Advances in Astronomy

Volume 2015, Article ID 473483, 21 pages

http://dx.doi.org/10.1155/2015/473483

## Equilibrium Points and Related Periodic Motions in the Restricted Three-Body Problem with Angular Velocity and Radiation Effects

^{1}Department of Electrical & Computer Engineering, University of Patras, 26500 Patras, Greece^{2}Department of Civil Engineering, University of Patras, 26500 Patras, Greece^{3}Department of Computer Engineering & Informatics, University of Patras, 26500 Patras, Greece

Received 24 April 2015; Accepted 2 June 2015

Academic Editor: Zdzislaw E. Musielak

Copyright © 2015 E. A. Perdios et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper deals with a modification of the restricted three-body problem in which the angular velocity variation is considered in the case where the primaries are sources of radiation. In particular, the existence and stability of its equilibrium points in the plane of motion of the primaries are studied. We find that this problem admits the well-known five planar equilibria of the classical problem with the difference that the corresponding collinear points may be stable depending on the parameters of the problem. For all planar equilibria, sufficient parametric conditions for their stability have been established which are used for the numerical determination of the stability regions in various parametric planes. Also, for certain values of the parameters of the problem for which the equilibrium points are stable, the short and long period families have been computed. To do so, semianalytical expressions have been found for the determination of appropriate initial conditions. Special attention has been given to the continuation of the long period family, in the case of the classical restricted three-body problem, where we show numerically that periodic orbits of the short period family, which are bifurcation points with the long period family, are connected through the characteristic curve of the long period family.

#### 1. Introduction

The restricted three-body problem is the most celebrated problem of Celestial Mechanics (see, among others, Szebehely [1], Marsden and Ross [2], and references therein) and due to its interdisciplinary applications it has also attracted the interest of many researchers from different fields of science (see, e.g., Prosmiti et al. [3], Vrahatis et al. [4], Sano [5]). During the last century, several modifications of this classical problem have been introduced in order to make it more relevant and applicable to certain systems of Dynamical Astronomy. In our present study, we will consider the combination of two such modifications in which the variation of the angular velocity (the Chermnykh’s problem) and the radiation effects of the primaries (the photogravitational problem) are considered.

The Chermnykh’s problem is a generalization of the Euler’s problem of two fixed gravitational centers and the restricted three-body problem, in which the third body of negligible mass moves in the orbital plane of a dumbbell which is rotating with constant angular velocity around the center of mass (Chermnykh [6]). This problem admits applications in both Dynamical Astronomy and Celestial Mechanics as well as in Chemistry (see Goździewski and Maciejewski [7] and references therein) and has been studied recently by several authors. In particular, Goździewski and Maciejewski [7] investigated the nonlinear Lyapunov stability of the triangular equilibrium points. Perdios and Ragos [8] studied asymptotic motions around the collinear equilibrium points while Papadakis [9] considered planar and three-dimensional periodic motions around them. Perdiou et al. [10] studied three-dimensional motions through bifurcations of the Sitnikov family. In recent years, certain variants of this problem have been also proposed by other authors. Specifically, Jiang and Yeh [11] investigated the equilibrium points of the Chermnykh’s problem in the case where the value of the angular velocity is determined by the gravitational potential of the belt which was considered to be a part of the potential function. For the same variant Abouelmagd et al. [12] studied the linear stability of the triangular equilibrium points as well as periodic solutions around them when the primaries are radiating oblate spheroids. Recently, Singh and Leke [13, 14] considered also the oblateness of the test particle as well as the effect of small perturbations in the Coriolis and centrifugal forces.

On the other hand, an interesting modification of the classical problem is the photogravitational restricted three-body problem in which the repulsive force of the radiation is also considered in the potential function and it was introduced for studying the specific three-body problem of Sun, planet, and a dust particle (Radzievskii [15]). This is a simplified model, since it does not take into account the perturbation force of the Poynting-Robertson effect but only the radiation pressure as a reduction factor of the magnitude of the inverse square law forces of the primaries upon the massless body. However, it may be useful for the study of motion of extremely small particles such as dust grains and interplanetary drifters (see, e.g., Kalvouridis [16], Bewick et al. [17], and references therein). In the framework of this simplified model many works have been done in the last years. For example, Simmons et al. [18] studied the existence and stability of the equilibrium points for all values of the radiation factors and the mass parameter, Goździewski et al. [19] considered the nonlinear stability of the triangular equilibrium points, Elipe and Lara [20] studied three-dimensional periodic orbits, and Perdios et al. [21] investigated the bifurcations of plane to three-dimensional periodic orbits, while Perdios [22] determined series of horizontally critical symmetric periodic orbits as well as their vertical stability. Also, Zimovshikov and Tkhai [23] studied the stability of the collinear equilibrium points in the elliptic case of the corresponding problem, Abouelmagd [24] considers the additional effects of the oblateness as well as the Coriolis and centrifugal forces on the location and stability of the equilibrium points, while Singh and Taura [25] investigated the linear stability of the triangular equilibrium points by taking into account the oblateness of the smaller primary and the gravitational potential from a circumbinary belt. Recently, Yárnoz et al. [26] examined the feasibility of manipulating asteroid material by means of solar radiation pressure, while Verrier et al. [27] studied the evolution of the halo family under the effects of radiation pressure in the Sun-Earth system.

In this paper, the equilibrium points as well as the periodic orbits around them are studied in the framework of the photogravitational Chermnykh’s restricted three-body problem with the aim to investigate the changes which may result in these basic dynamical features due to the three parameters of this model-problem (the angular velocity and the radiation factors and of the primaries). In particular, we study the existence and location of the planar equilibrium points as well as their linear stability. An interesting result is that there are combinations of the parameters of the problem for which the collinear equilibrium points may be stable. This is established by certain conditions which are determined when the triangular equilibrium points fall on the O*x*-axis and transfer their stability to the collinear points. Several stability regions are given to show this result. For all cases of stable collinear and triangular equilibrium points we compute the corresponding short and long period families. We also give emphasis on the computation of the long period family emanating from each equilateral equilibrium point, in the framework of the classical restricted three-body problem for the Sun-Jupiter system, showing numerically that the long period family constitutes a bridge connecting orbits of the short period family which are bifurcation points of these two families complementing thusly the results of Henrard [28].

Specifically, the paper is organized as follows. In Section 2, we introduce the equations of motion of the considered model-problem. In Section 3, the existence, location, and stability of planar equilibrium points are investigated while, in Section 4, periodic motions around stable collinear and triangular equilibrium points are studied. Finally, in Section 5, we summarize our results and conclude.

#### 2. Equations of Motion

We consider a barycentric, rotating, and dimensionless coordinate system O*xy* whose origin is at the center of mass of two bodies and , known as primaries, where is the mass parameter. The two bodies perform circular orbits around the center of mass with angular velocity and are always on the O*x*-axis (Figure 1). They are also considered to be radiating sources with mass reduction factors due to the radiation , , respectively (Chernikov [29]). Thus, the motion of the third body of negligible mass is influenced by the gravitational force as well as the repulsive force of light pressure. This additional force of radiation has applications in stellar dynamics, where it is a main component of the influence of a star on dust particles (Kunitsyn and Tureshbaev [30] and Das et al. [31]). Note that radiation factors may also have negative values meaning that gravity is strengthened by radiation, a case which will not be considered in our study. The equations of motion of the massless body are given by the following system (Schuerman [32], Chermnykh [6], and Perdios and Ragos [8]):where we have abbreviatedwhile is the potential function which is given by the following formula:are the distances of the third body of infinitesimal mass from the primary and secondary body, respectively. The system admits the Jacobi integral:where is the Jacobi constant.