetric homoclinic orbits to L 3, when varying the mass parameter μ. Concerning the symmetric homoclinic orbits (SHO), we study the multi-round, m-round, SHO for m ≥ 2. More precisely, given a transversal value of μ for which there is a 1-round SHO, say μ 1, we show that for any m ≥ 2, there are countable sets of values of μ, tending to μ 1, corresponding to m-round SHO. Some comments on related analytical results are also made.

The instability transition for the restricted 3-body problem

Aims. We study the onset of orbital instability for a small object, identified as a planet, that is part of a stellar binary system with properties equivalent to the restricted three body problem. Methods. Our study is based on both analytical and numerical means and makes use of a rotating (synodic) coordinate system keeping both binary stars at rest. This allows us to define a constant of motion (Jacobi’s constant), which is used to describe the permissible region of motion for the planet. We illustrate the transition to instability by depicting sets of time-dependent simulations with starplanet systems of different mass and distance ratios. Results. Our method utilizes the existence of an absolute stability limit. As the system parameters are varied, the permissible region of motion passes through the three collinear equilibrium points, which significantly changes the type of planetary orbit. Our simulations feature various illustrative examples of instability transitions. Conclusions. Our study allows us to identify systems of absolute stability, where the stability limit does not depend on the specifics or duration of time-dependent simulations. We also find evidence of a quasi-stability region, superimposed on the region of instability, where the planetary orbits show quasi-periodic behavior. The analytically deduced onset of instability is found to be consistent with the behavior of the depicted time-dependent models, although the manifestation of long-term orbital stability will require more detailed studies.