International Scholarly Research Notices

International Scholarly Research Notices / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 281849 | https://doi.org/10.1155/2013/281849

Maria N. Croustalloudi, Tilemahos J. Kalvouridis, "The Restricted 2+2 Body Problem: Parametric Variation of the Equilibrium States of the Minor Bodies and Their Attracting Regions", International Scholarly Research Notices, vol. 2013, Article ID 281849, 15 pages, 2013. https://doi.org/10.1155/2013/281849

The Restricted 2+2 Body Problem: Parametric Variation of the Equilibrium States of the Minor Bodies and Their Attracting Regions

Academic Editor: D. M. Christodoulou
Received29 Nov 2012
Accepted17 Dec 2012
Published04 Apr 2013

Abstract

The restricted 2+2 body problem was stated by Whipple (1984) as a particular case of the general n + v problem described by Whipple and Szebehely (1984). In this work we reconsider the problem by studying some aspects of the dynamics of the minor bodies, such as the parametric variation of their equilibrium positions, as well as the attracting regions formed by the initial approximations used for the numerical determination of these positions. In the latter case we describe the process to form these regions, and we numerically investigate their dependence on the parameters of the system. The results in many cases show a fractal-type structure of these regions. As test problems, we use the Sun-Jupiter-binary asteroids and the Earth-Moon-dual artificial satellites systems.

1. Introduction

After many years of thorough study of the restricted three-body problem and the new issues imposed by the flight missions in the 60s, 70s, and 80s, many investigators have focused their scientific interest to new N-body models () in order to approximate real celestial systems. The restricted body problem, which can be considered as a version of the restricted three-body problem and as a particular case of the bodies problem ( major bodies and ν minor ones) where and , belongs to this category.

The original configuration consists of two big spherical, homogeneous bodies P1 and P2, called hereafter the primaries, with masses and , respectively, which rotate around their common center of mass in circular orbits with a constant angular velocity under their mutual gravitational attraction (Figure 1). In the resultant gravitational field created by the primaries, two point-like small bodies, namely, and , move while mutually attracting each other without perturbing the primaries. According to the formulation given by Whipple [1], the problem is characterized by three parameters , , and which are the reduced masses of primary and of the two minor bodies, respectively.

Among the works treating this problem, we can mention the papers of Whipple and White [2] and Milani and Nobili [3] who applied this model to study the dynamics of binary asteroids in the solar system, as well as the paper of Thanos [4] who studied the case of possible collisions between the minor bodies and proposed a regularization formula for the transformation of the equations of motion similar to the one used in the restricted three-body problem. In the 90s some improved versions of the problem have appeared in the international bibliography. El-Shaboury [5] considered that the 2+2 bodies are homogenous, axisymmetric ellipsoids so that their equatorial planes coincide with the orbital plane of the centers of mass. Michalakis and Mavraganis [6] replaced the two minor point-like masses with two triaxial rigid bodies, Kalvouridis and Mavraganis [7] studied the case where the two primaries are radiating sources, and Mavraganis and Kalvouridis [8] considered the two minor bodies as gyrostats. A little later Kalvouridis [9] assumed that the two primaries are oblate spheroids. Finally, in a completely different version, Prasad and Ishwar [10] studied the same configuration by considering that the primaries are magnetic dipoles and the minor bodies are electric dipoles.

In what follows, we shall numerically investigate some new aspects of the 2+2 body problem including the parametric variation of the equilibrium states (Section 3), as well as the formation, the structure, and the parametric dependence of the attracting regions of these equilibrium states (Section 4). We shall use two test problems; the Sun-Jupiter-binary asteroids system (case A) and an imaginary flying formation mission of a dual satellite system in the Earth-Moon’s gravitational field (case B). Our choice is based upon the fact that the physical measures of these two systems present significant differences. All along the study of the attracting regions, and for comparison reasons, we shall also use the restricted three-body problem (RTBP). The data which are used in this work and concern the aforementioned cases are shown in an appendix at the end of the text.

2. Equations of Motion

As we have mentioned before, the system consists of two primaries , , with masses , (), which revolve in circular orbits around the center of mass, and of two minor bodies , , with masses , , such as (Figure 1). The minor bodies move under the combined action of the primaries and their mutual attraction. The aim of the problem is to describe the dynamical behavior of this pair of minor bodies.

By considering a synodic coordinate system Oxyz, the xy-plane of which coincides with the orbital plane of the primaries, and by taking the axis of syzygy of the primaries as the -axis, the dimensionless equations of motion of the minor bodies and in this system (without loss of generality we may assume that the dimensionless angular velocity of the synodic system is equal to unity) are (Whipple and Szebehely [11]), where is the reduced mass of the smaller primary P2 are the reduced masses of the minor bodies . Also, is the distance between the minor bodies and , are their distances from the two primaries, where

Here we note that for the normalization process we have used the total mass of the primaries and their distance .

There is a Jacobian-type integral of motion which resembles the one of the restricted three-body problem (Szebehely [12]).

3. Equilibrium Positions, Parametric Variation, and Stability

3.1. Distribution of the Equilibrium Positions

Whipple has proved that all equilibrium positions are located on the -plane of the synodic system. He also found that there are 14 different equilibrium positions of the minor bodies which evolve in the neighborhood of the Lagrangian equilibrium points of the respective restricted three-body problem with the same value of the reduced mass . More precisely, two equilibrium solutions evolve in the neighborhood of each collinear Lagrangian point and on both sides of it. Whipple called these solutions collinear, adopting the terminology used in the restricted three-body problem. We hereafter denote them with and , . The first index denotes the particular minor body while the second one depicts the respective collinear Lagrangian point , around which the equilibria of the minor bodies evolve. The double upper index used in this section (e.g., 12) describes the relative position of the minor bodies with respect to the Lagrangian point. For example, when body is located on the left side of the Lagrangian point and body on the right one, then the upper double index is (12). In the opposite case index (21) is used. In the first case the coordinates of the two minor bodies satisfy the inequality while in the second one, they satisfy the relation . Furthermore, four equilibrium solutions exist near a triangular Lagrangian point , , where the plus sign stands for and the minus for . Whipple also stated that two of these solutions lie on a straight line which connects the triangular Lagrangian point with the origin (inline equilibria). However, by making more precise calculations we have found that this line does not pass through the origin O but forms a small angle φ with the direction (or ). The remaining two equilibrium solutions are located on a line which is perpendicular to the previous one and crosses it at the triangular Lagrangian point. For this reason, they are called perpendicular. By using the same notation rules as before, we symbolize the inline and the perpendicular equilibria as and , respectively. Here we note that the equilibrium positions have been calculated by Whipple [1], under the approximated assumption that the center of mass of the minor bodies in an equilibrium configuration almost coincides with the corresponding Lagrangian point of the restricted three-body problem. This means that the coordinates of the minor bodies in an equilibrium position approximately satisfy the relations

We have made all the computations from scratch by considering the four coordinates of the minor bodies as independent variables, and we have adjusted the numerical methods used to a new and more general treatment of the problem. Obviously, this results in more complex and extended expressions as well. Further details will be given in a later section of this paper. Figure 2 shows the Lagrangian equilibrium positions of the restricted three-body problem, while Figures 3 and 4 are magnifications of the dotted small frames of Figure 2 showing the distribution of the equilibrium positions of the minor bodies in the neighborhood of the collinear Lagrangian points (Figure 3) and of the triangular one (Figure 4), for the two considered cases. Tables 1 and 2 contain the numerical results which concern the two considered cases, while Tables 3 and 4 contain the values of the Lagrangian points of the respective restricted three-body systems. In Table 5 we give the dimensionless and the physical values of the distances between the minor bodies (their data are given in Appendices A and B) in a pair of equilibria in the Sun-Jupiter-binary asteroids and the Earth-Moon-dual satellites systems.

(a) Collinear equilibrium solutions .

Jacobian constant  (10−15)Stability

0 03.037482U
0 03.037482U
0 03.038764U
0 03.038764U
0 03.000966U
0 03.000966U

(b) Inline ( , ) and perpendicular ( , ) equilibrium solutions,

Jacobian constant  (10−15)Stability

2.999844U
2.999844U
2.999844U
2.999844U

Jacobian constant  (10−15)Stability

2.999844S
2.999844S
2.999844S
2.999844S

(a) Collinear equilibrium solutions .

Jacobian constant  (10−20)Stability

0 03.772852U
0 03.772852U
0 03.792084U
0 03.792084U
0 03.582608U
0 03.582608U

(b) Inline ( , ) and perpendicular ( , ) equilibrium solutions,

Jacobian constant  (10−20)Stability

3.553904U
3.553904U
3.553904U
3.553904U

Jacobian constant  (10−20)Stability

3.553904S
3.553904S
3.553904S
3.553904S


: Jacobian constant

0
0


: Jacobian constant

0
0