Semi-Analytical Approach in BiER4BP for Exploring the Stable Positioning of the Elements of a Dyson Sphere

Abstract

In this study, we present a new approach with semi-analytical and numerical findings for solving equations of motion of small orbiter m, which is moving under the combined gravitational attraction of three primaries, M1, M2, and M3, in case of the bi-elliptic restricted problem of four bodies (BiER4BP), where three such primaries, M1, M2, and M3, are moving on elliptic orbits with hierarchical configuration M3 << M2 << M1 within one plane as follows: third primary body M3 is moving on elliptical orbit around second M2, and second primary M2 is moving on elliptical orbit around first M1. Our aim for constructing the aforementioned quasi-planar motion of planetoid m is obtaining its coordinates supporting its orbit in a regime of close motion to the plane of orbiting the main bodies M1, M2, and M3. Meanwhile, the system of equations of motion was successfully numerically explored with respect to the existence and stable positioning of approximate solution for a Dyson sphere. As a result, the concept of the Dyson sphere for possible orbiting variety of solar energy absorbers was transformed to the elongated Dyson space net with respect to their trajectories for the successful process of absorbing the energy from the Sun; this can be recognized as symmetry reduction. We obtain the following: (1) the solution for coordinates {x, y} is described by the simplified system of two nonlinear ordinary differential equations of second order, depending on true anomaly f; (2) the expression for coordinate z is given by an equation of Riccati-type where small orbiter that quasi-oscillates close to the fixed plane {x,y,0}.