Analysis of Linear Stability and Bifurcations of Central Configurations in the Planar Restricted Circular Four-Body Problem

The planar restricted four-body problem is considered. That is, motion of an infinitesimal body P in the gravitational field of three attracting bodies is studied. It is supposed, that the above three body form a stable equilateral Lagrange triangle and all four bodies move in a plane. In this case there are eight central configurations formed by the bodies. An analysis of stability and bifurcation of the central configurations is performed. In particular, it is shown that the bifurcation is only possible in cases of degeneracy, when mass of an attracting body vanishes. It is also established that in non-degenerate cases five central configurations are unstable and three central configurations can be both stable and unstable. Domains of stability in linear approximation are constructed in the plane of parameters.


Introduction
The restricted four-body problem is a simplest generalization of the classical restricted three-body problem. It is of great interest for celestial mechanics and dynamics of satellites. In this problem, the motion of four bodies is investigated, one of which has an infinitely small mass, three other bodies have finite masses and move under the influence of their mutual attraction according to Newton's law of universal gravitation. The infinitesimal body moves under gravitational attraction of other three bodies and does not affect their motion. There exist a partial type of motion of the bodies, when they form the so-called central configuration. It means that the resultant attracting force acting on a body is a central force directed to the system mass center.
A numerous papers are devoted to the study of the central configuration in the restricted planar four-body problem. The existence and bifurcation of central configurations were studied in [1][2][3][4][5]. In particular, it has been established [1] that there can be eight, nine, or ten configurations. A linear analysis of the stability of central configurations has been performed in [6][7][8][9]. In [10][11], conclusions on the Lyapunov stability of central configurations were obtained in the case of two equal masses of attracting bodies.
In what follows, we denote by masses of attracting bodies (i = 1, 2, 3). We assume that the attracting bodies form an equilateral triangle (Lagrangian triangle) and move in circular orbits. The infinitesimal body P moves in the plane of the Lagrangian triangle. For any values of the masses of IOP Publishing doi:10.1088/1757-899X/1191/1/012002 2 attracting bodies, we carry out a linear stability analysis and study possible scenarios of bifurcation for central configuration formed by small body P with attracting bodies .

Formulation of the problem
Let us introduce rotating coordinate system Oxyz, which is rigidly connected with the Lagrangian triangle. The origin of this system is located in the middle of the segment . The axis Ox passes through this segment. The axis Oy is perpendicular to the Ox and passes through the attracting body . The axis Oz complements the coordinate system to the right, orthogonal triple.
The dimensionless coordinates , are introduced by means of formulae = , = , where x and y are coordinates of the infinitesimal body in system Oxyz and = is the distance between the primaries. The dimensionless time is introduced by means of formula = , where = ( + + ) and f is the gravitational constant. Let us note that is the angular velocity of Lagrange triangular configuration.
The Hamiltonian function reads It is worth noting that the coordinates of the system mass center in dimensionless variables , read In terms of the stability of the central configurations, we are only interested in values of the parameters and , for which the Lagrangian triangular configuration of three bodies is stable. The necessary condition for the stability of such a configuration is the well-known Routh condition [12], which in terms of the parameters and reads Without loss of any generality, we assume that the mass of the body is greater than the masses of the bodies and . It follows from inequality (6) that the quantities and can take values from the interval If the Routh condition is not satisfied, then the central configuration of bodies (i = 1, 2, 3) is unstable. It yields the instability of the corresponding central configuration of four bodies. In what follows, we assume that the Routh condition is satisfied. The equations of motion (1) have the following stationary solution describing the relative equilibria of infinitesimal body P in the rotating coordinate system Oxyz.
where * and * satisfy the following equations The equilibrium of the body P in the rotating coordinate system Oxyz corresponds to its absolute motion in a circular orbit. In this motion the body P forms a constant central configuration with the bodies .
In the next sections for any parameters values we investigate linear stability of the corresponding central configuration and perform analysis of their bifurcations.

Bifurcation of relative equilibria
Let us first consider the limiting cases = 0 and = 0, when our problem degenerates into the restricted three-body problem. In this cases, the central configurations degenerate into Euler and Lagrangian libration points. The central configurations for the limiting cases are shown in figures 2 and 3. The positions of the body P is indicated by blue points. To designate the relative equilibrium P in limiting cases, we used the standard notation for libration points ( ) , where the superscript k corresponds to the case = 0. For parameter values satisfying the Routh condition, the regions of relative equilibria for the infinitesimal body P were numerically constructed (see figure 4). It turned out that for parameter values ≠ 0 and ≠ 0 there are exactly eight relative equilibria, which are located in narrow regions. These regions are located either along a circle of unit radius, whose center is in the body or along the straight lines passing through , and , respectively (see figure 4). The libration points ( ) are the limit points of the indicated regions, so that the relative equilibria pass into one of the libration points at = 0 or = 0.     Note also that the regions of existence of relative equilibria were constructed using the method of numerical continuation in the parameters. In the vicinity of the points ( ) and ( ) , the numerical analysis becomes more complicated and requires higher accuracy of calculations. Let us note that in the vicinity of the points ( ) and ( ) the study of the relative equilibria can be carried out analytically by using the small parameter method, which in this case is more efficient than the numerical analysis.

Stability of central configuration
In this section, we present the results of studying the stability of central configurations in linear approximation. If the Routh condition is satisfied, then the problem of stability of the central configuration is reduced to studying the stability of relative equilibria of the body P in the coordinate system rotating together with the attracting centers. In order to study the stability of equilibria we introduce local canonical variables , , and in a neighborhood of the above equilibria = * + , = * + , and expand the Hamiltonian (2) in a series with respect to powers of the new canonical variables The quadratic part of the Hamiltonian reads where the coefficients a, b, c have the following explicit form The question of the stability for this linear system with the Hamiltonian can be solved by analysing the roots of its characteristic equation In accordance with Lyapunov's theorem [13], the linear system is stable if all roots of the characteristic equation are simple and purely imaginary. This condition satisfied if the coefficients of equation (14) satisfy the following inequalities + + 2 > 0, − − − + 1 > 0, + + 4 − 2 + 8( + ) > 0. which is stable in linear approximation for all values of the parameter . The most interesting situation takes place for the relative equilibria , and . These relative equilibria, depending on the values of the parameters, can be both stable and unstable. For these relative equilibria stability diagrams were constructed in the plane of the parameters (see figures 7, 8, 9). The instability domains are indicated by red color and domains of stability the linear approximation are indicated by blue color.
Note that in the limiting case = 0 the relative equilibria , and degenerate into the libration point

Conclusions
Let us briefly formulate the results of the study. It has been numerically established that if Routh conditions are satisfied, then there exist exactly eight central configurations, which correspond to the relative equilibria of the infinitesimal body P in the coordinate system rotating together with the attracting centers , and . In the limiting cases, when an attracting center has zero mass, the relative equilibria degenerate into triangular (Lagrangian) or rectilinear (Euler) libration points. In this case, bifurcation is possible. It occurs according to the following scenarios. Four relative equilibria , , and pass into the libration point ( ) at = 0, and four other relative equilibria , , and pass into the libration point ( ) at = 0. There are no other bifurcations. The regions of existence of possible relative equilibria of the infinitesimal body are numerically constructed for all values of the parameters that satisfy the Routh conditions. By using the numerical analysis, it was established that the position of relative equilibrium is unstable for all values of the parameters. The relative equilibria and are linearly stable only in the limiting case = 0, when they pass into the libration point ( ) , and and are linearly stable only in the limiting case = 0, when they pass into the libration point ( ) . The relative equilibria , and can be both stable and unstable depending on value of parameters.