Abstract

The spatial equilateral restricted four-body problem (ERFBP) is a four body problem where a mass point of negligible mass is moving under the Newtonian gravitational attraction of three positive masses (called the primaries) which move on circular periodic orbits around their center of mass fixed at the origin of the coordinate system such that their configuration is always an equilateral triangle. Since fourth mass is small, it does not affect the motion of the three primaries. In our model we assume that the two masses of the primaries and are equal to and the mass is . The Hamiltonian function that governs the motion of the fourth mass is derived and it has three degrees of freedom depending periodically on time. Using a synodical system, we fixed the primaries in order to eliminate the time dependence. Similarly to the circular restricted three-body problem, we obtain a first integral of motion. With the help of the Hamiltonian structure, we characterize the region of the possible motions and the surface of fixed level in the spatial as well as in the planar case. Among other things, we verify that the number of equilibrium solutions depends upon the masses, also we show the existence of periodic solutions by different methods in the planar case.

1. Introduction

Dynamical systems with few bodies (three) have been extensively studied in the past, and various models have been proposed for research aiming to approximate the behavior of real celestial systems. There are many reasons for studying the four-body problem besides the historical ones, since it is known that approximately two-thirds of the stars in our Galaxy exist as part of multistellar systems. Around one-fifth of these is a part of triple systems, while a rough estimate suggests that a further one-fifth of these triples belongs to quadruple or higher systems, which can be modeled by the four-body problem. Among these models, the configuration used by Maranhão [1] and Maranhão and Llibre [2], where three point masses form at any time a collinear central configuration (Euler configuration, see [3]), is of particular interest not only for its simplicity but mainly because in the last 10 years, an increasing number of extrasolar systems have been detected, most of them consisting of a “sun" and a planet or of a “sun" and two planets.

We study the motion of a mass point of negligible mass under the Newtonian gravitational attraction of three mass points of masses , and (called primaries) moving in circular periodic orbits around their center of mass fixed at the origin of the coordinate system. At any instant of time, the primaries form an equilateral equilibrium configuration of the three-body problem which is a particular solution of the three-body problem given by Lagrange (see [4] or [3]). Two of these primaries have equal masses and are located symmetrically with respect to the third primary.

We choose the unity of mass in such a way that and are the masses of the primaries, where . Units of length and time are chosen in such a way that the distance between the primaries is one.

For studying the position of the infinitesimal mass, , in the plane of motion of the primaries, we use either the sideral system of coordinates, or the synodical system of coordinates (see [5] for details). In the synodical coordinates, the three point masses , and are fixed at , and , respectively. In this paper, the equilateral restricted four-body problem (shortly, ERFBP) consists in describing the motion of the infinitesimal mass, , under the gravitational attraction of the three primaries , and . Maranhão's PhD thesis [1] and the paper [2] by Maranhão and Llibre studied a restricted four body problem, where three primaries rotating in a fixed circular orbit define a collinear central configuration.

In the ERFBP, the equations of motion of in synodical coordinates are

where

We remark that the ERFBP becomes the central force problem when and is situated in the origin of the system, while results in the restricted three-body problem with the bodies and of mass

Our paper is organized as follows: Section 2 is devoted to describing the most important dynamical phenomena that governs the evolution of asteroid movement and states the problem under consideration in the present study. In Section 3 reductions of the problem are discussed and a comprehensive treatment of streamline analogies is given. Section 4 is devoted to the principal qualitative aspect of the restricted problem—the surfaces and curves of zero velocity, several uses of which are discussed. The regions of allowed motion and the location and properties of the equilibrium points are established. We describe the Hill region. The description of the number of equilibrium points is given in Section 5, and in the symmetrical case (i.e., ), we describe the kind of stability of each equilibrium. In Section 6, the planar case is considered. There, we prove the existence of periodic solutions as a continuation of periodic Keplerian orbits, and also when the parameter is small and when it is close to 1/2. Finally, in Section 8 we present the conclusions of the present work.

Next, we will enunciate some four-body problem that has been considered in the literature. Cronin et al. in [6, 7] considered the models of four bodies where two massive bodies move in circular orbits about their center of mass or barycenter. In addition, this barycenter moves in a circular orbit about the center of mass of a system consisting of these two bodies and a third massive body. It is assumed that this third body lies in the same plane as the orbits of the first two bodies. The authors studied the motion of a fourth body of small mass which moves under the combined attractions of these three massive bodies. This model is called bicircular four-body problem. Considering this restricted four-body problem consisting of Earth, Moon, Sun, and a massless particle, this problem can be used as a model for the motion of a space vehicle in the Sun-Earth-Moon system. Several other authors have considered the study of this problem, for example, [8–11] and references therein. The quasi-bicircular problem is a restricted four body problem where three masses, Earth-Moon-Sun, are revolving in a quasi-bicircular motion (i.e. a coherent motion close to bicircular) also has been studied, see [12] and references therein. The restricted four-body problem with radiation pressure was considered in [13], while the photogravitational restricted four body problem was considered in [14].

2. Statement of the Problem

It is known that equilateral configurations of three-bodies with arbitrary masses , and on the same plane, moving with the same angular velocity, form a relative equilibrium solution of the three-body problem (see e.g., [4] or [3]). More precisely, we consider three particles of masses , , and (called primaries) each describing, at any instant, a circle around their center of masses (which is fixed at the origin), with the same angular velocity and such that its configuration at any instant is an equilateral triangle (see Figure 1). Now, we consider an infinitesimal particle attracted by the primaries , , and according to Newton's gravitational law. Let be the position vector of .

Figure 1 

The equilateral restricted four body problem in inertial coordinates.

Figure 2 

The equilateral restricted four body problem in a rotating frame.

The equations of motion can be written as

where denotes derivative with respect to and

with and representing the position of each primary, respectively. To remove the time dependence of the system (2.1), we consider the orthonormal moving frame in , given by where

with . This orthonormal moving frame corresponds to the synodical system. Then, (2.1) can be written as where

where , with for . Applying the above notation, we can write , , , , and so for

We perform the reparametrization of time , then the system (2.4) is transformed into

where the dot denotes the derivative with respect to , and the potential is given by

with

If we define , and , where , the equations of motions (2.4) become

where

For simplicity, we will consider an equilateral triangle of side 1 and so we obtain that

3. Equations of Motion and Preliminary Results

From (2.9), we deduce that the equations of motion of the ERFBP in synodical coordinates are given by the system of differential equations

where

with

Analogously to the circular three-body problem, we can verify that the system (3.1) possesses a first Jacobi type integral given by

Thus we have the following result.

Proposition. The Jacobi-type function (3.4) is a first integral of the ERFBP for any value of .

Proof. Differentiating (3.4) with respect to the time, we get and using (2.9) we can reduce the obtained expression to Hence is a constant of motion.

In order to write the Hamiltonian formulation of the ERFBP we introduce the new variables

Hence, it is verified that system (3.1) is equivalent to an autonomous Hamiltonian system with three degrees of freedom with Hamiltonian function given by

Therefore, the Hamiltonian system associated is

Of course, the phase space where the equations of motion are well defined is

where the points that have been removed correspond to binary collisions between the massless particle and one of the primaries.

Additionally, the spatial ERFBP admits the planar case as a subproblem, that is, is invariant under the flow defined by (3.9).

On the other hand, we see that there are two limiting cases in the ERFBP, which we described below.

Note that corresponds to the symmetric case, that is, where the masses of the primaries are all equal to

It is easily seen that the equations of motion (3.9) are invariant by the symmetry

This means that if is a solution of the system (3.9), then is also a solution. We note that this symmetry corresponds to a symmetry with respect to the -plane. In the planar case, the symmetry corresponds to symmetry with respect to the -axis.

4. Permitted Regions of Motion

In this section, we will see that the function allows us to establish regions in the space, where the motion of the infinitesimal particle could take place. We will use similar ideas to those developing in [15, 16].

By using (3.4), the surface of zero velocity is defined by the set

This set corresponds to the so-called Hill region. We note that implies . That is, the region of all possible motions is given by the whole phase space and so the infinitesimal particle is free to move; in particular escape solutions are permitted.

In the spatial case, the surfaces that separate allowed and nonallowed motions are called zero-velocity surfaces, and for the planar case the set that separates the allowed and nonallowed motions is called zero-velocity curve. The shape and size of zero velocity sets depend on and . They correspond to the boundary of the Hill regions. The zero-velocity set () is defined by the equation

only for and any value of . Next, we give a list of all possible situations that may appear when this condition is fulfilled.

By simplicity, we will only show zero-velocity surfaces for the case and different values of the integral of motion . Figures 3, 4, 5, and 6 show evolution of zero-velocity surfaces for several values.

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Figure 3 

Evolution of zero-velocity surface in the three-dimensional ERFBP for . (a) . (b) . (c) .

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Figure 4 

Evolution of zero-velocity surface in the three dimensional ERFBP for . (a) . (b) and (c) under different points of view.

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Figure 5 

Evolution of zero-velocity surface in the three dimensional ERFBP for . All cases correspond to under different points of view.

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Figure 6 

Evolution of zero-velocity surface in the three dimensional ERFBP for . (a) . (b) . (c) .

4.1. The Planar Case

As we mentioned in last section, the set is invariant under the flow, and so the motion of the infinitesimal body lies on the plane that contains the primaries. In Figure 7, we show the evolution of the function in the planar case for different values of the parameter .

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Figure 7 

Evolution of the graph of on the plane for different values of .

Next we show the evolution of the Hill's regions as well as the zero velocity curves, for and many values of the Jacobian constant ; the permissible areas are shown on Figures 8, 9, 10, and 11 shading.

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Figure 8 

Evolution of zero-velocity curves and Hill's region in the planar ERFBP for , where shading represents permissible areas. (a) , (b) , (c) .

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Figure 9 

Evolution of zero-velocity curves and Hill's region in the planar ERFBP for , where shading are permissible areas. (a) , (b) , (c) .

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Figure 10 

Evolution of zero-velocity curves and Hill's region in the planar ERFBP for , where shading are permissible areas. (a) . (b) . (c) .

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Figure 11 

Evolution of zero-velocity curves and Hill's region in the planar ERFBP for , where shading are permissible areas. (a) , (b) and (c) both figures correspond to with different window size.

In Figure 12, we show the behavior of level curves in the planar case for some values of and for different energy levels.

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Figure 12 

Energy level curves for some values of the parameter in the planar case.

5. Equilibrium Solutions

It is verified that the equilibrium solutions of the system (3.9) or equivalently (3.1) are given by the critical points of the function or simply they are the solutions of the following system of equations:

From the last equation we see that the coordinate must be zero, so the critical points are restricted to the plane , and are given by the solutions of the first two equations.

It is known (see [17]) that the number of equilibrium solutions of the system (5.1) is 8, 9 or 10 depending on the values of the masses, , and which must be positive. Six of them are out of the symmetry axis (i.e., out of the -axis), therefore on the axis of symmetry we must have 2, 3 or 4. From the analysis done it follows that the number of the equilibrium solutions depends on the parameter . This implies that finding the critical points is a non-trivial problem, and this is one of the main differences with the problem studied by Maranhão in his doctoral thesis [1], because there, the number of critical points did not depend on the parameter .

The critical points on the axis are the zeros of the function

where

An explicit computation shows that in the limit case problems the number of equilibrium points corresponding to the system (5.1) is as follows.

From numerical simulations we get that the number of critical points along the –axis is given in Table 1. Observe that is the bifurcation value.

In the symmetric case when all the masses are equals (i.e., ) we have that the graph of is similar to the one shown in Figure 13. As a consequence, there are exactly 4 equilibrium solutions on the -axis, and therefore there are exactly 10 equilibrium solutions. Of course, is an equilibrium solution.

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Figure 13 

Graph of . (a) . (b).

In general for any equilibrium solution of the form , the linearized system (3.9) in the planar case give us that the characteristic polynomial is

whose roots are

where is given by

with , and . A very simple result is the following.

Lemma. The roots of are real and negative if and only if , and .

Associating to our characteristic polynomial (5.6) we have

Now, we remark that

Consequently we have the following result:

Corollary. In the spatial ERFBP for any equilibrium solution we have at least two pure imaginary eigenvalues associated to the linear part, which are given by .