Progression of bifurcated family f type periodic orbits in the circular restricted three-body problem

  • Authors

    • Nishanth Pushparaj Department of Aerospace Engineering, Karunya University, Coimbatore - 641114, Tamil Nadu, India
    • Ram Krishan Sharma Department of Aerospace Engineering, Karunya University, Coimbatore - 641114, Tamil Nadu, India
    2017-07-06
    https://doi.org/10.14419/ijaa.v5i2.7776
  • Restricted Three-Body Problem, Poincaré Surfaces of Section, Solar Radiation Pressure, Sun-Jupiter System, Direct and Retrograde Periodic Orbit.

  • Progression of f-type family of periodic orbits, their nature, stability and location nearer the smaller primary for different mass ratios in the framework of circular restricted three-body problem is studied using Poincaré surfaces of section. The orbits around the smaller primary are found to decrease in size with increase in Jacobian Constant C, and move very close towards the smaller primary. The orbit bifurcates into two orbits with the increase in C to 4.2. The two orbits that appear for this value of C belong to two adjacent separate families: one as direct orbit belonging to family g of periodic orbits and other one as retrograde orbit belonging to family f of periodic orbits. This bifurcation is interesting. These orbits increase in size with increase in mass ratio. The elliptic orbits found within the mass ratio 0 < µ ≤ 0.1 have eccentricity less than 0.2 and the orbits found above the mass ratio µ > 0.1 are elliptical orbits with eccentricity above 0.2. Deviations in the parameters: eccentricity, semi-major axis and time period of these orbits with solar radiation pressure q are computed in the frame work of photogravitational restricted Three-body problem in addition to the restricted three-body problem. These parameters are found to decrease with increase in the solar radiation pressure.

  • References

    1. [1] Beevi, A.S. & Sharma, R.K. (2011) Analysis of periodic orbits in the Saturn-Titan system using the method of Poincare section surfaces. Astrophysics and Space Science 333, 37-48. https://doi.org/10.1007/s10509-011-0630-0.

      [2] Beevi, A.S. & Sharma, R.K. (2012) Oblateness effect of Saturn on periodic orbits in Saturn- Titan restricted three-body problem. Astrophysics and Space Science 340, 245-261. https://doi.org/10.1007/s10509-012-1052-3.

      [3] Broucke, R.A. (1968) Periodic orbits in the restricted three-body problem with Earth-Moon masses. Pasadena, CA: Jet Propulsion Lab.

      [4] Bruno, A. (1990) The Restricted 3-Body Problem. Plane Periodic Orbits. Nauka, 296.

      [5] Bruno, A. & Varin, V. (2006) On families of periodic solutions of the restricted three-body problem. Celestial Mechanics & Dynamical Astron. 95, 27-54. https://doi.org/10.1007/s10569-006-9021-1.

      [6] Dutt, P. & Sharma, R.K. (2010) Analysis of periodic and quasi-periodic orbits in the Earth- Moon system. Journal of Guidance, Control and Dyn. 33, 1010-1017. https://doi.org/10.2514/1.46400.

      [7] Dutt, P. & Sharma, R.K. (2011) Evolution of periodic orbits near the Lagrangian point L2. Advances in Space Res. 47, 1894-1904. https://doi.org/10.1016/j.asr.2011.01.024.

      [8] Dutt, P. & Sharma, R.K. (2011) Evolution of Periodic Orbits in the Sun-Mars system. Journal of Guidance, Control and Dyn. 34, 635-644. https://doi.org/10.2514/1.51101.

      [9] Hénon, M. (1969) Numerical exploration of the restricted problem. V.Astron.Astrophys. 1, 223-238.

      [10] Jefferys, W. H. (1971) An Atlas of Surface of Section for the Restricted Problem of Three bodies. Austin,Texas: Dept. of Astronomy,University of Texas.

      [11] Kalvouridis, T., Arribas, M. & Elipe, A. (2007) Parametric Evolution of Periodic Orbits in Restricted Four-Body Problem with Radiation Pressure. Planetary and Space Sci. 55, 475-493. https://doi.org/10.1016/j.pss.2006.07.005.

      [12] Murray, C.D. & Dermott, S.F. (1999) Solar System Dynamics. Cambridge: Cambridge University Press.

      [13] Poincaré, H. (1892) Les Methodes Nouvellas de la Mechanique Celeste (Vol. 1). Paris: Gauthier-Villars.

      [14] Poynting, J. (1904) Radiation in the Solar system: It’s Effect on Temperature and its pressure on Small Bodies. Philosophical Transactions of the Royal Society of London.Series A, Physical Sciences and Engineering 202, 525-552. https://doi.org/10.1098/rsta.1904.0012.

      [15] Pushparaj, N. and Sharma, R.K. (2017) Interior Resonance Periodic Orbits in the Photogravitational Restricted Three-Body Problem. Advances in Astrophys. 2, 25-34.

      [16] Radzievskii, V. (1950) The Restricted problem of Three Bodies Taking Account of Light Pressure. Akademii Nauk USSR, Astronomicheskii Zhurnal 27, 250-256.

      [17] Robertson, H. (1937) Dynamical Effects of Radiation in the solar system. Monthly Noties of the Royal Astronomical Society 97, 423-438. https://doi.org/10.1093/mnras/97.6.423.

      [18] Sharma, R.K. (1975) Perturbations of Lagrangian points in the restricted three-body problem, Indian J Pure and Appl. Math. 6, 1099-1102.

      [19] Sharma, R.K. ( 1982) On linear stability of triangular libration points of the photogravitational three-body problem when the more massive primary is an oblate spheroid, Sun and Planetary System, W. Fricke and G. Teleki (Eds.), D. Reidel Publishing Co., Dordrecht, Holland, 435-436.

      [20] Sharma, R.K. (1987). The linear stability of libration points of the photogravitational restricted three-body problem when the smaller primary is an oblate spheroid, Astrophys. And Space Sci. 135, 271-281. https://doi.org/10.1007/BF00641562.

      [21] Sharma, R. K. & Subba Rao, P.V. (1976) Stationary solutions and their characteristic exponents in the restricted three-body problem when the more massive primary is an oblate spheroid. Celestial Mech. 13, 137-149. https://doi.org/10.1007/BF01232721.

      [22] Sharma, R. K. & Subba Rao, P.V. (1986) on finite periodic orbits around the equilateral solutions of the planar restricted three-body problem, International workshop on Space Dynamics and Celestial mechanics, November 1985, Delhi, India, K. B. Bhatnagar (Ed), D. Reidel Publishing Co., Dordrecht, Holland, pages 71-85. https://doi.org/10.1007/978-94-009-4732-0_8.

      [23] Smith, R. H., & Szebehely, V. (1993) the onset of chaotic motion in the restricted problem of three bodies. Celest. Mech. 56, 409-425. https://doi.org/10.1007/BF00691811.

      [24] Subba Rao, P. V. & Sharma R. K. (1988) Oblateness effect on finite periodic orbits at L4, IAF-88-300, 5 pages, 39th Congress of the International Astronautical Federation, October 1988, Bangalore, India.

      [25] Szebehely, V. (1967) Theory of Orbits. San Diego: Academic Press.

      [26] Voyatzis, G., Kotoulas, T. & Hadjidemetriou, J. D. (2005) Symmetric and nonsymmetric periodic orbits in the exterior mean motion resonances with Neptune. Celestial Mechanics and Dynamical Astronomy, 91, 191-202. https://doi.org/10.1007/s10569-004-0891-9.

      [27] Winter, O. C. & Murray, C. D. (1994b) Atlas of the Planar, Circular, Restricted Threebody Problem: External orbits. Queen Mary and Westfield College,London: QMW Maths Notes.

      [28] Winter, O. C. & Murray, C. D. (1994a) Atlas of the Planar, Circular, Restricted Threebody Problem: Internal orbits. Queen Mary and Westfield College, London: QMW Maths Notes.

      [29] Winter, O. C. & Murray, C. D. (1997a) Resonance and chaos. I. First-order interior resonances. Astronomy and Astrophys. 319, 290-304.

      [30] Winter, O. C. & Murray, C. D. (1997b). Resonance and chaos. II. Exterior resonances and asymmetric libration. Astronomy and Astrophys. 328, 399-408.

      [31] Zotos, E. E. (2015) Crash test for the Copenhagen problem with oblateness. Celest Mech & Dyn Astron. 122, 75-99. https://doi.org/10.1007/s10569-015-9611-x.

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    Pushparaj, N., & Sharma, R. K. (2017). Progression of bifurcated family f type periodic orbits in the circular restricted three-body problem. International Journal of Advanced Astronomy, 5(2), 69-78. https://doi.org/10.14419/ijaa.v5i2.7776