Abstract
In the studied axisymmetric case of the central four-body problem, the axis of symmetry is defined by two unequal-mass bodies, while the other two bodies are situated symmetrically with respect to this axis and have equal masses. Here, we consider a special case of the problem and assume that three of the masses are equal. Using a recently found analytical solution of the general case, we formulate the equations of condition for three equal masses analytically and solve them numerically. A complete description of the problem is given by providing both the coordinates and masses of the bodies. We show furthermore how the three-equal-mass solutions are related to the general case in the coordinate space. The physical aspects of the configurations are also studied and discussed.
Highlights
A fundamental problem in celestial mechanics is to describe the motion of n pointlike bodies, assuming only the mutual Newtonian gravitational forces acting between them
Where rij ≡ |r i − r j | is the distance between the i-th and j-th bodies
We aim to show how the analytical solutions given in [6] can be applied to systems with three equal masses
Summary
A fundamental problem in celestial mechanics is to describe the motion of n pointlike bodies, assuming only the mutual Newtonian gravitational forces acting between them. Using the analytical formulae given in [6], we first derive the analytical conditions of these cases; by numerical computations, represent them as solution curves in the allowed domains of the parameter plane, giving a clear view of how the three-equal-mass cases are related to the general case of the axisymmetric central configurations of four bodies. This latter relation was previously not known. Some of the analytical considerations are detailed in the Appendixes A–C
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