Effect of Perturbations in the Coriolis and Centrifugal Forces on the Stability of L 4 in the Relativistic R3BP

Transcription

1 J. Astrophys. Astr. 04 5, 70 7 c Indian Academy of Sciences Effect of Perturbations in the Coriolis and Centrifugal Forces on the Stability of L 4 in the Relativistic RBP Jagadish Singh & Nakone Bello, Department of Mathematics, Faculty of Science, Ahmadu Bello University, Zaria, Nigeria Department of Mathematics, Faculty of Science, Usmanu Danfodiyo University, Sokoto, Nigeria jgds004@yahoo.com; bnakone@yahoo.com Received 7 June 04; accepted 9 September 04 Abstract. The centrifugal and Coriolis forces do not appear as a result of physically imposed forces, but are due to a special property of a rotation. Thus, these forces are called pseudo-forces or fictitious forces. They are merely an artifact of the rotation of the reference frame adopted. This paper studies the motion of a test particle in the neighbourhood of the triangular point L 4 in the framework of the perturbed relativistic restricted three-body problem RBP when small perturbations are conferred to the centrifugal and Coriolis forces. It is found that the position and stability of the triangular point are affected by both the relativistic factor and small perturbations in the Coriolis and centrifugal forces. As an application, the Sun Earth system is considered. Key words. Celestial mechanics perturbations relativity RBP.. Introduction The general three-body problem is the problem of motion of three celestial bodies under their mutual gravitational attraction. The restricted three-body problem RBP is a simplified form of the general three-body problem, in which one of the bodies is of infinitesimal mass, and therefore does not influence the motion of the remaining two massive bodies called the primaries Bruno 994; Valtonen and Kartunen 006. The circular restricted three-body problem CRBP possesses five stationary solutions called Lagrangian points. Three are collinear with the primaries and the other two are in equilateral triangular configuration with the primaries. The three collinear points L,, are unstable, while the triangular points L 4,5 are stable for the mass ratio μ = m m +m < ,m m being the masses of the primaries Szebehely 967a. Wintner 94 and Contopoulos 00 haveshown that the stability of the triangular equilibrium points is due to the existence of the Coriolis terms in the equations of motion when these equations are recorded in rotating coordinate system. In the classical problem, the effects of the gravitational 70

2 70 Jagadish Singh & Nakone Bello attraction of the infinitesimal body and other perturbations have been ignored. Perturbations can be caused due to the lack of sphericity or triaxiality, oblateness and radiation forces of the bodies, variation of masses, atmospheric drag, solar wind, Poynting Robertson effect and the action of the other bodies. The most striking example of perturbations owing to oblateness in the solar system is the orbit of the fifth satellite of Jupiter, Amalthea. This planet is very oblate and the satellite s orbit is too small that its line of apsides advances about 900 in one year Moulton 94. Such oblateness driven effects have competing disturbing effects on qualitatively similar general relativistic effects Iorio 006; Iorio009; Iorioet al. 0; Renzetti 0b. The Kirkwood gaps in the ring of the asteroid s orbits lying between the orbits of the Mars and Jupiter are examples of the perturbations produced by Jupiter on an asteroid. This enables many researchers to study the restricted problem by taking into account the effects of small perturbations in the Coriolis and centrifugal forces radiation, oblateness and triaxiality of the bodies Sharma and Ishwar 995; Szebehely 967a; SubbaRao and Sharma 975; Bhatnagar and Hallan 978; Singh 0; Singh and Begha 0; Singh 0; AbdulRaheem and Singh 006; AbdulRaheem and Singh 008. Szebehely 967b investigated the stability of triangular points by keeping the centrifugal force constant and found that the Coriolis force is a stabilizing force, whereas SubbaRao and Sharma 975 observed that the oblateness of the primary resulted in an increase in both the Coriolis and the centrifugal forces, thereby concluding that the Coriolis force is not always a stabilizing force. This was confirmed by AbdulRaheem and Singh 006. The theory of general relativity is currently the most successful gravitational theory describing the nature of space and time, and well confirmed by observations Will 04. Regarding the three-body relativistic effects we may also cite that: The geodesic precession of the orbit of two-body system is about one-third the mass in general relativity Renzetti 0a and also for the post-newtonian tidal effects Iorio 04. Indeed, the Newtonian effects of the planet s gravitational fields are in order of magnitude greater than the first corrections due to general relativity, and completely swamp the higher corrections that in principle was provided by the exact Schwarzschild solution. Krefetz 967 computed the post-newtonian deviations of the triangular Lagrangian points from their classical positions in a fixed frame of reference for the first time, but without explicitly stating the equations of motion. After a decade, Contopoulos 976 dealt with the relativistic RBP in rotating coordinates. He derived the Lagrangian of the system along with the deviations of the triangular points. Brumberg 97, 99 studied the relativistic n-body problem of three bodies in more detail and gathered most of the important results on relativistic celestial mechanics. He not only obtained the equations of motion for the general problem of three bodies but also deduced the equations of motion for the restricted problem of three bodies. Maindl and Dvorak 994 derived the equations of motion for the relativistic RBP using the post-newtonian approximation of relativity. They applied this model to the computation of the advance of Mercury s perihelion in the solar system and found that they were compatible with the published data. Bhatnagar and Hallan 998 studied the existence and linear stability of the triangular points L 4,5 in the relativistic RBP, and found that L 4,5 are always unstable in the whole range 0 μ in comparison to the classical RBP which was stable for μ<μ 0,whereμis the mass ratio and μ 0 = is the Routh s value.

3 Perturbations in the Coriolis and Centrifugal Forces in the RBP 70 In the beginning of the st century, Ragos et al. 00 have investigated numerically, the linear stability of the collinear libration points L,, in the relativistic RBP for several cases in solar system, and found that the points L,, were unstable. Douskos and Perdios 00 examined the stability of the triangular points in the relativistic RBP and contrary to the result of Bhatnagar and Hallan 998, they obtained a region of linear stability in the parameter space as 0 μ<μ 0 769, 486c where μ 0 = is the Routh s value. They also determined the positions of the collinear points and showed that they were always unstable. Lucas 00 studied the chaotic amplification in the relativistic RBP and noticed that the difference between Newtonian and post-newtonian trajectories for the RBP is greater for chaotic trajectories than it is for trajectories that are not chaotic. He also discussed the possibility of using this chaotic amplification effect as a novel test of general relativity. Rodica and Vasile 006 investigated the existence and corresponding positions of the equilibrium points in orbital plane in the framework of the RBP in Schwarzchild s gravitational field. They observed that there are three collinear libration points, and if they exist, then only two points are triangular libration points situated in the orbital plane of the primaries. If triangular points exist, they may not form equilateral triangles; the triangles are isosceles for equal masses of the primaries, otherwise scalene. Ahmed et al. 006 also investigated the stability of the triangular points in the relativistic RBP. In contrast to the previous result of Bhatnagar and Hallan 998, they obtained a region of linear stability as 0 μ<μ ,whereμ 9c 0 is the Routh s value. Abd El-Salam and Abd El-Bar 0 derived the equations of motion of the relativistic three-body in the post-newtonian formalism. Yamada and Asada 0 discussed the post-newtonian effects on Lagrange s equilateral triangular solution for the three-body problem. For three finite masses, it was found that a triangular configuration satisfies the post-newtonian equation of motion in general relativity, if and only if, it has the relativistic corrections to each length side. This post-newtonian configuration for three finite masses was not always equilateral and it reclaims the previous results for the restricted-problem when one mass approches zero. They also found that for the same masses and angular velocity, the post-newtonian triangular is always smaller than the Newtonian. It is worth highlighting that so far all the Schwarzschild terms have been used in studies, but gravitomagnetic Lense-Thirring Renzetti 0; Iorio00; Linsen 99 associated with the rotation of the bodies have been neglected. From the previous works and also to the best of our knowledge, no work has been carried out on the combined effects of perturbations on the relativistic RBP, so this increased the curiosity to study the effects of small perturbations in the centrifugal and Coriolis forces on the RBP. The focus of this paper is to study the difference between Newtonian and general relativistic positions and stability of triangular points in the presence of small perturbations in the centrifugal and Coriolis forces. This paper is organized as follows. In section, the equations of motions are presented, section describes the positions of equilibrium points, while their linear stability is analysed 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.

4 704 Jagadish Singh & Nakone Bello. Equations of motion The system of coordinates ξ, η such that the ξ η plane rotates in the positive direction with angular velocity equal to that of the common velocity of one primary with respect to the other keeping the origin fixed, then that coordinate system is known as synodic system. The primaries appear at rest in the synodic or rotating frame ξ, η having its origin at the centre of mass and rotating along with them and are placed on the ξ- axis. The plane ξ η is the plane of the motion of the primaries. The coordinates ξ, η are sometimes called synodical. In this system,the primaries m and m are located at μ, 0, μ, 0, respectively and have zero velocity. The advantage of this system is that m and m have fixed positions, so that the equations of motion are time-independent and therefore it is the easiest way to obtain the stationary solutions. The pertinent equations of motion of an infinitesimal mass in the relativistic RBP in a barycentric synodic coordinate system ξ, η with origin at the centre of mass of the primaries with dimlensionless variables can be denoted as Brumberg 97; Bhatnagar and Hallan 998: ξ n η = W ξ d W, η + n ξ = W dt ξ η d W dt η with W = ξ + η + μ + μ ρ ρ μ μ ξ + η + 8 { ξ + η + ξ η η ξ + ξ + η } + μ + μ ξ + η ρ ρ + + ξ η η ξ+ ξ + η μ c ρ + μ ρ + μ μ {4 η + 7 }, ξ ρ ρ η μ ρ + μ ρ + + μ + μ ρ ρ ρ ρ n = c μ μ, ρ = ξ + μ + η, ρ = ξ + μ 4 + η, where 0<μ is the ratio of mass of the smaller primary to the total mass of the primaries, ρ and ρ are distances of the infinitesimal mass from the bigger and smaller primary, respectively, n the mean motion of the primaries and c is the velocity of light. We now introduce small perturbations in the centrifugal and Coriolis forces with the help of the parameters ψ = +ε ; ε <<,ϕ = +ε ; ε <<. The unperturbed value of each is unity. Consequently, the equations of motion can be taken as ξ ϕn η = W ξ d W, dt ξ

5 where Perturbations in the Coriolis and Centrifugal Forces in the RBP 705 η + ϕn ξ = W η d W, 5 dt η W = ψξ + η + μ + μ ρ ρ μ μ ξ + η ψ + 8 {ϕ ξ + ϕ η + ϕξ η η ξ+ ψξ + η } + μ + μ {ϕ ξ + ϕ η + ϕξ η η ξ + ρ ρ c + ψξ + η } μ ρ + μ ρ. + μ μ {4ϕ η + 7 ψξ ρ ρ } ψη μ ρ + μ ρ + ψ + μ + μ ρ ρ ρ ρ 6. Locations of triangular points The libration points are obtained from equations 5 on changing ξ = η = ξ = η = 0. These points are the solutions of the equations W ξ = 0 = W η, with ξ = η = 0, μξ + μ μξ + μ ψξ ρ ρ ψξ μ μ + ψ ξξ + η ψξ + η μξ + μ μξ + μ ρ + ρ μ +ψ + μ ρ ρ ξ + μ ξ + μ ρ 4 + μ ξ + μ ρ c ψ + 7 ρ ρ ψξ ξ + μ ξ + μ ρ + ρ =0, + μξ + μ μ ξ + μ ψη ρ +μ μ 5 + ρ 5 ξ + μ + ψ ρ ρ + ξ + μ μ ξ + μ ρ ρ ρ μξ + μ ρ ηf = 0, 7

6 706 Jagadish Singh & Nakone Bello with μ F = ψ ρ μ ρ ψ μ μ + μ ψ ξ +η + ψ + μ ρ ρ + ψξ + η μ ρ + μ μ ρ + ρ 4 + μ ρ 4 c { } 7 + μ μ ψξ ρ + μ ρ ψ ρ + μ. ρ + μ μ ψη ρ 5 + ρ 5 +ψ ρ ρ + ρ ρ μ ρ μ ρ The triangular points are the solutions of equations 7 with η = 0. Since << c and in the case 0, one can obtain ρ c = ρ = ; we assume in the relativistic RBP that ρ = + x and ρ ψ = + y, where x,y <<. Substituting these ψ values in equations 4, solving them for ξ,η and ignoring terms of second and higher orders of x and y, we get η =± ξ = x y ψ 4 ψ + ψ + μ, ψ x + y Now substituting the values of ρ,ρ,ξ,η, ψ = + ε,ϕ = + ε ε <<, ε << from the previous equations 7 with η = 0 and neglecting the second and higher order terms in x,y, c,ε,ε and their products, we have ψ 4. μx μy μ μ μ 8c = 0, μx + μy + 7μ μ 8c = 0. Solving these equations for x and y, we get μ + μ x = 8c, y = μ5 μ 8c. 8

7 Perturbations in the Coriolis and Centrifugal Forces in the RBP 707 Thus, the coordinates of the triangular points ξ, ±η denoted by L 4 and L 5, respectively are ξ = μ + 5 4c, { η =± 4ε 9 + } 9 c 5 + 6μ 6μ. 4. Stability of L 4 Let a, b be the coordinates of the triangular point L 4. We set ξ = a + α, η = b + β,α,β << in the equations 5 of motion. First, we compute the terms on their R.H.S, neglecting higher order terms, we get W = Aα + Bβ + C α + D β, ξ where A = 4 ξ=a+α,η=b+β B = 4 { + 5 ε + } c 9μ + 9μ, { μ + 9 ε } c, C = μ, c D = 5μ 5μ + 6 c. Similarly, we obtain W = A α + B β + C α + D β, η ξ=a+α,η=b+β where A = μ 4 B = 9 4 { + 9 ε } c, { ε μ μ 6c }, C = c 4 + μ + μ, D = μ. c d W = A α + B β + C α + D β, dt ξ ξ=a+α,η=b+β

8 708 Jagadish Singh & Nakone Bello where where A = μ, c B = 4 μ + μ c, 7 μ + μ C = 4c, D = μ. 4c d W = A α + B β + C α + D β, dt η ξ=a+α,η=b+β 6 5μ + 5μ A = c, B = μ, c C = μ, 4c D = 5 μ + μ 4c. Thus, the variational equations of motion corresponding to equations 5, on utilizing equation, can be obtained as p α + p β + p α + p 4 β + p 5 α + p 6 β = 0, 0 q α + q β + q α + q 4 β + q 5 α + q 6 β = 0, where p = { + C, p = D, p = A C, p 4 = B + ε μ + μ c q = C, q = + D, q = + ε q 4 = B D, q 5 = A, q 6 = B. } D, p 5 = A, p 6 = B C + A, c μ + μ Then, the characteristic equation is p q p q λ 4 + p q 6 + p 5 q + p q 4 p 6 q p q 5 p 4 q λ + p 5 q 6 p 6 q 5 = 0. Substituting the values of p i,q i,i =,,...,6 in, the characteristic equation becomes λ 4 + ε 9 c +8ε λ 95μ μ + 7μ 6μ 4 8 c 8μ 44με + 8μ + 44μ ε = 0.

9 Perturbations in the Coriolis and Centrifugal Forces in the RBP 709 For 0 and in the absence of small perturbations in the centrifugal Coriolis c forces i.e., ε = 0,ε = 0 this reduces to its well-known classical restricted problem form see Szebehely 967a: λ 4 + λ 7μ μ + = 0. 4 The discriminant of is = 54μ4 c + 08μ c + c +7+66ε μ + c 7 66ε μ 6 + c + 6ε + 6ε Its roots are λ = b ±, 4 where b = ε 9 c + 8ε. The discriminant of the original form of the characteristic equation, before normalizing to obtain is = 54 c μ4 08 c μ + 66ε c μ + 66ε 7 87 c μ + c 6ε + 6ε. 5 This is mathematically equivalent to Nowwehavefrom5, d dμ = 6 c μ 4 c μ + From 6, we get d dμ = 648 c μ 648 c μ + The discriminant of the equation d dμ 66ε c μ+ 66ε 7 87 c. 6 66ε c μ + 66ε c. 7 = 0is = ε + 7 < 0, indicating that d dμ function in [0, / ] But > 0forμ [0, / ], hence d dμ d dμ μ=0 is a monotone increasing = 66ε 7 87 < 0, 8 c d = 0. 9 dμ μ=

10 70 Jagadish Singh & Nakone Bello The equations 8and9 imply that d dμ 0forμ [0, / ] Hence is a monotone decreasing function in [0, / ].Also, μ=0 = c 6ε + 6ε > 0, μ= = 4 ε c + 6ε < 0. Since μ=0 and μ= are of opposite signs, and is monotone decreasing and continuous, there is only one value of μ, e.g., μ c in the interval [0, / ] for which vanishes. Solving the equation = 0, using or5, we obtain the critical value of the mass parameter as μ c = ε 9ε 486c c + 6ε 69 76ε, μ c = μ 0 769, where μ 0 = istheRouth s value. We consider the following three scenarios of the values of μ separately: When 0 μ<μ c, >0, the values of λ given by 4 are negative and therefore all the four characteristic roots are distinctly pure imaginary numbers. Hence, the triangular points are stable. When μ c <μ, < 0, the real parts of the characteristic roots are positive. Therefore, the triangular points are unstable. When μ = μ c, = 0; the values of λ givenby4 are the same. This induces instability of the triangular points. Hence the stability region is 0 μ<μ c + 4 6ε 9ε With the absence of small perturbations in the centrifugal and Coriolis forces i.e., ε = 0,ε = 0, μ c reduces to the critical mass value of the relativistic RBP. This confirms the result of Douskos and Perdios 00, but disagrees with that of Ahmed et al In the presence of the perturbations ε and ε and in the absence of relativistic term,μ c c verifies the result of Bhatnagar and Hallan 978. Hence, it can be noted that μ c < μ 0 for ε > 0andμ c > μ 0 for ε > 0, showing that the centrifugal force is a destabilizing force when the Coriolis force is kept constant and vice versa. This agrees with the results of Singh 0 and AbdulRaheem and Singh Discussion Equations 5 6 describe the motion of a test particle in the relativistic RBP with small perturbations ε andε in the centrifugal and Coriolis forces, respectively.

11 Perturbations in the Coriolis and Centrifugal Forces in the RBP 7 Equations 9 determine the positions of triangular points which are affected by the perturbations in the centrifugal force and the relativistic factor. In the absence of the perturbation ε, these positions correspond to those of Bhatnagar and Hallan 998, Douskos and Perdios 00 andahmedet al Equation 0 gives the critical value of the mass parameter which depends upon the small perturbations in the centrifugal, Coriolis forces and relativistic factor. This critical value is used to determine the size of the region of stability and also helps in analysing the behaviour of the parameters involved therein. It is obvious from 0 that the centrifugal force and the relativistic factor have destabilizing effects, while the Coriolis force shows stabilizing effect and thus when there is no perturbation in the Coriolis force, the region of stability decreases in size with the increase in the values of the parameters involved and increases with the increase in the values of the parameters when there is no perturbation in the centrifugal force. Equation describes the region of stability. In the absence of perturbations i.e., ε = 0,ε = 0,the stability results obtained in this study are in agreement with those of Douskos and Perdios 00 but disagrees with Ahmed et al. 006 and Bhatnagar & Hallan 998. In the absence of relativistic terms, the results compatible with those of AbdulRaheem and Singh 006 when the primaries are spherical darkbodies. 6. Numerical results In the following, the previous results are applied to the Sun Earth system. For the Sun Earth system: μ = and the corresponding dimensionless speed of light is c = Using equation 0 for arbitrary values of small perturbation parameters in centrifugal and Coriolis forces, we show their effects on the critical mass μ c as shown in Tables and respectively. Table. Effect of the centrifugal force. Perturbation Critical mass equation 0 Critical mass parameter ε μ 0 with ε = ε = 0 equation 0 with ε =

12 7 Jagadish Singh & Nakone Bello Table. Effect of the Coriolis force. Perturbation Critical mass equation 0 Critical mass equation 0 parameter ε μ 0 with ε = ε = 0 with ε = Conclusion By considering small perturbations in the centrifugal and Coriolis forces in the relativistic CRBP, we have determined the positions of the triangular points and have examined their linear stability. It is found that their positions are sligthly affected by a small change in the centrifugal force. It is also observed that the general destabilizing and stabilizing characteristics of the centrifugal and Coriolis forces, respectively remains unaltered, resulting in either decrease or increase of the region of stability, respectively. This is confirmed by Tables and. We have noticed that the expressions for A, D, A,C in Bhatnagar and Hallan 998 differ from the present unperturbed study. Consequently, the expression for p,p,p 4,p 5 and the characteristic equations are also different. This led Bhatnagar andhallan998 to infer that triangular points are unstable, contrary to Douskos and Perdios 00 and the present results. In future studies, it might be important to look into the combined effects of the relativistic and perturbations factors i.e., ε, ε c c and also the Lense Thirring effects connected with the rotation of the primaries. Acknowledgements The authors are thankful to the anonymous referee for critical remarks that helped improve the manuscript. References AbdulRaheem, A. R., Singh, J. 006, AJ,, 880. AbdulRaheem, A. R., Singh, J. 008, Ap.&SS., 7, 9. Ahmed, M. K., Abd El-Salam, F. A., Abd El-Bar, S. E. 006, Am.J.Appl.Sci.,, 99.

13 Perturbations in the Coriolis and Centrifugal Forces in the RBP 7 Abd El-Salam, F. A., Abd El-Bar, S. 0, Appl. Math.,, 55. Bhatnagar, K. B., Hallan, P. P. 978, Celest. Mech., 8, 05. Bhatnagar, K. B., Hallan, P. P. 998, Celest, Mech. Dyn. Astron., 69, 7. Brumberg, V. A. 97, Relativistic Celestial Mechanics, Press Science, Nauka, Moscow. Brumberg, V. A. 99, Essential Relativistic Celestial Mechanics, Adam Hilger Ltd, New York. Bruno, A. D. 994, The restricted -body: plane periodic orbits, Walter de Gruyter. Contopoulos, G. 976, in: In Memoriam D. Eginitis, ed. D. Kotsakis, 59. Contopoulos, G. 00, Order and Chaos in Dynamical Astronomy, p. 54, Springer, Berlin. Douskos, C. N., Perdios, E. A. 00, Celest. Mech. Dyn. Astron., 8, 7. Iorio, L. 00, Il Nuovo Cimento, 67, 777. Iorio, L. 006, Class. Quantum.Gravit., 7, 545. Iorio, L. 009, Space Sci. Rev., 48 4, 6. Iorio, L. et al. 0, Acta Astronautica,, 4. Iorio, L. 04, Front. Astron. Space Sci.,, article id., 04. Krefetz, E. 967, A.J., 7, 47. Linsen, Li. 99, Commun. Theor. Phys., 5, 5. Lucas, F. W. 00, Z. Naturforsch., 58a,. Maindl, T. I., Dvorak, R. 994, A&A, 90, 5. Moulton, F. R. 94, An Introduction to Celestial Mechanics, nd ed., New York: Dover. Ragos, O., Perdios, E. A., Kalantonis, V. S., Vrahatis, M. N. 00, Nonlin. Anal., 47, 4. Renzetti, G. 0a, Earth, Moon Planets, 09 4, 55. Renzetti, G. 0b, Can. J. Phys., 908, 88. Renzetti, G. 0, Central European Journal of Physics, 5, 5. Rodica, R., Vasile, M. 006, Ap.&S S., 04, 0. Sharma, R. K., Ishwar, B. 995, in: Proceedings of the workshop on Space Dynamics and Celestial Mechanics, eds Bhatnagar, K. & Ishwar, B., BRA University, India. Singh, J., Begha, J. M. 0, Ap.&S S.,, 5. Singh, J. 0, Ap.&S S.,, 6. Singh, J. 0, Ap.&S S., 46, 4. SubbaRao, P. V., Sharma, R. K. 975, A&A, 4, 8. Szebehely, V. 967a, Theory of orbits, The restricted problem of three bodies, Academic Press, New York. Szebehely, V. 967b, A.J., 7, 7. Valtonen, M., Kartunen, H. 006, The Three-Body Problem, Cambridge University Press. Will, C. M. 04, Living Rev. Relativity, 7,4. Wintner, A. 94, The analytical foundations of celestial mechanics, p. 7, Princeton University Press, Princeton. Yamada, K., Asada, H. 0, PRD, 86, 409.