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On the stability of triangular points in the relativistic R3BP with oblate primaries and bigger radiating
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On the stability of triangular points in the relativistic R3BP with oblate primaries and bigger radiating Nakone Bello a,⇑, Jagadish Singh b a
Department of Mathematics, Faculty of Science, Usmanu Danfodiyo University, Sokoto, Nigeria b Department of Mathematics, Faculty of Science, Ahmadu Bello University, Zaria, Nigeria Received 3 April 2015; received in revised form 28 October 2015; accepted 30 October 2015 Available online 26 November 2015
Abstract We consider a version of the relativistic R3BP which includes the effects of oblateness of the primaries and radiation of the bigger primary as well on the stability of triangular points. We observe that the positions of the triangular points and their stability are affected by the relativistic effect apart from the radiation and oblateness of the primaries. It is further seen for these points that the range of stability region increases or decreases according as the part of the critical mass value, depending upon relativistic terms, radiation and oblateness coefficients, is positive or negative. A numerical exploration shows that in the Sun–Saturn, Sun–Uranus, Sun–Neptune systems, the oblateness has no influence on their positions and range of stability region; whereas it has a little influence on the Sun–Mars, Sun–Jupiter systems. On the other hand, we found that radiation pressure has an observable effect on the solar system. Ó 2015 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Celestial mechanics; Oblateness; Radiation; Relativity; R3BP
1. Introduction The circular restricted three-body problem (CR3BP) is a famous classical problem and has been receiving considerable attention of scientists and astronomers because of its applications in the dynamics of the solar and stellar systems, lunar theory and artificial satellites. It possesses three collinear L1;2;3 and two triangular L4;5 points of equilibrium. The former points are in general unstable, while the latter points are stable for the mass ratio l < 0:03852 . . . (Szebehely, 1967). Over the years this problem has been unable to discuss the motion of a test particle when its bodies concerned are luminous or oblate spheroids or of variable mass. In the last several decades, there has been strong revival of ⇑ Corresponding author.
E-mail addresses: [email protected] (N. Bello), jgds2004@yahoo. com (J. Singh). http://dx.doi.org/10.1016/j.asr.2015.10.044 0273-1177/Ó 2015 COSPAR. Published by Elsevier Ltd. All rights reserved.
interest in the restricted problem. To make the problem more realistic several perturbing agents have been included in the study of the R3BP. Radiation and oblateness have been main subjects of various investigations (Radzievskii, 1950; Schuerman, 1972, 1980; SubbaRao and Sharma, 1975; Sharma and SubbaRao, 1975; Bhatnagar and Hallan, 1979; Kunitsyn and Polyakhova, 1995; Oberti and Vienne, 2003; AbdulRaheem and Singh, 2006; Singh, 2011; Abouelmagd, 2013). The relativistic effect also plays a key role in the CR3BP. Brumberg (1972, 1991) studied the relativistic problem of three bodies in more details and collected most of the important results on relativistic celestial mechanics. He not only obtained the equations of the motion for the general problem of three bodies but also deduced the equation of motion for the restricted problem of three bodies. Maindl and Dvorak (1994) derived the equations of motion for the relativistic R3BP using the postNewtonian approximation of relativity. Their equations
N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
depend on the mass parameter l and MR , where M is the mass of the system and R is the distance between the primaries. This parameter MR is thus a kind of measure for the role of relativistic effects in the system (the classical case is MR ¼ 0). Finally, they applied the model to the computation of the advance of Mercury’s perihelion in solar system and leads to results compatible with published data. Bhatnagar and Hallan (1998) investigated the existence and linear stability of the triangular points L4,5 in the relativistic R3BP, and found that L4,5 are always unstable in the whole range 0 6 l 6 12, in contrast to the classical R3BP in which they are stable for l < l0 , where l is the mass ratio and l0 ¼ 0:038520 . . . is the Routh’s value. Douskos and Perdios (2002) studied the stability of the triangular points in the relativistic R3BP and contrary to the result of Bhatnagar and Hallan (1998), they obtained a region of linear stability in the parameter space as pffiffiffiffi 69 0 6 l < l0 17 where l0 is Routh’s value. They also 486c2 determined the positions of the collinear points and showed that they are always unstable. Abd El-Bar and Abd El-Salam (2012) studied the effects of relativistic R3BP on both triangular and collinear equilibrium points. The approximate locations of the collinear and triangular points are determined. Recently, many perturbing forces, such as radiation, oblateness, centrifugal force have been included in the study of the relativistic R3BP. Abd El-Salam and Abd El-Bar (2014) studied the photogravitational restricted three-body problem within the framework of the post-Newtonian approximation. The mass of the primaries are assumed to change under the effect of continuous radiation process. The locations of the triangular points are computed. Series forms of the locations are obtained as new analytic results. Katour et al. (2014) extended this work by including the effect of oblateness of both primaries. They computed the new perturbed locations of the triangular points. Singh and Bello (2014a) investigated the motion of a test particle in the vicinity of the triangular points L4,5 by considering the more massive primary as a source of radiation in the framework of the relativistic restricted three-body problem (R3BP). They found that the position and stability of the triangular point are affected by both the relativistic factor and radiation pressure. In their further paper (Singh and Bello, 2014b) they studied the motion of a test particle (infinitesimal mass) in the neighborhood of the triangular point L4 in the framework of the perturbed relativistic restricted three-body (R3BP). The problem is perturbed in the sense that a small perturbation is given to the centrifugal force. They found that the position and stability of the triangular point are affected by both
577
the relativistic factor and a small perturbation in the centrifugal force. To the present authors’ Knowledge no investigation has been carried on the stability of equilibrium points in the relativistic R3BP with oblate primaries and the bigger one radiating. Hence, we thought to study the stability of triangular points of this problem. This model is more realistic for our solar system because the Sun is a source of radiation as well as oblate and the most of its planets are oblate. This paper proceeds as follows: in Section 2, the equations governing the motion are presented; Section 3 describes the positions of triangular points, while their linear stability is analyzed in Section 4; the discussion is given in Section 5, the numerical applications are given in Section 6, finally Section 7 conveys the main findings of this paper. 2. Equations of motion The pertinent equations of motion of an infinitesimal mass in the relativistic R3BP when the primaries are oblate spheroids as well as the bigger primary is a source of radiation, in a barycentric synodic coordinate system ðn; gÞ and dimensionless variables, can be written as Brumberg (1972) and Bhatnagar and Hallan (1998): €n 2ng_ ¼ @W d @W @n dt @ n_ ð1Þ @W d @W _ €g þ 2nn ¼ @g dt @ g_ where
q ð1 lÞ 1 A1 l A2 W ¼ n2 n2 þ g2 þ 1 1þ 2 þ 1þ 2 2 q1 q2 2q1 2q2 o2 1 1 n_2 _ þ n2 ðn2 þ g2 Þ n þ g_ 2 þ 2nðng_ gnÞ þ 2 c 8 3 q1 ð1 lÞ A1 l A2 þ 1þ 2 þ 1þ 2 2 q1 q2 2q1 2q n o 2 2 2 2 2 2 _ þ n ðn þ g Þ n_ þ g_ þ 2nðng_ gnÞ 2 2 ! 2 1 q21 ð1 lÞ A1 l2 A2 1þ 2 þ 2 1þ 2 2 q2 q21 2q1 2q2 7 1 A1 þq1 lð1 lÞ n 4g_ þ nn 1þ 2 2 q1 2q1 1 A2 n2 2 l q1 ð1 lÞ 1þ 2 þ g q2 2 2q2 q31 q32 1 l q ð1 lÞ q1 ð1 lÞ A1 þn2 þ þ 1 1þ 2 qq 2q1 2q2 q1 2q1 1 2
l A2 1þ 2 ð2Þ q2 2q2
578
N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
3 3 1 n ¼ 1 þ ðA1 þ A2 Þ 2 1 lð1 lÞ 4 2c 3
ð3Þ
2
q21 ¼ ðn þ lÞ þ g2 2
q22 ¼ ðn þ l 1Þ þ g2
ð4Þ
Here 0 < l 6 12 is the ratio of the mass of the smaller primary to the total mass of the primaries; q1 and q2 are distances of the infinitesimal mass from the bigger and smaller primary, respectively; n is the perturbed mean motion of the primaries; c is the velocity of light. AE2 AP 2
Ai ¼ i5R2 i ði ¼ 1; 2Þ; 0 < Ai 1 (McCuskey, 1963) characterizes the oblateness of the bigger and smaller primary respectively, AE1 and AE2 are the equatorial radii and AP 1 and AP 2 the polar radii of the bigger and smaller primary respectively, and R is the distance between the primaries, q1 is the radiation factor of the bigger primary and it is given by F p1 ¼ F g1 ð1 q1 Þ such that 0 < 1 q1 1 Radzievskii (1950), where F g1 and F p1 are respectively the gravitational and radiation pressure force. It should be noted here that the second and higher powers of Ai (i ¼ 1; 2) and c12 have been ignored in writing above equations. We notice here our Eq. (2) differs from Eqs. (3) and (4) of Katour et al. (2014) because they did not consider the coupling effects of oblateness together with relativistic terms. They also did not use the proper expression for the mean motion. We have only considered the bigger primary as radiating, while in their case both are radiating.
3. Location of triangular points Libration points are those points at which no resultant force acts on the third (infinitesimal) body. Therefore, if it is placed at any of these points with zero velocity, it will stay there. In fact, all derivatives of the coordinates with respect to the time are zero at these points. Hence, the libration points are obtained from Eqs. (1) after putting n_ ¼ g_ ¼ € n¼€ g ¼ 0. These points are the solutions of the equations @W @W ¼0¼ with n_ ¼ g_ ¼ 0: @n @g For simplicity putting q1 ¼ 1 d, 0 6 d 1 in above equations and neglecting second and higher order terms of d and Ai ði ¼ 1; 2Þ and their product, we get
ð1 lÞðn þ lÞ lðn 1 þ lÞ 3 ð1 lÞðn þ lÞ þ A1 n 5 3 3 2 q1 q q2 1 3 lðn 1 þ lÞ dð1 lÞðn þ lÞ þ þ A2 n 2 q52 q31 1 lð1 lÞ 1 2 3 þ nðn þ g2 Þ ðn2 þ g2 Þ þ 2 3n 1 c 3 2 2 ð1 lÞðn þ lÞ lðn 1 þ lÞ 1l l þ þ n q1 q2 q31 q32 ð1 lÞ2 ðn þ lÞ l2 ðn 1 þ lÞ 7 1 1 þ þ lð1 lÞ þ 2 q1 q2 q4 q42 1 7 ðn þ lÞ ðn 1 þ lÞ þ þ n 2 q31 q32 3 lðn þ lÞ ð1 lÞðn 1 þ lÞ nþl n1þl þ þ 3 þ þ g2 5 5 2 q1 q2 qq q1 q32 1 2 ð3l 2Þðn þ lÞ ð1 3lÞðn 1 þ lÞ 2q3 2q32 1 9 lð1 lÞ 3 n þ ðn2 þ g2 Þn þA1 1 4 3 2 9ð1 lÞðn þ lÞ 9ð1 lÞðn þ lÞ 9lðn 1 þ lÞ þ þ þ 4q51 4q31 4q32 ) 3ð1 lÞ 9ð1 lÞ 9l 2ð1 lÞ2 ðn þ lÞ n2 þ g2 Þ þ þ þ n þ 2q1 2q2 2q31 q61 7 21 1 1 7 3ðn þ lÞ þlð1 lÞA1 þ þ n 2 2q51 4q31 4 q1 q2 2 21 ðn þ lÞ ðn 1 þ lÞ 9g lðn þ lÞ þ þ þ n 4 q51 4 q31 q32 ð1 lÞðn 1 þ lÞ 3ð1 lÞðn þ lÞ 3ðn þ lÞ þ þ þ q52 2q51 2q31 q2 3ðn 1 þ lÞ 3lðn þ lÞ 3ð1 3lÞðn 1 þ lÞ þ 2q1 q32 4q3 4q32 1 3ð1 lÞðn þ lÞ 9 lð1 þ lÞ 3 n þ n2 þ g2 n þ þ A2 1 3 4 3 2 2q1 9lðn 1 þ lÞ 9ð1 lÞðn þ lÞ 9lðn 1 þ lÞ þ ðn2 þ g2 Þ þ 5 3 3 4q2 4q1 4q2 3l 9ð1 lÞ 9l 2l2 ðn 1 þ lÞ þ þ n þ þ lð1 lÞA2 þ 2q1 2q2 2q32 q62 7 21 1 1 7 3ðn 1 þ lÞ 21 ðn þ lÞ 3þ þ n þ n 5 2 4 4q2 4 q1 q2 2q2 q31 2 ðn 1 þ lÞ 9g lðn þ lÞ ð1 lÞðn 1 þ lÞ þ þ þ q51 q52 4 q32 3lðn 1 þ lÞ 3ðn þ lÞ 3ðn 1 þ lÞ 3ð3l 2Þðn þ lÞ þ þ þ 2q52 2q31 q2 2q1 q32 4q31 3ð1 lÞðn 1 þ lÞ 3lðn 1 þ lÞ þ 4q32 2q32 ( ) 3 2 3ð1 lÞn 2ð1 lÞ2 ðn þ lÞ 2 ð1 lÞðn þ lÞ þd ðn þ g Þ 2 q1 q41 q31 7 1 1 7 ðn þ lÞ ðn 1 þ lÞ lð1 lÞd þ þ n 3 3 2 q1 q2 2 q q2 1 ð1 lÞðn 1 þ lÞ nþl n1þl 2 lðn þ lÞ þ þ 3 þ þ3g 2q51 q52 qq q1 q32 1 2 ðn þ lÞð4 5lÞ ðn 1 þ lÞð4l 2Þ þ þ ¼0 2q31 2q32
n
ð5:1Þ
And gF ¼ 0;
ð5:2Þ
N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
with ð1 lÞ l 3 ð1 lÞ 3 l F ¼1 3 þ A1 1 þ A2 1 5 q51 2 q2 q31 q2 2 ð1 lÞ 1 lð1 lÞ 1 þ ðn2 þ g2 Þ þd þ 2 3 1 c 3 2 q31 3 1l l 1l l ðn2 þ g2 Þ þ þ þ3 2 q1 q2 q1 q2 ! ð1 lÞ2 l2 7 1 1 n þ lð1 lÞ þ þ þ 2 q41 q42 q31 q32
1 2l ; 2 ! pffiffiffi 3 xþy þ pffiffiffi : g¼ 2 3
n¼xyþ
Now substituting the values of q1 ; q2 ; n; g from the above equations in (5.1) and (5.2) with g – 0, and neglecting second and higher terms in x; y; c12 , A1 ; A2 ; d we have
3 3 5 3l 3 d ð1 lÞ þ A1 þ A2 ð1 lÞ þ ðl 1Þ x 2 2 2 2 2 2 3l 3 3 3 3l þ lA1 þ 1 þ l A2 þ dð1 lÞ y A1 2 2 2 2 4 3 d 1 9l 27l2 9l3 þ A2 ð1 lÞ ðl 1Þ þ 2 þ 4 2 c 16 16 8 11 125l 183l2 63l3 þ þ A1 8 32 32 16 3 137l 195l2 63l3 þ þ A2 4 32 32 16 7 51l 109l2 35l3 þ þ þ d ¼0 ð6:1Þ 4 8 16 16
15 3ð1 lÞ þ ð1 lÞA1 3ð1 lÞd x 2 15 3 3 þ 3l þ lA2 y þ lA1 þ A2 ð1 lÞ þ dð1 lÞ 2 2 2 1 21 11 53l 49l2 3l3 2 ðl l Þ þ þ þ 2 A1 c 8 4 16 16 2 3 117l 121l2 3l3 þ þ þ A2 2 16 16 2 7 3l 39l2 17l3 þ þ þ d ¼0 ð6:2Þ 2 4 8 8
l 1l 3 l 1l 1 1 3 þ 3 þ g2 5 þ 5 þ 3 þ 2 q1 q2 q1 q2 q1 q2 q1 q32 ð3l 2Þ ð1 3lÞ 9 lð1 lÞ þ A1 1 3 2q1 4 3 2q2 3 9ð1 lÞ 9ð1 lÞ 9l þ ðn2 þ g2 Þ þ ðn2 þ g2 Þ 2 4q51 4q31 4q32 ) 3ð1 lÞ 9l 9ð1 lÞ 2ð1 lÞ2 þ þ þ þ 2q2 2q1 q61 2q31 21 1 1 1 3 l 1l n 5 3þ 3 þ 3 þlð1 lÞA1 4 q1 q1 q2 2 q31 q2 9 l 1l 3 1l 1l 1 1 þ g2 5 þ 5 þ þ 3 þ 3 þ 4 q1 q2 2 q51 q1 q1 q2 q1 q32 l ð1 3lÞ 9 lð1 lÞ 3 þ ðn2 þ g2 Þ þ A2 1 3 3 4 3 2 2q1 2q2 9ð1 lÞ 9l 9l 3l 9ð1 lÞ 9l 2l2 2 2 þ g Þ þ þ þ þ ðn þ 2q1 2q2 q61 4q31 4q32 4q52 2q32 21 1 1 1 3 l 1l n 5þ 3 3 þlð1 lÞA2 þ 4 q2 q2 q1 2 q31 q32 9 l 1l 3 l l 1 1 þ g2 5 þ 5 þ þ þ þ 4 q1 q2 2 q52 q32 q1 q32 q31 q2 ð3l 2Þ ð1 lÞ 3ð1 lÞ 3 2 ð1 lÞ þ ðn þ g2 Þ þ d q1 2 2q31 2q32 q31 ) 2 2ð1 lÞ 7 1 1 l 2ð1 lÞ 3 lð1 lÞd þ n 2 q41 q31 q32 q1 q32 þ3g2
l 1l 1 1 ð4 5lÞ ð4l 2Þ þ þ þ þ þ 2q51 q52 q31 q2 q1 q32 2q31 2q31
The triangular points are the solutions of Eqs. (5.1) and (5.2) with g – 0: Since c12 1 and in the case c12 ! 0 and in the absence of oblateness and radiation (i.e. A1 ¼ A2 ¼ d ¼ 0), one can obtain q1 ¼ q2 ¼ 1 . We now assume in the relativistic R3BP when both primaries are oblate spheroids and the bigger one is radiating, that q1 ¼ 1 þ x and q2 ¼ 1 þ y where, x; y 1 may be depending upon the relativistic and oblateness factors. Substituting these values in the Eqs. (4), solving them for n; g and ignoring terms of second and higher powers of x and y , we get
579
Solving these equations for x and y , we get lð2 þ 3lÞ ð44 þ 51l 22l2 þ 30l3 Þ A1 8c2 48ðl 1Þc2 1 39 16l þ 51l2 1 23 22l þ þ þ A d; 2 2 3 24c2 48c2 ð1 lÞð5 3lÞ 1 74 86l þ 51l2 þ y¼ A1 8c2 2 48c2 15 þ 97l 68l2 þ 30l3 A2 48lc2 ( ) 33 64l þ l5 þ 26l2 þ d: ð7Þ 24c2
x¼
Thus, the coordinates of the triangular points ðn; gÞ denoted by L4 and L5 respectively are
580
N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
1 2l 5 1þ 2 2 4c 1 ð30 109l þ 115l2 21l3 Þ þ A1 2 48ðl 1Þc2 1 15 58l þ 52l2 þ 21l3 þ A2 2 48lc2 1 1 10 5 26 2 þ 2 14l þ þ l d 3 8c 3 3l 3 (pffiffiffi 1 3 2 1þ ð5 þ 6l 6l Þ g¼ 12c2 2 ! pffiffiffi pffiffiffi 3 3ð118 þ 211l 159l2 þ 81l3 Þ þ A1 144ðl 1Þc2 6 ! pffiffiffi pffiffiffi 3 3ð15 þ 136l 84l2 þ 81l3 Þ þ A2 6 144lc2 pffiffiffi ) 3 1 5 2 1 þ 2 28 43l þ þ 13l þ d : ð8Þ 9 4c 2l
n¼
4. Stability of triangular points We examine the linear stability of an equilibrium configuration that is its ability to restrain the body motion in its vicinity. To do so we displace the infinitesimal body a little from an equilibrium point with small velocity. If its motion is rapid departure from vicinity of the point, we call such a position of an equilibrium an unstable one. If the body oscillates about the point, it is said to be a stable position. Since the nature of linear stability about the point L5 will be similar to that about L4 , it will be sufficient to consider here the stability only near L4 . Let ða; bÞ be the coordinates of the triangular point L4 and ða; b 1Þ denote small displacements of the infinitesimal body from the equilibrium point ða; bÞ . First, we use _ gÞ _ about the point ða; b; 0; 0Þ to the expansion of W ðn; g; n; express the R:H :S: of Eq. (1), then after setting n ¼ a þ a; g ¼ b þ b; in the Eqs. (1) of motion and retaining only terms of first order on the R:H :S: , we get @W ¼ Aa þ Bb þ C a_ þ Db_ @n n¼aþa;g¼bþb where A¼
3 1 1 þ 2 ð2 19l þ 19l2 Þ 4 2c 3 ð226 1036l þ 1758l2 1065l3 þ 87l4 Þ þ ð8l 9Þ A1 8 32ðl 1Þc2 3 þ 24l 30 þ 367l 915l2 þ 717l3 þ 87l4 þ A2 þ 8 32lc2 1 1 2 3 4 þ ð30 61l þ 444l 615l þ 234l Þ þ ð3l 1Þ d; 48lc2 2
B¼
pffiffiffi 3 3 2 ð1 2lÞ 1 2 3c 4 ( pffiffiffi ) pffiffiffi 3ð26l 19Þ 3ð46 þ 393l 599l2 135l3 þ 417l4 Þ þ A1 8 96ðl 1Þc2 ( pffiffiffi ) pffiffiffi 3ð26l 7Þ 3ð30 458l þ 1498l2 1533l3 þ 417l4 Þ A2 þ 96lc2 8 pffiffiffi 1 3 2 3 4 þ ð10 26l þ 101l 9l þ 40l Þ d; l 1 þ 8lc2 6
( pffiffiffi ) pffiffiffi 3 3ð46l 35Þ C ¼ 2 ð1 2lÞ þ A1 2c 24c2 ( pffiffiffi ) 2 3ð46l 11Þ p ffiffi ffi A2 þ þ ðl 2Þ d; 24c2 3 3c 2 6 5l þ 5l2 22 33l þ 45l2 D¼ þ A1 2c2 8c2 34 57l þ 45l2 1 2 þ þ ð5 þ 9l 6l Þ d: A 2 3c2 8c2 Similarly, we obtain @W ¼ Ea þ B1 b þ C 1 a_ þ D1 b_ @g n¼aþa;g¼bþb where, E¼
pffiffiffi 3 3 2 ð1 2lÞ 1 2 4 3c ( pffiffiffi ) pffiffiffi 3ð26l 19Þ 3ð46 þ 393l 599l2 135l3 þ 417l4 Þ þ A1 þ 96ðl 1Þc2 8 ( pffiffiffi ) pffiffiffi 3ð26l 7Þ 3ð30 458l þ 1498l2 1533l3 þ 417l4 Þ þ A2 96lc2 8 ( pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi !) 1 13 3 101 3 3 3 2 5 3 5 3 3 3 þ l l þ l þ ð1 þ lÞþ 2 d; 2c 12 24 8 12l 3 6
9 7 1 þ 2 ð2 þ 3l 3l2 Þ 4 6c 33 290 1292l þ 2082l2 1221l3 þ 111l4 þ þ A1 8 32ðl 1Þc2 33 30 347l þ 915l2 777l3 111l4 þ þ A2 8 32lc2 1 3 1 45 5 193 2 59 3 lþ 2 11l þ þ l l þ d; 2 2 4c 4 2l 4 2 1 ð20 13l þ 9l2 Þ 2 C 1 ¼ 2 ð4 þ l l Þ þ A1 2c 8c2 ð16 5l þ 9l2 Þ 1 þ þ ð5 7lÞ d; A 2 8c2 3c2 (pffiffiffi ) pffiffiffi 3ð1 2lÞ 3ð46l 35Þ D1 ¼ þ A1 2c2 24c2 (pffiffiffi ) (pffiffiffi ) 3ð46l 11Þ 3 2 þ ð4 20l þ 18l Þ d: A2 þ 24c2 9c2
B1 ¼
N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
d @W € ¼ F a_ þ B2 b_ þ C 2 € a þ D2 b dt @ n_ n¼aþa;g¼bþb
€ þ P 3 a_ þ P 4 b_ þ P 5 a þ P 6 b ¼ 0; P 1 €a þ P 2 b € þ q a_ þ q b_ þ q a þ q b ¼ 0: q €a þ q b 1
where, ( pffiffiffi ) pffiffiffi 3ð46l 35Þ 3 F ¼ 2 ð1 2lÞ þ A1 24c2 2c ( pffiffiffi ) ( pffiffiffi ) 3ð46l 11Þ 2 3 þ ðl 2Þ d; A2 þ 24c2 9c2 4 l þ l2 ð20 13l þ 9l2 Þ þ B2 ¼ A1 8c2 2c2 ð16 5l þ 9l2 Þ 1 þ þ ð5 7lÞ d A 2 8c2 3c2 17 2l þ 2l2 23 10l þ 6l2 C2 ¼ þ A1 4c2 8c2 19 2l þ 6l2 1 þ þ ð7l 8Þ d A 2 3c2 8c2 (pffiffiffi ) pffiffiffi 3 3ð14l 13Þ D2 ¼ 2 ð1 2lÞ þ A1 4c 24c2 (pffiffiffi ) (pffiffiffi ) 3ð1 þ 14lÞ 3 þ ð2 lÞ d A2 þ 24c2 9c2 d @W € ¼ A3 a_ þ B3 b_ þ C 3 € a þ D3 b dt @ g_ n¼aþa;g¼bþb where,
2
6 5l þ 5l 22 33l þ 45l þ A1 2 2c 8c2 34 57l þ 45l2 1 þ A2 þ 2 5 þ 9l 6l2 d; 2 3c 8c (pffiffiffi ) pffiffiffi 3ð1 2lÞ 3ð35 þ 46lÞ B3 ¼ þ A1 2c2 24c2 ( pffiffiffi ) (pffiffiffi ) 2 3 3ð11 þ 46lÞ 2 þ ð2 10l þ 9l Þ d; A2 þ 9c2 24c2
A3 ¼
2
(pffiffiffi ) pffiffiffi 3ð1 2lÞ 3ð13 þ 14lÞ C3 ¼ þ A1 4c2 24c2 (pffiffiffi ) (pffiffiffi ) 3ð1 þ 14lÞ 3 þ ð2 lÞ d A2 þ 24c2 9c2 3ð5 2l þ 2l2 Þ 25 30l þ 18l2 þ D3 ¼ A1 4c2 8c2 13 6l þ 18l2 1 þ ð9l 8Þ d A2 þ 3c2 8c2 Thus, the variational equations of motion corresponding to Eqs. (1), on making use of Eq. (3), can be obtained as
581
2
3
4
5
ð9Þ
6
The system (9) can be written as:
P3 P4 P5 P6 P1 P2 X€ þ X_ þ X ¼0 q1 q2 q3 q4 q5 q6
ð9:1Þ
T
where X ¼ ða; bÞ , P 1 ¼ 1 þ C 2 ; P 2 ¼ D2 ; P 3 ¼ F C; 3 3 1 2 D ; P 4 ¼ B2 2 1 þ ðA1 þ A2 Þ 2 1 lð1 þ l Þ 4 2c 3 P 5 ¼ A; P 6 ¼ B q1 ¼ C 3 ; q2 ¼ 1 þ D3 ; 3 3 1 C þ A3 ; q3 ¼ 2 1 þ ðA1 þ A2 Þ 2 1 lð1 þ l2 Þ 4 2c 3 q4 ¼ B3 D1 ; q5 ¼ E; q6 ¼ B1 The characteristic equation of the system (9.1) is
P1 P2 2 P3 P4 P 5 P 6
¼0 ð9:2Þ kþ
q q k þ q q4 q5 q6 1 2 3 It is important to note that P 1 ¼ P 2 ; q3 ¼ P 4 ; q5 ¼ P 6 , hence (9.2) simplifies to ðP 1 q2 P 2 q1 Þk4 þ ðP 1 q6 þ P 5 q2 þ P 3 q4 P 6 q1 P 2 q5 P 4 q3 Þk2 þ P 5 q6 P 6 q5 ¼ 0
ð9:3Þ
Substituting the values of P i ; qi ; i ¼ 1; 2; . . . ; 6 in (9.3), the characteristic Eq. (9.3) after normalizing becomes k4 þ bk2 þ d ¼ 0
ð10Þ
where, 9 b¼ 1 2 c 3 80 108l þ 105l2 18l3 þ þ 3l þ A1 2 8c2 3 101 48l þ 51l2 þ 18l3 3l þ þ A2 2 8c2 61 6l 57l2 þ 12l3 þ d; 12c2
27lð1 lÞ 9lð65 þ 77l 24l2 þ 12l3 Þ d¼ þ 4 8c2 117 3ð80 7245l þ 9624l2 3366l3 þ 846l4 Þ A1 þ lð1 lÞ þ 4 64c2 117 3ð61 5289l þ 4602l2 18l3 þ 846l4 Þ lð1 lÞ þ þ A2 4 64c2 61 þ 397l 832l2 þ 240l3 þ 336l4 3 þ lð1 lÞ d þ 2 32c2
582
N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
When c12 ! 0 and in the absence of the oblateness and radiation ði:e: A1 ¼ A2 ¼ d ¼ 0Þ; (10) reduces to its wellknown classical restricted problem form (see e.g. Szebehely, 1967): k4 þ k 2 þ
27 lð1 lÞ ¼ 0 4
Its roots are pffiffiffiffi b D 2 k ¼ 2
8A1 653A2 427d 18 þ 24c2 c2 c2 16c2
20439 16539 405 A1 þ A2 2 16c2 16c2 8c
1 < 08l 2 0; ð15Þ 2 3525 3525 15d ¼ 6 A1 þ 6 þ A2 þ 32c2 32c2 24c2 þ
The discriminant of (10) is 54 1269A1 1269A2 42d 4 D¼ 2 l c2 c 8c2 8c2 108 5013A1 63A2 28d 3 þ þ þ 2 l c2 c 8c2 8c2 693 þ 27 2 þ 117A1 þ 6dþ117A2 2c 7113A1 6801A2 189d 2 þ l c2 8c2 8c2 585 þ 27 þ 2 111A1 123A2 6d 2c 20439A1 16539A2 405d þ þ lþ1 8c2 16c2 16c2 3A1 þ 3A2
is monotone increasing in 0; 12 . This implies that dD dl But dD 585 ¼ 27 þ 2 111A1 123A2 6d dl l¼0 2c
dD dl l¼1 2
ð16Þ ðDÞl¼0 ¼ 1 3A1 þ 3A2
18 8A1 653A2 427d 2 þ >0 c2 24c2 c 16c2 ð17Þ
23 117 207 32585 ðA1 þ A2 Þ þ 2 þ ðA1 þ A2 Þ 4 4 4c 128c2 3d 479d <0 ð18Þ þ 2 48c2
ðDÞl¼1 ¼ 2
Here in order to study the monotonicity of D , we have to considertwo cases: 60 Case 1: dD dl 1 ð11Þ
l¼2
For this case the table of variation of D is given in table below
ð12Þ
From (11), we have dD 54 1269A1 1269A2 42d 3 ¼4 l dl c2 c2 8c2 8c2 108 5013A1 63A2 28d 2 þ3 þ þ 2 l c2 c 8c2 8c2 693 7113A1 þ 2 27 2 þ 117A1 þ 117A2 þ 6d 2c 4c2 6801A2 189d 585 þ l þ 27 þ 2 111A1 123A2 2 2 2c 2c 8c 20439A1 16539A2 405d 6d þ þ ; ð13Þ 8c2 16c2 16c2 d 2D 54 1269 1269 42 ¼ 12 A A d l2 1 2 dl2 c2 8c2 8c2 c2 108 5013 63 28 þ6 þ A þ A d l 1 2 c2 8c2 8c2 c2 693 7113 þ 2 27 2 þ 117A1 þ 117A2 þ 6d A1 2c 4c2
6801 189 1 2 A2 þ 2 d > 08l 2 0; ð14Þ 8c 2c 2
From the above 1table it can be seen that D is monotone decreasing in 0; 2 . Since ðDÞl¼0 and ðDÞl¼1 are of opposite 2 signs, and D is monotone and continuous, there is one value of l, e.g. l0c in the interval 0; 12 for which D vanishes. >0 Case 2: dD dl l¼12 Since from (14), dD is monotone increasing in 0; 12 and dl dD dD < 0 and > 0, this implies that there exists dl dl l¼0 l¼12 l0 2 0; 12 such that dD ¼ 0, hence dD 6 08l 2 ð0; l0 dl dl l¼l0 and dD P 08l 2 l0 ; 12 dl Hence we have the following table of variation of D below
N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
583
We consider the following three regions of the values of l separately.
Since ðDÞl¼0 > 0 and ðDÞl¼1 < 0 ,it can be concluded from 2 the above table that ðDÞl¼l0 < 0 , hence since ðDÞl¼0 and ðDÞl¼l0 are of opposite signs, and D is monotone decreasing and continuous in ð0; l0 ,there is one value of l, e.g. l00c in ð0; l0 for which D vanishes. Solving the equation D ¼ 0 , using (11), we obtain only one value, say lc 2 0; 12 for which D vanishes. Hence lc ¼ l0c ¼ l00c . Therefore the critical value of the mass parameter is obtained as pffiffiffiffiffi 1 1 pffiffiffiffiffi 17 69 1 13 p ffiffiffiffiffi 69 1 þ lc ¼ A1 2 18 486c2 9 69 1 13 2 1 pffiffiffiffiffi A2 pffiffiffiffiffi d þ 9 69 27 69 pffiffiffiffiffi ð197133 þ 15493 69Þ þ A1 536544c2 pffiffiffiffiffi ð197133 þ 15493 69Þ þ A2 536544c2 pffiffiffiffiffi ð327543 þ 20267 69Þ þ d ð19Þ 804816c2 pffiffiffiffiffi 17 69 1 13 p ffiffiffiffiffi 1 þ lc ¼ l0 A1 486c2 9 69 1 13 2 1 pffiffiffiffiffi A2 pffiffiffiffiffi d þ 9 69 27 69 pffiffiffiffiffi 197133 þ 15493 69 þ A1 536544c2 pffiffiffiffiffi 197133 þ 15493 69 þ A2 536544c2 pffiffiffiffiffi 327543 þ 20267 69 þ d ð20Þ 804816c2 where l0 ¼ 0:03852 . . . is the Routh’s value.
i. When 0 6 l < lc ,D > 0 , the values of k2 given by (12) are negative and therefore all the four characteristic roots are distinct pure imaginary numbers. Hence, the triangular points are stable. ii. When lc < l 6 12 ; D < 0 , the real parts of the characteristic roots are positive. Therefore, the triangular points are unstable. iii. When l ¼ lc ; D ¼ 0 , the values of k2 given by (12) are the same. So the solutions of the equations of motion contain secular terms. This induces instability of the triangular points.
5. Discussion This section discusses the triangular libration points in the relativistic restricted three-body problem under the assumption that both primaries are oblate and the bigger one radiating. The positions of triangular libration points (8) are obtained. These locations points are affected by the oblateness coefficients, radiation and relativistic factors. It is important to note that these triangular libration points (8) cease to be classical ones In the absence of oblateness and radiation ði:e: A1 ¼ A2 ¼ d ¼ 0Þ these locations coincide with those of Bhatnagar and Hallan (1998) and Douskos and Perdios (2002). Regarding the stability, Eq. (19) shows that the purely relativistic, oblateness, and radiation effects all reduce the size of the stability region. Moreover, an increase in any of the oblateness and radiating factors results a decrease
Table 2 Critical mass Sun–Mars system: l ¼ 0:0000003222700, 4:859829444 109 , A2 ¼ 0:000000000 108 , c = 12424.24.
A1 ¼
d
lc Classical
Relativistic
Eq. (19)
0.0 0.0001 0.001 0.01 0.02
0.0385208965 0.0385208965 0.0385208965 0.0385208966 0.0385208965
0.0385208946 0.0385208946 0.0385208946 0.0385208946 0.0385208946
0.0385208932 0.0385200015 0.0385119757 0.0384317185 0.0383425439
Table 1 Locations of triangular points Sun–Mars system: c = 12424.24, l ¼ 0:0000003222700, A1 ¼ 4:859829444 109 , A2 ¼ 0:0000000001 108 . d
0.0 0.0001 0.001 0.01 0.02
g
n Classical
Relativistic
Eq. (8)
Classical
Relativistic
Eq. (8)
0.4999996773 0.4999996773 0.4999996773 0.4999996773 0.4999996773
0.4999996813 0.4999996813 0.4999996813 0.4999996813 0.4999996813
0.4999996837 0.4999659321 0.4996621680 0.4966245269 0.4932493700
0.8660254011 0.8660254016 0.8660254040 0.8660254040 0.8660254040
0.8660254016 0.8660254040 0.8660254016 0.8660254016 0.8660254016
0.8660254002 0.8660063970 0.8658353681 0.8641250784 0.8622247566
584
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Table 3 location of triangular points Sun–Jupiter system: l ¼ 0:000953692200, A1 ¼ 4:168135294 1010 , A2 ¼ 0:000019288 108 , c = 22947.35. d
0.0 0.0001 0.001 0.01 0.02
g
n Classical
Relativistic
Eq. (8)
Classical
Relativistic
Eq. (8)
0.4990463078 0.4990463078 0.4990463078 0.4990463078 0.4990463078
0.4990463088 0.4990463088 0.4990463088 0.4990463088 0.4990463088
0.4990463090 0.4990129756 0.4987129753 0.4957129715 0.4923796340
0.8660254040 0.8660254040 0.8660254040 0.8660254040 0.8660254040
0.8660254033 0.8660254033 0.8660254033 0.8660254033 0.8660254033
0.8660254031 0.8660061581 0.8658329532 0.8641009047 0.8621764062
Table 4 Critical mass Sun–Jupiter system: l ¼ 0:000953692200, 4:168135294 1010 , A2 ¼ 0:000019288 108 , c = 22947.35. d
0.0 0.0001 0.001 0.01 0.02
A1 ¼
lc Classical
Relativistic
Eq. (19)
0.0385208965 0.0385208965 0.0385208965 0.0385208965 0.0385208965
0.0385208959 0.0385208959 0.0385208959 0.0385208959 0.0385208959
0.0385208958 0.0385200041 0.0385119783 0.0384317211 0.0383425464
in the size of stability region. This indicates that the oblateness and radiation have destabilizing characteristic behavior. However from mathematical point of view it can be seen from (19) that the joint effects of relativistic term with oblateness of the smaller primary and radiation both expand the size of stability region; whereas the joint effect of relativistic term with oblateness of the bigger primary shrinks it. This can be explained by the presence of positive expressions containing the coupling terms Ac22 and cd2 for the former ones and negative expression containing the coupling term Ac21 for the latter one. However, it can be seen that the overall effect decreases the size of the stability region. In the absence of oblateness and radiation factors ði:e: A1 ¼ A2 ¼ d ¼ 0Þ , the stability results obtained in this study are in agreement with those of Douskos and Perdios (2002) and disagree with those of Bhatnagar and Hallan (1998). The expressions for A; D; A2 ; C 2 in Bhatnagar and Hallan (1998) differ from those of the present study in the absence of oblateness and radiation factors
Table 6 Critical mass Sun–Saturn system: l ¼ 0:000285726000, 1:244960783 1010 , A2 ¼ 0:0000018690 108 , c = 31050.90.
A1 ¼
d
lc Classical
Relativistic
Eq. (19)
0.0 0.0001 0.001 0.01 0.02
0.0385208965 0.0385208965 0.0385208965 0.0385208965 0.0385208965
0.0385208961 0.0385208961 0.0385208961 0.0385208961 0.0385208961
0.0385208961 0.0385200044 0.0385119786 0.0384317215 0.0383425467
ði:e: A1 ¼ A2 ¼ d ¼ 0Þ . Consequently, the expressions for P 1 , P 3 , P 4 , P 5 and the characteristic equation are also different. This led them (Bhatnagar and Hallan 1998) to infer that the triangular points are unstable, contrary to Douskos and Perdios (2002) and our results. In the absence of the relativistic terms, our results coincide with those of AbdulRaheem and Singh (2006) in the absence of small perturbations in the Coriolis and centrifugal forces and the bigger primary is luminous only. When in the case of spherical primaries ði:e: A1 ¼ A2 ¼ 0Þ the result of the present study are in accordance with those of Singh and Bello (2014a) when the bigger primary is only luminous. However, regarding the positions of triangular libration points, when the bigger primary is only luminous, the results of the present study disagree apparently with those of Katour et al. (2014) due to our Eq. (2) differing from their Eqs. (3) and (4) [as mentioned in Section 2]
Table 5 Locations of triangular points Sun–Saturn system: l ¼ 0:000285726000, A1 ¼ 1:244960783 1010 , A2 ¼ 0:0000018690 108 , c = 31050.90. d
0.0 0.0001 0.001 0.01 0.02
g
n Classical
Relativistic
Eq. (8)
Classical
Relativistic
Eq. (8)
0.4997142740 0.4997142740 0.4997142740 0.4997142740 0.4997142740
0.4997142745 0.4997142745 0.4997142745 0.4997142745 0.4997142745
0.4997142746 0.4996809412 0.4993809405 0.4963809337 0.4930475928
0.8660254040 0.8660254040 0.8660254040 0.8660254040 0.8660254040
0.8660254037 0.8660254037 0.8660254037 0.8660254037 0.8660254037
0.8660254037 0.8660061587 0.8658329539 0.8641009071 0.8621764106
N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
585
Table 7 Locations of triangular points Sun–Uranus system: l ¼ 0:000043548000, A1 ¼ 3:073529178 1011 , A2 ¼ 0:0000000070 108 , c = 44056.13. d
0.0 0.0001 0.001 0.01 0.02
g
n Classical
Relativistic
Eq. (8)
Classical
Relativistic
Eq. (8)
0.4999564520 0.4999564520 0.4999564520 0.4999564520 0.4999564520
0.4999564525 0.4999564525 0.4999564525 0.4999564525 0.4999564525
0.4999564525 0.4999231189 0.4996231167 0.4966230945 0.4923897365
0.8660254040 0.8660254040 0.8660254040 0.8660254040 0.8660254040
0.8660254038 0.8660254038 0.8660254038 0.8660254038 0.8660254038
0.8660254038 0.8660061590 0.8658329551 0.8641009172 0.8621764305
Table 8 Critical mass Sun–Uranus system: l ¼ 0:000043548000, A1 ¼ 3:073529178 1011 , A2 ¼ 0:0000000070 108 , c = 44056.13.
Table 10 Critical mass Sun–Neptune system: c = 55148.85, l ¼ 0:000051668900, A1 ¼ 3:073480978 1011 , A2 ¼ 0:0000000010 108 , c = 55148.85.
d
d
0.0 0.0001 0.001 0.01 0.02
lc Classical
Relativistic
Eq. (19)
0.0385208965 0.0385208965 0.0385208965 0.0385208966 0.0385208965
0.0385208963 0.0385208963 0.0385208963 0.0385208963 0.0385208963
0.0385208963 0.0385200046 0.0385119788 0.0384317216 0.0383425469
6. Numerical applications We have used Eqs. (8) and (19) to obtain the positions and critical mass for various systems of our solar system. The necessary data has been borrowed from Ragos et al. (2001), Sharma and SubbaRao (1975) and Murray and Dermott (1999). We have also included the corresponding positions and critical mass of the classical problem for comparison purpose (see Fig. 2). It is seen from Table 1 and Table 4 that the oblateness has a little impact on the positions and stability region, whereas from Tables 5–10 that it has no influence. It is clear from Tables 1–10 that the radiation factor has a remarkable effect. It is also observed that, in all figures of the critical mass with varying radiation pressure, the lines lclassical and lrelativistic coincide and the line leq:ð19Þ is below them. It is also seen that in all those figures leq:ð19Þ decreases with an increase in the radiation pressure parameter. With the above analysis one can infer that the relativistic terms and oblateness coefficients have negligible effect on the stability region whereas the radiation pressure has an observable effect (see Figs. 4–7).
lc
0.0 0.0001 0.001 0.01 0.02
Classical
Relativistic
Eq. (19)
0.0385208965 0.0385208965 0.0385208965 0.0385208966 0.0385208965
0.0385208964 0.0385208964 0.0385208964 0.0385208964 0.0385208964
0.0385208964 0.0385200047 0.0385119789 0.0384317217 0.0383425469
Fig. 1 and Fig. 3 show that for all values of the radiation pressure, the locations of the triangular points coincide. This indicates that the relativistic terms, oblateness coefficients and radiation pressure have negligible effect on the locations of the considered systems.
Fig. 1. Effect of varying radiation on the positions of triangular points of the Sun–Mars system.
Table 9 Locations of triangular points Sun–Neptune system: l ¼ 0:000051668900, A1 ¼ 3:073480978 1011 , A2 ¼ 0:0000000010 108 , c = 55148.85. d
0.0 0.0001 0.001 0.01 0.02
g
n Classical
Relativistic
Eq. (8)
Classical
Relativistic
Eq. (8)
0.4999483311 0.4999483311 0.4999483311 0.4999483311 0.4999483311
0.4999483311 0.4999483311 0.4999483311 0.4999483311 0.4999483311
0.4999483311 0.4999149976 0.4996149964 0.4966149845 0.4932816379
0.8660254040 0.8660254040 0.8660254040 0.8660254040 0.8660254040
0.8660254040 0.8660254040 0.8660254040 0.8660254040 0.8660254040
0.8660254040 0.8660061590 0.8658329546 0.8641009107 0.8621764175
586
N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
Fig. 2. Effect of varying radiation on the critical mass of the Sun–Mars system.
Fig. 3. Effect of varying radiation on the positions of triangular points of Sun–Jupiter system.
Fig. 6. Effect of varying radiation on the critical mass of the Sun–Uranus system.
Fig. 7. Effect of varying radiation on the critical mass of the Sun–Neptune system.
7. Conclusion
Fig. 4. Effect of varying radiation on the critical mass on Sun–Jupiter system.
Fig. 5. Effect of varying radiation on the critical mass of the Sun–Saturn system.
We study the linear stability of the triangular points under the assumption that both primaries are oblate spheroids and bigger one is radiating in the relativistic R3BP. It is found that their positions and stability are affected by the relativistic, oblateness and radiation factors. It is seen that the range of stability increases or decreases according as the part of the critical mass value , depending upon the relativistic, oblateness and radiation factors, is positive or negative. The results of the present study differ from those of Katour et al. (2014) when the bigger primary is luminous only. Our major difference is that the expression of the perturbed mean motion (3) in this study differs from their own analytically although it will not affect the results numerically in a big scale. It seems that there is an error in the expression of the mean motion which they have used [ see SubbaRao and Sharma 1975]. In addition to that they did not incorporate the coupling term Ac2i ði ¼ 1; 2Þ in their study, while we do and did not study stability, while we do. From numerical exploration it is seen that for our solar system the relativistic terms, oblateness , radiation have a little or no impact on the positions of the triangular points. This is shown in Fig. 1 and Fig. 3. It is also observed that the relativistic terms and oblateness have no impact on the stability, whereas the radiation has a remarkable impact as shown in all figures of the critical mass parameter with varying radiation.
N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
References Abd.El-Bar, S.E., Abd El-Salam, F.A., 2012. Computation of the locations of the libration points in the relativistic restricted threebody problem. Am. J. Appl. Sci. 9 (5), 659–665. Abd El-Salam, F.A., Abd El-Bar, S.E., 2014. On the triangular equilibrium points in the photogravitational relativistic restricted three-body problem. Astrophys. Space Sci. 349, 125–135. AbdulRaheem, A., Singh, J., 2006. Combined effects of perturbations, radiation, and oblateness. Astron. J. 131, 1880–1885. Abouelmagd, E.I., 2013. Stability of triangular points under the combined effects of radiation and oblateness in the restricted three-body problem. Earth Moon Planets 110, 143–155. Bhatnagar, K.B., Hallan, P.P., 1979. Effect of perturbed potentials on the stability of libration points in the restricted problem. Celest. Mech. 20, 95–103. Bhatnagar, K.B., Hallan, P.P., 1998. Existence and stability of L4,5 in the relativistic restricted three-body problem. Celest. Mech. Dyn. Astron. 69 (3), 271–281. Brumberg, V.A., 1972. Relativistic Celestial Mechanics Press (Science). Nauka, Moscow. Brumberg, V.A., 1991. Essential Relativistic Celestial Mechanics. Adam Hilger Ltd, New York. Douskos, C.N., Perdios, E.A., 2002. On the stability of equilibrium points in the relativistic restricted three-body problem. Celest. Mech. Dyn. Astron. 82, 317–321. Katour, D.A., Abd El-Salam, F.A., Shaker, M.O., 2014. Relativistic restricted three-body with oblateness and photo-gravitational corrections to triangular equilibrium points. Astrophys. Space Sci. 351 (1), 143–149. Kunitsyn, A.L., Polyakhova, E.N., 1995. The restricted photogravitational three-body problem: a modern state. Astron. Astrophys. Trans. 6, 283–293.
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Maindl, T.I., Dvorak, R., 1994. On the dynamics of the relativistic restricted three-body problem. Astron. Astrophys. 290, 335–339. McCuskey, S.W., 1963. Introduction to Celestial Mechanics. AddisonWesley. Murray, C.D., Dermott, S.F., 1999. Solar System Dynamics. Cambridge University Press. Oberti, P., Vienne, A., 2003. An upgraded theory for Helene, Telesto and Calypso. A&A 397, 353–359. Radzievskii, V.V., 1950. The restricted problem of three bodies taking account of light pressure. Astron. Z. 27, 250–256. Ragos, O., Perdios, E.A., Kalantonis, V.S., Vrahatis, M.N., 2001. On the equilibrium points of the relativistic restricted three-body problem. Nonlinear Anal. 47, 3413–3418. Schuerman, D.W., 1972. Roche potential including radiation effects. Astrophys. Space Sci. 19, 351–358. Schuerman, D.W., 1980. The restricted three-body problem including radiation pressure. Astron. J. 238, 337–342. Sharma, R.K., SubbaRao, P.V., 1975. Collinear equilibria and their characteristic exponents in the restricted three-body problem when the primaries are oblate spheroids. Celest. Mech. 12, 189–201. Singh, J., 2011. Combined effects of perturbations, radiation, and oblateness on the nonlinear stability of triangular points in the restricted three-body problem. Astrophys. Space Sci. 322, 331–339. Singh, J., Bello, N., 2014a. Effect of radiation pressure on the stability of L4,5 in the relativistic R3BP. Astrophys. Space Sci. 351 (2), 483–490. Singh, J., Bello, N., 2014b. Motion around L4 in the perturbed relativistic R3BP. Astrophys. Space Sci. 35 (2), 491–497. SubbaRao, P.V., Sharma, R.K., 1975. A note on the stability of the triangular points of equilibrium in the restricted three-body problem. Astron. Astrophys. 43, 381–383. Szebehely, V., 1967. The Restricted Problem of Three-bodies. Academic Press, New York.
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On the stability of triangular points in the relativistic R3BP with oblate primaries and bigger radiating Nakone Bello a,⇑, Jagadish Singh b a
Department of Mathematics, Faculty of Science, Usmanu Danfodiyo University, Sokoto, Nigeria b Department of Mathematics, Faculty of Science, Ahmadu Bello University, Zaria, Nigeria Received 3 April 2015; received in revised form 28 October 2015; accepted 30 October 2015 Available online 26 November 2015
Abstract We consider a version of the relativistic R3BP which includes the effects of oblateness of the primaries and radiation of the bigger primary as well on the stability of triangular points. We observe that the positions of the triangular points and their stability are affected by the relativistic effect apart from the radiation and oblateness of the primaries. It is further seen for these points that the range of stability region increases or decreases according as the part of the critical mass value, depending upon relativistic terms, radiation and oblateness coefficients, is positive or negative. A numerical exploration shows that in the Sun–Saturn, Sun–Uranus, Sun–Neptune systems, the oblateness has no influence on their positions and range of stability region; whereas it has a little influence on the Sun–Mars, Sun–Jupiter systems. On the other hand, we found that radiation pressure has an observable effect on the solar system. Ó 2015 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Celestial mechanics; Oblateness; Radiation; Relativity; R3BP
1. Introduction The circular restricted three-body problem (CR3BP) is a famous classical problem and has been receiving considerable attention of scientists and astronomers because of its applications in the dynamics of the solar and stellar systems, lunar theory and artificial satellites. It possesses three collinear L1;2;3 and two triangular L4;5 points of equilibrium. The former points are in general unstable, while the latter points are stable for the mass ratio l < 0:03852 . . . (Szebehely, 1967). Over the years this problem has been unable to discuss the motion of a test particle when its bodies concerned are luminous or oblate spheroids or of variable mass. In the last several decades, there has been strong revival of ⇑ Corresponding author.
E-mail addresses: [email protected] (N. Bello), jgds2004@yahoo. com (J. Singh). http://dx.doi.org/10.1016/j.asr.2015.10.044 0273-1177/Ó 2015 COSPAR. Published by Elsevier Ltd. All rights reserved.
interest in the restricted problem. To make the problem more realistic several perturbing agents have been included in the study of the R3BP. Radiation and oblateness have been main subjects of various investigations (Radzievskii, 1950; Schuerman, 1972, 1980; SubbaRao and Sharma, 1975; Sharma and SubbaRao, 1975; Bhatnagar and Hallan, 1979; Kunitsyn and Polyakhova, 1995; Oberti and Vienne, 2003; AbdulRaheem and Singh, 2006; Singh, 2011; Abouelmagd, 2013). The relativistic effect also plays a key role in the CR3BP. Brumberg (1972, 1991) studied the relativistic problem of three bodies in more details and collected most of the important results on relativistic celestial mechanics. He not only obtained the equations of the motion for the general problem of three bodies but also deduced the equation of motion for the restricted problem of three bodies. Maindl and Dvorak (1994) derived the equations of motion for the relativistic R3BP using the postNewtonian approximation of relativity. Their equations
N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
depend on the mass parameter l and MR , where M is the mass of the system and R is the distance between the primaries. This parameter MR is thus a kind of measure for the role of relativistic effects in the system (the classical case is MR ¼ 0). Finally, they applied the model to the computation of the advance of Mercury’s perihelion in solar system and leads to results compatible with published data. Bhatnagar and Hallan (1998) investigated the existence and linear stability of the triangular points L4,5 in the relativistic R3BP, and found that L4,5 are always unstable in the whole range 0 6 l 6 12, in contrast to the classical R3BP in which they are stable for l < l0 , where l is the mass ratio and l0 ¼ 0:038520 . . . is the Routh’s value. Douskos and Perdios (2002) studied the stability of the triangular points in the relativistic R3BP and contrary to the result of Bhatnagar and Hallan (1998), they obtained a region of linear stability in the parameter space as pffiffiffiffi 69 0 6 l < l0 17 where l0 is Routh’s value. They also 486c2 determined the positions of the collinear points and showed that they are always unstable. Abd El-Bar and Abd El-Salam (2012) studied the effects of relativistic R3BP on both triangular and collinear equilibrium points. The approximate locations of the collinear and triangular points are determined. Recently, many perturbing forces, such as radiation, oblateness, centrifugal force have been included in the study of the relativistic R3BP. Abd El-Salam and Abd El-Bar (2014) studied the photogravitational restricted three-body problem within the framework of the post-Newtonian approximation. The mass of the primaries are assumed to change under the effect of continuous radiation process. The locations of the triangular points are computed. Series forms of the locations are obtained as new analytic results. Katour et al. (2014) extended this work by including the effect of oblateness of both primaries. They computed the new perturbed locations of the triangular points. Singh and Bello (2014a) investigated the motion of a test particle in the vicinity of the triangular points L4,5 by considering the more massive primary as a source of radiation in the framework of the relativistic restricted three-body problem (R3BP). They found that the position and stability of the triangular point are affected by both the relativistic factor and radiation pressure. In their further paper (Singh and Bello, 2014b) they studied the motion of a test particle (infinitesimal mass) in the neighborhood of the triangular point L4 in the framework of the perturbed relativistic restricted three-body (R3BP). The problem is perturbed in the sense that a small perturbation is given to the centrifugal force. They found that the position and stability of the triangular point are affected by both
577
the relativistic factor and a small perturbation in the centrifugal force. To the present authors’ Knowledge no investigation has been carried on the stability of equilibrium points in the relativistic R3BP with oblate primaries and the bigger one radiating. Hence, we thought to study the stability of triangular points of this problem. This model is more realistic for our solar system because the Sun is a source of radiation as well as oblate and the most of its planets are oblate. This paper proceeds as follows: in Section 2, the equations governing the motion are presented; Section 3 describes the positions of triangular points, while their linear stability is analyzed in Section 4; the discussion is given in Section 5, the numerical applications are given in Section 6, finally Section 7 conveys the main findings of this paper. 2. Equations of motion The pertinent equations of motion of an infinitesimal mass in the relativistic R3BP when the primaries are oblate spheroids as well as the bigger primary is a source of radiation, in a barycentric synodic coordinate system ðn; gÞ and dimensionless variables, can be written as Brumberg (1972) and Bhatnagar and Hallan (1998): €n 2ng_ ¼ @W d @W @n dt @ n_ ð1Þ @W d @W _ €g þ 2nn ¼ @g dt @ g_ where
q ð1 lÞ 1 A1 l A2 W ¼ n2 n2 þ g2 þ 1 1þ 2 þ 1þ 2 2 q1 q2 2q1 2q2 o2 1 1 n_2 _ þ n2 ðn2 þ g2 Þ n þ g_ 2 þ 2nðng_ gnÞ þ 2 c 8 3 q1 ð1 lÞ A1 l A2 þ 1þ 2 þ 1þ 2 2 q1 q2 2q1 2q n o 2 2 2 2 2 2 _ þ n ðn þ g Þ n_ þ g_ þ 2nðng_ gnÞ 2 2 ! 2 1 q21 ð1 lÞ A1 l2 A2 1þ 2 þ 2 1þ 2 2 q2 q21 2q1 2q2 7 1 A1 þq1 lð1 lÞ n 4g_ þ nn 1þ 2 2 q1 2q1 1 A2 n2 2 l q1 ð1 lÞ 1þ 2 þ g q2 2 2q2 q31 q32 1 l q ð1 lÞ q1 ð1 lÞ A1 þn2 þ þ 1 1þ 2 qq 2q1 2q2 q1 2q1 1 2
l A2 1þ 2 ð2Þ q2 2q2
578
N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
3 3 1 n ¼ 1 þ ðA1 þ A2 Þ 2 1 lð1 lÞ 4 2c 3
ð3Þ
2
q21 ¼ ðn þ lÞ þ g2 2
q22 ¼ ðn þ l 1Þ þ g2
ð4Þ
Here 0 < l 6 12 is the ratio of the mass of the smaller primary to the total mass of the primaries; q1 and q2 are distances of the infinitesimal mass from the bigger and smaller primary, respectively; n is the perturbed mean motion of the primaries; c is the velocity of light. AE2 AP 2
Ai ¼ i5R2 i ði ¼ 1; 2Þ; 0 < Ai 1 (McCuskey, 1963) characterizes the oblateness of the bigger and smaller primary respectively, AE1 and AE2 are the equatorial radii and AP 1 and AP 2 the polar radii of the bigger and smaller primary respectively, and R is the distance between the primaries, q1 is the radiation factor of the bigger primary and it is given by F p1 ¼ F g1 ð1 q1 Þ such that 0 < 1 q1 1 Radzievskii (1950), where F g1 and F p1 are respectively the gravitational and radiation pressure force. It should be noted here that the second and higher powers of Ai (i ¼ 1; 2) and c12 have been ignored in writing above equations. We notice here our Eq. (2) differs from Eqs. (3) and (4) of Katour et al. (2014) because they did not consider the coupling effects of oblateness together with relativistic terms. They also did not use the proper expression for the mean motion. We have only considered the bigger primary as radiating, while in their case both are radiating.
3. Location of triangular points Libration points are those points at which no resultant force acts on the third (infinitesimal) body. Therefore, if it is placed at any of these points with zero velocity, it will stay there. In fact, all derivatives of the coordinates with respect to the time are zero at these points. Hence, the libration points are obtained from Eqs. (1) after putting n_ ¼ g_ ¼ € n¼€ g ¼ 0. These points are the solutions of the equations @W @W ¼0¼ with n_ ¼ g_ ¼ 0: @n @g For simplicity putting q1 ¼ 1 d, 0 6 d 1 in above equations and neglecting second and higher order terms of d and Ai ði ¼ 1; 2Þ and their product, we get
ð1 lÞðn þ lÞ lðn 1 þ lÞ 3 ð1 lÞðn þ lÞ þ A1 n 5 3 3 2 q1 q q2 1 3 lðn 1 þ lÞ dð1 lÞðn þ lÞ þ þ A2 n 2 q52 q31 1 lð1 lÞ 1 2 3 þ nðn þ g2 Þ ðn2 þ g2 Þ þ 2 3n 1 c 3 2 2 ð1 lÞðn þ lÞ lðn 1 þ lÞ 1l l þ þ n q1 q2 q31 q32 ð1 lÞ2 ðn þ lÞ l2 ðn 1 þ lÞ 7 1 1 þ þ lð1 lÞ þ 2 q1 q2 q4 q42 1 7 ðn þ lÞ ðn 1 þ lÞ þ þ n 2 q31 q32 3 lðn þ lÞ ð1 lÞðn 1 þ lÞ nþl n1þl þ þ 3 þ þ g2 5 5 2 q1 q2 qq q1 q32 1 2 ð3l 2Þðn þ lÞ ð1 3lÞðn 1 þ lÞ 2q3 2q32 1 9 lð1 lÞ 3 n þ ðn2 þ g2 Þn þA1 1 4 3 2 9ð1 lÞðn þ lÞ 9ð1 lÞðn þ lÞ 9lðn 1 þ lÞ þ þ þ 4q51 4q31 4q32 ) 3ð1 lÞ 9ð1 lÞ 9l 2ð1 lÞ2 ðn þ lÞ n2 þ g2 Þ þ þ þ n þ 2q1 2q2 2q31 q61 7 21 1 1 7 3ðn þ lÞ þlð1 lÞA1 þ þ n 2 2q51 4q31 4 q1 q2 2 21 ðn þ lÞ ðn 1 þ lÞ 9g lðn þ lÞ þ þ þ n 4 q51 4 q31 q32 ð1 lÞðn 1 þ lÞ 3ð1 lÞðn þ lÞ 3ðn þ lÞ þ þ þ q52 2q51 2q31 q2 3ðn 1 þ lÞ 3lðn þ lÞ 3ð1 3lÞðn 1 þ lÞ þ 2q1 q32 4q3 4q32 1 3ð1 lÞðn þ lÞ 9 lð1 þ lÞ 3 n þ n2 þ g2 n þ þ A2 1 3 4 3 2 2q1 9lðn 1 þ lÞ 9ð1 lÞðn þ lÞ 9lðn 1 þ lÞ þ ðn2 þ g2 Þ þ 5 3 3 4q2 4q1 4q2 3l 9ð1 lÞ 9l 2l2 ðn 1 þ lÞ þ þ n þ þ lð1 lÞA2 þ 2q1 2q2 2q32 q62 7 21 1 1 7 3ðn 1 þ lÞ 21 ðn þ lÞ 3þ þ n þ n 5 2 4 4q2 4 q1 q2 2q2 q31 2 ðn 1 þ lÞ 9g lðn þ lÞ ð1 lÞðn 1 þ lÞ þ þ þ q51 q52 4 q32 3lðn 1 þ lÞ 3ðn þ lÞ 3ðn 1 þ lÞ 3ð3l 2Þðn þ lÞ þ þ þ 2q52 2q31 q2 2q1 q32 4q31 3ð1 lÞðn 1 þ lÞ 3lðn 1 þ lÞ þ 4q32 2q32 ( ) 3 2 3ð1 lÞn 2ð1 lÞ2 ðn þ lÞ 2 ð1 lÞðn þ lÞ þd ðn þ g Þ 2 q1 q41 q31 7 1 1 7 ðn þ lÞ ðn 1 þ lÞ lð1 lÞd þ þ n 3 3 2 q1 q2 2 q q2 1 ð1 lÞðn 1 þ lÞ nþl n1þl 2 lðn þ lÞ þ þ 3 þ þ3g 2q51 q52 qq q1 q32 1 2 ðn þ lÞð4 5lÞ ðn 1 þ lÞð4l 2Þ þ þ ¼0 2q31 2q32
n
ð5:1Þ
And gF ¼ 0;
ð5:2Þ
N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
with ð1 lÞ l 3 ð1 lÞ 3 l F ¼1 3 þ A1 1 þ A2 1 5 q51 2 q2 q31 q2 2 ð1 lÞ 1 lð1 lÞ 1 þ ðn2 þ g2 Þ þd þ 2 3 1 c 3 2 q31 3 1l l 1l l ðn2 þ g2 Þ þ þ þ3 2 q1 q2 q1 q2 ! ð1 lÞ2 l2 7 1 1 n þ lð1 lÞ þ þ þ 2 q41 q42 q31 q32
1 2l ; 2 ! pffiffiffi 3 xþy þ pffiffiffi : g¼ 2 3
n¼xyþ
Now substituting the values of q1 ; q2 ; n; g from the above equations in (5.1) and (5.2) with g – 0, and neglecting second and higher terms in x; y; c12 , A1 ; A2 ; d we have
3 3 5 3l 3 d ð1 lÞ þ A1 þ A2 ð1 lÞ þ ðl 1Þ x 2 2 2 2 2 2 3l 3 3 3 3l þ lA1 þ 1 þ l A2 þ dð1 lÞ y A1 2 2 2 2 4 3 d 1 9l 27l2 9l3 þ A2 ð1 lÞ ðl 1Þ þ 2 þ 4 2 c 16 16 8 11 125l 183l2 63l3 þ þ A1 8 32 32 16 3 137l 195l2 63l3 þ þ A2 4 32 32 16 7 51l 109l2 35l3 þ þ þ d ¼0 ð6:1Þ 4 8 16 16
15 3ð1 lÞ þ ð1 lÞA1 3ð1 lÞd x 2 15 3 3 þ 3l þ lA2 y þ lA1 þ A2 ð1 lÞ þ dð1 lÞ 2 2 2 1 21 11 53l 49l2 3l3 2 ðl l Þ þ þ þ 2 A1 c 8 4 16 16 2 3 117l 121l2 3l3 þ þ þ A2 2 16 16 2 7 3l 39l2 17l3 þ þ þ d ¼0 ð6:2Þ 2 4 8 8
l 1l 3 l 1l 1 1 3 þ 3 þ g2 5 þ 5 þ 3 þ 2 q1 q2 q1 q2 q1 q2 q1 q32 ð3l 2Þ ð1 3lÞ 9 lð1 lÞ þ A1 1 3 2q1 4 3 2q2 3 9ð1 lÞ 9ð1 lÞ 9l þ ðn2 þ g2 Þ þ ðn2 þ g2 Þ 2 4q51 4q31 4q32 ) 3ð1 lÞ 9l 9ð1 lÞ 2ð1 lÞ2 þ þ þ þ 2q2 2q1 q61 2q31 21 1 1 1 3 l 1l n 5 3þ 3 þ 3 þlð1 lÞA1 4 q1 q1 q2 2 q31 q2 9 l 1l 3 1l 1l 1 1 þ g2 5 þ 5 þ þ 3 þ 3 þ 4 q1 q2 2 q51 q1 q1 q2 q1 q32 l ð1 3lÞ 9 lð1 lÞ 3 þ ðn2 þ g2 Þ þ A2 1 3 3 4 3 2 2q1 2q2 9ð1 lÞ 9l 9l 3l 9ð1 lÞ 9l 2l2 2 2 þ g Þ þ þ þ þ ðn þ 2q1 2q2 q61 4q31 4q32 4q52 2q32 21 1 1 1 3 l 1l n 5þ 3 3 þlð1 lÞA2 þ 4 q2 q2 q1 2 q31 q32 9 l 1l 3 l l 1 1 þ g2 5 þ 5 þ þ þ þ 4 q1 q2 2 q52 q32 q1 q32 q31 q2 ð3l 2Þ ð1 lÞ 3ð1 lÞ 3 2 ð1 lÞ þ ðn þ g2 Þ þ d q1 2 2q31 2q32 q31 ) 2 2ð1 lÞ 7 1 1 l 2ð1 lÞ 3 lð1 lÞd þ n 2 q41 q31 q32 q1 q32 þ3g2
l 1l 1 1 ð4 5lÞ ð4l 2Þ þ þ þ þ þ 2q51 q52 q31 q2 q1 q32 2q31 2q31
The triangular points are the solutions of Eqs. (5.1) and (5.2) with g – 0: Since c12 1 and in the case c12 ! 0 and in the absence of oblateness and radiation (i.e. A1 ¼ A2 ¼ d ¼ 0), one can obtain q1 ¼ q2 ¼ 1 . We now assume in the relativistic R3BP when both primaries are oblate spheroids and the bigger one is radiating, that q1 ¼ 1 þ x and q2 ¼ 1 þ y where, x; y 1 may be depending upon the relativistic and oblateness factors. Substituting these values in the Eqs. (4), solving them for n; g and ignoring terms of second and higher powers of x and y , we get
579
Solving these equations for x and y , we get lð2 þ 3lÞ ð44 þ 51l 22l2 þ 30l3 Þ A1 8c2 48ðl 1Þc2 1 39 16l þ 51l2 1 23 22l þ þ þ A d; 2 2 3 24c2 48c2 ð1 lÞð5 3lÞ 1 74 86l þ 51l2 þ y¼ A1 8c2 2 48c2 15 þ 97l 68l2 þ 30l3 A2 48lc2 ( ) 33 64l þ l5 þ 26l2 þ d: ð7Þ 24c2
x¼
Thus, the coordinates of the triangular points ðn; gÞ denoted by L4 and L5 respectively are
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N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
1 2l 5 1þ 2 2 4c 1 ð30 109l þ 115l2 21l3 Þ þ A1 2 48ðl 1Þc2 1 15 58l þ 52l2 þ 21l3 þ A2 2 48lc2 1 1 10 5 26 2 þ 2 14l þ þ l d 3 8c 3 3l 3 (pffiffiffi 1 3 2 1þ ð5 þ 6l 6l Þ g¼ 12c2 2 ! pffiffiffi pffiffiffi 3 3ð118 þ 211l 159l2 þ 81l3 Þ þ A1 144ðl 1Þc2 6 ! pffiffiffi pffiffiffi 3 3ð15 þ 136l 84l2 þ 81l3 Þ þ A2 6 144lc2 pffiffiffi ) 3 1 5 2 1 þ 2 28 43l þ þ 13l þ d : ð8Þ 9 4c 2l
n¼
4. Stability of triangular points We examine the linear stability of an equilibrium configuration that is its ability to restrain the body motion in its vicinity. To do so we displace the infinitesimal body a little from an equilibrium point with small velocity. If its motion is rapid departure from vicinity of the point, we call such a position of an equilibrium an unstable one. If the body oscillates about the point, it is said to be a stable position. Since the nature of linear stability about the point L5 will be similar to that about L4 , it will be sufficient to consider here the stability only near L4 . Let ða; bÞ be the coordinates of the triangular point L4 and ða; b 1Þ denote small displacements of the infinitesimal body from the equilibrium point ða; bÞ . First, we use _ gÞ _ about the point ða; b; 0; 0Þ to the expansion of W ðn; g; n; express the R:H :S: of Eq. (1), then after setting n ¼ a þ a; g ¼ b þ b; in the Eqs. (1) of motion and retaining only terms of first order on the R:H :S: , we get @W ¼ Aa þ Bb þ C a_ þ Db_ @n n¼aþa;g¼bþb where A¼
3 1 1 þ 2 ð2 19l þ 19l2 Þ 4 2c 3 ð226 1036l þ 1758l2 1065l3 þ 87l4 Þ þ ð8l 9Þ A1 8 32ðl 1Þc2 3 þ 24l 30 þ 367l 915l2 þ 717l3 þ 87l4 þ A2 þ 8 32lc2 1 1 2 3 4 þ ð30 61l þ 444l 615l þ 234l Þ þ ð3l 1Þ d; 48lc2 2
B¼
pffiffiffi 3 3 2 ð1 2lÞ 1 2 3c 4 ( pffiffiffi ) pffiffiffi 3ð26l 19Þ 3ð46 þ 393l 599l2 135l3 þ 417l4 Þ þ A1 8 96ðl 1Þc2 ( pffiffiffi ) pffiffiffi 3ð26l 7Þ 3ð30 458l þ 1498l2 1533l3 þ 417l4 Þ A2 þ 96lc2 8 pffiffiffi 1 3 2 3 4 þ ð10 26l þ 101l 9l þ 40l Þ d; l 1 þ 8lc2 6
( pffiffiffi ) pffiffiffi 3 3ð46l 35Þ C ¼ 2 ð1 2lÞ þ A1 2c 24c2 ( pffiffiffi ) 2 3ð46l 11Þ p ffiffi ffi A2 þ þ ðl 2Þ d; 24c2 3 3c 2 6 5l þ 5l2 22 33l þ 45l2 D¼ þ A1 2c2 8c2 34 57l þ 45l2 1 2 þ þ ð5 þ 9l 6l Þ d: A 2 3c2 8c2 Similarly, we obtain @W ¼ Ea þ B1 b þ C 1 a_ þ D1 b_ @g n¼aþa;g¼bþb where, E¼
pffiffiffi 3 3 2 ð1 2lÞ 1 2 4 3c ( pffiffiffi ) pffiffiffi 3ð26l 19Þ 3ð46 þ 393l 599l2 135l3 þ 417l4 Þ þ A1 þ 96ðl 1Þc2 8 ( pffiffiffi ) pffiffiffi 3ð26l 7Þ 3ð30 458l þ 1498l2 1533l3 þ 417l4 Þ þ A2 96lc2 8 ( pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi !) 1 13 3 101 3 3 3 2 5 3 5 3 3 3 þ l l þ l þ ð1 þ lÞþ 2 d; 2c 12 24 8 12l 3 6
9 7 1 þ 2 ð2 þ 3l 3l2 Þ 4 6c 33 290 1292l þ 2082l2 1221l3 þ 111l4 þ þ A1 8 32ðl 1Þc2 33 30 347l þ 915l2 777l3 111l4 þ þ A2 8 32lc2 1 3 1 45 5 193 2 59 3 lþ 2 11l þ þ l l þ d; 2 2 4c 4 2l 4 2 1 ð20 13l þ 9l2 Þ 2 C 1 ¼ 2 ð4 þ l l Þ þ A1 2c 8c2 ð16 5l þ 9l2 Þ 1 þ þ ð5 7lÞ d; A 2 8c2 3c2 (pffiffiffi ) pffiffiffi 3ð1 2lÞ 3ð46l 35Þ D1 ¼ þ A1 2c2 24c2 (pffiffiffi ) (pffiffiffi ) 3ð46l 11Þ 3 2 þ ð4 20l þ 18l Þ d: A2 þ 24c2 9c2
B1 ¼
N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
d @W € ¼ F a_ þ B2 b_ þ C 2 € a þ D2 b dt @ n_ n¼aþa;g¼bþb
€ þ P 3 a_ þ P 4 b_ þ P 5 a þ P 6 b ¼ 0; P 1 €a þ P 2 b € þ q a_ þ q b_ þ q a þ q b ¼ 0: q €a þ q b 1
where, ( pffiffiffi ) pffiffiffi 3ð46l 35Þ 3 F ¼ 2 ð1 2lÞ þ A1 24c2 2c ( pffiffiffi ) ( pffiffiffi ) 3ð46l 11Þ 2 3 þ ðl 2Þ d; A2 þ 24c2 9c2 4 l þ l2 ð20 13l þ 9l2 Þ þ B2 ¼ A1 8c2 2c2 ð16 5l þ 9l2 Þ 1 þ þ ð5 7lÞ d A 2 8c2 3c2 17 2l þ 2l2 23 10l þ 6l2 C2 ¼ þ A1 4c2 8c2 19 2l þ 6l2 1 þ þ ð7l 8Þ d A 2 3c2 8c2 (pffiffiffi ) pffiffiffi 3 3ð14l 13Þ D2 ¼ 2 ð1 2lÞ þ A1 4c 24c2 (pffiffiffi ) (pffiffiffi ) 3ð1 þ 14lÞ 3 þ ð2 lÞ d A2 þ 24c2 9c2 d @W € ¼ A3 a_ þ B3 b_ þ C 3 € a þ D3 b dt @ g_ n¼aþa;g¼bþb where,
2
6 5l þ 5l 22 33l þ 45l þ A1 2 2c 8c2 34 57l þ 45l2 1 þ A2 þ 2 5 þ 9l 6l2 d; 2 3c 8c (pffiffiffi ) pffiffiffi 3ð1 2lÞ 3ð35 þ 46lÞ B3 ¼ þ A1 2c2 24c2 ( pffiffiffi ) (pffiffiffi ) 2 3 3ð11 þ 46lÞ 2 þ ð2 10l þ 9l Þ d; A2 þ 9c2 24c2
A3 ¼
2
(pffiffiffi ) pffiffiffi 3ð1 2lÞ 3ð13 þ 14lÞ C3 ¼ þ A1 4c2 24c2 (pffiffiffi ) (pffiffiffi ) 3ð1 þ 14lÞ 3 þ ð2 lÞ d A2 þ 24c2 9c2 3ð5 2l þ 2l2 Þ 25 30l þ 18l2 þ D3 ¼ A1 4c2 8c2 13 6l þ 18l2 1 þ ð9l 8Þ d A2 þ 3c2 8c2 Thus, the variational equations of motion corresponding to Eqs. (1), on making use of Eq. (3), can be obtained as
581
2
3
4
5
ð9Þ
6
The system (9) can be written as:
P3 P4 P5 P6 P1 P2 X€ þ X_ þ X ¼0 q1 q2 q3 q4 q5 q6
ð9:1Þ
T
where X ¼ ða; bÞ , P 1 ¼ 1 þ C 2 ; P 2 ¼ D2 ; P 3 ¼ F C; 3 3 1 2 D ; P 4 ¼ B2 2 1 þ ðA1 þ A2 Þ 2 1 lð1 þ l Þ 4 2c 3 P 5 ¼ A; P 6 ¼ B q1 ¼ C 3 ; q2 ¼ 1 þ D3 ; 3 3 1 C þ A3 ; q3 ¼ 2 1 þ ðA1 þ A2 Þ 2 1 lð1 þ l2 Þ 4 2c 3 q4 ¼ B3 D1 ; q5 ¼ E; q6 ¼ B1 The characteristic equation of the system (9.1) is
P1 P2 2 P3 P4 P 5 P 6
¼0 ð9:2Þ kþ
q q k þ q q4 q5 q6 1 2 3 It is important to note that P 1 ¼ P 2 ; q3 ¼ P 4 ; q5 ¼ P 6 , hence (9.2) simplifies to ðP 1 q2 P 2 q1 Þk4 þ ðP 1 q6 þ P 5 q2 þ P 3 q4 P 6 q1 P 2 q5 P 4 q3 Þk2 þ P 5 q6 P 6 q5 ¼ 0
ð9:3Þ
Substituting the values of P i ; qi ; i ¼ 1; 2; . . . ; 6 in (9.3), the characteristic Eq. (9.3) after normalizing becomes k4 þ bk2 þ d ¼ 0
ð10Þ
where, 9 b¼ 1 2 c 3 80 108l þ 105l2 18l3 þ þ 3l þ A1 2 8c2 3 101 48l þ 51l2 þ 18l3 3l þ þ A2 2 8c2 61 6l 57l2 þ 12l3 þ d; 12c2
27lð1 lÞ 9lð65 þ 77l 24l2 þ 12l3 Þ d¼ þ 4 8c2 117 3ð80 7245l þ 9624l2 3366l3 þ 846l4 Þ A1 þ lð1 lÞ þ 4 64c2 117 3ð61 5289l þ 4602l2 18l3 þ 846l4 Þ lð1 lÞ þ þ A2 4 64c2 61 þ 397l 832l2 þ 240l3 þ 336l4 3 þ lð1 lÞ d þ 2 32c2
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When c12 ! 0 and in the absence of the oblateness and radiation ði:e: A1 ¼ A2 ¼ d ¼ 0Þ; (10) reduces to its wellknown classical restricted problem form (see e.g. Szebehely, 1967): k4 þ k 2 þ
27 lð1 lÞ ¼ 0 4
Its roots are pffiffiffiffi b D 2 k ¼ 2
8A1 653A2 427d 18 þ 24c2 c2 c2 16c2
20439 16539 405 A1 þ A2 2 16c2 16c2 8c
1 < 08l 2 0; ð15Þ 2 3525 3525 15d ¼ 6 A1 þ 6 þ A2 þ 32c2 32c2 24c2 þ
The discriminant of (10) is 54 1269A1 1269A2 42d 4 D¼ 2 l c2 c 8c2 8c2 108 5013A1 63A2 28d 3 þ þ þ 2 l c2 c 8c2 8c2 693 þ 27 2 þ 117A1 þ 6dþ117A2 2c 7113A1 6801A2 189d 2 þ l c2 8c2 8c2 585 þ 27 þ 2 111A1 123A2 6d 2c 20439A1 16539A2 405d þ þ lþ1 8c2 16c2 16c2 3A1 þ 3A2
is monotone increasing in 0; 12 . This implies that dD dl But dD 585 ¼ 27 þ 2 111A1 123A2 6d dl l¼0 2c
dD dl l¼1 2
ð16Þ ðDÞl¼0 ¼ 1 3A1 þ 3A2
18 8A1 653A2 427d 2 þ >0 c2 24c2 c 16c2 ð17Þ
23 117 207 32585 ðA1 þ A2 Þ þ 2 þ ðA1 þ A2 Þ 4 4 4c 128c2 3d 479d <0 ð18Þ þ 2 48c2
ðDÞl¼1 ¼ 2
Here in order to study the monotonicity of D , we have to considertwo cases: 60 Case 1: dD dl 1 ð11Þ
l¼2
For this case the table of variation of D is given in table below
ð12Þ
From (11), we have dD 54 1269A1 1269A2 42d 3 ¼4 l dl c2 c2 8c2 8c2 108 5013A1 63A2 28d 2 þ3 þ þ 2 l c2 c 8c2 8c2 693 7113A1 þ 2 27 2 þ 117A1 þ 117A2 þ 6d 2c 4c2 6801A2 189d 585 þ l þ 27 þ 2 111A1 123A2 2 2 2c 2c 8c 20439A1 16539A2 405d 6d þ þ ; ð13Þ 8c2 16c2 16c2 d 2D 54 1269 1269 42 ¼ 12 A A d l2 1 2 dl2 c2 8c2 8c2 c2 108 5013 63 28 þ6 þ A þ A d l 1 2 c2 8c2 8c2 c2 693 7113 þ 2 27 2 þ 117A1 þ 117A2 þ 6d A1 2c 4c2
6801 189 1 2 A2 þ 2 d > 08l 2 0; ð14Þ 8c 2c 2
From the above 1table it can be seen that D is monotone decreasing in 0; 2 . Since ðDÞl¼0 and ðDÞl¼1 are of opposite 2 signs, and D is monotone and continuous, there is one value of l, e.g. l0c in the interval 0; 12 for which D vanishes. >0 Case 2: dD dl l¼12 Since from (14), dD is monotone increasing in 0; 12 and dl dD dD < 0 and > 0, this implies that there exists dl dl l¼0 l¼12 l0 2 0; 12 such that dD ¼ 0, hence dD 6 08l 2 ð0; l0 dl dl l¼l0 and dD P 08l 2 l0 ; 12 dl Hence we have the following table of variation of D below
N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
583
We consider the following three regions of the values of l separately.
Since ðDÞl¼0 > 0 and ðDÞl¼1 < 0 ,it can be concluded from 2 the above table that ðDÞl¼l0 < 0 , hence since ðDÞl¼0 and ðDÞl¼l0 are of opposite signs, and D is monotone decreasing and continuous in ð0; l0 ,there is one value of l, e.g. l00c in ð0; l0 for which D vanishes. Solving the equation D ¼ 0 , using (11), we obtain only one value, say lc 2 0; 12 for which D vanishes. Hence lc ¼ l0c ¼ l00c . Therefore the critical value of the mass parameter is obtained as pffiffiffiffiffi 1 1 pffiffiffiffiffi 17 69 1 13 p ffiffiffiffiffi 69 1 þ lc ¼ A1 2 18 486c2 9 69 1 13 2 1 pffiffiffiffiffi A2 pffiffiffiffiffi d þ 9 69 27 69 pffiffiffiffiffi ð197133 þ 15493 69Þ þ A1 536544c2 pffiffiffiffiffi ð197133 þ 15493 69Þ þ A2 536544c2 pffiffiffiffiffi ð327543 þ 20267 69Þ þ d ð19Þ 804816c2 pffiffiffiffiffi 17 69 1 13 p ffiffiffiffiffi 1 þ lc ¼ l0 A1 486c2 9 69 1 13 2 1 pffiffiffiffiffi A2 pffiffiffiffiffi d þ 9 69 27 69 pffiffiffiffiffi 197133 þ 15493 69 þ A1 536544c2 pffiffiffiffiffi 197133 þ 15493 69 þ A2 536544c2 pffiffiffiffiffi 327543 þ 20267 69 þ d ð20Þ 804816c2 where l0 ¼ 0:03852 . . . is the Routh’s value.
i. When 0 6 l < lc ,D > 0 , the values of k2 given by (12) are negative and therefore all the four characteristic roots are distinct pure imaginary numbers. Hence, the triangular points are stable. ii. When lc < l 6 12 ; D < 0 , the real parts of the characteristic roots are positive. Therefore, the triangular points are unstable. iii. When l ¼ lc ; D ¼ 0 , the values of k2 given by (12) are the same. So the solutions of the equations of motion contain secular terms. This induces instability of the triangular points.
5. Discussion This section discusses the triangular libration points in the relativistic restricted three-body problem under the assumption that both primaries are oblate and the bigger one radiating. The positions of triangular libration points (8) are obtained. These locations points are affected by the oblateness coefficients, radiation and relativistic factors. It is important to note that these triangular libration points (8) cease to be classical ones In the absence of oblateness and radiation ði:e: A1 ¼ A2 ¼ d ¼ 0Þ these locations coincide with those of Bhatnagar and Hallan (1998) and Douskos and Perdios (2002). Regarding the stability, Eq. (19) shows that the purely relativistic, oblateness, and radiation effects all reduce the size of the stability region. Moreover, an increase in any of the oblateness and radiating factors results a decrease
Table 2 Critical mass Sun–Mars system: l ¼ 0:0000003222700, 4:859829444 109 , A2 ¼ 0:000000000 108 , c = 12424.24.
A1 ¼
d
lc Classical
Relativistic
Eq. (19)
0.0 0.0001 0.001 0.01 0.02
0.0385208965 0.0385208965 0.0385208965 0.0385208966 0.0385208965
0.0385208946 0.0385208946 0.0385208946 0.0385208946 0.0385208946
0.0385208932 0.0385200015 0.0385119757 0.0384317185 0.0383425439
Table 1 Locations of triangular points Sun–Mars system: c = 12424.24, l ¼ 0:0000003222700, A1 ¼ 4:859829444 109 , A2 ¼ 0:0000000001 108 . d
0.0 0.0001 0.001 0.01 0.02
g
n Classical
Relativistic
Eq. (8)
Classical
Relativistic
Eq. (8)
0.4999996773 0.4999996773 0.4999996773 0.4999996773 0.4999996773
0.4999996813 0.4999996813 0.4999996813 0.4999996813 0.4999996813
0.4999996837 0.4999659321 0.4996621680 0.4966245269 0.4932493700
0.8660254011 0.8660254016 0.8660254040 0.8660254040 0.8660254040
0.8660254016 0.8660254040 0.8660254016 0.8660254016 0.8660254016
0.8660254002 0.8660063970 0.8658353681 0.8641250784 0.8622247566
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Table 3 location of triangular points Sun–Jupiter system: l ¼ 0:000953692200, A1 ¼ 4:168135294 1010 , A2 ¼ 0:000019288 108 , c = 22947.35. d
0.0 0.0001 0.001 0.01 0.02
g
n Classical
Relativistic
Eq. (8)
Classical
Relativistic
Eq. (8)
0.4990463078 0.4990463078 0.4990463078 0.4990463078 0.4990463078
0.4990463088 0.4990463088 0.4990463088 0.4990463088 0.4990463088
0.4990463090 0.4990129756 0.4987129753 0.4957129715 0.4923796340
0.8660254040 0.8660254040 0.8660254040 0.8660254040 0.8660254040
0.8660254033 0.8660254033 0.8660254033 0.8660254033 0.8660254033
0.8660254031 0.8660061581 0.8658329532 0.8641009047 0.8621764062
Table 4 Critical mass Sun–Jupiter system: l ¼ 0:000953692200, 4:168135294 1010 , A2 ¼ 0:000019288 108 , c = 22947.35. d
0.0 0.0001 0.001 0.01 0.02
A1 ¼
lc Classical
Relativistic
Eq. (19)
0.0385208965 0.0385208965 0.0385208965 0.0385208965 0.0385208965
0.0385208959 0.0385208959 0.0385208959 0.0385208959 0.0385208959
0.0385208958 0.0385200041 0.0385119783 0.0384317211 0.0383425464
in the size of stability region. This indicates that the oblateness and radiation have destabilizing characteristic behavior. However from mathematical point of view it can be seen from (19) that the joint effects of relativistic term with oblateness of the smaller primary and radiation both expand the size of stability region; whereas the joint effect of relativistic term with oblateness of the bigger primary shrinks it. This can be explained by the presence of positive expressions containing the coupling terms Ac22 and cd2 for the former ones and negative expression containing the coupling term Ac21 for the latter one. However, it can be seen that the overall effect decreases the size of the stability region. In the absence of oblateness and radiation factors ði:e: A1 ¼ A2 ¼ d ¼ 0Þ , the stability results obtained in this study are in agreement with those of Douskos and Perdios (2002) and disagree with those of Bhatnagar and Hallan (1998). The expressions for A; D; A2 ; C 2 in Bhatnagar and Hallan (1998) differ from those of the present study in the absence of oblateness and radiation factors
Table 6 Critical mass Sun–Saturn system: l ¼ 0:000285726000, 1:244960783 1010 , A2 ¼ 0:0000018690 108 , c = 31050.90.
A1 ¼
d
lc Classical
Relativistic
Eq. (19)
0.0 0.0001 0.001 0.01 0.02
0.0385208965 0.0385208965 0.0385208965 0.0385208965 0.0385208965
0.0385208961 0.0385208961 0.0385208961 0.0385208961 0.0385208961
0.0385208961 0.0385200044 0.0385119786 0.0384317215 0.0383425467
ði:e: A1 ¼ A2 ¼ d ¼ 0Þ . Consequently, the expressions for P 1 , P 3 , P 4 , P 5 and the characteristic equation are also different. This led them (Bhatnagar and Hallan 1998) to infer that the triangular points are unstable, contrary to Douskos and Perdios (2002) and our results. In the absence of the relativistic terms, our results coincide with those of AbdulRaheem and Singh (2006) in the absence of small perturbations in the Coriolis and centrifugal forces and the bigger primary is luminous only. When in the case of spherical primaries ði:e: A1 ¼ A2 ¼ 0Þ the result of the present study are in accordance with those of Singh and Bello (2014a) when the bigger primary is only luminous. However, regarding the positions of triangular libration points, when the bigger primary is only luminous, the results of the present study disagree apparently with those of Katour et al. (2014) due to our Eq. (2) differing from their Eqs. (3) and (4) [as mentioned in Section 2]
Table 5 Locations of triangular points Sun–Saturn system: l ¼ 0:000285726000, A1 ¼ 1:244960783 1010 , A2 ¼ 0:0000018690 108 , c = 31050.90. d
0.0 0.0001 0.001 0.01 0.02
g
n Classical
Relativistic
Eq. (8)
Classical
Relativistic
Eq. (8)
0.4997142740 0.4997142740 0.4997142740 0.4997142740 0.4997142740
0.4997142745 0.4997142745 0.4997142745 0.4997142745 0.4997142745
0.4997142746 0.4996809412 0.4993809405 0.4963809337 0.4930475928
0.8660254040 0.8660254040 0.8660254040 0.8660254040 0.8660254040
0.8660254037 0.8660254037 0.8660254037 0.8660254037 0.8660254037
0.8660254037 0.8660061587 0.8658329539 0.8641009071 0.8621764106
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585
Table 7 Locations of triangular points Sun–Uranus system: l ¼ 0:000043548000, A1 ¼ 3:073529178 1011 , A2 ¼ 0:0000000070 108 , c = 44056.13. d
0.0 0.0001 0.001 0.01 0.02
g
n Classical
Relativistic
Eq. (8)
Classical
Relativistic
Eq. (8)
0.4999564520 0.4999564520 0.4999564520 0.4999564520 0.4999564520
0.4999564525 0.4999564525 0.4999564525 0.4999564525 0.4999564525
0.4999564525 0.4999231189 0.4996231167 0.4966230945 0.4923897365
0.8660254040 0.8660254040 0.8660254040 0.8660254040 0.8660254040
0.8660254038 0.8660254038 0.8660254038 0.8660254038 0.8660254038
0.8660254038 0.8660061590 0.8658329551 0.8641009172 0.8621764305
Table 8 Critical mass Sun–Uranus system: l ¼ 0:000043548000, A1 ¼ 3:073529178 1011 , A2 ¼ 0:0000000070 108 , c = 44056.13.
Table 10 Critical mass Sun–Neptune system: c = 55148.85, l ¼ 0:000051668900, A1 ¼ 3:073480978 1011 , A2 ¼ 0:0000000010 108 , c = 55148.85.
d
d
0.0 0.0001 0.001 0.01 0.02
lc Classical
Relativistic
Eq. (19)
0.0385208965 0.0385208965 0.0385208965 0.0385208966 0.0385208965
0.0385208963 0.0385208963 0.0385208963 0.0385208963 0.0385208963
0.0385208963 0.0385200046 0.0385119788 0.0384317216 0.0383425469
6. Numerical applications We have used Eqs. (8) and (19) to obtain the positions and critical mass for various systems of our solar system. The necessary data has been borrowed from Ragos et al. (2001), Sharma and SubbaRao (1975) and Murray and Dermott (1999). We have also included the corresponding positions and critical mass of the classical problem for comparison purpose (see Fig. 2). It is seen from Table 1 and Table 4 that the oblateness has a little impact on the positions and stability region, whereas from Tables 5–10 that it has no influence. It is clear from Tables 1–10 that the radiation factor has a remarkable effect. It is also observed that, in all figures of the critical mass with varying radiation pressure, the lines lclassical and lrelativistic coincide and the line leq:ð19Þ is below them. It is also seen that in all those figures leq:ð19Þ decreases with an increase in the radiation pressure parameter. With the above analysis one can infer that the relativistic terms and oblateness coefficients have negligible effect on the stability region whereas the radiation pressure has an observable effect (see Figs. 4–7).
lc
0.0 0.0001 0.001 0.01 0.02
Classical
Relativistic
Eq. (19)
0.0385208965 0.0385208965 0.0385208965 0.0385208966 0.0385208965
0.0385208964 0.0385208964 0.0385208964 0.0385208964 0.0385208964
0.0385208964 0.0385200047 0.0385119789 0.0384317217 0.0383425469
Fig. 1 and Fig. 3 show that for all values of the radiation pressure, the locations of the triangular points coincide. This indicates that the relativistic terms, oblateness coefficients and radiation pressure have negligible effect on the locations of the considered systems.
Fig. 1. Effect of varying radiation on the positions of triangular points of the Sun–Mars system.
Table 9 Locations of triangular points Sun–Neptune system: l ¼ 0:000051668900, A1 ¼ 3:073480978 1011 , A2 ¼ 0:0000000010 108 , c = 55148.85. d
0.0 0.0001 0.001 0.01 0.02
g
n Classical
Relativistic
Eq. (8)
Classical
Relativistic
Eq. (8)
0.4999483311 0.4999483311 0.4999483311 0.4999483311 0.4999483311
0.4999483311 0.4999483311 0.4999483311 0.4999483311 0.4999483311
0.4999483311 0.4999149976 0.4996149964 0.4966149845 0.4932816379
0.8660254040 0.8660254040 0.8660254040 0.8660254040 0.8660254040
0.8660254040 0.8660254040 0.8660254040 0.8660254040 0.8660254040
0.8660254040 0.8660061590 0.8658329546 0.8641009107 0.8621764175
586
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Fig. 2. Effect of varying radiation on the critical mass of the Sun–Mars system.
Fig. 3. Effect of varying radiation on the positions of triangular points of Sun–Jupiter system.
Fig. 6. Effect of varying radiation on the critical mass of the Sun–Uranus system.
Fig. 7. Effect of varying radiation on the critical mass of the Sun–Neptune system.
7. Conclusion
Fig. 4. Effect of varying radiation on the critical mass on Sun–Jupiter system.
Fig. 5. Effect of varying radiation on the critical mass of the Sun–Saturn system.
We study the linear stability of the triangular points under the assumption that both primaries are oblate spheroids and bigger one is radiating in the relativistic R3BP. It is found that their positions and stability are affected by the relativistic, oblateness and radiation factors. It is seen that the range of stability increases or decreases according as the part of the critical mass value , depending upon the relativistic, oblateness and radiation factors, is positive or negative. The results of the present study differ from those of Katour et al. (2014) when the bigger primary is luminous only. Our major difference is that the expression of the perturbed mean motion (3) in this study differs from their own analytically although it will not affect the results numerically in a big scale. It seems that there is an error in the expression of the mean motion which they have used [ see SubbaRao and Sharma 1975]. In addition to that they did not incorporate the coupling term Ac2i ði ¼ 1; 2Þ in their study, while we do and did not study stability, while we do. From numerical exploration it is seen that for our solar system the relativistic terms, oblateness , radiation have a little or no impact on the positions of the triangular points. This is shown in Fig. 1 and Fig. 3. It is also observed that the relativistic terms and oblateness have no impact on the stability, whereas the radiation has a remarkable impact as shown in all figures of the critical mass parameter with varying radiation.
N. Bello, J. Singh / Advances in Space Research 57 (2016) 576–587
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