International Frontier Siene Letters Submitted: 6-- ISSN: 9-8, Vol., pp -6 Aepted: -- doi:.8/www.sipress.om/ifsl.. Online: --8 SiPress Ltd., Switzerland Collinear Equilibrium Points in the Relativisti RBP when the Bigger Primary is a Triaxial Rigid Body Nakone Bello,a and Aminu Abubakar Hussain,b Department of Mathematis, Faulty of Siene, Usmanu Danfodiyo University, Sokoto, Nigeria Department of Mathematis, Faulty of Siene, Ahmadu Bello University, Zaria, Nigeria a bnakone@yahoo.om orresponding author, b dadinduniya@gmail.om Keywords: Celestial Mehanis, Triaxiality, Relativity, RBP. Abstrat. This study examines the effet of the relativisti fator as well as the triaxiality effet of the bigger primary on the positions and stability of the ollinear points in the frame work of the post-newtonian approximation. Using semi-analytial and numerial approah the ollinear points are found to be unstable. A numerial exploration in this onnetion, with the Earth-Moon system, reveals that the relativisti fator has an effet on these positions. It is also found that under the ombined effet of relativisti fator and triaxiality, the ollinear point L moves towards the primaries with the inrease in triaxiality, while L and L move away from the bigger primary. It is also seen that in most of the ases in the presene of triaxiality, the effet of relativisti fator on the positions of L and L is not observable; however it has an observable effet on the position of L in the presene of triaxiality exept for the ase.. Introdution In the restrited three-body problem RBP, two massive bodies of finite masses m and m alled bigger and smaller primary respetively having spherial symmetry move about their enter of mass in irular orbits. A third mass m, the infinitesimal one, moves under the ombined gravitational attration of the two bodies but does not influene their motion. This problem possesses five equilibrium points, three ollinear points L, L and L whih are in general unstable and two triangular points L and Lwhih are stable for the mass ratio <.8... Szebehely []. The relativisti restrited three-body problem was originally studied by Brumberg []. Bhatnagar and Hallan [] were the first to study the stability of triangular points of the same model problem and found that the triangular point are unstable in the whole region ontrary to the lassial ase where they are stable for < where is the mass ratio and.8... Douskos and Perdios [] reinvestigated the same model problem and found the region of 69 stability of triangular points as < <, where is the dimensionless speed of light and 86.8... is the Routh s value. Later on, Ahmed et al. []reexamined the same model problem and found the region of stability of triangular points as <.8. Ragos et al. [6] studied the existene, position and stability of ollinear points in the relativisti RBP. In reent years, there has been a strong revival of interest in the relativisti RBP. Many perturbing fores i.e. radiation, oblateness, perturbations in the entrifugal and Coriolis fores et. have been inluded in the study of relativisti RBP. Several authors Abd El-Bar and Abd El- Salam []; Abd El-Bar and Abd El-Salam [8]; Abd El-Salam and Abd El-Bar [9]; Katour et al. []; SiPress applies the CC-BY. liense to works we publish: https://reativeommons.org/lienses/by/./
Abd El-Bar et al. [] have foused their study on the loations of equilibrium points of relativisti RBP under some various aspets of the above mentioned perturbing fores. Singh and Bello [-] have studied the loations and stability of triangular points of the relativisti RBP under some of the above mentioned perturbing fores. In general, the elestial bodies are not perfet spheres. They are either oblate or triaxial. From authors knowledge no work has been attempted yet by any researher on the loations and stability of the ollinear points when the bigger primary is a triaxial rigid body in the relativisti RBP. Hene it raised a uriosity in our minds to study the effet of triaxiality of the bigger primary on the loation and stability of ollinear equilibrium points in the relativisti RBP. This paper is organized as follows: In Setion, the equations governing the motion are presented; Setion desribes the positions of ollinear points, while their linear stability is analyzed in Setion. A numerial appliation of these results and disussion are given in Setion, and Setion 6, respetively. Finally, Setion onveys the main findings of this paper.. Equations of motion The pertinent equations of motion of the infinitesimal mass in the relativisti RBP when the bigger primary is a triaxial rigid body, in a baryentri synodi oordinate system, and dimensionless variables are given by Brumberg [] and Bhatnagar and Hallan [] as: dt d n dt d n where is the potential-like funtion of the relativisti RBP. As Katour et al. [] we do not inlude the triaxiality parameters, i i in the relativisti part of sine the magnitude of these terms is so small due to where is the speed of light. { } 8 6 Volume
and, n the perturbed mean motion of the primaries is given by n where < is the ratio of the mass of the smaller primary to the total mass of the primaries; and are distanes of the infinitesimal mass from the bigger and smaller primary, respetively., << i, i haraterize the triaxiality of the bigger primary are given by, R a h R b h MCuskey [] with h b a,, as lengths of its semi-axes.. Loations of ollinear points Equilibrium points are those points at whih no resultant fore ats on the third infinitesimal body. Therefore, if it is plaed at any of these points with zero veloity, it will stay there. In fat, all derivatives of the oordinates with respet to the time are zero at these points. Therefore, the equilibrium points are solutions of equations and and may be written as and F, with. F International Frontier Siene Letters Vol.
8 Volume In order to find the ollinear points, we put in equation. Their absissae are the roots of the equation f 6 with,. To loate the ollinear points on the axis, we divide the orbital plane into three parts: <, < < and < with respet to the primaries. L a,, C,,, L b,, C,,, L, Case : Position of > Figure. Referene parameter for ollinear Lagrangian points. L see Fig. a,, Let ; ; sine the distane between the primaries is unity, i.e. and then C ; ;,,
International Frontier Siene Letters Vol. 9 Now substituting equation in 6, we obtain 9 8 6 6 6 8 9 8 6 6 8 6 6 6 8 6 6 9 6 9 8 6 6 In the presene of triaxiality only, we have 6 6 8 6 6 6 8 8 6 6 8 Case : Position of L < < see Fig. b Let ; ; ; 9 Substituting equation 9 in 6, we obtain 9 8 6 6 8 6 8 6 6 8 6 6 6 8 8 8 9 6 66 6 6 9 8 9 6 6 In the presene of triaxiality only, we have 6 6 6 8 6 6 8 8 8 6 6 8 Case : Position of L < see Fig. b Let the distane of the point L from the bigger primary be. Sine ; and 8 8a a ; ;
Volume substituting equation in 6, we obtain 6 6 96 6 9 8 8 8 8 9 68 6 6 8 6 8 8 9 9 6 6 9 96 8 6 96 9 6 6 6 6 8 6 8 6 In the presene of triaxiality only, we have 6 6 8 6 8 68 8 6 8 6 9 8 96 6 8 6 96 6 6 6 8 It is notied that in eah ase there exists only one physially reasonable root.. Stability of ollinear points a e examine the stability of an equilibrium onfiguration that is its ability to restrain the body motion in its viinity. To do so we displae the infinitesimal body a little from an equilibrium point with small veloity. If its motion is rapid departure from viinity of the point, we all suh a position of equilibrium an unstable one. If the body osillates about the point, it is said to be a stable position. As Singh and Bello [, ] to study the stability of the ollinear points we linearize the equations of motion by using Taylor s expansion and negleting seond and higher order terms of,, and also their produts, the harateristi equation is given by where, Pq Pq ω Pq Pq Pq Pq Pq Pq ω Pq Pq 6 6 6 6 P, P, P, P n, P, P, q, q, q n, 6 q, q, q. 6 The seond order partial derivative of are denoted by subsripts. The supersripts indiates that the derivative is to be evaluated at the ollinear equilibrium points, under onsideration.
International Frontier Siene Letters Vol. The seond order derivatives are 9 9 9 6 6 9 9 9 9 6 6 6 9 9 9 6 6 a a 6a
Volume a 8a 9a a a a 6 6 a 8 8 In order to study the stability of the ollinear points we have to study the motion in the proximity of these points, hene in this ase a-a an be written as: 9 9 6 6 b b
International Frontier Siene Letters Vol. 6b b 8b 9b b b 6 6 b b 8 Now we will show the disriminant of is positive at the ollinear points i,, It is notied that L i. K p q pq p q p q > 6 6 as shown below K an also be written as K From b and 8b it is lear that > and > Now we will study the signs of Firstly we will do this at, and at the ollinear points L i i,, L sine the oordinate of this point is,, then r and r where < <<, hene we an write and as a funtion in say g and h, respetively. Therefore, in the ase from b, g g and from b, h h, hene < and > in whih < and onsequently K >. Hene, the disriminant of the harateristi equation is positive, and the harateristi roots an be written as ω, ± s, ω, ± is. where s and s are real. Thus, ω, are real and ω, are pure imaginary, hene the motion around the ollinear point L is unbounded and the solution is unstable. Similarly, it an be shown that the points L, L are also unstable.
Volume. Numerial results Earth-Moon System.. Loations For the effet of triaxiality fator on the positions of the ollinear points of the Earth-Moon systems, we take five different ases of different set of semi-axes in Km. a, b, h of the bigger primary i.e. 6, 6, 6, 6, 69, 68, 6, 68, 66, 6, 6, 6 and 6, 66, 6. e have alulated the loations of the three ollinear libration points in all the five ases by using the method in the above mentioned analysis. The seond entries in table orrespond to the positions of ollinear points in the presene of triaxiality only. Some of the data has been borrowed from Sharma and SubbaRao [6] and Dermott & Murray []. Table. Collinear libration points Earth-Moon system, m.9, 96.88, R 8km. Parameter Classial Case Case Case Case Case.66968x - 6.96x -.688x -.88x -.x -.968x -.x - 6.898x - L.68888.68.686.686.68.689.686.686.689. 689 L.89996.896.89998.89998 L -.8 -.668 -.669 -.6689.89.89 -.699 -.699.8966.8966 -.696 -.696.899.899 -.668 -.668.. Stability at Point L for the Earth-Moon System e numerially investigate the stability of the ollinear equilibrium point L for the Earth- Moon System. For this, we ompute the value of using equation 8 and roots of harateristi equation for varying the triaxiality parameters, and list them in table. Case Table. Roots of the harateristi equation. Case.66968. Case 6.96.968 Case.688. Case.88 6.898.698.698.6999.696.69688 p.99.99.99.99.99 p p p -. -.8 -.86 -.9 -.86 p -.89689 -.868986 -.866886 -.8668 -.8699 p 6 q q..... q..8.86.9.86 q q q 6.9989.9886.98999.989.86 ω, ±.888888 ±.8896 ±.886698 ±.886 ±.88686 ω, ±.869988i ±.8698i ±.8666i ±.866i ±.86i
International Frontier Siene Letters Vol. It is lear from table that for a speifi set of parameters, at least one of the roots among all has a positive real root. Thus we onlude that the equilibrium point L is unstable. e may examine the stability of other equilibrium points L, L in the same manner as L. e will see that L, L are also unstable. 6. Disussion Equations - desribe the motion of a third body under the influene of the triaxiality of the bigger primary. Equations 8,, give respetive positions of the ollinear equilibrium points L, L, L whih are dependent of the relativisti terms and triaxiality oeffiients while equations 8a, a, a, give their positions in the presene of triaxiality fator only. It is notied in setion that the relativisti terms and triaxiality oeffiients are unable to alter the instability harateristi behavior of the ollinear points. This is onfirmed numerially from setion by the presene of positive real roots of the harateristi equation as shown in table. In the absene of triaxiality oeffiients, it is observed from ase in table that the positions of L, L L deviate from the lassial ones. This indiates that the relativisti fator has an effet on, these positions. It an also observed from ases - that the position of L dereases with the inrease of triaxiality oeffiients while the position of L and L inreases with the inrease of triaxiality oeffiients. This indiates that with the ombined effet of relativisti and triaxiality, L omes nearer to the primaries with the inrease in triaxiality oeffiient while L and L moves away from the more massive primary. By omparing first and seond entries of eah ase of table, it an be said that in the presene of triaxiality, the effet of relativisti fator does not show physially in most of the ases on the positions of L and L whereas it is shown on the positions of L exept for the ase. It an be seen from table that L moves towards the origin from the lassial positions due to relativisti effet. The triaxiality also shifts L towards the origin from the lassial position. The similar shift is also seen due to the joint effet. L and L move towards the origin from the lassial position due to relativisti or triaxiality or joint effet.. Conlusion A study of the effet of triaxiality on the loations and stability of ollinear points is arried out. It is notied that in spite of the presene of relativisti terms and triaxiality oeffiient, the instability harateristi behavior of the ollinear points remains unhanged. This is onfirmed using numerial approah as it reveals the existene of at least a positive root. Consequently, the motion is unbounded and we onlude that the equilibrium points are unstable due to positive roots. A numerial survey of the Earth-Moon system showed that in the absene of triaxiality oeffiients, the relativisti terms have an effet on the positions of L, L and L. It is also notied that L omes nearer to the primaries with the inrease in triaxiality oeffiients while L, L move away from the more massive primary with the inrease in triaxiality oeffiients. It is also found that for the Earth-Moon system, in the presene of triaxiality, the relativisti fator has no observable effet in most of the ases on the positions of L and L whereas it has a notieable effet on the position of L exept for the ase. For the future work, the study of the effet of mass ratios on the loations and stability of ollinear points is suggested.
6 Volume Referenes [] V. Szebehely, Theory of orbits. The restrited problem of three- bodies, Aademi Press, New York, 96. [] V. A. Brumberg, Relativisti Celestial Mehanis, Nauka, Mosow, USSR, 9. [] K.B. Bhatnagar, P.P. Hallan, Existene and stability of L, in the relativisti restrited threebody problem, Celest. Meh. Dyn. Astron. 69 998-8. [] C.N. Douskos, E.A. Perdios, On the stability of equilibrium points in the relativisti restrited three-body problem, Celest. Meh. Dyn. Astron. 8 -. [] M.K. Ahmed, F.A. Abd El-Salam, S.E. Abd El-Bar, On the stability of triangular Lagrangian equilibrium points in the relativisti restrited three-body problem, Amerian Journal of Applied Sienes. 6 99-998. [6] O. Ragos et al., On the equilibrium points of the relativisti restrited three-body problem, Nonlinear Analysis. -8. [] S.E. Abd El-Bar, F.A. Abd El-Salam, Computation of the loations of the libration points in the relativisti restrited three-body problem, Amerian Journal of Applied sienes. 9 69-66. [8] S.E Abd El-Bar, F.A. Abd El-Salam, Analytial and semi analytial treatment of the ollinear points in the photogravitational relativisti RBP, Mathematial Problems in Engineering.. [9] F.A. Abd El-Salam, S.E. Abd El-Bar, On the triangular equilibrium points in the photogravitational relativisti restrited three-body problem, Astrophys. Spae Si. 9 -. [] D.A. Katour, F.A. Abd El-Salam, M.O. Shaker, Relativisti restrited three-body problem with oblateness and photo-gravitational orretions to triangular equilibrium points, Astrophys. Spae Si. -9. [] S.E. Abd El- Bar, F.A. Abd El Salam, M. Rasseem, Perturbed loation of L point in the photogravitational relativisti restrited three-body problem RBP with oblate primaries, Canadian journal of physis. 9 -. [] J. Singh, N. Bello, Motion around L in the perturbed relativisti RBP, Astrophys. Spae Si. 9-9. [] J. Singh, N. Bello, Effet of radiation pressure on the stability of L, in the RBP, Astrophys. Spae Si. 8-9. [] J. Singh, N. Bello, Existene and stability of triangular points in the relativisti RBP when the primaries are triaxial rigid bodies and soures of radiation, Journal of Astrophysis and Aerospae Tehnology. 6. [] S.. MCuskey, Introdution to elestial mehanis, Addision-esley, 96. [6] R.K. Sharma, P.V.S. Rao, Collinear equilibria and their harateristi exponents in the restrited three-body problem when the primaries are oblate spheroids, Celest. Meh. 9 89-. [] C.D. Murray, S.F. Dermott, Solar System Dynamis, Cambridge University Press, 999.