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

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## Abstract

**:**

## 1. Introduction

## 2. Description of the Model, Equations of Motion

_{i}= (${M}_{i}$/$M$) (i = 1, 2, 3), ${r}_{i}$ is the nondimensional distance between orbiter m and i-th primary [20], which are to be determined as follows:

_{1}is the semimajor axis of the binary system {${M}_{1}$, ${M}_{2}$}, ${M}_{2}$ << ${M}_{1}$; e

_{1}is the eccentricity of elliptic orbit of the rotating primary ${M}_{2}$ around primary ${M}_{1}$, f is the true anomaly for this orbital primary motion; a

_{2}is the semimajor axis of the binary system {${M}_{2}$, ${M}_{3}$}, ${M}_{3}$ << ${M}_{2}$; e

_{2}is the eccentricity of elliptic orbit of the rotating primary ${M}_{3}$ around the barycenter of primaries {${M}_{2}$, ${M}_{3}$}, and f

_{2}is the true anomaly for this orbital secondary motion. The parameter θ is determined as in the work [20] by

_{0}is initial value of θ. Meanwhile, from (4) and (5), we obtain

## 3. Semi-Analytical Approximate Solution to the System of Equation (7)

_{3}<< μ

_{2}⇒ (μ

_{3}· μ

_{2})/(μ

_{3}+ μ

_{2}) ≅ μ

_{3}.

_{3}<< μ

_{2}, e

_{2}<< 1, and by three primaries, we mean “Sun–Earth–Moon” in BiER4BP):

## 4. Families of Quasi-Stable Plane Orbits $\mathbf{\{}\mathit{x}\mathbf{,}\mathit{y}\mathbf{\}}\mathbf{,}\mathit{z}\mathbf{\to}\mathbf{0}$ of System (7) and Equation (11), Their Graphical and Numerical Solutions

## 5. Discussion

- We consider in (11) two primaries of masses {${M}_{2}$, ${M}_{3}$}, μ
_{3}<< μ_{2}, rotating on elliptic orbits, whereas their barycenter is Kepler-rotating around the main primary M_{1}, μ_{2}<< μ_{1}. - The motions of the primaries are preferably coplanar (while it is a well-known fact that the orbit of the Moon is inclined on circa 5 degrees with respect to the invariable plane of rotation of Earth around the Sun).
- Eccentricity e
_{2}of orbit ${M}_{3}$ around ${M}_{2}$ is negligible, e_{2}<< 1. - Orbiter m is moving outside the Hill sphere [15] of second primary ${M}_{2}$ with radius (minimal distance) ~ 0.101 AU for the system “Sun–Earth”.
- Small orbiter m is assumed to oscillate nearby plane $\{x,y\},\hspace{1em}z\to 0$.
- Masses of all primaries are constant.

_{2}(f), and r

_{3}(f) shown in Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17, and graphical solutions for r

_{1}(f) demonstrate to us that the orbiter will move outside the most active sphere of influence by the Sun and Earth. However, this does not mean there will be a divergence for the solution (as soon as true anomaly moves to infinity), or this does not relate with respect to a correction to the position, or any numerical error associated with the solution to the equations. This means that there are not any stable orbits (which were supposed to be existing in the concept of Dyson sphere), or at least quasi-stable orbits, within formulation of the aforementioned bi-elliptic restricted four-body problem.

## 6. Conclusions

_{3}, in the case of a bi-elliptic restricted four-body problem, where three primaries, ${M}_{1}$, M

_{2}, and M

_{3}, are Kepler-moving on their orbits (M

_{3}<< ${M}_{2}$ << ${M}_{1}$) in one plane as follows: the third primary body, ${M}_{3}$, is moving on an elliptical orbit around the second, ${M}_{2}$, the second primary, ${M}_{2}$, is moving on an elliptical orbit around the first, ${M}_{1}$. Our aim for constructing the aforementioned quasi-planar motion of a planetoid or an orbiter m is to obtain its coordinates supporting its orbit in a regime of close motion to the plane of orbiting the main bodies ${M}_{1}$, ${M}_{2}$, and ${M}_{3}$. We considered stable positioning of approximate solution for elements of the Dyson sphere (Dyson swarm) for the aforeformulated problem from the point of view of equations of motion in celestial mechanics formulated in the case of BiER4BP.

- Elegant ansatz is developed for analysis of motion of a small mass in BiER4BP.
- Three primaries rotate around a barycenter on bi-elliptic orbits: ${M}_{3}$ << ${M}_{2}$, ${M}_{2}$ << ${M}_{1}$.
- Coordinate z is considered to be stable in oscillating close to fixed plane {x, y, 0}.
- The planar bi-elliptic restricted four-body problem (BiER4BP) is investigated well.
- Stable drift dynamics of solutions for analogue of a Dyson sphere are analyzed.
- No stable solutions for a Dyson swarm were found, but an orbiter will flyby near Earth.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Schematic plot for x(f) with the help of Equation (11) (true anomaly f is depicted as abscissa).

**Figure 3.**Schematic plot for y(f) with the help of Equation (11) (true anomaly f is depicted as abscissa).

**Figure 4.**Schematic plot for distance r

_{1}(f) with the help of Equation (11) (true anomaly f is depicted as abscissa), ${r}_{1}^{}=\sqrt{{(x+{\mu}_{3}+{\mu}_{2})}^{2}+{y}^{2}+{z}^{2}}$.

**Figure 5.**Schematic plot for distance r

_{2}(f) from Earth with the help of Equation (11) (true anomaly f is depicted as abscissa), ${r}_{2}\cong \sqrt{{(x-{\mu}_{1}-\frac{{\mu}_{3}}{{\mu}_{2}}r\mathrm{cos}\theta )}^{2}+{(y-\frac{{\mu}_{3}}{{\mu}_{2}}r\mathrm{sin}\theta )}^{2}+{z}^{2}},\hspace{1em}{\mu}_{3}<<{\mu}_{2}$.

**Figure 6.**Schematic plot for distance r

_{3}(f) from Moon with the help of Equation (11) (true anomaly f is depicted as abscissa), ${r}_{3}\cong \sqrt{{(x-{\mu}_{1}+r\mathrm{cos}\theta )}^{2}+{(y+r\mathrm{sin}\theta )}^{2}+{z}^{2}},\hspace{1em}{\mu}_{3}<<{\mu}_{2}$.

**Figure 7.**Numerically obtained solution of Equation (11) for dependence y(x) (which means the actual trajectory of the small orbiter starting from initial conditions in a plane {x, y, 0}).

**Figure 8.**Schematic plot for x(f) with the help of Equation (11) (true anomaly f is depicted as abscissa).

**Figure 9.**Schematic plot for y(f) with the help of Equation (11) (true anomaly f is depicted as abscissa).

**Figure 10.**Schematic plot for distance r

_{1}(f) with the help of Equation (11) (true anomaly f is depicted as abscissa), ${r}_{1}=\sqrt{{(x+{\mu}_{3}+{\mu}_{2})}^{2}+{y}^{2}+{z}^{2}}$.

**Figure 11.**Schematic plot for distance r

_{2}(f) from Earth with the help of Equation (11) (true anomaly f is depicted as abscissa), ${r}_{2}\cong \sqrt{{(x-{\mu}_{1}-\frac{{\mu}_{3}}{{\mu}_{2}}r\mathrm{cos}\theta )}^{2}+{(y-\frac{{\mu}_{3}}{{\mu}_{2}}r\mathrm{sin}\theta )}^{2}+{z}^{2}},\hspace{1em}{\mu}_{3}<<{\mu}_{2}$.

**Figure 12.**Schematic plot for distance r

_{3}(f) from Moon with the help of Equation (11) (true anomaly f is depicted as abscissa), ${r}_{3}\cong \sqrt{{(x-{\mu}_{1}+r\mathrm{cos}\theta )}^{2}+{(y+r\mathrm{sin}\theta )}^{2}+{z}^{2}},\hspace{1em}{\mu}_{3}<<{\mu}_{2}$.

**Figure 13.**Schematic plot for x(f) with the help of Equation (11) (true anomaly f is depicted as abscissa).

**Figure 14.**Schematic plot for y(f) with the help of Equation (11) (true anomaly f is depicted as abscissa).

**Figure 15.**Schematic plot for distance r

_{1}(f) with the help of Equation (11) (true anomaly f is depicted as abscissa), ${r}_{1}=\sqrt{{(x+{\mu}_{3}+{\mu}_{2})}^{2}+{y}^{2}+{z}^{2}}$.

**Figure 16.**Schematic plot for distance r

_{2}(f) from Earth with the help of Equation (11) (true anomaly f is depicted as abscissa), ${r}_{2}\cong \sqrt{{(x-{\mu}_{1}-\frac{{\mu}_{3}}{{\mu}_{2}}r\mathrm{cos}\theta )}^{2}+{(y-\frac{{\mu}_{3}}{{\mu}_{2}}r\mathrm{sin}\theta )}^{2}+{z}^{2}},\hspace{1em}{\mu}_{3}<<{\mu}_{2}$.

**Figure 17.**Schematic plot for distance r

_{3}(f) from Moon with the help of Equation (11) (true anomaly f is depicted as abscissa), ${r}_{3}\cong \sqrt{{(x-{\mu}_{1}+r\mathrm{cos}\theta )}^{2}+{(y+r\mathrm{sin}\theta )}^{2}+{z}^{2}},\hspace{1em}{\mu}_{3}<<{\mu}_{2}$.

**Figure 18.**Numerical solution for coordinate z(f) using Equation (10) (depending on true anomaly f, depicted on abscissa axis).

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**MDPI and ACS Style**

Ershkov, S.; Leshchenko, D.; Prosviryakov, E.Y.
Semi-Analytical Approach in BiER4BP for Exploring the Stable Positioning of the Elements of a Dyson Sphere. *Symmetry* **2023**, *15*, 326.
https://doi.org/10.3390/sym15020326

**AMA Style**

Ershkov S, Leshchenko D, Prosviryakov EY.
Semi-Analytical Approach in BiER4BP for Exploring the Stable Positioning of the Elements of a Dyson Sphere. *Symmetry*. 2023; 15(2):326.
https://doi.org/10.3390/sym15020326

**Chicago/Turabian Style**

Ershkov, Sergey, Dmytro Leshchenko, and Evgeniy Yu. Prosviryakov.
2023. "Semi-Analytical Approach in BiER4BP for Exploring the Stable Positioning of the Elements of a Dyson Sphere" *Symmetry* 15, no. 2: 326.
https://doi.org/10.3390/sym15020326