A Study of the Elliptic Restricted Three-Body Problem with Triaxial and Radiating Primaries Surrounded by a Belt

Jagadish Singh; Ndaman Isah

doi:doi:10.11648/j.ijass.20210901.11

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A Study of the Elliptic Restricted Three-Body Problem with Triaxial and Radiating Primaries Surrounded by a Belt

Jagadish Singh, Ndaman Isah

Published in International Journal of Astrophysics and Space Science (Volume 9, Issue 1)

Received: 28 September 2020 Accepted: 30 November 2020 Published: 22 January 2021

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Abstract

This paper examines the effects of radiation pressure and triaxiality of two stars (primaries) surrounded by a belt (circumbinarydisc)onthepositionsandstabilityofathirdbodyofaninfinitesimalmassintheframeworkoftheEllipticrestricted three body problem (ER3BP). We have obtained analytical solutions to the triangular equilibrium points and their stability and have also investigated these solutions numerically and graphically using radiating binary system (Xi- Bootis and Kruger 60). It is observed that their positions and stability are affected by semi-major axis, eccentricity of the orbit, triaxiality, radiation pressure of the primaries and potential from the belt. The perturbed parameters show the destabilizing tendency by decreasing the range of stability. The triangular points are found to bestable for 0 < µ < µ_c where µ_c is the critical mass parameter. The stability analysis for the binary system yielded a stable outcome when we consider the range of mass parameterµ in the region of the Routhonian critical mass ratio (0.03852) when the effect of circumbinary disc is dominant. We found triaxiality and radition factors inducing instability even within this range.

Published in	International Journal of Astrophysics and Space Science (Volume 9, Issue 1)
DOI	10.11648/j.ijass.20210901.11
Page(s)	1-16
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

ER3BP, Triaxiality, Radiation Pressure, Pontetial from the Belt, Stability, Binary System, Routhonian Mass Ratio, Oblateness, Eccentricity, Semi-Major Axis, CR3BP

References

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[3]	Zimovschikov, A. S. and Tkhai, V. N (2004) ‘Instability of libration points and resonance phenomena in the photogravitational Elliptic restricted three- body problem’, Solar Systems Research, Vol. 38, p155.
[4]	Pandey, K. K, Narayan, A. and Shivastava, S. K (2019) ‘Effect of Triaxiality and radiation pressure on Existence of Resonance Stability of Triangular Equilibrium Points in Elliptical Restricted Three-Body Problem’, Journal of Informatics and Mathematical Sciences Vol. 11, pp 3-4.
[5]	Singh, J. and Haruna, S. (2020) ¡®Motion around triangular points in the Restricted Three-Body problem with radiating heterogeneous primaries surrounded by a belt¡¯, Science Reports, Vol 10,18861.Doi/10.1038/S41598-020-7517 pp 4-7
[6]	Narayan, A. and Nutan, S.(2014)‘Stability of triangular Lagragian points in the elliptic restricted three-body problem under radiating binary system’ , Astrophysics and Space Science Vol. 353, No. 2, pp 457-464
[7]	Narayan,A. and Ramesh, C. (2011) ‘Stability of Triangular Equilibrium points in elliptical restricted three-body problem under effects of photogravitational and oblateness of primaries’, International journal of pure and applied Mathematics, Vol. 70 pp 735.
[8]	Selim, H. H.,Guirao,J. L. G. and Abouelmagd, E. I. (2019) ¡®Libration Points in the Restricted Three Body- Problem: Euler angles, existence and stability¡¯, Discrete and Continous Dynamical systems 12 ( 4 and 5) pp 703- 710.
[9]	Singh, J. and Umar, A. (2012a) ‘Motion in the photo gravitational elliptic restricted three- body problem under oblate primaries’, The Astronomical Journal, Vol. 143 pp 109-131.
[10]	Singh, J. and Umar, A. (2012b) ‘On the stability of triangular points in the elliptic restricted three- body problem under radiating and oblate primaries’, Astrophysics and Space Science, Vol. 341, pp 349-358.
[11]	Arif, M and Sagar, R. K (2018)¡¯ Study of the Restricted Three-Body Problem when the Primary is a uniform circularDisk¡¯Application and Applied Mathematics Vol. 13, 1 pp 160-172.
[12]	Khanna, M. and Bhatnagar, K. B. (1999) ‘Existence and Stability of libration points in the stricted three-body problem when the smaller primary is a triaxial rigid body and the Bigger one an oblate spheroids’ ,India journal of pure and applied Mathematics, Vol. 30No. 7pp721-733.
[13]	Singh, J. and Taura, J. J. (2014b) ‘Combine effect of oblateness, radiation and a cluster of material points on the stability of triangular libration points in the restricted three-body problem’, Journal of Astrophysics and Astronomy, Vol. 351, No. 2. Doi 10.1007/s 10509- 014-1860-8
[14]	Narayan, A., Pandey, K. K. and Shrivastava, S. K. (2015)‘ Effects of radiation and triaxiality of triangular points in elliptical restricted three-body problem’, International Journal of Advanced Astronomy, Vol. 3, No. 2, pp 97-106 16 Jagadish Singh and Ndaman Isah: A Study of the Elliptic Restricted Three-Body Problem with Triaxial and Radiating Primaries Surrounded by a Belt
[15]	Umar, A. and Hussain, A. A (2016) ‘Motion in the elliptic restricted three- body problem with an oblate primary and triaxial stellar companion’, Astrophysics and Space Science, Vol. 361, No. 344
[16]	Singh, J. and Taura, J. J. (2014c) ‘Effects of triaxiality, oblateness and gravitational pontetial from a belt on the linear stability of L4,5 in the restricted three-body problem’, Journal of Astrophysics and Astronomy, Vol. 35, No. 4 pp 729-743
[17]	Singh, J. and Taura, J. J. (2014a) ‘Stability of triangular equilibrium points in the photogravitational restricted three-body problem with oblateness and pontetial from a belt’, Journal of Astrophysics and Astronomy, Vol. 35 pp 107-109.
[18]	Singh, J. and Amuda, T. O. (2019)‘ Stability Analysis of Triangular equilibrium points inrestricted three-body problem under effect of circumbinary disc, radiation and drag forces’, Astrophysics and astronomy, Vol. 40, No. 5 http://doi.org/10.1007/sDoi10.1007/s 12036-019-9537-6
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[23]	Singh, J. and Tokyaa, R. K. (2016)‘Stability of Triangular points in the ellipticrestricted three-body problem with oblateness up to zonal harmonics J4 of both primaries’, The European physical Journal plus, Vol. 131, pp 1-11.

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APA Style

Jagadish Singh, Ndaman Isah. (2021). A Study of the Elliptic Restricted Three-Body Problem with Triaxial and Radiating Primaries Surrounded by a Belt. International Journal of Astrophysics and Space Science, 9(1), 1-16. https://doi.org/10.11648/j.ijass.20210901.11

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ACS Style

Jagadish Singh; Ndaman Isah. A Study of the Elliptic Restricted Three-Body Problem with Triaxial and Radiating Primaries Surrounded by a Belt. Int. J. Astrophys. Space Sci. 2021, 9(1), 1-16. doi: 10.11648/j.ijass.20210901.11

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AMA Style

Jagadish Singh, Ndaman Isah. A Study of the Elliptic Restricted Three-Body Problem with Triaxial and Radiating Primaries Surrounded by a Belt. Int J Astrophys Space Sci. 2021;9(1):1-16. doi: 10.11648/j.ijass.20210901.11

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@article{10.11648/j.ijass.20210901.11,
  author = {Jagadish Singh and Ndaman Isah},
  title = {A Study of the Elliptic Restricted Three-Body Problem with Triaxial and Radiating Primaries Surrounded by a Belt},
  journal = {International Journal of Astrophysics and Space Science},
  volume = {9},
  number = {1},
  pages = {1-16},
  doi = {10.11648/j.ijass.20210901.11},
  url = {https://doi.org/10.11648/j.ijass.20210901.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijass.20210901.11},
  abstract = {This paper examines the effects of radiation pressure and triaxiality of two stars (primaries) surrounded by a belt (circumbinarydisc)onthepositionsandstabilityofathirdbodyofaninfinitesimalmassintheframeworkoftheEllipticrestricted three body problem (ER3BP). We have obtained analytical solutions to the triangular equilibrium points and their stability and have also investigated these solutions numerically and graphically using radiating binary system (Xi- Bootis and Kruger 60). It is observed that their positions and stability are affected by semi-major axis, eccentricity of the orbit, triaxiality, radiation pressure of the primaries and potential from the belt. The perturbed parameters show the destabilizing tendency by decreasing the range of stability. The triangular points are found to bestable for 0 µ µc where µc is the critical mass parameter. The stability analysis for the binary system yielded a stable outcome when we consider the range of mass parameterµ in the region of the Routhonian critical mass ratio (0.03852) when the effect of circumbinary disc is dominant. We found triaxiality and radition factors inducing instability even within this range.},
 year = {2021}
}

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TY  - JOUR
T1  - A Study of the Elliptic Restricted Three-Body Problem with Triaxial and Radiating Primaries Surrounded by a Belt
AU  - Jagadish Singh
AU  - Ndaman Isah
Y1  - 2021/01/22
PY  - 2021
N1  - https://doi.org/10.11648/j.ijass.20210901.11
DO  - 10.11648/j.ijass.20210901.11
T2  - International Journal of Astrophysics and Space Science
JF  - International Journal of Astrophysics and Space Science
JO  - International Journal of Astrophysics and Space Science
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PB  - Science Publishing Group
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AB  - This paper examines the effects of radiation pressure and triaxiality of two stars (primaries) surrounded by a belt (circumbinarydisc)onthepositionsandstabilityofathirdbodyofaninfinitesimalmassintheframeworkoftheEllipticrestricted three body problem (ER3BP). We have obtained analytical solutions to the triangular equilibrium points and their stability and have also investigated these solutions numerically and graphically using radiating binary system (Xi- Bootis and Kruger 60). It is observed that their positions and stability are affected by semi-major axis, eccentricity of the orbit, triaxiality, radiation pressure of the primaries and potential from the belt. The perturbed parameters show the destabilizing tendency by decreasing the range of stability. The triangular points are found to bestable for 0 µ µc where µc is the critical mass parameter. The stability analysis for the binary system yielded a stable outcome when we consider the range of mass parameterµ in the region of the Routhonian critical mass ratio (0.03852) when the effect of circumbinary disc is dominant. We found triaxiality and radition factors inducing instability even within this range.
VL  - 9
IS  - 1
ER  -

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Author Information

Jagadish Singh

Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University, Zaria, Kaduna State, Nigeria
Ndaman Isah

Department of Mathematics, Faculty of Sciences, Niger State College of Education, Minna, Niger State, Nigeria

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APA Style

Jagadish Singh, Ndaman Isah. (2021). A Study of the Elliptic Restricted Three-Body Problem with Triaxial and Radiating Primaries Surrounded by a Belt. International Journal of Astrophysics and Space Science, 9(1), 1-16. https://doi.org/10.11648/j.ijass.20210901.11

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ACS Style

Jagadish Singh; Ndaman Isah. A Study of the Elliptic Restricted Three-Body Problem with Triaxial and Radiating Primaries Surrounded by a Belt. Int. J. Astrophys. Space Sci. 2021, 9(1), 1-16. doi: 10.11648/j.ijass.20210901.11

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AMA Style

Jagadish Singh, Ndaman Isah. A Study of the Elliptic Restricted Three-Body Problem with Triaxial and Radiating Primaries Surrounded by a Belt. Int J Astrophys Space Sci. 2021;9(1):1-16. doi: 10.11648/j.ijass.20210901.11

Copy | Download

@article{10.11648/j.ijass.20210901.11,
  author = {Jagadish Singh and Ndaman Isah},
  title = {A Study of the Elliptic Restricted Three-Body Problem with Triaxial and Radiating Primaries Surrounded by a Belt},
  journal = {International Journal of Astrophysics and Space Science},
  volume = {9},
  number = {1},
  pages = {1-16},
  doi = {10.11648/j.ijass.20210901.11},
  url = {https://doi.org/10.11648/j.ijass.20210901.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijass.20210901.11},
  abstract = {This paper examines the effects of radiation pressure and triaxiality of two stars (primaries) surrounded by a belt (circumbinarydisc)onthepositionsandstabilityofathirdbodyofaninfinitesimalmassintheframeworkoftheEllipticrestricted three body problem (ER3BP). We have obtained analytical solutions to the triangular equilibrium points and their stability and have also investigated these solutions numerically and graphically using radiating binary system (Xi- Bootis and Kruger 60). It is observed that their positions and stability are affected by semi-major axis, eccentricity of the orbit, triaxiality, radiation pressure of the primaries and potential from the belt. The perturbed parameters show the destabilizing tendency by decreasing the range of stability. The triangular points are found to bestable for 0 µ µc where µc is the critical mass parameter. The stability analysis for the binary system yielded a stable outcome when we consider the range of mass parameterµ in the region of the Routhonian critical mass ratio (0.03852) when the effect of circumbinary disc is dominant. We found triaxiality and radition factors inducing instability even within this range.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - A Study of the Elliptic Restricted Three-Body Problem with Triaxial and Radiating Primaries Surrounded by a Belt
AU  - Jagadish Singh
AU  - Ndaman Isah
Y1  - 2021/01/22
PY  - 2021
N1  - https://doi.org/10.11648/j.ijass.20210901.11
DO  - 10.11648/j.ijass.20210901.11
T2  - International Journal of Astrophysics and Space Science
JF  - International Journal of Astrophysics and Space Science
JO  - International Journal of Astrophysics and Space Science
SP  - 1
EP  - 16
PB  - Science Publishing Group
SN  - 2376-7022
UR  - https://doi.org/10.11648/j.ijass.20210901.11
AB  - This paper examines the effects of radiation pressure and triaxiality of two stars (primaries) surrounded by a belt (circumbinarydisc)onthepositionsandstabilityofathirdbodyofaninfinitesimalmassintheframeworkoftheEllipticrestricted three body problem (ER3BP). We have obtained analytical solutions to the triangular equilibrium points and their stability and have also investigated these solutions numerically and graphically using radiating binary system (Xi- Bootis and Kruger 60). It is observed that their positions and stability are affected by semi-major axis, eccentricity of the orbit, triaxiality, radiation pressure of the primaries and potential from the belt. The perturbed parameters show the destabilizing tendency by decreasing the range of stability. The triangular points are found to bestable for 0 µ µc where µc is the critical mass parameter. The stability analysis for the binary system yielded a stable outcome when we consider the range of mass parameterµ in the region of the Routhonian critical mass ratio (0.03852) when the effect of circumbinary disc is dominant. We found triaxiality and radition factors inducing instability even within this range.
VL  - 9
IS  - 1
ER  -

Copy | Download