# Revisiting the Dynamics of Two-Body Problem in the Framework of the Continued Fraction Potential

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description and Equations of Motion

_{1}m

_{2}(m

_{1}is the mass of the primary body, m

_{2}is the mass of the secondary body), G is the universal gravitation constant, r denotes the separation distance, and ε signifies the parameter of the perturbed force originating from the continued fraction potential.

**r**and

_{1}**r**be the position vectors of the two bodies with respect to the origin of the inertial reference frame, while

_{2}**r**=

**r**−

_{2}**r**is the relative position vector of the body m

_{1}_{2}with respect to the body m

_{1}. Thus, with the help of Equation (1), the vectorial equation of motion of the second body around the first is given by (2)

**r**.

_{0}, coinciding with the center of mass of the celestial body m

_{1}. This body is either in dynamical equilibrium with outer space or moves uniformly in a straight line (without rotation) at a constant velocity. Let us then examine Equations (11A) and (11B), which describe the dynamics of the mass point m

_{2}in the aforementioned formulation from the work [8]. These equations are treated here as a Cauchy problem with given initial conditions and are represented in the forms (3) and (4) below:

_{1}m

_{2}λ

_{1}is a parameter that represents the perturbed force of the continued fraction potential in the problem, and |λ

_{1}| = const « 1.

## 3. Solving Procedure for the System of Equations (3) and (4)

^{2}r/dt

^{2}= y dy/dr), we thus have

_{1}, C

_{2}, C

_{3}are the constants, determined by the initial conditions)

_{1}+ C

_{2}+ C

_{3}):

## 4. Analytical Results

_{2}, in orbit around a more massive primary body, m

_{1}, (within the framework of the restricted two-body problem, R2BP) has been introduced. Such an approach involves the consideration of a continued fraction potential instead of the classical potential function in Kepler’s formulation of the R2BP. Simultaneously, a system of equations of motion has been successfully explored to identify an analytical means of representing the solution in polar coordinates, {r(t), φ(t)}. An analytical approach for obtaining the function t = t(r), incorporating an elliptic integral, was developed. By establishing the inverse function r = r(t), a further solution can be derived via quasi-period cycles. Consequently, the previously mentioned restricted two-body problem (R2BP) with a continued fraction potential is fully and analytically solved.

_{0}}, and the right part of the quadrature in (9) equals (t − t

_{0}), where t

_{0}= 0. Let us re-inverse this expression into the dependence r = r(t), which can be obtained by numerical methods only (by an appropriate approximation technique or, e.g., by a series of Taylor expansions), and then afterwards, a solution can be obtained to be presented in terms of quasi-periodic cycles.

_{0}), B(r

_{0}) are the constants, determined by the initial conditions)

_{0}> 0. Thus, (11) can be simplified using the expressions in (12) as follows:

## 5. Discussion

## 6. Conclusions

## 7. Remarks (with Highlights)

- The novel approach is applied to obtain the stable orbit of a planetoid in a modified ER2BP.
- A continued fraction potential is considered here instead of Kepler’s formulation.
- The perturbation effect on the Kepler gravitational potential is tackled in this scheme.
- The modified ER2BP with a continued fraction potential has been fully solved.
- The orbiter demonstrates stable dynamics, as shown by the numerical findings presented in graphical plots.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Periodic results of semi-analytical calculations for the radius of orbit, r(t) = 0.01t + 4, schematically presented by graphical approximation with the help of (13) in polar coordinates.

**Figure 2.**Periodic results of semi-analytical calculations for the radius of orbit of the mass point, r(t) = 0.005t − 4, schematically presented by graphical approximation with the help of another type of Formula (13) in polar coordinates.

**Figure 3.**Periodic results of semi-analytical calculations for the radius of orbit of the mass point, r(t) = 0.01t − 100, schematically presented by graphical approximation with the help of the classical variant of Formula (13) in polar coordinates (when we choose ε → 0 in (13)).

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**MDPI and ACS Style**

Ershkov, S.; Mohamdien, G.F.; Idrisi, M.J.; Abouelmagd, E.I.
Revisiting the Dynamics of Two-Body Problem in the Framework of the Continued Fraction Potential. *Mathematics* **2024**, *12*, 590.
https://doi.org/10.3390/math12040590

**AMA Style**

Ershkov S, Mohamdien GF, Idrisi MJ, Abouelmagd EI.
Revisiting the Dynamics of Two-Body Problem in the Framework of the Continued Fraction Potential. *Mathematics*. 2024; 12(4):590.
https://doi.org/10.3390/math12040590

**Chicago/Turabian Style**

Ershkov, Sergey, Ghada F. Mohamdien, M. Javed Idrisi, and Elbaz I. Abouelmagd.
2024. "Revisiting the Dynamics of Two-Body Problem in the Framework of the Continued Fraction Potential" *Mathematics* 12, no. 4: 590.
https://doi.org/10.3390/math12040590