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 forsolving equations of motion of small orbiter m, which is moving under the combined gravitationalattraction of three primaries, M1, M2, and M3 , in case of the bi-elliptic restricted problemof four bodies (BiER4BP), where three such primaries, M1, M2, and M3, are moving on ellipticorbits with hierarchical configuration M3 << M2 << M1 within one plane as follows: thirdprimary 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 closemotion to the plane of orbiting the main bodies M1, M2, and M3. Meanwhile, the system ofequations of motion was successfully numerically explored with respect to the existence and stablepositioning of approximate solution for a Dyson sphere. As a result, the concept of the Dyson spherefor possible orbiting variety of solar energy absorbers was transformed to the elongated Dysonspace net with respect to their trajectories for the successful process of absorbing the energy fromthe Sun; this can be recognized as symmetry reduction. We obtain the following: (1) the solution forcoordinates {x, y} is described by the simplified system of two nonlinear ordinary differentialequations of second order, depending on true anomaly f; (2) the expression for coordinate z is givenby an equation of Riccati-type where small orbiter that quasi-oscillates close to the fixed plane{x, y, 0}.

Solvability analysis for the Boussinesq model of heat transfer under the nonlinear Robin boundary condition for the temperature

“…the papers [2–5] and the review article [6]). The second group contains papers devoted to using the Lie–Ovsyannikov symmetry method to study qualitative properties of solutions to heat and mass transfer equations in heat-conducting and/or viscous binary fluids (see more details in the book [7] and in the review articles [8,9]). Another group contains papers in which mathematical modelling of hydrodynamic processes in heat-conducting and/or binary fluids takes into account the thermodiffusion (or Sorét) effect and/or concentration diffusion (or Dufour) effect.…”