Non-linear stability zones around triangular equilibria in the plane circular restricted three-body problem with oblateness

Non-linear stability zones of the triangular Lagrangian points are computed numerically in the case of oblate larger primary in the plane circular restricted three-body problem. It is found that oblateness has a noticeable effect and this is identified to be related to the resonant cases and the associated curves in the mass parameter versus oblateness coefficientA ₁ parameter space.     

The motion around the libration points in the restricted three-body problem with the effect of radiation and oblateness

This paper deals with the existence of libration points and their linear stability when the more massive primary is radiating and the smaller is an oblate spheroid. Our study includes the effects of oblateness of $\bar{J}_{2i}$ (i=1,2) with respect to the smaller primary in the restricted three-body problem. Under combining the perturbed forces that were mentioned before, the collinear points remain unstable and the triangular points are stable for 0<μ<μ _c , and unstable in the range $\mu_{c} \le\mu\le\frac{1}{2}$ , where $\mu_{c} \in(0,\frac{1}{2})$ , it is also observed that for these points the range of stability will decrease. The relations for periodic orbits around five libration points with their semimajor, semiminor axes, eccentricities, the frequencies of orbits and periods are found, furthermore for the orbits around the triangular points the orientation and the coefficients of long and short periodic terms also are found in the range 0<μ<μ _c .     

Periodic orbits of the third kind in the restricted three-body problem with oblateness

Recent calculations (analytical and semi-analytical), indicating that thermal effects in the relativistic regime may enhance the stability of spherically-symmetric models, are confirmed in a new example.     

Equilibrium points and stability in the restricted three-body problem with oblateness and variable masses

The existence and stability of a test particle around the equilibrium points in the restricted three-body problem is generalized to include the effect of variations in oblateness of the first primary, small perturbations ϵ and ϵ′ given in the Coriolis and centrifugal forces α and β respectively, and radiation pressure of the second primary; in the case when the primaries vary their masses with time in accordance with the combined Meshcherskii law. For the autonomized system, we use a numerical evidence to compute the positions of the collinear points L _2κ , which exist for 0<κ<∞, where κ is a constant of a particular integral of the Gylden-Meshcherskii problem; oblateness of the first primary; radiation pressure of the second primary; the mass parameter ν and small perturbation in the centrifugal force. Real out of plane equilibrium points exist only for κ>1, provided the abscissae x < \fracn(k-1)b\xi<\frac{\nu(\kappa-1)}{\beta}. In the case of the triangular points, it is seen that these points exist for ϵ′<κ<∞ and are affected by the oblateness term, radiation pressure and the mass parameter. The linear stability of these equilibrium points is examined. It is seen that the collinear points L _2κ are stable for very small κ and the involved parameters, while the out of plane equilibrium points are unstable. The conditional stability of the triangular points depends on all the system parameters. Further, it is seen in the case of the triangular points, that the stabilizing or destabilizing behavior of the oblateness coefficient is controlled by κ, while those of the small perturbations depends on κ and whether these perturbations are positive or negative. However, the destabilizing behavior of the radiation pressure remains unaltered but grows weak or strong with increase or decrease in κ. This study reveals that oblateness coefficient can exhibit a stabilizing tendency in a certain range of κ, as against the findings of the RTBP with constant masses. Interestingly, in the region of stable motion, these parameters are void for k = \frac43\kappa=\frac{4}{3}. The decrease, increase or non existence in the region of stability of the triangular points depends on κ, oblateness of the first primary, small perturbations and the radiation pressure of the second body, as it is seen that the increasing region of stability becomes decreasing, while the decreasing region becomes increasing due to the inclusion of oblateness of the first primary.     

Halo orbit transfer trajectory design using invariant manifold in the Sun-Earth system accounting radiation pressure and oblateness

In this paper, we study the invariant manifold and its application in transfer trajectory problem from a low Earth parking orbit to the Sun-Earth \(L_{1}\) and \(L_{2}\)-halo orbits with the inclusion of radiation pressure and oblateness. Invariant manifold of the halo orbit provides a natural entrance to travel the spacecraft in the solar system along some specific paths due to its strong hyperbolic character. In this regard, the halo orbits near both collinear Lagrangian points are computed first. The manifold’s approximation near the nominal halo orbit is computed using the eigenvectors of the monodromy matrix. The obtained local approximation provides globalization of the manifold by applying backward time propagation to the governing equations of motion. The desired transfer trajectory well suited for the transfer is explored by looking at a possible intersection between the Earth’s parking orbit of the spacecraft and the manifold.     

The effect of photogravitational force and oblateness in the perturbed restricted three-body problem

The model of restricted three-body is generalized to include the effects of the oblateness, the radiation pressure and fictitious forces. The positions of libration points, their stability, the critical mass ratio and periodic orbits emanating from these points are analyzed under the influence of these effects. The results obtained are more generalized. In addition the locations of the out of plane equilibrium points are studied. We also observe that there is no explicit effect for the perturbation of Coriolis force on the positions of the out of plane equilibrium points. It is worth mentioning that this model can be degraded into 128 special cases.     

Parameter values for stable low-inclination periodic motion in the restricted three-body problem with oblateness

The known intervals of possible stability, on the mgr-axis, of basicfamilies of 3D periodic orbits in the restricted three-body problem areextended into -A₁ regions for oblate larger primary, A ₁ beingthe oblateness coefficient. Eight regions, corresponding to the basicstable bifurcation orbits l1v, l1v, l2v, l3v, m1v, m1v,m2v, i1v are determined and related branching 3D periodic orbits arecomputed systematically and tested for stability. The regions for l1v,m1v and m2v survive the test emerging as the regions allowing thesimplest types of stable low inclination 3D motion. For l1v, l2v,l3v, m1v and m2v oblateness seems to have a stabilising effect,while stability of i1v survives only for a very small range of A ₁values.     

The circular restricted three-body problem in curvilinear coordinates

We derive the exact equations of motion for the circular restricted three-body problem in cylindrical curvilinear coordinates together with a number of useful analytical relations linking curvilinear coordinates and classical orbital elements. The equations of motion can be seen as a generalization of Hill’s problem after including all neglected nonlinear terms. As an application of the method, we obtain a new expression for the averaged third-body disturbing function including eccentricity and inclination terms. We employ the latter to study the dynamics of the guiding center for the problem of circular coorbital motion providing an extension of some results in the literature.     

On the restricted circular conservative three-body problem with variable masses

We consider the restricted circular three-body problem in which the main bodies have variable masses but the sum of their masses always remains constant. For this problem, we have obtained the possible regions of motions of the small body and the previously unknown surfaces of minimum energy that bound them using the Jacobi quasi-integral. For constant masses, these surfaces transform into the well-known surfaces of zero velocity. We consider the applications of our results to close binary star systems with conservative mass transfer.     

On the modified circular restricted three-body problem with variable mass

In the present work, we shall show an exhaustive numerical analysis of the dynamical behavior of the test particle in the modified restricted three-body problem with variable mass in which the extra effect of the three-body interaction is also taken into account. This additional force ingredient appears in the potential of the classical problem as a new extra term. As the main results, we determine the motion for test particle, under the effect of three-body interaction, which varies its mass according to Jeans’ law (Jeans (1928)). The number and existence of the libration points along with their stability have been investigated as the function of value of the parameters which occur due to variable mass of the test particle. Moreover, the regions of the possible motion are also unveiled where the test particle is free to move. Furthermore, the multivariate version of the Newton-Raphson (NR) iterative scheme is used to determine the outcomes of the used parameters on the topology of the basins of convergence (BoC) linked to the libration points. The numerical analysis shows that the topology of the basins of convergence linked with the libration points is highly influenced by the used parameters. Moreover, we perform a systematic analysis to unveil how the regions of convergence are related with the number of required iterations and also with the corresponding probability distributions.     

Sitnikov five-body problem with combined effects of radiation pressure and oblateness

We study the motion of negligible mass in the frame work of Sitnikov five-body problem where four equal oblate spheroids known as primaries symmetrical in all respect are placed at the vertices of square. These primaries are also considered as source of radiations moving in a circular orbit around their common center of mass. The fifth negligible mass performs oscillations along z-axis which is perpendicular to the orbital plane of motion of the primaries and passes through the center of mass of the primaries. Under the combined effects of radiation pressure and oblateness, we have developed the series solution by the Lindstedt-Poincare technique and established averaged Hamiltonians by applying the Van der Pol transformation and averaging technique of Guckenheimer and Holmes (1983). The orbits such as regular, periodic, quasi-periodic, chaotic, or stochastic have been examined with the help of Poincare surfaces of section.     

Stability of generalized elliptic restricted four body problem with radiation and oblateness effects

This paper studies the existence and stability of non-collinear equilibrium points in the elliptic restricted four body problem with bigger primary as a source of radiation and other two primaries having equal masses as oblate spheroid. In the elliptic restricted four body problem, three of the bodies are moving in elliptical orbit around their common centre of mass fixed at the origin of the coordinate system, while the fourth one is infinitesimal. Three pairs of non-collinear points are obtained symmetric with respect to x-axis. We found the equilibrium points are stable in linear sense. We also investigate the pulsating zero velocity surfaces and basin of attraction for varying value of oblateness coefficient and radiation pressure parameter.     

Second-species solutions in the circular and elliptic restricted three-body problem

An analytical proof of the existence of some kinds of periodic orbits of second species of Poincaré, both in the Circular and Elliptic Restricted three-body problem, is given for small values of the mass parameter. The proof uses the asymptotic approximations for the solutions and the matching theory developed by Breakwell and Perko. In the paper their results are extended to the Elliptic problem and applied to prove the existence of second-species solutions generated by rectilinear ellipses in the Circular problem and nearly-rectilinear ones in the Elliptic case.     

Some results on the global dynamics of the regularized restricted three-body problem with dissipation

We perform an analysis of the dynamics of the circular, restricted, planar three-body problem under the effect of different kinds of dissipation (linear, Stokes and Poynting–Robertson drags). Since the problem is singular, we implement a regularization technique in the style of Levi–Civita. The effect of the dissipation is often to decrease the semi-major axis; as a consequence the minor body collides with one of the primaries. In general, it is quite difficult to find non-collision orbits using random initial conditions. However, by means of the computation of the Fast Lyapunov Indicators (FLI), we obtain a global view of the dynamics. Precisely, we detect the regions of the phase space potentially belonging to basins of attraction. This investigation provides information on the different regions of the phase space, showing both collision and non-collision trajectories. Moreover, we find periodic orbit attractors for the case of linear and Stokes drags, while in the case of the Poynting–Robertson effect no other attractors are found beside the primaries, unless a fourth body is added to counterbalance the dissipative effect.     

Minimum fuel control of the planar circular restricted three-body problem

The circular restricted three-body problem is considered to model the dynamics of an artificial body submitted to the attraction of two planets. Minimization of the fuel consumption of the spacecraft during the transfer, e.g. from the Earth to the Moon, is considered. In the light of the controllability results of Caillau and Daoud (SIAM J Control Optim, 2012), existence for this optimal control problem is discussed under simplifying assumptions. Thanks to Pontryagin maximum principle, the properties of fuel minimizing controls is detailed, revealing a bang-bang structure which is typical of L¹-minimization problems. Because of the resulting non-smoothness of the Hamiltonian two-point boundary value problem, it is difficult to use shooting methods to compute numerical solutions (even with multiple shooting, as many switchings on the control occur when low thrusts are considered). To overcome these difficulties, two homotopies are introduced: One connects the investigated problem to the minimization of the L²-norm of the control, while the other introduces an interior penalization in the form of a logarithmic barrier. The combination of shooting with these continuation procedures allows to compute fuel optimal transfers for medium or low thrusts in the Earth–Moon system from a geostationary orbit, either towards the L ₁ Lagrange point or towards a circular orbit around the Moon. To ensure local optimality of the computed trajectories, second order conditions are evaluated using conjugate point tests.     

Stability of motion of the restricted circular and charged three-body problem

Stabiliity is applied to characterize type of motion in which the moving body is confined to certain limited regions and in this sense we may say that the motion of the body in question is stable. This method has been used in the past chiefly in connection with the classical restricted problem of three bodies.In this paper we consider a dynamical system defined by the Lagrangian

The stability of the triangular libration points for the plane circular restricted three-body problem with light pressure

We study the fourth-order stability of the triangular libration points in the absence of resonance for the three-body problem when the infinitesimal mass is affected not only by gravitation but also by light pressure from both primaries. A comprehensive summary of previous results is given, with some inaccuracies being corrected. The Lie triangle method is used to obtain the fourth-order Birkhoff normal form of the Hamiltonian, and the corresponding complex transformation to pre-normal form is given explicitly. We obtain an explicit expression for the determinant required by the Arnold-Moser theorem, and show that it is a rational function of the parameters, whose numerator is a fifth-order polynomial in the mass parameter. Particular cases where this polynomial reduces to a quartic are described. Our results reduce correctly to the purely gravitational case in the appropriate limits, and extend numerical work by previous authors.     

Effect of oblateness, perturbations, radiation and varying masses on the stability of equilibrium points in the restricted three-body problem

This paper investigates the combined effect of small perturbations ε,ε′ in the Coriolis and centrifugal forces, radiation pressure q _i , and changing oblateness of the primaries A _i (t) (i=1,2) on the stability of equilibrium points in the restricted three body problem in which the primaries is a supergiant eclipsing binary system which consists of a pair of bright oblate stars having the appearance of a giant peanut in space and their masses assumed to vary with time in the absence of reactive forces. The equations of motion are derived and the equilibrium points are obtained. For the autonomized system, it is seen that there are more than a pair of the triangular points as κ→∞; κ being the arbitrary sum of the masses of the primaries. In the case of the collinear points, two additional equilibrium points exist on the line joining the primaries when simultaneously κ+ε′<0 and both primaries are oblate, i.e., 0<α _i ?1. So there are five collinear equilibrium points in this case. Two non-planar equilibrium points exist for κ>1. Hence, there are at least nine equilibrium points of the system. The stability of these points is explored analytically and numerically. It is seen that the collinear and triangular points are stable with respect to certain conditions controlled by κ while the non-planar equilibrium points are unstable.