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Long-term perturbations due to a disturbing body in elliptic inclined orbit

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Abstract

In the current study, a double-averaged analytical model including the action of the perturbing body’s inclination is developed to study third-body perturbations. The disturbing function is expanded in the form of Legendre polynomials truncated up to the second-order term, and then is averaged over the periods of the spacecraft and the perturbing body. The efficiency of the double-averaged algorithm is verified with the full elliptic restricted three-body model. Comparisons with the previous study for a lunar satellite perturbed by Earth are presented to measure the effect of the perturbing body’s inclination, and illustrate that the lunar obliquity with the value 6.68∘ is important for the mean motion of a lunar satellite. The application to the Mars-Sun system is shown to prove the validity of the double-averaged model. It can be seen that the algorithm is effective to predict the long-term behavior of a high-altitude Martian spacecraft perturbed by Sun. The double-averaged model presented in this paper is also applicable to other celestial systems.

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References

  • Ash, M.E.: Double averaged effect of the Moon and Sun on a high altitude Earth satellite orbit. Celest. Mech. Dyn. Astron. 14(2), 209–238 (1976). doi:10.1007/BF01376321

    MathSciNet  Google Scholar 

  • Broucke, R.A.: Long-term third-body effects via double averaging. J. Guid. Control Dyn. 26(1), 27–32 (2003). doi:10.2514/2.5041

    Article  Google Scholar 

  • Carvalho, J.P.S., Moraes, R.V., Prado, A.F.B.A.: Semi-analytic theory of a Moon artificial satellite considering lunar oblateness and perturbations due to a third-body in elliptic orbit. In: Proceedings of the 7th Brazilian Conference on Dynamics, Control and Applications, Presidente Prudente, Brazil, pp. 51–57 (2008)

    Google Scholar 

  • Carvalho, J.P.S., Moraes, R.V., Prado, A.F.B.A.: Nonsphericity of the Moon and near sun-synchronous polar lunar orbits. Math. Probl. Eng. 2009, 740460 (2009a). doi:10.1155/2009/740460

    Article  Google Scholar 

  • Carvalho, J.P.S., Moraes, R.V., Prado, A.F.B.A.: Non-sphericity of the Moon and critical inclination. In: Proceedings of the 32nd Congresso Nacional de Matemática Aplicada e Computacional. Cuiabá, Brazil, vol. 2, pp. 464–470 (2009b)

    Google Scholar 

  • Carvalho, J.P.S., Elipe, A., Moraes, R.V., Prado, A.F.B.A.: Frozen orbits around the Europa satellite. In: International Conference on Chaos and Nonlinear Dynamics, Brazil, July 26–30 (2010a)

    Google Scholar 

  • Carvalho, J.P.S., Moraes, R.V., Prado, A.F.B.A.: Some orbital characteristics of lunar artificial satellites. Celest. Mech. Dyn. Astron. 108(4), 371–388 (2010b). doi:10.1007/s10569-010-9310-6

    Article  ADS  MATH  Google Scholar 

  • Carvalho, J.P.S., Moraes, R.V., Prado, A.F.B.A.: Planetary satellite orbiters: applications for the Moon. Math. Probl. Eng. 2011, 187478 (2011). doi:10.1155/2011/187478

    Article  Google Scholar 

  • Chobotov, V.A.: Orbital Mechanics, 3rd edn. AIAA, Reston (2002)

    Google Scholar 

  • Collins, S.K., Cefola, P.J.: Double averaged third body model for prediction of super-synchronous orbits over long time spans. In: Astrodynamics Specialist Conference (AAS), pp. 79–135 (1979)

    Google Scholar 

  • Cook, G.E.: Luni-solar perturbations of the orbit of an Earth satellite. Geophys. J. Int. 6(3), 271–291 (1962). doi:10.1111/j.1365-246X.1962.tb00351.x

    Article  ADS  MATH  Google Scholar 

  • Delhaise, F., Morbidelli, A.: Luni-solar effects of geosynchronous orbits at the critical inclination. Celest. Mech. Dyn. Astron. 57(1–2), 155–173 (1993). doi:10.1007/BF00692471

    ADS  MATH  Google Scholar 

  • Domingos, R.C., Moraes, R.V., Prado, A.F.B.A.: Third-body perturbation in the case of elliptic orbits for the disturbing body. Math. Probl. Eng. 2008, 763654 (2008). doi:10.1155/2008/763654

    Article  Google Scholar 

  • Ely, T.A.: Stable constellations of frozen elliptical inclined lunar orbits. J. Astronaut. Sci. 53(3), 301–316 (2005)

    MathSciNet  Google Scholar 

  • Ely, T.A., Lieb, E.: Constellations of elliptical inclined lunar orbits providing polar and global coverage. J. Astronaut. Sci. 54(1), 53–67 (2006)

    MathSciNet  Google Scholar 

  • Ferrer, S., Osacar, C.: Harrington’s Hamiltonian in the stellar problem of three bodies: reductions, relative equilibria and bifurcations. Celest. Mech. Dyn. Astron. 58(3), 245–275 (1994). doi:10.1007/BF00691977

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Folta, D., Quinn, D.: Lunar frozen orbits. In: AIAA/AAS Astrodynamics Specialist Conference and Exhibit (AIAA 2006-6749) (2006)

    Google Scholar 

  • Harrington, R.S.: The stellar three-body problem. Celest. Mech. Dyn. Astron. 1(2), 200–209 (1969). doi:10.1007/BF01228839

    Google Scholar 

  • Hough, M.E.: Orbits near critical inclination, including lunisolar perturbations. Celest. Mech. Dyn. Astron. 25(2), 111–136 (1981). doi:10.1007/BF01230514

    MATH  Google Scholar 

  • Kinoshita, H., Nakai, H.: Secular perturbations of fictitious satellites of Uranus. Celest. Mech. Dyn. Astron. 52(3), 293–303 (1991). doi:10.1007/BF00048489

    Article  ADS  Google Scholar 

  • Kwok, J.H.: Long-term orbit prediction using an averaging method. AIAA 84-1985 (1985)

  • Kwok, J.H.: Doubly averaging method for third body perturbations. AAS 91-464 (1991)

  • Lane, M.T.: On analytic modeling of lunar perturbations of artificial satellites of the earth. Celest. Mech. Dyn. Astron. 46(4), 287–305 (1989). doi:10.1007/BF00051484

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Lara, M.: Three-body dynamics around the smaller primary. Application to the design of science orbits. J. Aerospace Eng. Sci. Appl. 2(1), 53–65 (2010)

    Google Scholar 

  • Lara, M.: Design of long-lifetime lunar orbits: a hybrid approach. Acta Astronaut. 69(3–4), 186–199 (2011). doi:10.1016/j.actaastro.2011.03.009

    Article  ADS  Google Scholar 

  • Lara, M., Saedeleer, B., Ferrer, S.: Preliminary design of low lunar orbits. In: Proceedings of the 21st International Symposium on Space Flight Dynamics, Toulouse, France, pp. 1–15 (2009)

    Google Scholar 

  • Lidov, M.L., Ziglin, S.L.: The analysis of restricted circular twice-averaged three body problem in the case of close orbits. Celest. Mech. Dyn. Astron. 9(2), 151–173 (1974). doi:10.1007/BF01260510

    MATH  Google Scholar 

  • Lorell, J.: Long term behavior of artificial satellite orbits due to third body perturbations. J. Astronaut. Sci. 12, 142–152 (1965)

    ADS  Google Scholar 

  • Murray, C.D., Dermott, S.F.: Solar System Dynamics. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  • Musen, P., Bailie, A., Upton, E.: Development of the lunar and solar perturbations in the motion of an artificial satellite. NASA Goddard Spaceflight Center technical note D-494 (1961)

  • Paskowitz, M.E., Scheeres, D.J.: Orbit mechanics about planetary satellites including higher order gravity fields. In: Proceedings of the AAS/AIAA 15th Space Flight Mechanics Meetings. AAS 2005-190 (2005)

    Google Scholar 

  • Paskowitz, M.E., Scheeres, D.J.: Robust capture and transfer trajectories for planetary satellite orbiters. J. Guid. Control Dyn. 29(2), 342–353 (2006a). doi:10.2514/1.13761

    Article  Google Scholar 

  • Paskowitz, M.E., Scheeres, D.J.: Design of science orbits about planetary satellites: application to Europa. J. Guid. Control Dyn. 29(5), 1147–1158 (2006b). doi:10.2514/1.19464

    Article  Google Scholar 

  • Prado, A.F.B.A.: Third-body perturbation in orbits around natural satellites. J. Guid. Control Dyn. 26(1), 33–40 (2003). doi:10.2514/2.5042

    Article  Google Scholar 

  • Radwan, M.: Resonance caused by the luni-solar attractions on a satellite of the oblate Earth. Astrophys. Space Sci. 282(3), 551–562 (2002). doi:10.1007/s10509-005-6563-8

    Article  MathSciNet  ADS  Google Scholar 

  • Russell, R.P., Brinckerhoff, A.T.: Circulating eccentric orbits around planetary moons. J. Guid. Control Dyn. 32(2), 423–435 (2009). doi:10.2514/1.38593

    Article  Google Scholar 

  • Scheeres, D.J., Guman, M.D., Villac, B.F.: Stability analysis of planetary satellite orbiters: application to the Europa orbiter. J. Guid. Control Dyn. 24(4), 778–787 (2001). doi:10.2514/2.4778

    Article  Google Scholar 

  • Šidlichovský, M.: On the double averaged three-body problem. Celest. Mech. Dyn. Astron. 29(3), 295–305 (1983). doi:10.1007/BF01229141

    MATH  Google Scholar 

  • Solórzano, C.R.H., Prado, A.F.B.A.: Third-body perturbation using single averaged model: Application to lunisolar perturbations. Nonlinear Dyn. Syst. Theory 7(4), 409–417 (2007)

    MathSciNet  MATH  Google Scholar 

  • Williams, J.G., Benson, G.S.: Resonances in the Neptune-Pluto system. Astron. J. 76, 167–176 (1971). doi:10.1086/111100

    Article  ADS  Google Scholar 

  • Winter, O.C., Mourão, D.C., Melo, C.F., Macau, E.N., Ferreira, J.L., Carvalho, J.P.S.: Controlling the eccentricity of polar lunar orbits with low-thrust propulsion. Math. Probl. Eng. 2009, 159287 (2009). doi:10.1155/2009/159287

    Article  Google Scholar 

  • Yokoyama, T.: Dynamics of some fictitious satellites of Venus and Mars. Planet. Space Sci. 47(5), 619–627 (1999). doi:10.1016/S0032-0633(98)00110-X

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work was supported by National Basic Research Program of China (973 Program) (2012CB720000) and the National Natural Science Foundation of China (No. 11072122).

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  1. School of Aerospace, Tsinghua University, 100084, Bei**g, China

    **aodong Liu, Hexi Baoyin & **ngrui Ma

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  1. **aodong Liu
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  2. Hexi Baoyin
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  3. **ngrui Ma
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Correspondence to **aodong Liu.

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Liu, X., Baoyin, H. & Ma, X. Long-term perturbations due to a disturbing body in elliptic inclined orbit. Astrophys Space Sci 339, 295–304 (2012). https://doi.org/10.1007/s10509-012-1015-8

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  • DOI: https://doi.org/10.1007/s10509-012-1015-8

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