The masses in an isosceles solution of the three-body problem

A proof is given that the masses at the base of the triangle formed by the three particles of a three-body problem are necessarily equal. The proof uses only basic calculus.     

Relative equilibrium solutions in the four body problem

Beyond the casen=3 little was known about relative equilibrium solutions of then-body problem up to recent years. Palmore's work provides in the general case much useful information. In the casen=4 he gives the totality of solutions when the four masses are equal and studies some degeneracies. We present here a survey of solutions for arbitrary masses, discussing the manifolds of degeneracy. The ordering of restricted potentials allows a counting of the number of bifurcation sets and different invariant manifolds. An analysis of linear stability is done in the restricted and general cases. As a result, values of the masses ensuring linear stability are given.     

Low-thrust propulsion in a coplanar circular restricted four body problem

This paper formulates a circular restricted four body problem (CRFBP), where the three primaries are set in the stable Lagrangian equilateral triangle configuration and the fourth body is massless. The analysis of this autonomous coplanar CRFBP is undertaken, which identifies eight natural equilibria; four of which are close to the smaller body, two stable and two unstable, when considering the primaries to be the Sun and two smaller bodies of the Solar System. Following this, the model incorporates ‘near term’ low-thrust propulsion capabilities to generate surfaces of artificial equilibrium points close to the smaller primary, both in and out of the plane containing the celestial bodies. A stability analysis of these points is carried out and a stable subset of them is identified. Throughout the analysis the Sun-Jupiter-asteroid-spacecraft system is used, for conceivable masses of a hypothetical asteroid set at the libration point L ₄. It is shown that eight bounded orbits exist, which can be maintained with a constant thrust less than 1.5 × 10⁻⁴ N for a 1000 kg spacecraft. This illustrates that, by exploiting low-thrust technologies, it would be possible to maintain an observation point more than 66% closer to the asteroid than that of a stable natural equilibrium point. The analysis then focusses on a major Jupiter Trojan: the (624) Hektor asteroid. The thrust required to enable close asteroid observation is determined in the simplified CRFBP model. Finally, a numerical simulation of the real Sun-Jupiter-(624) Hektor-spacecraft is undertaken, which tests the validity of the stability analysis of the simplified model.     

The restricted problem of a tri-axial rigid body and two spherical bodies with variable masses

The restricted problem of a tri-axial rigid body and two spherical bodies with variable masses be considered. The general solution of the equations of motion of the tri-axial body be obtained in which the motion of the spherical bodies is determined by the classic nonsteady Gyldén-Meshcherskii problem.     

Motions close to escapes in the rhomboidal four body problem

In this work we study escape and capture orbits in the planar rhomboidal 4-body problem in a level of constant negative energy. There are only two different values of the masses here. By using numerical analysis, we show certain transversal intersections of the invariant manifolds of parabolic orbits. We then introduce Symbolic Dynamics when the mass ratio is small, and when it is close to one. In the first case the escapes or captures predominate in the direction of one of the diagonals of the rhombus, while in the second case we find solutions escaping or being captured in the direction of both possible diagonals.     

The rhomboidal symmetric four-body problem

We consider the planar symmetric four-body problem with two equal masses m ₁?=?m ₃?>?0 at positions (±x ₁(t),?0) and two equal masses m ₂?=?m ₄?>?0 at positions (0, ±x ₂(t)) at all times t, referred to as the rhomboidal symmetric four-body problem. Owing to the simplicity of the equations of motion this problem is well suited to study regularization of the binary collisions, periodic solutions, chaotic motion, as well as the four-body collision and escape manifolds. Furthermore, resonance phenomena between the two interacting rectilinear binaries play an important role.     

The existence of three-dimensional symmetric orbits in the magnetic-binary problem

We study the existence of three-dimensional symmetric orbits in a magnetic-binary system. We point out that only two kinds of such orbits exist, depending on the orientation of both magnetic momentsM _i,i=1, 2; one with respect to the plane,y=0 and one with respect to thex-axis of the rotating-coordinate system.     

The solution of a time-dependent problem in radiative transfer

The time-dependent equation of radiative transfer for a finite, plane-parallel, non-radiating, and isotropically scattering atmosphere of arbitrary stratification is solved by using the integral equation method. The medium is taken to be inhomogeneous. The Laplace transform is used in the time domain. It is seen that the obtained solutions are reducible to the corresponding ones for steady-state problems by simply changing the Laplace transform parameter to zero.     

Families of symmetric relative periodic orbits originating from the circular Euler solution in the isosceles three-body problem

We study symmetric relative periodic orbits in the isosceles three-body problem using theoretical and numerical approaches. We first prove that another family of symmetric relative periodic orbits is born from the circular Euler solution besides the elliptic Euler solutions. Previous studies also showed that there exist infinitely many families of symmetric relative periodic orbits which are born from heteroclinic connections between triple collisions as well as planar periodic orbits with binary collisions. We carry out numerical continuation analyses of symmetric relative periodic orbits, and observe abundant families of symmetric relative periodic orbits bifurcating from the two families born from the circular Euler solution. As the angular momentum tends to zero, many of the numerically observed families converge to heteroclinic connections between triple collisions or planar periodic orbits with binary collisions described in the previous results, while some of them converge to “previously unknown” periodic orbits in the planar problem.     

An analytic solution to the hypersonic, radiative blunt body problem

We present an analytic model for the thin-shell, radiative interaction between a hypersonic, plane-parallel wind and a rigid, spherical obstacle. This problem has clear applications, e.g., to the interaction of winds from young stars and dense cloudlets, and to the interaction of the wind from a binary partner with the photosphere of the second star. We also present a comparison of the analytic model with a full, axisymmetric numerical simulation. We find only a partial agreement between the numerical simulation and the analytic model, apparently as a result of the very strong 'thin-shell' instabilities of the post-bow shock flow. Our analytic model predicts the surface density, flow velocity and the energy radiated per unit area, as well as the total luminosity of the bow shock. The model can therefore be used directly for carrying out comparisons with observations of different astrophysical objects.     

Gravitational lunar capture based on bicircular model in restricted four body problem

Gravitational capture is a useful phenomenon in the design of the low energy transfer (LET) orbit for a space mission. In this paper, gravitational lunar capture based on the Sun–Earth–Moon bicircular model (BCM) in the restricted four body problem is studied. By the mechanical analysis in the space near the Moon, we first propose a new parameter \(k\) , the corrected ratio of the radial force, to investigate the influence of the radial force on the capture eccentricity in the BCM. Then, a parametric analysis is performed to detect the influences on the corrected ratio \(k\) . Considering the restriction of time-of-flight and corrected ratio, we investigate, respectively, the minimum capture eccentricity and the corrected minimum capture eccentricity. Via numerical analysis, we discover two special regions on the sphere of capture, in which the capture point possesses the global minimum capture eccentricity and corrected capture eccentricity. They denote the optimal capture regions in terms of minimizing the fuel consumption of the maneuver. According to the results obtained, some suggestions on the design of the LET orbit are given.     

The planetary masses and the orbits of the first four minor planets

Numerical tests are the basis of a study about the effects caused in the orbits of the planets (1)–(4) by possible errors in the system of planetary masses. The masses of five major and three minor planets are considered. Especially, the effects caused by (1) Ceres in the orbit of (2) Pallas since the time of discovery are found to be large enough for a determination of the mass of Ceres. A first result for this mass is (6.7±0.4)×10^–10 solar masses.     

The relativistic dynamics of a thin spherically symmetric radiating shell in the presence of a central body

We construct an idealized spherically symmetric relativistic model of an exploding object within the framework of the theory of surface layers in GR. A Vaidya solution for a radially radiating star is matched through a spherical shell of dust to a Schwarzschild solution. The (incomplete) equations for the motion of the spherical shell of dust and the radiation density of the Vaidya solution, as given by the matching conditions, are reduced to a first-order system and a general analysis of the characteristics of the motion is given. This system of differential equations is completed, adding a relation between the unknowns which represents the simplest way to avoid an unphysical singularity in the motion. The results of a numerical integration of the equations are presented in two cases which we think may have some relationship to stellar explosions. A comparative set of results for other solutions is also given, and some possible generalizations of the model are pointed out.     

The global solution of the N-body problem

The problem of finding a global solution for systems in celestial mechanics was proposed by Weierstrass during the last century. More precisely, the goal is to find a solution of the n-body problem in series expansion which is valid for all time. Sundman solved this problem for the case of n = 3 with non-zero angular momentum a long time ago. Unfortunately, it is impossible to directly generalize this beautiful theory to the case of n > 3 or to n = 3 with zero-angular momentum.A new blowing up transformation, which is a modification of McGehee's transformation, is introduced in this paper. By means of this transformation, a complete answer is given for the global solution problem in the case of n > 3 and n = 3 with zero angular momentum.The main result in this paper has appeared in Chinese in Acta Astro. Sinica. 26 (4), 313–322. In this version some mistakes have been rectified and the problems we solved are now expressed in a much clearer fashion.     

The periodic solution of a rigid body in a central newtonian field

The equations of motion of a rigid body about a fixed point in a central Newtonian field is reduced to the equation of plane motion under the action of potential and gyroscopic forces, using the isothermal coordinates on the inertia ellipsoid.The construction of periodic solutions nearby equilibrium points, by using the Liapunov theorem of holomorphic integral are obtained and the necessary and sufficient conditions for the stability of the system are given.     

The angle of escape in the three body problem

In the three body problem, an upper bound is found for the angle defined by the invariable plane and the position vector of an escaping particle.This work was partially supported by NSF GP-32116.     

The two-body problem of a pseudo-rigid body and a rigid sphere

In this paper we consider the two-body problem of a spherical pseudo-rigid body and a rigid sphere. Due to the rotational and “re-labelling” symmetries, the system is shown to possess conservation of angular momentum and circulation. We follow a reduction procedure similar to that undertaken in the study of the two-body problem of a rigid body and a sphere so that the computed reduced non-canonical Hamiltonian takes a similar form. We then consider relative equilibria and show that the notions of locally central and planar equilibria coincide. Finally, we show that Riemann’s theorem on pseudo-rigid bodies has an extension to this system for planar relative equilibria.     

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.