A new regularization of the planar problem of three bodies

A new method of simultaneously regularizing the three types of binary collisions in the planar problem of three bodies is developed: The coordinates are transformed by means of certain fourth degree polynomials, and a new independent variable is introduced, too. The proposed transformation is in each binary collision locally equivalent to Levi-Civita's transformation, whereas the singularity corresponding to a triple collision is mapped into infinity. The transformed Hamiltonian is a polynomial of degree 12 in the regularized variables.Presented before the Division of Dynamical Astronomy at the 133rd meeting of the American Astronomical Society, Tampa, Florida, December 6–9, 1970.Department of Aerospace Engineering and Engineering Mechanics.     

The formal integrals of the restricted planar problem of the three bodies

The first integrals of motion of the restricted planar circular problem of three bodies are constructed as the formal power series in r^1/2, r being the distance of a moving particle from the primary. It is shown that the coefficients of these series are trigonometric polynomials of an angular variable. Some particular solutions have been found in a closed form. The proposed method for constructing the formal integrals can be generalized to a spatial problem of three bodies.     

The formal integrals of the regularized restricted planar problem of three bodies

The algorithm for constructing the first integrals of motion of the regularized restricted planar problem of three bodies is proposed. The integrals are constructed as the formal power series in one from variables. It is shown that coefficients of these series are trigonometric polynomials of the other variable. The proposed algorithm can be realized on a computer both analytically and numerically.     

On the equilibrium solutions in the circular planar restricted three rigid bodies problem

In this paper, the restricted problem of three rigid bodies under central forces is considered, and the collinear and triangular equilibrium solutions are obtained. Finally, an application to the case of axisymmetric ellipsoids is made.     

Motion near the triangular points in the elliptic restricted problem of three bodies

In this paper the first variational equations of motion about the triangular points in the elliptic restricted problem are investigated by the perturbation theories of Hori and Deprit, which are based on Lie transforms, and by taking the mean equations used by Grebenikov as our upperturbed Hamiltonian system instead of the first variational equations in the circular restricted problem. We are able to remove the explicit dependence of transformed Hamiltonian on the true anomaly by a canonical transformation. The general solution of the equations of motion which are derived from the transformed Hamiltonian including all the constant terms of any order in eccentricity and up to the periodic terms of second order in eccentricity of the primaries is given.     

Existence of libration points in the generalised photogravitational restricted problem of three bodies

In this paper we have proved the existence of libration points for the generalised photogravitational restricted problem of three bodies. We have assumed the infinitesimal mass of the shape of an oblate spheroid and both of the finite masses to be radiating bodies and the effect of their radiation pressure on the motion of the infinitesimal mass has also been taken into account. It is seen that there is a possibility of nine libration points for small values of oblateness, three collinear, four coplanar and two triangular.     

Equilibrium points in the planar magnetic-binary problem

This article deals with the analytical investigation of the number and location of equilibrium points, in the planar magnetic-binary problem, by taking into consideration the influence of the parameters which exactly determine the model of such a system.     

A note on the equilibrium points of the restricted problem of three bodies

Using Sylvester's theorem on matrices, an elegant expression is obtained for the solutions of the restricted problem of three bodies in the neighborhood of the equilibrium points.     

Existence of libration points in the restricted problem of three bodies with radiation pressure

In this paper the photogravitational forces restricted of three bodies be considered. We have assumed the infinitesimal mass of the shape of an axisymmetric body and one the finite masses be spherical luminous body while the other be an axisymmetric body non-luminous body. It is seen that there is a possibility of nine libration points for small values of oblatenesses.     

On the stability of the lagrangian points in the spatial restricted problem of three bodies

The problem of stability of the Lagrangian equilibrium point of the circular restricted problem of three bodies is investigated in the light of Nekhoroshev-like theory. Looking for stability over a time interval of the order of the estimated age of the universe, we find a physically relevant stability region. An application of the method to the Sun-Jupiter and the Earth-Moon systems is made. Moreover, we try to compare the size of our stability region with that of the region where the Trojan asteroids are actually found; the result in such case is negative, thus leaving open the problem of the stability of these asteroids.     

The libration points of a spherical satellite in the photogravitational restricted problem of three bodies

In this paper the photogravitational circular restricted problem of three bodies is considered. We have assumed that one of the finite bodies be a spherical luminous and the other be a triaxial nonluminous body. The possibility of existence of the libration points be studied.     

Periodic motion around the triangular equilibrium points of the photogravitational restricted problem of three bodies

In this article the effect of radiation pressure on the periodic motion of small particles in the vicinity of the triangular equilibrium points of the restricted three body problem is examined. Second order parametric expansions are constructed and the families of periodic orbits are determined numerically for two sets of values of the mass and radiation parameters corresponding to the non-resonant and the resonant case. The stability of each orbit is also studied.     

Three-dimensional periodic orbits about the triangular equilibrium points of the restricted problem of three bodies

The third-order parametric expansions given by Buck in 1920 for the three-dimensional periodic solutions about the triangular equilibrium points of the restricted Problem are improved by fourthorder terms. The corresponding family of periodic orbits, which are symmetrical w.r.t. the (x, y) plane, is computed numerically for =0.00095. It is found that the family emanating from L₄ terminates at the other triangular point L₅ while it bifurcates with the family of three-dimensional periodic orbits originating at the collinear equilibrium point L₃. This family consists of stable and unstable members. A second family of nonsymmetric three-dimensional periodic orbits is found to bifurcate from the previous one. It is also determined numerically until a collision orbit is encountered with the computations.     

The problem of three bodies

The gravitational problem of three bodies is presented in the general case, without restrictions on the distances and masses of the participating bodies. Recent advances are discussed and the consequences of the Laplacean instability in stellar dynamics are described.     

Mass effects in the problem of three bodies

This is a study of the dynamical behavior of three point masses moving under their mutual gravitational attraction in a plane. The initial positions and velocities are identical for all cases studied and only the masses of the participating bodies change in the series of numerical experiments. In this way the effect of the coupling terms in the differential equations of motion are investigated. The motion in all 125 cases begins with an interplay between the three bodies, followed by temporary ejections or by an eventual escape. The total mass of the system is kept constant while the massratios change from 1 to 5. The initial velocities being zero, the total energy is negative in all cases.Approximately 74% of the cases disintegrated (i.e. two bodies formed a binary and the third body escaped) in less than 140 time units, 47% in less than 50 time units and 10% ended in escape in less than 10 time units. Considering three stars with total mass 12M , initially placed at 3, 4 and 5 parsec distances (or three galaxies with mass 2.4×10¹² M , initially placed 30, 40 and 50 kpc apart), the unit of time (approximately the crossing time) becomes 1.5×10⁷ y (3.2×10⁷ y). The average time of disintegration was found to be of the order of 10⁹ y. The average semi-major axis of the binaries left behind after disintegration was 0.7 parsec and the average value of the eccentricity was 0.76. The effect of the masses on the escapes was established and it was found that the bodynot with the smallest mass escaped in 13% of the disintegrated cases. The cases which did not disintegrate in 150 time units were analyzed in detail and the time of their eventual escape was estimated.The numerical results are tabulated regarding escape time, ejection period, total energy, escape energy, terminal velocity, semi-major axis, and eccentricity.The evolution of triple systems is followed from interplays through ejections to escapes and the orbital parameters for the separation of these classes are estimated.     

Nonlinear stability of the triangular libration points for the photo gravitational elliptic restricted problem of three bodies

In the present paper we have studied the stability of the triangular libration points for the doubly photogravitational elliptic restricted problem of three bodies under the presence of resonances as well as under their absence. Here we have found the conditions for stability.     

Stability of triangular equilibrium points in the elliptic restricted problem of three bodies with radiating and triaxial primaries

This paper studies the stability of infinitesimal motions about the triangular equilibrium points in the elliptic restricted three body problem assuming bigger primary as a source of radiation and the smaller one a triaxial rigid body. The perturbation technique developed by Bennet (Icarus 4:177, 1965b) has been used for determination of characteristic exponents. This technique is based on Floquet’s Theory for determination of characteristic exponents in the system with periodic coefficients. The results of the study are analytical and numerical expressions are simulated for the transition curves bounding the region of stability in the μ–e plane, accurate to O(e ²). The unstable region is found to be divided into three parts. The effect of radiation parameter is significant. For small values of e, the results are in favor with the numerical analysis of Danby (Astron. J. 69:166, 1964), Bennet (Icarus 4:177, 1965b), Alfriend and Rand (AIAA J. 6:1024, 1969). The effect of radiation pressure is significant than the oblateness and triaxiality of the primaries.     

Effect of peturbations on the location of equilibrium points in the restricted problem of three bodies with variable mass

The effect of small perturbations in the coriolis and the centrifugal forces on the location of equilibrium points in the restricted problems of three bodies with variable mass has been studied. It is found that the points L₄ and L₅ form nearly equilateral triangles with the primaries and the points L₁, L₂, L₃ remain collinear and lie on the line joining the primaries.     

The effect of oblateness of the bigger primary on collinear libration points in the restricted problem of three bodies

In the restricted problem of three bodies, the effect of oblateness of the bigger primary appears as an additional term in the potential. As a result, the location of libration points and the roots of the characteristic equation at these points depend not only upon the mass parameter but also on the oblateness termI of the bigger primary. Series solutions are developed in terms of andI which are used for locating the collinear libration points and for determining the mean motions and characteristic exponents at these points.The work is supported by a fellowship awarded to the second author by University Grant Commission, India.     

Evolution of a ‘Class Two’ family of periodic orbits in the general planar problem of three bodies

The elliptic restricted problem of three bodies with unit eccentricity of the primaries is used to generate a family of periodic orbits in the general problem of three bodies. The parameter of the family is the mass of one of the participating bodies. This varies from zero to a termination value. The mass ratio of the primaries of the unperturbed problem (three to five) is maintained throughout the generation of the family. In this way an asymmetry is introduced generalizing the Copenhagen elliptic problem as the generating model. All members of the family experience a close approach and a collision between the primaries during half of the period of the orbit, therefore, the family is classified as Class Two.