Collinear Libration Points in the Photogravitational CR3BP with Zonal Harmonics and Potential from a Belt

Affiliation(s)

^{1}
Department of Mathematics, Faculty of Science, Ahmadu Bello University, Zaria, Nigeria.

^{2}
Department of Mathematics and Computer Science, Federal University, Kashere, Nigeria.

ABSTRACT

We have studied a reformed type of the classic restricted three-body problem where the bigger primary is radiating and the smaller primary is oblate; and they are encompassed by a homogeneous circular cluster of material points centered at the mass center of the system (belt). In this dynamical model, we have derived the equations that govern the motion of the infinitesimal mass under the effects of oblateness up to the zonal harmonics*J*_{4} of the smaller primary, radiation of the bigger primary and the gravitational potential generated by the belt. Numerically, we have found that, in addition to the three collinear libration points *L*_{i} (*i* = 1, 2, 3) in the classic restricted three-body problem, there appear four more collinear points *L*_{ni} (*i* = 1, 2, 3, 4). *L*_{n1} and *L*_{n2} result due to the potential from the belt, while *L*_{n3} and *L*_{n4} are consequences of the oblateness up to the zonal harmonics *J*_{4} of the smaller primary. Owing to the mutual effect of all the perturbations, *L*_{1} and *L*_{3} come nearer to the primaries while *L*_{n3} advances away from the primaries; and *L*_{2} and *L*_{n1} tend towards the smaller primary whereas *L*_{n2} and *L*_{n4} draw closer to the bigger primary. The collinear libration points *L*_{i} (*i *= 1, 2, 3) and *L*_{n2} are linearly unstable whereas the *L*_{n1}, *L*_{n3} and *L*_{n4} are linearly stable. A practical application of this model could be the study of motion of a dust particle near a radiating star and an oblate body surrounded by a belt.

We have studied a reformed type of the classic restricted three-body problem where the bigger primary is radiating and the smaller primary is oblate; and they are encompassed by a homogeneous circular cluster of material points centered at the mass center of the system (belt). In this dynamical model, we have derived the equations that govern the motion of the infinitesimal mass under the effects of oblateness up to the zonal harmonics

KEYWORDS

Circular Restricted Three-Body Problem, Photogravitational, Zonal Harmonic Effect, Potential from the Belt

Circular Restricted Three-Body Problem, Photogravitational, Zonal Harmonic Effect, Potential from the Belt

Cite this paper

Singh, J. and Taura, J. (2015) Collinear Libration Points in the Photogravitational CR3BP with Zonal Harmonics and Potential from a Belt.*International Journal of Astronomy and Astrophysics*, **5**, 155-165. doi: 10.4236/ijaa.2015.53020.

Singh, J. and Taura, J. (2015) Collinear Libration Points in the Photogravitational CR3BP with Zonal Harmonics and Potential from a Belt.

References

[1] Szebehely, V. (1967) Theory of Orbits: The Restricted Problem of Three Bodies. Academic Press, New York.

[2] Valtonen, M. and Karttunen, H. (2006) The Three-Body Problem. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511616006

[3] Poynting, J.H. (1903) Radiation in the Solar System: Its Effect on Temperature and Its Pressure on Small Bodies. Philosophical Transactions of the Royal Society of London A, 202, 525-552.

http://dx.doi.org/10.1098/rsta.1904.0012

[4] Radzievskii, V.V. (1950) The Restricted Problem of Three-Body Taking Account of Light Pressure. Astronomicheskii-Zhurnal, 27, 250-256.

[5] Radzievskii, V.V. (1953) The Space Photogravitational Restricted Three-Body Problem. Astronomicheskii-Zhurnal, 30, 225.

[6] Bhatnagar, K.B. and Chawla, J.M. (1979) A Study of the Lagrangian Points in the Photogravitational Restricted Three-Body Problem. Indian Journal of Pure and Applied Mathematics, 10, 1443-1451.

[7] Simmons, J.F.L., McDonald, J.C. and Brown, J.C. (1985) The Three-Body Problem with Radiation Pressure. Celestial Mechanics, 35, 145-187.

http://dx.doi.org/10.1007/BF01227667

[8] Das, M.K., Narang, P., Mahajan, S. and Yuasa, M. (2008) Effect of Radiation on the Stability of Equilibrium Points in the Binary Stellar Systems: RW-Monocerotis, Krüger 60. Astrophysics Space Science, 314, 261.

http://dx.doi.org/10.1007/s10509-008-9765-z

[9] Singh, J. and Taura, J.J. (2014) Stability of Triangular Libration Points in the Photogravitational Restricted Three-Body Problem with Oblateness and Potential from a Belt. Journal of Astrophysics and Astronomy, 35, 107-119.

http://dx.doi.org/10.1007/s12036-014-9299-4

[10] Singh, J. and Taura, J.J. (2015) Triangular Libration Points in the CR3BP with Radiation, Triaxiality and Potential from a Belt. Differential Equations and Dynamical System.

http://dx.doi.org/10.1007/s12591-015-0243-0

[11] Jiang, I.G. and Yeh, L.C. (2004) The Drag-Induced Resonant Capture for Kuiper Belt Objects. Monthly Notices of the Royal Astronomical Society, 355, L29-L32.

http://dx.doi.org/10.1111/j.1365-2966.2004.08504.x

[12] Jiang, I.G. and Yeh, L.C. (2004) On the Chaotic Orbits of Disk-Star-Planet Systems. The Astronomical Journal, 128, 923-932.

http://dx.doi.org/10.1086/422018

[13] Jiang, I.G. and Yeh, L.C. (2003) Bifurcation for Dynamical Systems of Planet-Belt Interaction. International Journal of Bifurcation and Chaos, 13, 617-630.

http://dx.doi.org/10.1142/s0218127403006807

[14] Yeh, L.C. and Jiang, I.G. (2006) On the Chermnykh-Like Problems: II. The Equilibrium Points. Astrophysics and Space Science, 306, 189-200.

http://dx.doi.org/10.1007/s10509-006-9170-4

[15] Kushvah, B.S. (2008) Linear Stability of Equilibrium Points in the Generalized Photogravitational Chermnykh’s Problem. Astrophysics and Space Science, 318, 41-50.

[16] Singh, J. and Taura, J.J. (2013) Motion in the Generalized Restricted Three-Body Problem. Astrophysics and Space Science, 343, 95-106.

http://dx.doi.org/10.1007/s10509-012-1225-0

[17] Moulton, F.R. (1914) An Introduction to Celestial Mechanics. 2nd Edition, Dover, New York.

[18] 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. Astrophysics and Space Science, 135, 271-281.

http://dx.doi.org/10.1007/BF00641562

[19] Kalvouridis, T.J. (1997) The Oblate Spheroids Version of the Photo-Gravitational 2+2 Body Problem. Astrophysics and Space Science, 246, 219-227.

http://dx.doi.org/10.1007/BF00645642

[20] Singh, J. and Umar, A. (2013) Application of Binary Pulsars to Axisymmetric Bodies in the Elliptic R3BP. Astrophysics and Space Science, 348, 393-402.

http://dx.doi.org/10.1007/s10509-013-1585-0

[21] Abdul Raheem, A. and Singh, J. (2006) Combined Effects of Perturbations, Radiation and Oblateness on the Stability of Equilibrium Points in the Restricted Three-Body Problem. Astronomical Journal, 131, 1880-1885.

http://dx.doi.org/10.1086/499300

[22] Abouelmagd, E.I. (2012) Existence and Stability of Triangular Points in the Restricted Three-Body Problem with Numerical Applications. Astrophysics and Space Science, 342, 45-53.

http://dx.doi.org/10.1007/s10509-012-1162-y

[23] Singh, J. and Taura, J.J. (2014) Effects of Triaxiality, Oblateness and Gravitational Potential from a Belt on the Linear Stability of L4,5 in the Restricted Three-Body Problem. Journal of Astrophysics and Astronomy, 35, 729-743.

http://dx.doi.org/10.1007/s12036-014-9308-7

[24] Singh, J. and Taura, J.J. (2014) Effects of Zonal Harmonics and a Circular Cluster of Material Points on the Stability of Triangular Equilibrium Points in the R3BP. Astrophysics and Space Science, 350, 127-132.

http://dx.doi.org/10.1007/s10509-013-1719-4

[25] Singh, J. and Taura, J.J. (2014) Combined Effect of Oblateness, Radiation and a Circular Cluster of Material Points on the Stability of Triangular Libration Points in the R3BP. Astrophysics and Space Science, 351, 499-506.

http://dx.doi.org/10.1007/s10509-014-1860-8

[26] Renzetti, G. (2013) Satellite Orbital Precessions Caused by the Octupolar Mass Moment of a Non-Spherical Body Arbitrarily Oriented in Space. Journal of Astrophysics and Astronomy, 34, 341-348.

http://dx.doi.org/10.1007/s12036-013-9186-4

[27] Abouelmagd, E.I. (2013) The Effect of Photogravitational Force and Oblateness in the Perturbed Restricted Three-Body Problem. Astrophysics and Space Science, 346, 51-69.

http://dx.doi.org/10.1007/s10509-013-1439-9

[28] Peter, I.D. and Lissauer, J.J. (2007) Planetary Science. Cambridge University Press, New York.

[29] Miyamoto, M. and Nagai, R. (1975) Three-Dimensional Models for the Distribution of Mass in Galaxies. Publications of the Astronomical Society of Japan, 27, 533-543.

[1] Szebehely, V. (1967) Theory of Orbits: The Restricted Problem of Three Bodies. Academic Press, New York.

[2] Valtonen, M. and Karttunen, H. (2006) The Three-Body Problem. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511616006

[3] Poynting, J.H. (1903) Radiation in the Solar System: Its Effect on Temperature and Its Pressure on Small Bodies. Philosophical Transactions of the Royal Society of London A, 202, 525-552.

http://dx.doi.org/10.1098/rsta.1904.0012

[4] Radzievskii, V.V. (1950) The Restricted Problem of Three-Body Taking Account of Light Pressure. Astronomicheskii-Zhurnal, 27, 250-256.

[5] Radzievskii, V.V. (1953) The Space Photogravitational Restricted Three-Body Problem. Astronomicheskii-Zhurnal, 30, 225.

[6] Bhatnagar, K.B. and Chawla, J.M. (1979) A Study of the Lagrangian Points in the Photogravitational Restricted Three-Body Problem. Indian Journal of Pure and Applied Mathematics, 10, 1443-1451.

[7] Simmons, J.F.L., McDonald, J.C. and Brown, J.C. (1985) The Three-Body Problem with Radiation Pressure. Celestial Mechanics, 35, 145-187.

http://dx.doi.org/10.1007/BF01227667

[8] Das, M.K., Narang, P., Mahajan, S. and Yuasa, M. (2008) Effect of Radiation on the Stability of Equilibrium Points in the Binary Stellar Systems: RW-Monocerotis, Krüger 60. Astrophysics Space Science, 314, 261.

http://dx.doi.org/10.1007/s10509-008-9765-z

[9] Singh, J. and Taura, J.J. (2014) Stability of Triangular Libration Points in the Photogravitational Restricted Three-Body Problem with Oblateness and Potential from a Belt. Journal of Astrophysics and Astronomy, 35, 107-119.

http://dx.doi.org/10.1007/s12036-014-9299-4

[10] Singh, J. and Taura, J.J. (2015) Triangular Libration Points in the CR3BP with Radiation, Triaxiality and Potential from a Belt. Differential Equations and Dynamical System.

http://dx.doi.org/10.1007/s12591-015-0243-0

[11] Jiang, I.G. and Yeh, L.C. (2004) The Drag-Induced Resonant Capture for Kuiper Belt Objects. Monthly Notices of the Royal Astronomical Society, 355, L29-L32.

http://dx.doi.org/10.1111/j.1365-2966.2004.08504.x

[12] Jiang, I.G. and Yeh, L.C. (2004) On the Chaotic Orbits of Disk-Star-Planet Systems. The Astronomical Journal, 128, 923-932.

http://dx.doi.org/10.1086/422018

[13] Jiang, I.G. and Yeh, L.C. (2003) Bifurcation for Dynamical Systems of Planet-Belt Interaction. International Journal of Bifurcation and Chaos, 13, 617-630.

http://dx.doi.org/10.1142/s0218127403006807

[14] Yeh, L.C. and Jiang, I.G. (2006) On the Chermnykh-Like Problems: II. The Equilibrium Points. Astrophysics and Space Science, 306, 189-200.

http://dx.doi.org/10.1007/s10509-006-9170-4

[15] Kushvah, B.S. (2008) Linear Stability of Equilibrium Points in the Generalized Photogravitational Chermnykh’s Problem. Astrophysics and Space Science, 318, 41-50.

[16] Singh, J. and Taura, J.J. (2013) Motion in the Generalized Restricted Three-Body Problem. Astrophysics and Space Science, 343, 95-106.

http://dx.doi.org/10.1007/s10509-012-1225-0

[17] Moulton, F.R. (1914) An Introduction to Celestial Mechanics. 2nd Edition, Dover, New York.

[18] 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. Astrophysics and Space Science, 135, 271-281.

http://dx.doi.org/10.1007/BF00641562

[19] Kalvouridis, T.J. (1997) The Oblate Spheroids Version of the Photo-Gravitational 2+2 Body Problem. Astrophysics and Space Science, 246, 219-227.

http://dx.doi.org/10.1007/BF00645642

[20] Singh, J. and Umar, A. (2013) Application of Binary Pulsars to Axisymmetric Bodies in the Elliptic R3BP. Astrophysics and Space Science, 348, 393-402.

http://dx.doi.org/10.1007/s10509-013-1585-0

[21] Abdul Raheem, A. and Singh, J. (2006) Combined Effects of Perturbations, Radiation and Oblateness on the Stability of Equilibrium Points in the Restricted Three-Body Problem. Astronomical Journal, 131, 1880-1885.

http://dx.doi.org/10.1086/499300

[22] Abouelmagd, E.I. (2012) Existence and Stability of Triangular Points in the Restricted Three-Body Problem with Numerical Applications. Astrophysics and Space Science, 342, 45-53.

http://dx.doi.org/10.1007/s10509-012-1162-y

[23] Singh, J. and Taura, J.J. (2014) Effects of Triaxiality, Oblateness and Gravitational Potential from a Belt on the Linear Stability of L4,5 in the Restricted Three-Body Problem. Journal of Astrophysics and Astronomy, 35, 729-743.

http://dx.doi.org/10.1007/s12036-014-9308-7

[24] Singh, J. and Taura, J.J. (2014) Effects of Zonal Harmonics and a Circular Cluster of Material Points on the Stability of Triangular Equilibrium Points in the R3BP. Astrophysics and Space Science, 350, 127-132.

http://dx.doi.org/10.1007/s10509-013-1719-4

[25] Singh, J. and Taura, J.J. (2014) Combined Effect of Oblateness, Radiation and a Circular Cluster of Material Points on the Stability of Triangular Libration Points in the R3BP. Astrophysics and Space Science, 351, 499-506.

http://dx.doi.org/10.1007/s10509-014-1860-8

[26] Renzetti, G. (2013) Satellite Orbital Precessions Caused by the Octupolar Mass Moment of a Non-Spherical Body Arbitrarily Oriented in Space. Journal of Astrophysics and Astronomy, 34, 341-348.

http://dx.doi.org/10.1007/s12036-013-9186-4

[27] Abouelmagd, E.I. (2013) The Effect of Photogravitational Force and Oblateness in the Perturbed Restricted Three-Body Problem. Astrophysics and Space Science, 346, 51-69.

http://dx.doi.org/10.1007/s10509-013-1439-9

[28] Peter, I.D. and Lissauer, J.J. (2007) Planetary Science. Cambridge University Press, New York.

[29] Miyamoto, M. and Nagai, R. (1975) Three-Dimensional Models for the Distribution of Mass in Galaxies. Publications of the Astronomical Society of Japan, 27, 533-543.