Geometry of halo and Lissajous orbits in the circular restricted three-body problem with drag forces

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1 MNRAS 446, (5) doi:.93/mnras/stu Geometry of halo Lissajous orbits in the irular restrited three-body problem with drag fores Ashok Kumar Pal Badam Singh Kushvah Department of Applied Mathematis, Indian Shool of Mines, Dhanbad-864, Jharkh, India Aepted 4 Otober 8. Reeived 4 Otober ; in original form 4 May 9 INTRODUCTION During the last few years, many researhers have studied the effet of drag fores beause of the signifiant role they have in the dynamial system. For eample, Murray (994) studied the dynamial effets of drag fore in the irular restrited three-body problem found the approimate loation stability properties of the Lagrangian points. Liou, Zook & Jakson (995) eamined the effets of radiation pressure, Poynting Robertson (P R) drag solar wind drag on dust grains trapped in mean motion resonane with the Sun Jupiter restrited three-body problem. Ishwar & Kushvah (6) studied the linear stability of triangular equilibrium points in the generalized photogravitational restrited three-body problem with P R drag found that the triangular equilibrium points are unstable. Also, Kushvah (8) determined the effet of radiation pressure on the equilibrium points in the generalized photogravitational restrited three-body problem, notied that the ollinear points deviate from the ais joining the two primaries, whereas the triangular points are not symmetri beause of the presene of radiation pressure. Moreover, Kumari & Kushvah (3) studied the motion of the infinitesimal mass in the restrited fourbody problem with solar wind drag found the range of the radiation fator of the equilibrium points. We know that the Lagrangian points are important for mission design transfer of trajetories. A number of missions have been suessfully operated in the viinity of the Sun Earth ashokpalism@gmail.om ABSTRACT In this paper, we determine the effet of radiation pressure, Poynting Robertson drag solar wind drag on the Sun (Earth Moon) restrited three-body problem. Here, we take the larger body of the Sun as a larger primary, the EarthMoon as a smaller primary. With the help of the perturbation tehnique, we find the Lagrangian points, see that the ollinear points deviate from the ais joining the primaries, whereas the triangular points remain unhanged in their onfiguration. We also find that Lagrangian points move towards the Sun when radiation pressure inreases. We have also analysed the stability of the triangular equilibrium points have found that they are unstable beause of the drag fores. Moreover, we have omputed the halo orbits in the third-order approimation using the Lindstedt Poinaré method have found the effet of the drag fores. Aording to this prevalene, the Sun (Earth Moon) model is used to design the trajetory for spaeraft travelling under drag fores. Key words: elestial mehanis solar wind planets satellites: dynamial evolution stability. Earth Moon ollinear Lagrangian points. In this regard, the International Sun Earth Eplorer (ISEE) programme was established as a joint projet of the National Aeronautis Spae Administration (NASA) the European Spae Ageny (ESA). ISEE-3 was launhed into a halo orbit around the Sun Earth L point in 978, allowing it to ollet data on solar wind onditions upstream from the Earth. Farquhar, Muhonen & Rihardson (977) had designed the ISEE-3 sientifi satellite in the viinity of the Sun Earth interior Lagrangian point to ontinuously monitor the spae between the Sun the Earth. WIND was launhed on 994 November was positioned in a sunward orbit. The Solar Heliospheri Observatory (SOHO) projet was launhed in 995 Deember to study the internal struture of the Sun. The Advaned Composition Eplorer (ACE) was launhed in 997 orbits the L Lagrangian point, whih is a point of the Sun Earth gravitational equilibrium about.5 million km from the Earth 48.5 million km from the Sun. ARTHEMIS was the first spaeraft to be in the viinity of Earth Moon Lagrangian point. In August, the ARTHEMIS P spaeraft entered an orbit near the Earth Moon L point for approimately 3 d, before transferring to an L quasi-halo orbit where it remained for an additional 85 d. On July 7, the ARTHEMIS P spaeraft was suessfully inserted into the Earth Moon Lagrangian point orbit with an arrival near the L point in Otober (Pavlak & Howell ). The ARTHEMIS Lagrangian point orbit design features quasi-halo orbits demonstrated reently by Folta et al. (3). To our knowledge, there have been numerous published papers that have etensively overed topis related to halo orbits, Lagrangian point satellite operations in general, their Downloaded from by guest on 8 November 8 C 4 The Authors Published by Oford University Press on behalf of the Royal Astronomial Soiety

2 96 A. K. Pal B. S. Kushvah appliations within the Earth Moon Sun Earth system. For more information on halo orbits, we refer to the three-dimensional periodi halo orbits near the ollinear Lagrangian points in the restrited three-body problem obtained by Howell (984). She obtained orbits that inrease in size when inreasing the mass parameter μ. Clarke(3, 5) disussed a disovery mission onept that utilizes oultations from a lunar halo orbit by the Moon to enable detetion of terrestrial planets. Breakwell & Brown (979) has omputed halo orbits around the Earth Moon L point. Calleja et al. () omputed the unstable manifolds of seleted vertial halo orbits, whih in several ases have led to the detetion of heterolini onnetions from suh a periodi orbit to invariant tori. Other authors have arried out similar work (Di Giamberardino & Monao 99; Farquhar ; Kim & Hall ; Junge et al. ; Kolemen, Kasdin & Gurfil 7; Hill & Born 8). However, they ignored the drag fore, although Eapen & Sharma (4) have studied the halo orbits at the Sun Mars L Lagrangian point in the photogravitational restrited three-body problem have found that as the radiation pressure inreases, the transition from the Mars-entri path to the helioentri path is delayed. In this paper, we study the effet of radiation pressure, P R drag, solar wind drag on the Lagrangian points use the Lindstedt Poinaré method to ompute halo orbits in viinity of the L point of the Sun Earth Moon system. This paper is organized as follows. In Setion, we reall some well-known fats about the irular restrited three-body problem with drag fores (i.e. its equations of motion, equilibrium points stability). In Setion 3, we desribe the motion near the Lagrangian point L, use the Lindstedt Poinaré method to ompute the halo orbits. In Setion 4, we disuss our results. Finally, in Appendi A, we provide all the oeffiients. MATHEMATICAL FORMULATION OF THE PROBLEM AND EQUATIONS OF MOTION We formulate the Sun Earth Moon system with the radiation pressure, P R drag solar wind drag (Parker 965) proeed with the problem as follows (Liou et al. 995). The equations of motion in the rotating referene frame are ẍ ẏ = U ( sw)f, () ÿ ẋ = U y ( sw)f y, () z = U z ( sw)f z, (3) where ( β)( μ) U = μ r r ( y ), (4) the drag fore omponents are β( μ) ( μ)ẋ (ẏ μ)y zż( μ) F = F y = F z = r } (ẋ y), (5) β( μ) r β( μ) r r ( μ)ẋ (ẏ μ)y zży r ( μ)ẋ (ẏ μ)y zżz r } (ẏ ), (6) } ż. (7) It is supposed that m is the mass of the Sun m is the mass of the Earth plus Moon, hene the mass parameter μ = m /(m m ). However, the definition of the mass parameter μ is different from that of Liou et al. (995). They have used μ μ to represent the masses of the Sun Jupiter, respetively, whereas in the present problem the masses are represented by m m. Here, β is the ratio of the radiation pressure fore to the solar gravitation fore, sw is the ratio of solar wind drag to P R drag (Burns, Lamy & Soter 979; Gustafson 994) is the unitless speed of light. The unit of mass is taken in suh a way that G(m m ) = ; the unit of distane is taken as the distane of the entre of mass (of the Earth Moon) to the Sun, whereas the unit of time is taken to be the time period of the rotating frame. In order to alulate the Lagrangian equilibrium points, we solve equations () (3) with the ondition that all derivatives are zero, we obtain ( β)( μ) 3 ( μ) μ r 3 ( μ ) ( sw)β( μ) μy( μ) y =, (8) ( β)( μ) y r 3 μy μ r 3 ( β)( μ) z 3 μ r 3 ( sw)β( μ) =, (9) ( sw)β( μ)μy 4 =. () From equation (), we obtain two possible solutions, either z = or z. If z, then ( β)( μ) r 3 μ r 3 = ( sw)β( μ)μy 4. () From equation (), with β = (i.e. no radiation pressure is taken into aount), we obtain μ r 3 = μ r 3. () From the right-h side of equation (), beause μ>, also r is the distane of the infinitesimal body from the seond primary, we find that μ 3 = μ r 3 <. (3) This gives μ>, whih is never possible. Therefore, at β =, there are no equilibrium points outside the y-plane. Again, with z if <β<, then it is obvious from equation () that the only possible solution is to have y >. From equations (9) (), we obtain ( sw)β( μ) y =. (4) With the ondition that y >, equation (4) gives one possible solution >, therefore μ>. Now, we divide both sides of equation (8) by ( μ), using equation (), we obtain μ μ ( sw)β( μ)y r 3 ( μ) ( μ) =. (5) Downloaded from by guest on 8 November 8 MNRAS 446, (5)

3 Geometry of halo Lissajous orbits in the CRTBP 96 For > y >, the left-h side of equation (5) has a non-zero quantity. Therefore, z must be zero have no equilibrium points outside the y-plane. Simmons, MDonald & Brown (985) have shown that out-ofplane equilibrium points our if ( β )/( β ) <, where β, are the ratios of the magnitudes of radiation to the gravitational fores from m,. However, in the present Sun Earth Moon system, only the Sun is a radiating body. Therefore, β = β = β>, whih is not possible. Consequently, out-of-plane equilibrium points do not eist in the present model. Thus, for z =, we solve equations (8) (9) using Taylor series epansions in r r some approimations, as in Murray & Dermott (999). The general solutions are given by =, y = y y. (6) Here, y are the small quantities that are introdued for drag fores (, y ) is a solution of the equations when there is no drag fore. The orresponding distanes of the infinitesimal body to the masses m m are given by r = ( μ) y z (7) r = ( μ ) y z. (8) We use the Taylor series epansions around (, y ) neglet seond- higher-order terms in y. Then, we solve the simultaneous equations in y ( z = for all Lagrangian points). We obtain following epressions of y for all the Lagrangian points with a fied value of sw =.35 (Gustafson 994): for L, = β, y =.4776( β) ; (4.455 β)(.978 β) for L, =.5537( β), β y =.53994( β)β ; (4.576 β)(3.33 β) for L 3, =.5β.5 β, y =.54( )β ; (.5 β)( β) for L 4, 5, =.5β(.6643 β)(3. β) (.5 β)( β)(3. β), (9) Figure. Effet of β on the oordinate in the L,, 3 points. is inreasing negatively (i.e. both the L L points tend towards the Sun with inreasing radiation pressure). For the L 3 point, inreases positively when β inreases, therefore the Lagrangian point L 3 also tends towards the Sun. y orresponds to the ase when L,, 3 inrease negatively, for an inreasing value of β, whih is shown in Fig.. Therefore, when the radiation pressure inreases, the ollinear points perturb from their ollinearity tend towards the radiating body of the Sun. The different Lagrangian points L i,(i =,, 3) at different values of β are presented in Table. The effet of radiation pressure in of L 4, 5 is the same, as shown in Fig. 3, while Fig. 4 shows that there is a symmetrial hange in y of L 4 L 5. These equilibrium points also tend towards the Sun symmetrially when β inreases. The different triangular points L 4, 5 at different values of β are shown in Table. Downloaded from by guest on 8 November 8 y =±.5( β)β. () (.5 β)( β) The effet of β in on the L L points is shown in the top panel of Fig., whereas the bottom panel is for the L 3 point. For the L L points, the effet of β is approimately equal Figure. Effet of β on the y oordinate in the L,, 3 points. MNRAS 446, (5)

4 96 A. K. Pal B. S. Kushvah Table. Lagrangian equilibrium points L i,(i =,, 3) with sw =.35. β L L L 3. (.98999, ) (.7, ) (.36495, ). ( ,.4554 ) ( ,.435 ) (.96489, 4.45 ). (.96656,.9474 ) ( , 3.5 ) (.938, ).3 ( , ) ( , ) (.8753, ).4 (.9485, ) (.96696, ) (.8885, ).5 (.9694, ) (.9489, ) (.754, ).6 (.9355,.8874 ) (.93386,. ) (.66667, ).7 ( ,.3454 ) (.98864,.3596 ) (.5654, ).8 (.88766,.6335 ) (.9374, ) (.48576, ).9 (.86367,.955 ) (.88645,.9685 ) (.56, 7.67 ). ( ,.397 ) ( ,.3354 ) ( , ) Figure 3. Effet of β on the oordinate of triangular points. Figure 4. Effet of β on the y oordinate of triangular points. Table. Triangular equilibrium points with sw =.35. β L (4, 5). ( , ±.8665). (.4839, ±.83596). (.46534, ±.79948).3 ( , ± ).4 (.4986, ±.78567).5 ( , ±.6495).6 (.33339, ±.57735).7 (.845, ±.4874).8 (.48, ±.3755).9 (.4995, ±.659). ( , ± ) viinity of an equilibrium point. Our approah is based on that of Shuerman (98) Murray (994). Consider a small displaement from the equilibrium position (, y ), let the solution for the subsequent motion be of the form = X, y = y Y,where X = X e λt, Y = Y e λt, () X, Y λ, are onstants. Using these substitutions in equations (8) (9), using Taylor series epansion, the following simultaneous linear equations in X Y are given as X λ ( β)( μ) 3( μ) μ 3 3( μ ) ( ) F r 3 } Y 3μy ( μ ) ( Fy y r 5 r ( sw) r 3 ( λ F ẋ λ 3( β)( μ)y ( μ) ( ( sw) λ F ẏ 5 ) } = () X λ 3( β)( μ)y ( μ) 3μy ( μ ) ( ( sw) λ F y ẋ Y λ ( sw) λ 5 ) ( μ)( β) 3 ) ( Fy ẏ ( )} Fy ( 3y r ( Fy y ) ) μ r 3 r 5 ) ) ( 3y r ) } =, (3) where ( ) denotes the evaluation of a partial derivative at the displaed equilibrium point. We an now rewrite equations () (3) as Xλ a d ( sw)(λk,ẋ K, ) Downloaded from by guest on 8 November 8. Stability of the equilibrium points The loations of equilibrium points do not affet our knowledge of their stability. We now onsider the stability property by using the stard tehnique of linearizing the perturbation equations in the Y λ ( sw)(λk,ẏ K,y ) =, (4) Xλ ( sw)(λk y,ẋ K y, ) Y λ a b ( sw)(λk y,ẏ K y,y ) =, (5) MNRAS 446, (5)

5 Geometry of halo Lissajous orbits in the CRTBP 963 where the onstants are a ( β)( μ) = 3 μ r 3, (6) ( β)( μ) b = 3 5 ( β)( μ)( μ) = 3 r 5 ( β)( μ)( d μ) = 3 5 K, = K,y = K y, = K y,y = ( ) F, K,ẋ = ( ) F, K,ẏ = y ( ) Fy, K y,ẋ = ( ) Fy, K y,ẏ = y μ y, (7) r 5. μ( μ ) y, (8) r 5 μ( μ ), (9) r 5 ( ) F, (3) ẋ ( ) F, (3) ẏ ( ) Fy, (3) ẋ ( ) Fy. (33) ẏ The ondition for the determinant of the linear equations defined by equations (4) (5) needs to be zero. Negleting the terms of O(K ), we obtain the harateristi equation as λ 4 a 3 λ 3 (a a )λ a λ (a a ) =, (34) where the approimate epressions for onstants a a, the drag fore terms a i (i =,,, 3) are given in Appendi A. If we take K =, then the harateristi equation redues to the following form λ 4 a λ a =, (35) whih gives the lassial solutions. The four roots in the lassial problem our as real pure imaginary pairs, as in the ase of eah L i,(i =,, 3), whereas two pure imaginary pairs our in the ase of L 4 L 5. The stability of the given equilibrium point depends on the sign value of the roots of the harateristi equation. If any of the roots has a positive real part, the motion is unstable to small displaements with eponential growth, whereas in the lassial ase, the L 4 L 5 points are linearly stable to small displaements in the system. If we restrit our analysis to L 4 L 5, the four roots of the harateristi equation, without drag fores, are given as λ n =±Zi, (n =,, 3, 4), (36) where a ± a Z = 4a. (37) In the presene of drag fores, we assume that the roots are in the following form λ = ηz ± i( γ )Z, (38) that is, λ, = ηz ( γ )Zi, (39) λ 3,4 = ηz ( γ )Zi, (4) where γ η are small real quantities. With the help of equations (34), (39) (4), negleting the produt of γ then η with a i, solving the real imaginary parts, we obtain η = (a 3Z a ) (4) Z(Z a ) γ = (a a ) (a a )Z Z 4. (4) Z (a Z ) In all the ases, the real part of at least one harateristi root is positive. Therefore, the equilibrium point is a saddle point. 3 COMPUTATION OF HALO ORBIT In order to disuss the motion near the Lagrangian point of the system, we hoose a oordinate system entred at the Lagrangian point L in the rotating referene frame. The equations of motion of an infinitesimal body are obtained by translating the origin to the loation of L. The referred translation is given as = X μ γ, y = Y l z = Z. (43) In this new oordinate system, the variables, y z are saled. The distanes between L the smaller primary are γ in the -ais l in the y-ais. Using these fats in equations () (3), we obtain the equations of motion with the help of Legendre polynomials for eping the non-linear terms onsidering the linear parts, we an write ẍ ẏ ν yν =, (44) ÿ ẋ yν ν =, (45) z ν z =. (46) All the oeffiients are given in Appendi A. Here, the z-ais solution is simple harmoni, beause ν >. However, the motion in the y-plane is oupled. Solving equations (44) (45), the harateristi equation has two real two omple roots. The omple roots are not pure imaginary but the real parts of the omple roots are very small with respet to the age of the Solar system. Therefore, we neglet them the roots are ±α ±iλ,where 4 ν ν (4 ν ν ) α =± 4ζ (47) 4 (ν ν ) (4 ν ν ) λ =± 4ζ. (48) All the oeffiients are given in Appendi A. Beause the two real roots are opposite in sign, arbitrarily hosen initial onditions give rise to unbounded solutions as time inreases. If, however, the initial onditions are restrited only a non-divergent mode is allowed, the y-solution will be bounded. In this ase, the linearized equations have solutions of the form (t) =A os(λt φ), (49) y(t) = κa λ sin(λt φ) ν os(λt φ), (5) Downloaded from by guest on 8 November 8 MNRAS 446, (5)

6 964 A. K. Pal B. S. Kushvah Figure 5. Effet of β in one period. z(t) = A z sin( ν t ψ), (5) with κ = λ ν 4λ ν. (5) The in-plane out-of-plane frequenies are not equal, they are inommensurable. Then, the linearized motion produes the Lissajous-type trajetories for the Sun (Earth Moon) system around L. When the radiation pressure inreases, the phase differene of the trajetories dereases. When there is no drag fore (i.e. β = ), the trajetory of the Lissajous orbit ompletes one period approimately at t = 3., but when the radiation pressure fore inreases (i.e. β =.), it does not omplete its period at that time, asshowninfig.5. Therefore, the period inreases with an inrease in the value of β. Also, beause of the inreasing value of β, the trajetories shrink in its amplitude. Fig. 6 shows the Lissajous orbits with β = β =. shown in the upper lower panels, respetively. Figs 7, 8 9 show the projetions in the y-, z-yz-plane, respetively, with the different values of β. Here, blak, blue red oloured orbits are shown with β =,.5., respetively. Clearly, the orbits shrink when β inreases. Figure 6. Effet of β on the Lissajous orbits: upper panel, β = ; lower panel, β =.. 3. Periodi orbits using the Lindstedt Poinarémethod The equations of motions an be written using the Legendre polynomials P n. A third-order approimation was used in the irular restrited three-body problem without any drag fore by Rihardson (98). Here, we inlude P R drag solar wind drag, then find the third-order approimation as desribed by Thurman & Worfolk (996): ẍ ẏ = ν yν ν y ν 3 z ν 4 yν 5 3 ν 6 y 3 ν 7 y ν 8 z ν 9 yν 9 yz ν 9 ν O(4), (53) ÿ ẋ = ν yν ν y ν 3 z ν 4 yν 5 3 ν 6 y 3 ν 7 y ν 9 z ν 9 yν 9 yz ν 9 ν O(4), (54) z ν z = zν 3 yzν 3 zν 3 yzν 33 y zν 34 z 3 ν 35 O(4). (55) All the oeffiients are given in Appendi A. Downloaded from by guest on 8 November 8 MNRAS 446, (5)

7 Geometry of halo Lissajous orbits in the CRTBP 965 Figure 7. Projetion on the y plane. Figure 8. Projetion on the z plane. 3. Corretion term For the onstrution of higher-order appliations, we linearize equation (55) introdue a orretion term = λ ν, (56) on the right-h side. The new third-order z equation beomes z λ z = zν 3 yzν 3 zν 3 yzν 33 y zν 34 z 3 ν 35 z O(4). (57) Figure 9. Projetion on the yz plane. Rihardson (98) developed a third-order periodi solution using the Lindstedt Poinaré type of suessive approimations. We follow their work with the P R drag solar wind drag, by removing seular terms. A new independent variable τ a frequeny onnetion ω are introdued via τ = ωt. Then, the equations of motion at the seond degree inluding the P R drag solar wind drag are given as ω ωy = ν yν ν y ν 3 z ν 4 yν 5 3 ν 6 y 3 ν 7 y ν 8 z ν 9 yν 9 yz ν 9 ν O(4), (58) ω y ω = ν yν ν y ν 3 z ν 4 yν 5 3 ν 6 y 3 ν 7 y ν 9 z ν 9 yν 9 yz ν 9 ν O(4), (59) ω z λ z = zν 3 yzν 3 yν 3 ν 33 zν 3 yzν 33 y zν 34 z 3 ν 35 z O(4), (6) where a prime denotes d/dτ. Beause of perturbation analysis, we assume solutions in the following forms: (τ) = ɛ (τ) ɛ ɛ ; (6) y(τ) = ɛy (τ) ɛ y ɛ 3 y 3...; (6) z(τ) = ɛz (τ) ɛ z ɛ 3 z 3...; (63) ω = ɛω (τ) ɛ ω ɛ 3 ω 3... (64) We substitute these into the equations of motion equate omponents of the same order of ɛ. We find the first-, seond third-order equations their solutions. Downloaded from by guest on 8 November 8 MNRAS 446, (5)

8 966 A. K. Pal B. S. Kushvah 3.3 First-order equations To find the first-order approimation, we use equations (58) (64) ompare the oeffiients of the first-order terms in ɛ on both sides. We obtain the following equations: y = ν yν ν ; (65) y = ν yν ; (66) z λ z =. (67) From the above equations, we find the following bounded solutions: (τ) =A os(λτ φ); (68) y (τ) = κa λ sin(λτ φ) ν os(λτ φ); (69) z (τ) = A z sin(λτ ψ). (7) In order to avoid seular solutions, we need onstraints on the onstants A, A z, φ ψ, but for now they are arbitrary. 3.4 Seond-order equations The order of the O(ɛ ) equations depend on the first-order solutions for, y, z. Colleting only non-seular terms, we obtain y ν ν y = α os τ α os τ α 3 sin τ α 4, (7) y ν y ν =δ sin τ δ os τ δ 3 os τ δ 4, (7) z λ z =h sin(τ τ ) sin(τ τ ) h os(τ τ ) os(τ τ ), (73) where all the oeffiients are given in Appendi A. We remove all the seular terms by setting ω =. Thus, we find the following solutions, = ρ ρ os τ ρ os τ ρ 3 sin τ ρ 4 sin τ, (74) y = ρ ρ sin τ ρ sin τ ρ 3 os τ ρ 4 os τ, (75) z = ρ 3 sin(τ τ ) ρ 3 sin(τ τ ) where ρ 3 os(τ τ ) ρ 33 os(τ τ ), (76) τ = λτ φ, τ = λτ ψ. (77) All the oeffiients are given in Appendi A. 3.5 Third-order equations The O(ɛ 3 ) equations are obtained by setting ω = substituting in the solutions for, y, z,, y z. Thus, we obtain 3 y 3 ν 3 ν y 3 = α ω λ A (κ ) os τ α κλa ν ω sinτ α 3 os 3τ α 4 os(τ τ ) α 5 os(τ τ ) α 6 sin(τ τ ) α 7 sin(τ τ ) α 8 sin 3τ, (78) y 3 3 ν 3 ν y 3 = α κλ A ν ω osτ α λa ω (κλ ) sin τ α 3 os 3τ α 4 os(τ τ ) α 5 os(τ τ ) α 6 sin(τ τ ) α 7 sin(τ τ ) α 8 sin 3τ, (79) z 3 λ z 3 = α 3 A z (ω λ ɛ ) sin τ α 3 sin(τ τ ) α 33 sin(τ τ ) α 34 sin 3τ α 35 os τ α 36 os 3τ α 37 os(τ τ ) α 38 os(τ τ ), (8) where the epressions of the oeffiients are given in Appendi A. There are seular terms. We start by eamining the seular terms in the z 3 equation, by simply setting a value for the frequeny orretion ω. To remove the seular terms α 33 sin (τ τ ) α 37 os (τ τ ), we need the oeffiients of these terms to be zero. Thus φ = ψ n π, where n =,,, 3. The solution will be bounded if ( α 3 A z ω λ ) ζα ɛ 33 =, (8) ζα 37 =, (8) where ζ = ( ) n. This phase onstraint affets the 3 y 3 equations; now eah ontains a seular term. The requirement of another onstraint is from the simultaneous equations (78) (79): (ν λ )α ω λ A (κ ) λα λa ω (κλ ) ν (α κλ A ν ω ) ζ α 5 (λ ν ) λ(α 7 ν α 5 ) =, (83) (ν λ )(α κλa ν ω ) λ(α κλ A ν ω ) ν α λa ω (κλ ) ζ α 7 (λ ν ) λ(α 5 ν α 7 ) =. (84) From these equations we find ω = λ α ζλ α 5 λα ζλα 7 α ν α ν ζ (λα 5 ν α 5 ν )λ A λ (κ ) ν κν κν }. (85) Downloaded from by guest on 8 November 8 MNRAS 446, (5)

9 Geometry of halo Lissajous orbits in the CRTBP 967 Table 3. At β =.3. n A z ω ± ± ± ± Table 4. At β = Final approimation Using all the first-, seond- third-order approimate solutions with the mapping A A /ɛ A z A z /ɛ, whih remove ɛ from all the equations, we obtain (τ) = ρ (A σ )osτ σ sin τ (ρ ζρ )osτ (ρ 4 ζρ 3 )sinτ σ os 3τ σ 3 sin 3τ, (9) n A z ω ± ± ± ± Table 5. At β =.8. n A z ω ± ± ± ± The amplitude relation is obtained by putting the value of equation (85) into equation (8). Thus, we an satisfy this onstraint by letting one amplitude be determined by the other. Different amplitudes A z at A = 6 /( ) with sw =.35 at different β n are shown in Tables 3, 4 5. From these tables, we see that the amplitudes ontrat, whereas ω dereases negatively when β inreases. Using these onstraints, the third-order equations are redued to 3 y 3 ν 3 ν y 3 = s os τ s sin τ (α 3 ζα 4 )os3τ (α 8 ζα 6 ) sin 3τ, (86) y 3 3 ν 3 ν y 3 = s os τ s sin τ (α 3 ζα 4 )os3τ (α 4 ζα 6 ) sin 3τ, (87) z 3 λ z 3 = ζ (α 3 sin 3τ α 38 os 3τ ) α 34 () n/ sin 3τ, n =, () (n)/ os 3τ, n =, 3 α 36 () n/ os 3τ, n =, () (n)/ sin 3τ, n =, 3. (88) The solutions are 3 = σ os τ σ sin τ σ os 3τ σ 3 sin 3τ, (89) y 3 = σ os τ σ sin τ σ os 3τ σ 3 sin 3τ, (9) z 3 = ζ (α 8λ 3 sin 3τ α 38 os 3τ ) () n/ (α 34 sin 3τ α 36 os 3τ ),n=, () (n)/ (α 34 os 3τ α 36 sin 3τ ),n=, 3, (9) where the epressions of the oeffiients are given in Appendi A. y(τ) = ρ (λκa σ )sinτ (κa ν σ )osτ (ρ ζρ )sinτ (ρ 4 ζρ 3 )osτ σ os 3τ σ 3 sin 3τ (93) z(τ) = A z () (n4)/ (A z sin τ ρ 3 sin τ ),n=, () (n)/ (A z os τ ρ 3 os τ ),n=, 3 ρ 3,n=, () (n)/,n=, 3 () n/,n=, ρ 3,n=, 3 ζ (α 8λ 3 sin 3τ α 38 os 3τ ) () n/ (α 34 sin 3τ α 36 os 3τ ),n=,. () (n)/ (α 34 os 3τ α 36 sin 3τ ),n=, 3 (94) The epressions of the oeffiients are given in Appendi A. For the Sun (Earth Moon) L, the differene of both frequenies of the z-plane y-plane is quite small. The values of A A z are found using the relation of equations (85) (8). In the final approimation, the halo orbits at n = with sw =.35 are shown in Figs, 4. Forn = 3, the halo orbits are depited in Figs, 3 5. The amplitude frequeny ω are both equal for eah ase, n =, n =, 3. When β inreases, then the trajetory shrinks. In the previous subsetions, we have omputed the first, seond, third final approimations for the halo orbits we have seen the effet of radiation pressure, with P R drag solar wind drag. In the absene of drag fores, the results agree with those of the lassial ase (Thurman & Worfolk 996) 55 z 5 y 5 55 z 5 y 5 Figure. Halo orbit at n = n = atβ =.3. Downloaded from by guest on 8 November 8 MNRAS 446, (5)

10 968 A. K. Pal B. S. Kushvah z z..5 5 y z.5. y 5 Figure. Halo orbit at n = n = 3atβ =.3. y.. z.5.5. z z. y. 5 5 Figure. Halo orbit at n = n = atβ =.5. y 5.5. z.5 y Figure 3. Halo orbit at n = n = 3atβ =.5. y 5.5. z.5 y 5 Figure 4. Halo orbit at n = n = atβ =.8. 4 CONCLUSIONS In this paper, we have studied the irular restrited three-body problem of the Sun Earth Moon system by assuming the effet of radiation pressure, P R drag solar wind drag. We have found that the ollinear Lagrangian points deviate from their ais joining z.5 y z.5 y Figure 5. Halo orbit at n = n = 3atβ =.8. the primaries, whereas the triangular points remain unhanged in their onfiguration. However, all points lie in a plane. If we inrease the value of β, with a fied value of sw =.35, the Lagrangian points L, L L 3 tend towards the radiating body (the Sun), whereas L 4, 5 have symmetrial hanges with the inreasing value of β. We have eamined the linear stability of the equilibrium points with the help of harateristi roots. It is observed that the Lagrangian points are unstable beause of the drag fores. Furthermore, we have omputed the orbit around the L point have seen that when radiation pressure inreases, the phase differene of the trajetory dereases. If there is no drag fore (i.e. β = ), then the trajetory of the Lissajous orbit ompletes one period approimately at t = 3., whereas if β inreases with sw =.35, it does not omplete its period at the same time. Also, beause of the inreasing value of β, the trajetories shrink in its amplitude. In this study, we have used the Lindstedt Poinaré method to ompute the halo orbits in the third-order approimation with the radiation pressure, P R drag solar wind drag. In this analysis, we have fied the value of sw =.35, whih is the ratio of solar wind drag to P R drag, we have varied the value of β (i.e. the ratio of radiation pressure fore to solar gravitation fore). This model an be used to ompute the higher-order approimation (four, fifth, et.) epressions for halo orbits. Moreover, stable unstable manifolds of the halo orbits, trajetory transfer, would be interesting topis of future researh with similar dissipative fores. ACKNOWLEDGEMENTS We are grateful to Inter-University Centre for Astronomy Astrophysis (IUCAA), Pune for supporting library visits for the use of omputing failities. BSK is also grateful to the Indian Spae Researh Organization (ISRO), Department of Spae, Government of India, for providing finanial support through the RESPOND Programme (Projet No -ISRO/RES//383/-3). REFERENCES Breakwell J., Brown J., 979, Celestial Mehanis,, 389 Burns J. A., Lamy P. L., Soter S., 979, Iarus, 4, Calleja R. C., Doedel E. J., Humphries A. R., Lemus-Rodríguez A., Oldeman E. B.,, Celestial Mehanis Dynamial Astronomy, 4, 77 Clarke T. L., 3, Bull. Am. Astron. So., 35, 4 Clarke T., 5, APS April Meeting, Abstrat C97 Di Giamberardino P., Monao S., 99, in Pro. 3st IEEE Conf. on Deision Control. IEEE, New York, p. 536 Eapen R. T., Sharma R. K., 4, Ap&SS, 35, 437 Farquhar R. W.,, Journal of the Astronautial Sienes, 49, 3 Farquhar R. W., Muhonen D. P., Rihardson D. L., 977, Journal of Spaeraft Rokets, 4, 7 5 Downloaded from by guest on 8 November 8 MNRAS 446, (5)

11 Folta D., Pavlak T., Haapala A., Howell K., 3, in 3rd AAS/AIAA Spae Flight Mehanis Meeting Gustafson B. A. S., 994, Annual Review of Earth Planetary Sienes,, 553 Hill K. A., Born G. H., 8, Journal of Spaeraft Rokets, 45, 548 Howell K. C., 984, Celestial Mehanis, 3, 53 Ishwar B., Kushvah B., 6, Journal of Dynamial Systems Geometri Theories, 4, 79 Junge O., Levenhagen J., Seifried A., Dellnitz M.,, Earth,, L3 Kim M., Hall C. D.,, in Pro. AAS/AIAA Astrodynamis Speialist Conferene, Vol. 9. Amerian Astronautial Soiety, Springfield, VA, p. 349 Kolemen E., Kasdin N. J., Gurfil P., 7, in Belbruno E., ed, AIP Conf. Pro. Vol. 886, New Trends in Astrodynamis Appliations III. Amerian Institute of Physis, New York, p. 68 Kumari R., Kushvah B. S., 3, Ap&SS, 344, 347 Kushvah B. S., 8, Ap&SS, 35, 3 Liou J-C., Zook H. A., Jakson A. A., 995, Iarus, 6, 86 Murray C. D., 994, Iarus,, 465 Murray C., Dermott S., 999, Solar System Dynamis. Cambridge Univ. Press, Cambridge Parker E. N., 965, Spae Si. Rev., 4, 666 Pavlak T., Howell K.,, in AIAA/AAS Astrodynamis Speialist Conferene Rihardson D. L., 98, Celestial Mehanis,, 4 Shuerman D. W., 98, ApJ, 38, 337 Simmons J. F. L., MDonald A. J. C., Brown J. C., 985, Celestial Mehanis, 35, 45 Thurman R., Worfolk P. A., 996, Tehnial report, The Geometry of Halo Orbits in the Cirular Restrited Three-Body Problem APPENDIX A: COEFFICIENTS a = (a d )(a b ), a = ( a ) b d, a 3 =( sw)(k,ẋ K y,ẏ ), a =( sw)k, K y,y (K,ẏ K y,ẋ ), a = ( sw)( a b )K,ẋ ( a d )K y,ẏ Geometry of halo Lissajous orbits in the CRTBP 969 (A) (A) (A3) (A4) (K,y K y, ) (K,ẏ K y,ẋ ), (A5) a = ( sw)( a b )K, ( a d )K y,y (K,y K y,y ), (A6) ζ = ν ν ν ν, (A7) (γ ) ν = ( d ) 3 d γ D ( sw)β( μ) μl l(γ ) 4 4μ(γ ) l 6, (A8) (γ ) ν = 3l d γ D ( sw)β( μ) μ(γ ) 4, (A9) ν = 5 ν 3 = 3 ν 4 = 3 3 (γ ) 3 3 d 3γ 3 D 3 9 d 3γ ( sw)β( μ) D D 3 (γ ) 6μ(γ )l 4l(γ ) 3 (γ ) D 4μ(γ )l ν 7 = 5l 3 (γ ) D ( sw)β( μ) 4 (γ ) D ( sw)β( μ) l d 3γ ( sw)β( μ) D μl(γ )3 8, (A) l 4, (A) d 3γ D μl(γ ) d 4γ D l μ(γ ) 4 3 (γ ) ν 5 = 5l 3 d 3γ ( 3 D 3 3l d ) 3 D D 5l 4 (γ ) 3 ( sw)β( μ) d 4γ 3 D 4 4 (γ ) ν 6 =5 d 4γ D 35 ν 8 = 5 4 (γ ) 4 ( sw)β( μ) 8(γ )4 lμ D 4 (γ ) D μ (γ ) d 4(γ ) D 4 3 ( 4 d 4 ) ( sw)β( μ), (A), (A3) 4μ(γ ) 6, (A4) 4lμ(γ ) 8l(γ ) 3 D 8 } μl 4(γ )l 6, (A5) d 4γ D 3 ( 4 d 4 ) 6μl l(γ ) (4μl μ)(γ ) 8, (A6) Downloaded from by guest on 8 November 8 MNRAS 446, (5)

12 97 A. K. Pal B. S. Kushvah ν 9 = 5 4 (γ ) d 4γ D ( sw)β( μ) D 3 ( 4 d 4 ) μl 4(γ )l 6 μ(γ ) l 8, (A7) ν = μ γ (γ ) D ( sw)β( μ) (γ ) ν =3l D l d γ D D μl(γ ) d γ ( sw)β( μ) D (γ )( μ γ ) 4 ν = d ν = 5 l 3 (γ ) D 3, (A8), (A9) ( sw)β( μ)(γ )l 4, (A) d 3γ D 3 3l ( 3 d ) 3 ( sw)β( μ) D D 3( γ ) μ 4(γ ) ( μ γ ), (A) ν 3 = 9l ( 3 d ) 3 ( sw)β( μ) D D ν 4 = 3l ( 3 d ) 3 D D 3 (γ ) ν 5 =3 D (γ ), ( sw)β( μ)( μ γ ) 4, d 3γ ( sw)β( μ) D 8l(γ ) l3(μ γ ) 3 (γ ) ν 3 =3 d 3γ D D l (A) (A3) }, (A4) 4( sw)βμ( μ)(γ )l 6, (A5) ( 3 ν 3 = 3l d ) 3 D D ν 9 = 45l 4 (γ ) D ( sw)βμ( μ) 4, (A6) d 4γ ( sw)β D 6μ(γ ) 4(γ ) 6 4 μ(γ )3 8, (A7) ν 9 = 5l 4 (γ ) d 4γ D ( sw)β( μ) D μ(γ ) 4 ν = mu γ (γ ) d γ ν 6 = 35l D ( sw)β( μ) D μl( γ ) 4 (γ ) 3 d 4γ 3 D 4 5l 4 (γ ) d 4γ D 4 8(γ ) 4μ(γ ) 8(γ )3 ( μ γ ) D 8 ν 7 = 3 ( 4 d 4 ) ν = d ν 9 = 45l ( sw)β( μ), (A8) l, (A9) ( sw)β( μ), (A3) 4l(γ μ) 6, (A3) ( sw)β( μ)μl 4, (A3) 4 (γ ) d 4γ D ( sw)β( μ) ν 9 = 5l 4 (γ ) ν 9 = 5 D 4(γ ) 6, (A33) d 4γ ( sw)β( μ) D 4(γ )( μ γ ) 4 (γ ) D d 4γ D 3 ( 4 d 4 ), (A34) ( sw)β( μ) l(γ ) 6 4l(γ ) (μ ) 8, (A35) ν 9 = 3 ( ( sw)β( μ) 4l(γ ) 4 d 4 ) 6, (A36) ( ν = l d ) ( sw)β( μ) (γ μ) D D, (A37) ν 3 = 5 4 (γ ) d 4γ D ( sw)β( μ)μl D 3 ( 4 d 4 ) (γ ) 4, (A38) Downloaded from by guest on 8 November 8 MNRAS 446, (5)

13 Geometry of halo Lissajous orbits in the CRTBP 97 4 (γ ) ν 33 = 5l d 4γ D 4( sw)β( μ)μ(γ ) 6, (A39) ν 34 = 3 ( 4 d 4 ) 6( sw)β( μ)μl 6, (A4) A 3 = 4λ α 3 ν α 3 4δ λ δ ν, A 4 = α ν ν δ 4, B = 6λ 4 4λ (4 ν ν ) (ν ν ν ν ), (A58) (A59) (A6) ν 35 = 3 ( ( sw)β( μ)μl 4 d 4 ) 6, (A4) A 4 ρ =, (A4) ν ν ν ν ρ = A B 4λA 3 (ν ν ) B 6λ (ν ν ), (A43) ρ = B A B 6λ (ν ν ), (A44) ρ 3 = B A 3 4A λ(ν ν ) B 6λ (ν ν ), (A45) ρ 4 = 4A λ(ν ν ) B 6λ (ν ν ), (A46) ρ = δ 4 ν 3 ρ ν, (A47) ρ = 4λρ ν 3 ρ 4 δ 4λ ν, (A48) ρ = 4λρ ν 3 ρ 3 4λ ν, (A49) ρ 3 = 4λρ 3 ν 3 ρ δ 3 4λ ν, (A5) ρ 4 = 4λ ν 3ρ δ 4λ ν, (A5) ρ 3 = h 3λ, ρ 3 = h λ, (A5) (A53) s = α ω λ A (κ ), s = α κλa ν ω, (A6) (A6) σ = β β β β β, (A63) β σ = β β β β β, (A64) β σ = β 3β 3 β 4 β 3 β3, (A65) β 3 σ 3 = β 4β 3 β 3 β 3 β 3 β3, (A66) β n = ( β)( μ) D n D = (γ ) l, D = γ l,, d n = μ D n β =s λ ν s s λ ν s, β =s λ ν s s λ ν s, β 3 =9λ (α 3 ζα 4 ) ν (α 3 ζα 4 ), (A67) (A68) (A69) (A7) (A7) 3λ(α 4 ζα 6 ) ν (α 3 ζα 4 ), (A7) β 4 =9λ (α 8 ζα 6 ) ν (α 8 ζα 6 ) 3λ(α 3 ζα 4 ) ν (α 4 ζα 6 ), (A73) Downloaded from by guest on 8 November 8 ρ 3 = h λ, (A54) β = λ 4 λ (4 ν ν ) ν ν ν ν, (A74) ρ 33 = h 3λ, (A55) β = λ(ν ν ), (A75) A = 4λ α αν 4λδ δ ν, (A56) β 3 = 8λ 4 9λ (4 ν ν ) ν ν ν ν, (A76) A = 4λ α α ν δ 3 ν, (A57) β 3 = 6λ(ν ν ), (A77) MNRAS 446, (5)

14 97 A. K. Pal B. S. Kushvah σ = σ λ ν σ s λ ν, (A78) δ = ν A (ν 5 ν 3 κ)κa ν λ ν 3 κ A, (A85) σ = σ λ ν σ s λ ν, (A79) δ 3 = ν 4A z, (A86) σ = 6σ 3λ ν σ α 3 ζα 4 9λ ν, (A8) α = ν A ν 3κ A ν ν 3 κ A λ, (A8) α = ν 4A z, α 3 = ν 3 κ A λν, (A8) α 4 = ν A ν 3κ A ν ν 4A z δ = ν 5 κa λ λν ν 3 κ A, ν 3 κ A λ, (A83) (A84) δ 4 = (ν ν 5 κν ν 3 κ ν λ κ ν 3 )A ν 4A z, (A87) h = (ν 3 ν 3 )A A z, (A88) h = λν 3 κa A z. (A89) This paper has been typeset from a TEX/LATEX file prepared by the author. Downloaded from by guest on 8 November 8 MNRAS 446, (5)