# The Axisymmetric Central Configurations of the Four-Body Problem with Three Equal Masses

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## Abstract

**:**

## 1. Introduction

## 2. Equations of Condition for Three Equal Masses

#### 2.1. Case $\mu ={\mu}_{1}$

#### 2.2. Case $\mu ={\mu}_{2}$

#### 2.3. Exceptional Cases

#### 2.3.1. ${b}_{1}+{a}_{0}-{b}_{0}=0$

#### 2.3.2. ${a}_{1}+{b}_{0}-{a}_{0}=0$

## 3. Solutions in the $\beta $, $\alpha $ Plane

#### 3.1. Convex Case

#### 3.2. Concave Cases

## 4. Computing the Nondimensional Masses

#### 4.1. Convex Case

#### 4.2. Concave Cases

## 5. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Coefficients a_{0}, a_{1}, b_{0}, b_{1}

## Appendix B. Properties of the Functions f_{i} (i = 1, …, 4)

#### Appendix B.1. Function f_{1} (Convex Case, μ = μ_{1})

#### Appendix B.2. Function f_{2} (Convex Case, μ = μ_{2})

#### Appendix B.3. Function f_{3} (Concave Cases, μ = μ_{1})

#### Appendix B.4. Function f_{4} (Concave Cases, μ = μ_{2})

## Appendix C. The Minimum Point of Function M

## References

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**Figure 1.**Convex central configuration. The four bodies, forming a convex deltoid, are labeled by A, B, E, and ${E}^{\prime}$, and their nondimensional masses are ${\mu}_{1}$, ${\mu}_{2}$, $\mu $, and $\mu $, respectively. O indicates the center of mass of the system. The two independent parameters are the angle coordinates $\alpha $ and $\beta $.

**Figure 2.**Concave central configurations. The denotations are the same as in Figure 1. The two independent parameters are the angle coordinates $\alpha $ and $\beta $. (

**a**) First concave case. The center of mass O is inside the deltoid. (

**b**) Second concave case. O is outside of the deltoid.

**Figure 3.**Solutions in the convex case. The colored area (bordered by critical lines) refers to those $\beta $, $\alpha $ pairs that allow convex central configurations of four bodies. Letter S at $\alpha =\beta ={30}^{\circ}$ indicates a singular point where ${\mu}_{1}+{\mu}_{2}=1$. Letter G at $\alpha =\beta ={45}^{\circ}$ corresponds to the square configuration with four equal masses. Along the blue curve, the bodies A, E, and ${E}^{\prime}$ have equal masses, while the masses of B, E, and ${E}^{\prime}$ become equal along the red curve. Letters ${P}_{1}$ and ${P}_{2}$ indicate the endpoints of the solutions curves on critical lines. Note the reversed direction of the horizontal axis.

**Figure 4.**Solutions in the concave cases. The colored areas (bordered by critical lines) refer to those $\beta $, $\alpha $ pairs that allow concave central configurations of four bodies. The triangles of the first and second concave cases are denoted by $C1$ and $C2$, respectively. Letter S at $\alpha ={60}^{\circ}$, $\beta ={30}^{\circ}$ indicates a singular point where $3{\mu}_{1}+{\mu}_{2}=1$. Along the blue curve, the bodies A, E, and ${E}^{\prime}$ have equal masses, while the masses of B, E, and ${E}^{\prime}$ become equal along the red curve. The two curves have two intersections, one at the singular point S, the other one at $\alpha =61.\phantom{\rule{-0.2cm}{0ex}}{}^{\circ}177$, $\beta =33.\phantom{\rule{-0.2cm}{0ex}}{}^{\circ}039$ (see a magnification in Figure 5). Letters ${P}_{3}$, ${P}_{4}$, ${P}_{5}$, and ${P}_{7}$ indicate the endpoints of the solutions curves on critical lines.

**Figure 5.**Magnification of the intersections of the blue $\mu ={\mu}_{1}$ and red $\mu ={\mu}_{2}$ curves, indicated by black dots. Both occur in the second concave case ($C2$); the first at $\alpha ={60}^{\circ}$, $\beta ={30}^{\circ}$ and the second at $\alpha =61.\phantom{\rule{-0.2cm}{0ex}}{}^{\circ}177$, $\beta =33.\phantom{\rule{-0.2cm}{0ex}}{}^{\circ}039$.

**Figure 6.**Normalized masses in the function of the angle coordinate $\beta $ (convex case). Both the $\mu ={\mu}_{1}$ (blue) and $\mu ={\mu}_{2}$ (red) solutions are displayed. The black solid and dashed curves refer to the residual mass ${\mu}_{2}$ or ${\mu}_{1}$ as opposed to the three equal masses. The generating point at $\beta ={45}^{\circ}$ is emphasized by a black dot and indicates the common mass $\mu ={\mu}_{1}={\mu}_{2}=1/4$. Note the reversed direction of the horizontal axis.

**Figure 7.**Normalized masses in the function of the angle coordinate $\beta $ (concave cases, $\mu ={\mu}_{1}$). The black solid line refers to the residual mass ${\mu}_{2}$. The pink dots at $\beta ={30}^{\circ}$ indicate the masses corresponding to Palmore’s constant. The intersection at $\beta =33.\phantom{\rule{-0.2cm}{0ex}}{}^{\circ}039$ is emphasized by a black dot and indicates the common mass $\mu ={\mu}_{1}={\mu}_{2}=1/4$.

**Figure 8.**(

**a**) Normalized masses in the function of the angle coordinate $\beta $ (concave cases, $\mu ={\mu}_{2}$). The black dashed line refers to the residual mass ${\mu}_{1}$. The intersections at $\beta ={30}^{\circ}$ and at $\beta =33.\phantom{\rule{-0.2cm}{0ex}}{}^{\circ}039$ are emphasized by black dots and indicate the common mass $\mu ={\mu}_{1}={\mu}_{2}=1/4$. (

**b**) A magnification of panel (

**a**) in the region between the intersections. The ratio of the masses ${\mu}_{1}=0.250499$ and $\mu ={\mu}_{2}=0.249834$ at the green dots corresponds to the maximum ${m}^{*}$ of [11].

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**MDPI and ACS Style**

Kővári, E.; Érdi, B.
The Axisymmetric Central Configurations of the Four-Body Problem with Three Equal Masses. *Symmetry* **2020**, *12*, 648.
https://doi.org/10.3390/sym12040648

**AMA Style**

Kővári E, Érdi B.
The Axisymmetric Central Configurations of the Four-Body Problem with Three Equal Masses. *Symmetry*. 2020; 12(4):648.
https://doi.org/10.3390/sym12040648

**Chicago/Turabian Style**

Kővári, Emese, and Bálint Érdi.
2020. "The Axisymmetric Central Configurations of the Four-Body Problem with Three Equal Masses" *Symmetry* 12, no. 4: 648.
https://doi.org/10.3390/sym12040648