alexa
Reach Us +44-1904-929220
Existence and Stability of Triangular Points in the Relativistic R3bp When the Primaries are Triaxial Rigid Bodies and Sources of Radiation | OMICS International
ISSN: 2329-6542
Journal of Astrophysics & Aerospace Technology
Make the best use of Scientific Research and information from our 700+ peer reviewed, Open Access Journals that operates with the help of 50,000+ Editorial Board Members and esteemed reviewers and 1000+ Scientific associations in Medical, Clinical, Pharmaceutical, Engineering, Technology and Management Fields.
Meet Inspiring Speakers and Experts at our 3000+ Global Conferenceseries Events with over 600+ Conferences, 1200+ Symposiums and 1200+ Workshops on
Medical, Pharma, Engineering, Science, Technology and Business

Existence and Stability of Triangular Points in the Relativistic R3bp When the Primaries are Triaxial Rigid Bodies and Sources of Radiation

Jagadish Singh1 and Nakone Bello2*
1Department of Mathematics, Faculty of Science, Ahmadu Bello University, Zaria, Nigeria
2Department of Mathematics, Faculty of Science, Usmanu Danfodiyo University, Sokoto, Nigeria
*Corresponding Author : Nakone Bello
Department of Mathematics
Faculty of Science
Usmanu Danfodiyo University, Sokoto, Nigeria
E-mail: [email protected]
Received: April 01, 2016 Accepted: April 14, 2015 Published: April 18, 2016
Citation: Singh J, Bello N (2016) Existence and Stability of Triangular Points in the Relativistic R3bp When the Primaries are Triaxial Rigid Bodies and Sources of Radiation. J Astrophys Aerospace Technol 4:131. doi:10.4172/2329-6542.1000131
Copyright: © 2016 Singh J, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Visit for more related articles at Journal of Astrophysics & Aerospace Technology

Abstract

This paper deals with the triangular points and their linear stability in the relativistic R3BP when the primaries are triaxial rigid bodies and sources of radiation. It is observed that the locations of the triangular points are affected by the relativistic terms, radiation pressure forces and the triaxiality of the primaries. It is also seen that for these points the range of stability region increases or decreases according as p >0 or p<0 where p depends upon the relativistic terms, the radiation and triaxiality coefficients.

Keywords
Celestial mechanics; Radiation; Triaxiality; Relativity; R3BP
Introduction
The restricted three-body problem possesses five stationary solutions called Lagrangian points, three of which called collinear equilibria lie on the line joining the primaries and the other two called equilateral equilibria make equilateral triangles with primaries.
In general, the collinear equilibria are unstable while equilateral ones are stable, in the Lyapunov sense, only in a certain region for the mass parameter. Various authors have made studies on Lagrangian points in the restricted three-body problem by considering the more massive primary or both primaries as source of radiation.
Some of the important contributions are by Radzievskii [1,2] Simmons et al. [3] Kunitsyn and Tureshbaev [4] Singh and Ishwar [5] Singh [6]. Some of the significant studies, by considering the triaxiality of one or both primaries, are Khanna and Bhatnagar [7] and Singh [8]. Sharma et al. [9,10] have studied the stationary solutions of the planar restricted three-body problem when one or both primaries are sources of radiation and triaxial rigid bodies with one of the axes as the axis of symmetry and the equatorial plane coinciding with the plane of motion.
The theory of the general relativity is currently the most successful gravitational theory describing the nature of space and time and well confirmed by observations [11]. Brumberg [12,13] studied the problem in more details and collected most of the important results on relativistic celestial mechanics. He did not obtain only the equations of motion for the general problem of the three bodies but also deduced the equation of the motion for the restricted problem of three bodies. Bhatnagar and Hallan [14] investigated the existence and linear stability of the triangular point image in the relativistic R3BP. They concluded that imageare always unstable in the whole range image in contrast to the classical R3BP where they are stable for image being the mass ratio and image is the Routh’s value.
Douskos and Perdios [15] investigated the stability of the triangular points in the relativistic R3BP and contrary to the results of Bhatnagar and Hallan [14], they obtained a region of linear stability in the parameter space image where imageis Routh’s value.
Katour et al. [16] obtained new locations of the triangular points in the framework of relativistic R3BP with oblateness and photogravitional corrections to triangular points.
Singh and Bello [17,18] studied the motion of a test particle in the vicinity of the triangular points image by considering one or both primaries as the sources of radiation in the framework of the relativistic restricted three-body problem (R3BP).
In the present work, we study the existence of the triangular points and their linear stability by considering both primaries as triaxial rigid bodies and sources of radiation.
This paper is organized as follows: In Sect. 2, the equations governing the motion are presented; Sect. 3 describes the positions of triangular points, while their linear stability is analyzed in Sect.4; The discussion of the results is given in Sect. 5. In Sect. 6, we conclude our work and highlight the differences and similarities between our work and previous works.
Equations of Motion
In our recent papers Singh and Bello [17,18] we have studied the effect of radiation pressure of the primaries in the relativistic R3BP and found that the positions of triangular points and their stability are affected by both relativistic and radiation factors. In this paper we extend this work by considering both primaries as triaxial rigid bodies as well as sources of radiation.
The pertinent equations of motion of an infinitesimal mass in the relativistic R3BP in a barycentric synodic coordinate system imageand dimensionless variables can be written as Singh and Bello [17,18]
image(1)
with
image(2)
image (3)
image (4)
where image is the ratio of the mass of the smaller primary to the total mass of the primaries, image are distances of the infinitesimal mass from the bigger and smaller primary, respectively; n is the mean motion of the primaries; cis the velocity of light.
image [19] and image characterize the triaxiality of the bigger and smaller primary with a, b, c as lengths of the semi-axes of the bigger primary and image′ as those of the smaller primary. The radiation factor image is given by image such that image Radzievskii [1] where image are respectively the gravitational and radiation pressure.
Here as Katour et al. [16] we do not include the parameters imagein the relativistic part of image since the magnitude of these terms is so small due to imagewhere c is the speed of light.
Location of Triangular Points
The libration points are obtained from equation (1) after putting image
These points are the solutions of the equations
image
That is
image(5)
and
image
with
image
For simplicity, putting image and neglecting second and higher order terms inimage c and their products. Following as our papers [17,18] we have obtained from the system (5) with η ≠ 0, the coordinates of the triangular points image as:
image (6)
Stability of L4,5
Since the nature of the linear stability about the pointimage will be similar to that aboutimage, it will be sufficient to consider here the stability only nearimage.
Let (a,b) be the coordinates of the triangular pointimage
We set image, in the equations (1) of motion.
First, we compute the terms of their R.H.S, neglecting second and higher order terms, we get
image
where,
image
image
image
image
Similarly, we obtain
image
where,
image
image
image
image
image
where,
image
image
image
image
image
where,
image
image
image
image
The characteristic equation of the variational equations of motion corresponding to (1) can be expressed as
image (7)
where,
image
image
For image when the primaries are non-luminous and spherical image Eq. (7) reduces to its well-known classical restricted problem form (See e.g. Szebehely [20]).
image
The discriminant of (7) is
image (8)
Its roots are
image (9)
where,
image
From (8), we have
image (10)
From (10), it can be easily seen that Δ is monotone decreasing in image
But
image (11)
image
Since imageare of opposite signs, andΔ is monotone decreasing and continuous, there is one value of image in the interval image for which image vanishes.
Solving the equation image using (8), we obtain the critical value of the mass parameter as
image(12)
image
where image is the Routh’s value.
We consider the following three regions of the values of μ separately.
When image the values ofimagegiven by (9) are negative and therefore all the four characteristic roots are distinct pure imaginary numbers. Hence, the triangular points are stable.
When image the real parts of the characteristic roots are positive. Therefore, the triangular points are unstable.
When image the values of image given by (9) are the same. This induces instability of the triangular points.
Hence, the stability region is
image (13)
with
image
Discussion
In this section, we discuss the triangular libration points in the relativistic restricted three-body problem, under the assumption that the primaries are luminous and triaxial. The positions of the triangular points in equation (6) are obtained. It can be seen that they are affected by the relativistic, radiation and triaxiality factors. It is important to note that these triangular libration points cease to be classical one i.e., they no longer form equilateral triangles with the primaries. Rather they form scalene triangles with the primaries. Equation (12) gives the critical value of the mass parameter image of the system which depends upon relativistic factor, triaxiality parameters image and radiation image. In the absence of relativistic factor, the results obtained in this study are in agreement with those of Sharma et al. [10] Singh [8] when there is no perturbations in the Coriolis and centrifugal forces image When the primaries are non triaxial, the results of the present study tally with those of Singh and Bello [18].
It is noticeable from (13) that that radiation and triaxiality all have destabilizing effects, and therefore the size of the range of stability decreases with increase of the values of these parameters. Evidently, it can also seen that the relativistic factor reduces the size of stability region. When the primaries are non-luminous and non-triaxial, the stability results obtained in this study are in accordance with those of Douskos and Perdios [15] and disagree with Bhatnagar and Hallan [14]. In the absence of relativistic factor, the results obtained in this study are in agreement with those of Sharma [14] and those of Singh [8] when the perturbations are absent. When the primaries are oblate spheroids image , the results of equation (6) in this study differ from those of katour et al. [16] when the radiation terms are absent in the realistic part of the potential W
Conclusion
By considering the primaries as triaxial rigid bodies and sources of radiation in the relativistic CR3BP, we have determined the positions of the triangular points and their linear stability. It is found that their positions and stability region are affected by relativistic, triaxiality and radiation factors. It is further observed that the relativistic, triaxiality and radiation factors have destabilizing tendencies resulting in a decrease in the size of the region of stability. We have noticed that the expressions for A, D, A2, C2 in Bhatnagar and Hallan [14] differ from the present study when the radiation pressure factors are absent and the primaries are spherical imageConsequently, the characteristic equation is also different. This led them Bhatnagar and Hallan [14] to infer that the triangular points are unstable, contrary to Douskos and Perdios and our results. Our results are also in disagreement with those of Katour et al. [16]. One major distinction is that the expression of the mean motion which they used in their study differ from our own. It seems that there is an error in their expression.
References
Select your language of interest to view the total content in your interested language
Post your comment

Share This Article

Article Usage

  • Total views: 8647
  • [From(publication date):
    December-2016 - Dec 16, 2018]
  • Breakdown by view type
  • HTML page views : 8536
  • PDF downloads : 111
 

Post your comment

captcha   Reload  Can't read the image? click here to refresh

Peer Reviewed Journals
 
Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals
International Conferences 2018-19
 
Meet Inspiring Speakers and Experts at our 3000+ Global Annual Meetings

Contact Us

Agri and Aquaculture Journals

Dr. Krish

[email protected]

+1-702-714-7001Extn: 9040

Biochemistry Journals

Datta A

[email protected]

1-702-714-7001Extn: 9037

Business & Management Journals

Ronald

[email protected]

1-702-714-7001Extn: 9042

Chemistry Journals

Gabriel Shaw

[email protected]

1-702-714-7001Extn: 9040

Clinical Journals

Datta A

[email protected]

1-702-714-7001Extn: 9037

Engineering Journals

James Franklin

[email protected]

1-702-714-7001Extn: 9042

Food & Nutrition Journals

Katie Wilson

[email protected]

1-702-714-7001Extn: 9042

General Science

Andrea Jason

[email protected]

1-702-714-7001Extn: 9043

Genetics & Molecular Biology Journals

Anna Melissa

[email protected]

1-702-714-7001Extn: 9006

Immunology & Microbiology Journals

David Gorantl

[email protected]

1-702-714-7001Extn: 9014

Materials Science Journals

Rachle Green

[email protected]

1-702-714-7001Extn: 9039

Nursing & Health Care Journals

Stephanie Skinner

[email protected]

1-702-714-7001Extn: 9039

Medical Journals

Nimmi Anna

[email protected]

1-702-714-7001Extn: 9038

Neuroscience & Psychology Journals

Nathan T

[email protected]

1-702-714-7001Extn: 9041

Pharmaceutical Sciences Journals

Ann Jose

[email protected]

1-702-714-7001Extn: 9007

Social & Political Science Journals

Steve Harry

streamtajm

[email protected]

1-702-714-7001Extn: 9042

 
© 2008- 2018 OMICS International - Open Access Publisher. Best viewed in Mozilla Firefox | Google Chrome | Above IE 7.0 version
Leave Your Message 24x7