# Monitoring Jovian Orbital Resonances of a Spacecraft: Classical and Relativistic Effects

## Abstract

**:**

## 1. Introduction

## 2. Modelling Jovian Resonances: The Newtonian Approach

## 3. Relativistic Effects in Resonant Orbits

#### Numerical Methods

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Ideal circular orbits for Jupiter and the spacecraft in the 2:1 resonant configuration. The position vector of Jupiter with respect to the Sun is denoted by $\mathbf{R}$ and $\mathbf{r}$ for the spacecraft. Initially, there is an angular difference of $\varphi $ radians in their orbital positions.

**Figure 2.**Distance of the spacecraft to the Sun in the resonance 2:1 with Jupiter. The initial phase is $\varphi =\pi /8$ (solid line) and $\varphi =-\pi /8$ (dotted line). Units of distance and time are scaled.

**Figure 3.**Argument of the perihelion for a spacecraft in the resonance the resonance 1:1 with Jupiter and initial phase $\varphi =\pi /8$.

**Figure 4.**Relative differences in the distances of the spacecraft to the Sun in the resonance 1:1 with Jupiter with initial phase $\varphi =\pi /8$. The pattern is not periodic, and most likely, is chaotic in the scale of several centuries.

**Figure 5.**Distances of the spacecraft to Jupiter in the same conditions as those of Figure 4. The unit of distance is ${R}_{0}$ (Sun–Jupiter average distance).

**Figure 6.**Relative differences of the distances of the spacecraft to Jupiter (relativistic prediction minus Newtonian prediction) in the same conditions as those of Figure 4.

**Figure 7.**Distance of the spacecraft to the Sun for $\varphi =-\pi /32$: Newtonian prediction (black line) and the relativistic one (blue line).

**Figure 8.**Difference between the distances of the spacecraft to the Sun as predicted for the post-Newtonian and Newtonian approximations in a resonant 1:1 orbit with $\varphi =-\pi /32$. Integration by the Runge–Kutta order four method (solid line) and the fourth order Adams predictor-corrector method (open circles).

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Acedo, L.
Monitoring Jovian Orbital Resonances of a Spacecraft: Classical and Relativistic Effects. *Universe* **2019**, *5*, 222.
https://doi.org/10.3390/universe5120222

**AMA Style**

Acedo L.
Monitoring Jovian Orbital Resonances of a Spacecraft: Classical and Relativistic Effects. *Universe*. 2019; 5(12):222.
https://doi.org/10.3390/universe5120222

**Chicago/Turabian Style**

Acedo, Luis.
2019. "Monitoring Jovian Orbital Resonances of a Spacecraft: Classical and Relativistic Effects" *Universe* 5, no. 12: 222.
https://doi.org/10.3390/universe5120222