#
Periodic Orbits of Third Kind in the Zonal J_{2} + J_{3} Problem

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction and Theoretical Background

## 2. Main Results

#### 2.1. Hamiltonian Description of the Model: The Zonal ${J}_{2}+{J}_{3}$ Problem

#### 2.2. Averaged Potential $\u2329{\mathcal{H}}_{1}\u232a$

#### 2.3. Equilibrium Points of the Differential System

**Theorem**

**1**

**.**For the system (4), there are two families of equilibrium points:

- (a)
- $K=0$ (polar orbits). This family is parameterized by:$$\begin{array}{cc}\hfill L& =\sqrt{\frac{1+4{e}^{2}}{2e(1-{e}^{2})}},\hfill \\ \hfill G& =\sqrt{\frac{1+4{e}^{2}}{2e}},\hfill \\ \hfill g& =\pm \frac{\pi}{2}\hfill \end{array}$$
- (b)
- $K\ne 0$. This family is parameterized by:$$\begin{array}{cc}\hfill G& ={\left(\frac{\sqrt{1875-1125{cos}^{2}\left(i\right)-935{sec}^{2}\left(i\right)+121{sec}^{4}\left(i\right)}}{2\sqrt{10\tau}}\right)}^{1/2},\hfill \\ \hfill L& =G{\left(\frac{3+5{cos}^{4}\left(i\right)}{1+12{cos}^{2}\left(i\right)+5{cos}^{4}\left(i\right)}\right)}^{1/2},\hfill \\ \hfill K& =Gcos\left(i\right),\hfill \\ \hfill g& =\pm \frac{\pi}{2},\hfill \end{array}$$

**Proof.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**2**

**.**For each energy level ${h}^{*}<0$ such that:

**Proof.**

## 3. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Theorem**

**A1.**

## Appendix B. Explicit Expression of the Hamiltonian ${\mathcal{H}}_{1}$ Written as a Function of the Variables E, g, L, G, and K

## References

- Poincaré, H. Les Méthodes Nouvelles de la Mécanique Céleste; Dover Publications Inc.: New York, NY, USA, 1957. [Google Scholar]
- Abouelmagd, E.I.; Guirao, J.L.G.; Llibre, J. Periodic Orbits for the Perturbed Planar Circular Restricted 3-Body Problem. Discrete Contin. Dyn. Syst.-Ser. B
**2019**, 24, 1007–1020. [Google Scholar] [CrossRef] - NPOESS (National Polar-Orbiting Operational Environmental Satellite System). Available online: https://directory.eoportal.org/web/eoportal/satellite-missions/n/npoess (accessed on 15 September 2018).
- Products. All Lockheed Martin Products. Available online: http://www.lockheedmartin.com/us/products/dmsp.html (accessed on 15 September 2018).
- Coffey, S.L.; Deprit, A.; Deprit, E. Frozen orbits for satellites close to an Earth-like planet. Celest. Mech. Dyn. Astron.
**1994**, 59, 37–72. [Google Scholar] [CrossRef] - Rosborough, G.W.; Ocampo, C.A. Influence of higher degree zonals on the frozen orbit geometry. Astrodynamics
**1992**, 76, 1291–1304. [Google Scholar] - Tzirti, S.; Tsiganis, K.; Varvoglis, H. Effect of 3rd-degree gravity harmonics and earth perturbations on lunar artificial satellite orbits. Celest. Mech. Dyn. Astron.
**2010**, 108, 405–416. [Google Scholar] [CrossRef] - Vallado, D. Fundamentals of Astrodynamics and Applications; Springer: New York, NY, USA, 2007. [Google Scholar]
- Brouwer, D. Solution of the problem of artificial satellite theory without drag. Astron. J.
**1959**, 64, 378–397. [Google Scholar] [CrossRef] - Abouelmagd, E.I.; Guirao, J.L.G. On the perturbed restricted three-body problem. Appl. Math. Nonlinear Sci.
**2016**, 1, 123–144. [Google Scholar] [CrossRef] - Abouelmagd, E.I.; Guirao, J.L.G.; Vera, J.A. Dynamics of a Dumbbel satellite under the zonal harmonic effect of an oblate body. Commun. Nonlinear Sci. Numer. Simul.
**2015**, 20, 1057–1069. [Google Scholar] [CrossRef] - Fernández, M.; López, M.A.; Vera, J.A. On the dynamics of planar oscillations for a dumbbell satellite in J
_{2}problem. Nonlinear Dyn.**2016**, 84, 143–151. [Google Scholar] [CrossRef] - Guirao, J.L.G.; Llibre, J.; Vera, J.A. Periodic orbits of Hamiltonian systems: Applications to perturbed Kepler problems. Chaos Solitons Fract.
**2013**, 57, 105–111. [Google Scholar] [CrossRef] [Green Version] - Carvalho, J.P.S.; Elipe, A.; de Moraes, R.V.; Prado, A.F.B.A. Low-altitude, near-polar and near-circular orbits around Europa. Adv. Space Rese
**2012**, 49, 994–1006. [Google Scholar] [CrossRef] - Cordani, B. The Kepler problem. In Progress in Mathematical Physics; Birkhäuser Basel: Basilea, Switzerland, 2003. [Google Scholar]
- Kaula, W.M. Theory of Satellite Geodesy; Blaisdell Publ. Co.: Waltham, MA, USA, 1966. [Google Scholar]
- Meyer, K.R.; Hall, G.R.; Offin, D. Introduction to Hamiltonian Dynamical Systems and the N-Body Problem; Springer: New York, NY, USA, 2009. [Google Scholar]
- Vinti, J.P. Zonal harmonic perturbations of an accurate reference orbit of an artificial satellite. J. Natl. Bur. Stand.
**1963**, 67, 191–222. [Google Scholar] [CrossRef] - Giacaglia, G.E.O.; Murphy, J.P.; Felsentreger, T.L. A semi-analytic theory for the motion of a lunar satellite. Celest. Mech.
**1970**, 3, 3–66. [Google Scholar] [CrossRef] [Green Version] - Coffey, S.L.; Deprit, A.; Miller, B.R. The critical inclination in artificial satellite theory. Celest. Mech.
**1986**, 39, 365–406. [Google Scholar] [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

de Bustos, M.T.; Fernández, A.; López, M.A.; Martínez, R.; Vera, J.A.
Periodic Orbits of Third Kind in the Zonal *J*_{2} + *J*_{3} Problem. *Symmetry* **2019**, *11*, 111.
https://doi.org/10.3390/sym11010111

**AMA Style**

de Bustos MT, Fernández A, López MA, Martínez R, Vera JA.
Periodic Orbits of Third Kind in the Zonal *J*_{2} + *J*_{3} Problem. *Symmetry*. 2019; 11(1):111.
https://doi.org/10.3390/sym11010111

**Chicago/Turabian Style**

de Bustos, M. Teresa, Antonio Fernández, Miguel A. López, Raquel Martínez, and Juan A. Vera.
2019. "Periodic Orbits of Third Kind in the Zonal *J*_{2} + *J*_{3} Problem" *Symmetry* 11, no. 1: 111.
https://doi.org/10.3390/sym11010111