Restricted Three Body Problem with Stokes Drag Effect
Affiliation(s)
1 Department of Mathematics, Shri Venkateshwara University, Gajraula, India. 2 Department of Mathematics, Sri Aurobindo College, University of Delhi, Delhi, India.
1 Department of Mathematics, Shri Venkateshwara University, Gajraula, India. 2 Department of Mathematics, Sri Aurobindo College, University of Delhi, Delhi, India.
ABSTRACT
The existence and stability of stationary solutions of the restricted three body problem under the effect of the dissipative force, Stokes drag, are investigated. It is observed that there exist two non collinear stationary solutions. Further, it is also found that these stationary solutions are unstable for all values of the parameters.
The existence and stability of stationary solutions of the restricted three body problem under the effect of the dissipative force, Stokes drag, are investigated. It is observed that there exist two non collinear stationary solutions. Further, it is also found that these stationary solutions are unstable for all values of the parameters.
KEYWORDS
Restricted Three Body Problem, Libration Points, Linear Stability, Dissipative Forces, Stokes Drag
Restricted Three Body Problem, Libration Points, Linear Stability, Dissipative Forces, Stokes Drag
Cite this paper
Jain, M. and Aggarwal, R. (2015) Restricted Three Body Problem with Stokes Drag Effect. International Journal of Astronomy and Astrophysics, 5, 95-105. doi: 10.4236/ijaa.2015.52013.
Jain, M. and Aggarwal, R. (2015) Restricted Three Body Problem with Stokes Drag Effect. International Journal of Astronomy and Astrophysics, 5, 95-105. doi: 10.4236/ijaa.2015.52013.
References
[1] Euler, L. (1772) Theoria Motuum Lunae, Typis Academiar Imperialis Seientiarum, Petropoli. Reprinted in Opera Omnia, Series 2, Courvoisier, L., Ed., Vol. 22, Orell, Fussli Turici, Lausanjse, 1958. [2] Szebehely, V. (1967) Theory of Orbits, the Restricted Problem of Three Bodies. Academic Press, New York and London [3] Celletti, A., Stefanelli, L., Lega, E. and Froeschle, C. (2011) Some Results on the Global Dynamics of the Regularized Restricted Three-Body Problem with Dissipation. Celestial Mechanics and Dynamical Astronomy, 109, 265-284.
http://dx.doi.org/10.1007/s10569-010-9326-y [4] Murray, C.D. and Dermott, S.F. (1999) Solar System Dynamics. Cambridge University Press, Cambridge. [5] Jeffreys, H. (1929) The Earth. 2nd Edition, Cambridge University Press, Cambridge. [6] Colombo, G.D., Lautman, I. and Shapiru, I. (1966) The Earth’s Dust Belt: Fact or Fiction? Gravitational Focusing and Jacobi Capture. Journal of Geophysical Research, 71, 5705-5717.
http://dx.doi.org/10.1029/JZ071i023p05705 [7] Schuerman, D. (1980) The Restricted Three-Body Problem Including Radiation Pressure. Astrophysical Journal, 238, 337-342.
http://dx.doi.org/10.1086/157989 [8] Simmons, J.F.L., Mcdonald, A.J.C. and Brown, J.C. (1985) The Restricted Three-Body Problem with Radiation Pressure. Celestial Mechanics, 35, 145-187.
http://dx.doi.org/10.1007/BF01227667 [9] Murray, C.D. (1994) Dynamical Effects of Drag in the Circular Restricted Three Body Problems: 1. Location and Stability of the Lagrangian Equilibrium Points. Icarus, 112, 465-184.
http://dx.doi.org/10.1006/icar.1994.1198 [10] Beauge, C. and Ferraz-Mello, S. (1993) Resonance Trapping in the Primordial Solar Nebula: The Case of a Stokes Drag Dissipation. Icarus, 103, 301-318.
http://dx.doi.org/10.1006/icar.1993.1072 [11] Beauge, C. and Ferraz-Mello, S. (1994) Capture in Exterior Mean Motion Resonance Due to Poynting Robertson Drag. Icarus, 110, 239-260.
http://dx.doi.org/10.1006/icar.1994.1119 [12] Beauge, C., Aarseth, S.J. and Ferraz-Mello, S. (1994) Resonance Capture and the Formation of the Outer Planets. Monthly Notices of the Royal Astronomical Society, 270, 21-34.
http://dx.doi.org/10.1093/mnras/270.1.21 [13] Sicardy, B., Beauze, C., Ferraz-Mello, S., Lazzaro, D. and Roques, F. (1993) Capture of Grains into Resonances through Poynting-Robertson Drag. Celestial Mechanics & Dynamical Astronomy, 57, 373-390.
http://dx.doi.org/10.1007/BF00692487 [14] Liou, J.C., Zook, H.A. and Jackson, A.A. (1995) Radiation Pressure, Poynting Robertson Drag and Solar Wind Drag in the Restricted Three Body Problem. Icarus, 116, 186-201.
http://dx.doi.org/10.1006/icar.1995.1120 [15] Ishwar, B. and Kushvah, B.S. (2006) Linear Stability of Triangular Equilibrium Points in the Generalized Restricted Three Body Problem with Poynting Robertson Drag. Journal of Dynamical Systems and Geometric Theories, 4, 79-86.
http://dx.doi.org/10.1080/1726037X.2006.10698504 [16] Kushvah, B.S., Sharma, J.P. and Ishwar, B. (2007) Nonlinear Stability in the Generalised Photogravitational Restricted Three Body Problem with Poynting-Robertson Drag. Astrophysics and Space Science, 312, 279-293.
http://dx.doi.org/10.1007/s10509-007-9688-0 [17] Singh, J. and Emmanuel, A.B. (2014) Stability of Triangular Points in the Photogravitational CR3BP with Poynting Robertson Drag and a Smaller Triaxial Primary. Astrophysics and Space Science, 353, 97-103.
http://dx.doi.org/10.1007/s10509-014-2023-7 [18] Jain, M. and Aggarwal, R. (2015) Existence and Stability of Non-Collinear Librations Points in the Restricted Problem with Poynting Robertson Light Drag Effect. International Journal of Mathematics Trends and Technology, 19, 20-33. [19] Brouwer, D. and Clemence, G.M. (1961) Methods of Celestial Mechanics. Academic Press, New York.
[1] Euler, L. (1772) Theoria Motuum Lunae, Typis Academiar Imperialis Seientiarum, Petropoli. Reprinted in Opera Omnia, Series 2, Courvoisier, L., Ed., Vol. 22, Orell, Fussli Turici, Lausanjse, 1958. [2] Szebehely, V. (1967) Theory of Orbits, the Restricted Problem of Three Bodies. Academic Press, New York and London [3] Celletti, A., Stefanelli, L., Lega, E. and Froeschle, C. (2011) Some Results on the Global Dynamics of the Regularized Restricted Three-Body Problem with Dissipation. Celestial Mechanics and Dynamical Astronomy, 109, 265-284.
http://dx.doi.org/10.1007/s10569-010-9326-y [4] Murray, C.D. and Dermott, S.F. (1999) Solar System Dynamics. Cambridge University Press, Cambridge. [5] Jeffreys, H. (1929) The Earth. 2nd Edition, Cambridge University Press, Cambridge. [6] Colombo, G.D., Lautman, I. and Shapiru, I. (1966) The Earth’s Dust Belt: Fact or Fiction? Gravitational Focusing and Jacobi Capture. Journal of Geophysical Research, 71, 5705-5717.
http://dx.doi.org/10.1029/JZ071i023p05705 [7] Schuerman, D. (1980) The Restricted Three-Body Problem Including Radiation Pressure. Astrophysical Journal, 238, 337-342.
http://dx.doi.org/10.1086/157989 [8] Simmons, J.F.L., Mcdonald, A.J.C. and Brown, J.C. (1985) The Restricted Three-Body Problem with Radiation Pressure. Celestial Mechanics, 35, 145-187.
http://dx.doi.org/10.1007/BF01227667 [9] Murray, C.D. (1994) Dynamical Effects of Drag in the Circular Restricted Three Body Problems: 1. Location and Stability of the Lagrangian Equilibrium Points. Icarus, 112, 465-184.
http://dx.doi.org/10.1006/icar.1994.1198 [10] Beauge, C. and Ferraz-Mello, S. (1993) Resonance Trapping in the Primordial Solar Nebula: The Case of a Stokes Drag Dissipation. Icarus, 103, 301-318.
http://dx.doi.org/10.1006/icar.1993.1072 [11] Beauge, C. and Ferraz-Mello, S. (1994) Capture in Exterior Mean Motion Resonance Due to Poynting Robertson Drag. Icarus, 110, 239-260.
http://dx.doi.org/10.1006/icar.1994.1119 [12] Beauge, C., Aarseth, S.J. and Ferraz-Mello, S. (1994) Resonance Capture and the Formation of the Outer Planets. Monthly Notices of the Royal Astronomical Society, 270, 21-34.
http://dx.doi.org/10.1093/mnras/270.1.21 [13] Sicardy, B., Beauze, C., Ferraz-Mello, S., Lazzaro, D. and Roques, F. (1993) Capture of Grains into Resonances through Poynting-Robertson Drag. Celestial Mechanics & Dynamical Astronomy, 57, 373-390.
http://dx.doi.org/10.1007/BF00692487 [14] Liou, J.C., Zook, H.A. and Jackson, A.A. (1995) Radiation Pressure, Poynting Robertson Drag and Solar Wind Drag in the Restricted Three Body Problem. Icarus, 116, 186-201.
http://dx.doi.org/10.1006/icar.1995.1120 [15] Ishwar, B. and Kushvah, B.S. (2006) Linear Stability of Triangular Equilibrium Points in the Generalized Restricted Three Body Problem with Poynting Robertson Drag. Journal of Dynamical Systems and Geometric Theories, 4, 79-86.
http://dx.doi.org/10.1080/1726037X.2006.10698504 [16] Kushvah, B.S., Sharma, J.P. and Ishwar, B. (2007) Nonlinear Stability in the Generalised Photogravitational Restricted Three Body Problem with Poynting-Robertson Drag. Astrophysics and Space Science, 312, 279-293.
http://dx.doi.org/10.1007/s10509-007-9688-0 [17] Singh, J. and Emmanuel, A.B. (2014) Stability of Triangular Points in the Photogravitational CR3BP with Poynting Robertson Drag and a Smaller Triaxial Primary. Astrophysics and Space Science, 353, 97-103.
http://dx.doi.org/10.1007/s10509-014-2023-7 [18] Jain, M. and Aggarwal, R. (2015) Existence and Stability of Non-Collinear Librations Points in the Restricted Problem with Poynting Robertson Light Drag Effect. International Journal of Mathematics Trends and Technology, 19, 20-33. [19] Brouwer, D. and Clemence, G.M. (1961) Methods of Celestial Mechanics. Academic Press, New York.