Stability of Triangular Libration Points in the Sun - Jupiter System under Szebehely s Criterion

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1 Available at Appl. Appl. Math. ISSN: Vl. 1, Issue 1 (June 017), pp Appliatins and Applied Mathematis: An Internatinal Jurnal (AAM) Stability f Triangular Libratin Pints in the Sun - Jupiter System under Szebehely s Criterin Abstrat M. R. Hassan 1, Md. Aminul Hassan and M. Z. Ali 3 1 P. G. Department f Mathematis S. M. Cllege, T. M. Bhagalpur University Bhagalpur-81001, India hassansm@gmail.m GTE, Yeshwanthpur Bangalre-5600, India mahassan01@gmail.m 3 Millia Cnvent Shl Bhagalpur-8100, India Reeived July, 016; Aepted February 1, 017 In the present study, the lassial furth-rder Runge-Kutta methd with seventh-rder autmati step-size ntrl has been arried ut t examine the stability f triangular libratin pints in the Sun-Jupiter system. The Sun is a highly luminus bdy and Jupiter is a highly spinning bdy, s radiatin pressure f the Sun and blateness f the Jupiter annt be negleted. These fatrs must have sme effets n the mtin f the infinitesimal mass (spaeraft) and nsequent effets n the stability f the triangular libratin pints. It is t be nted that in ur prblem, infinitesimal mass exerts n influene f attratin n the primaries (Sun and Jupiter) but its mtin is influened by the primaries. Therefre, the equatins f mtin f the infinitesimal mass mving in the gravitatinal field f the radiating Sun and blate Jupiter have been established fr numerial integratin. T hek the stability f the libratin pints, the infinitesimal mass is allwed t librate fr trajetry generatin in the viinity f ne f the triangular libratin pints. Using duble-preisin mputatin, the Jabian nstant was alulated in rder t bserve the validity f the trajetry generatin thrughut the numerial integratin. This nstant f integratin was heked t make sure that it remained nstant at least t eight deimal plaes, s that ther data may be aurate. Fllwing all the abve mputatinal tehniques, the maximum displaement and maximum velity envelpes were nstruted in the light f previus authrs. The reasn behind the assumptin f the maximum displaement and maximum velity envelpes is that the spaeraft (infinitesimal mass) will librate fr a lng time within the regin f the envelpes withut rssing the x-axis. If the area f the envelpe is nt maximum within the given time limit and the infinitesimal mass rsses the x-axis, then by hanging the initial nditins; we attempt t nstrut the envelpes f maximum area fllwing previus authrs. If the area f the envelpe is maximum it means spaeraft 15

2 16 M. R. Hassan et al. (infinitesimal mass) will librate in wider area fr a lng time withut rssing the x-axis and lngtime libratin will give the higher range f stability. Frm ur bservatin, it is fund that due t the blateness f Jupiter, the range f stability is redued but phtgravitatin f the Sun has n signifiant effet n the triangular libratin pints. Keywrds: Restrited Three-bdy prblem; Libratin Pints; Phtgravitatin; Oblateness; Critial mass; Pinare Surfae f Setin; Stability; Cmmensurability MSC 010 N.: 37M05, 70F07, 70F15 1. Intrdutin Reently sme spae researhers are engaged in searh f stable libratin pints t set a spae statin and parking zne fr the spae vyage and hene they are heking the stability f triangular libratin pints. In the Classial Restrited Three-bdy Prblem, all llinear libratin pints are unstable but arding t sme authrs, the triangular libratin pints are linearly stable whereas sme authrs established stability riteria as, where is the ritial mass f the restrited three-bdy prblem. A series f wrks has been perfrmed by Deprit et al. (1967). Markeev (1969), Alfriend (1970). Henrard (1970) established that in the viinity f L, family f peridi rbits des nt evlve in a ntinuus manner with the mass rati. Nayfeh (1971) studied the prblem with the help f mmensurability :1 and 3:1. Markeev (1973) and Sklski (1975) als studied the stability f the Langrage slutin. MKenzie and Szebehely (1981) defined a new riterin fr the stability f the third bdy in the neighburhd f the triangular libratin pints. Fr this, they defined maximum velity and maximum displaement envelpes within whih the third bdy remains fr a lng time starting frm the suitable initial nditins, s that the third bdy may nt rss the x axis. This nditin f nt rssing the x axis was intrdued as the stability riteria fr the third bdy. Tukness (1995) investigated the sensitiveness f the third bdy numerially in the neighburhd f L by giving psitinal and velity deviatins frm L under sme suitable initial nditins. He used Pinare s surfae f setins t mpare the peridi, quasi-peridi and hati regins f the trajetries with the definitins f stability given by MKenzie and Szebehely (1981). Mrever, he als investigated the value f (the mass rati) ranging frm zer t ritial mass Using the stability riteria, he determined sme values f fr whih libratin pints are mre stable in mparisn with the ther values f. Markells et al. (1996) and Papadakis (1998) studied the different aspets f nn-linear stability f Lagrangian pints in the plane Cirular Restrited Three-bdy Prblem. Zslt Sandar et al. (000) disussed phase-spae struture in the viinity f triangular libratin pints in the Restrited Three-bdy Prblem. Hassan et al. (013) extended the wrk f Tukness (1995) by nsidering the effet f blateness f the bigger primary and shwed that with the inrease f blateness and mmensurability, the ritial mass redues and the range f stability dereases ardingly. Tukness (005) established the valid stability riteria in the light f Syzygies espeially in the Newtnian time dmain. It prjets new infrmatin abut the stable mtin f the third bdy.

3 AAM: Intern. J., Vl 1, Issue 1 (June 017) 17 In the present wrk, we prpse t extend the wrk f Hassan et al. (013) in the Sun-Jupiter system in the Restrited Three-bdy prblem. Stability riteria given by MKenzie and Szebehely (1981) and mputatinal tehniques f Tukness (1995) and Hassan et al. (013) have been nsidered fr disussing the prblem.. Equatins f Mtin f the Infinitesimal Mass (Third Bdy) In dimensinless variables, the equatins f mtin f the third bdy, in syndi -rdinate system, in the gravitatinal field f the Sun and Jupiter are x ny, x y nx, y (1) where p n 1 1 3I 1 r1 r. 3 r1 r r () The mean mtin " n " f the syndi frame is given by 3I n 1, I A A Oblateness parameter f the smaller primar y, e p (3) b Ae mment f inertia f the blate bdy abut the equatrial radius b, 5R Ap mment f inertia f the blate bdy abut the plar radius, 5R R the dimensinal distane between the primaries, p Radiatin pressure f the Sun n the third bdy, 1 r1 x y the distane f the infinitesimal mass frm the first primary and 1 r x 1 y the distane f the infinitesimal mass frm the send primary. The equatins in the System (1) an be redued t a single equatin i.e., x y x, y C, F x. y, x, y x y x, y C 0, ()

4 18 M. R. Hassan et al. where C is alled Jabi nstant and F x, y, x, y is alled Jabi s manifld. 3. Triangular Libratin Pints Sine the triangular libratin pints are the singularities f the manifld F x, y, x, y 0, hene the libratin pints are the slutins f the equatins i.e., and 0 and 0, x y 1 x 1 p x 1 9I x 1 nx x r1 r r 0 (5) p I yn y r1 r r Equatin (5) Equatin (6) x 1 gives 0. (6) n 1 p. (7) r 3 1 Equatin (5) Equatin (6) x μ gives 9I n 0. (8) r r 3 5 Finally, the triangular libratin pints are the slutins f the equatins 1 p 9I n 0and n 0. (9) r1 r r As 0 pi, 1, hene fr the first apprximatin p0 I and the slutins f Equatin (9) are given by 1 1 n 0 and n r1 r i.e., r1 r n 1.

5 AAM: Intern. J., Vl 1, Issue 1 (June 017) 19 This is pssible nly when bth the primaries are f equal masses. But the masses f the Sun and Jupiter are nt equal s r 1 r. Fr better apprximatin we assume p 0and I 0, hene the slutin f Equatin (9) may be suppsed t be r 1and r 1, (10) 1 where, are distint but very small quantities, i.e., 0, 1. Negleting higher rder terms f, and upling terms, ne an find the values f xand y; by using the Equatin (10) as 1 x, 3 1 y, 3 (11) i.e., L,5 1 3, 1. 3 Putting the values f nand r1 in the first equatin f System (9) and negleting the higher I rder and upling terms, we get. Similarly, frm the send equatin f System (9), we get I p 3I. 3 Hene, the -rdinates f triangular libratin pints L,5 are given by L,5 1 p 3I 3 p 3 I, Critial Mass The harateristi equatin fr equilateral triangular libratin pints an be written as n xx yy xx yy xy , (1) where xx p I, (13) 8

6 0 M. R. Hassan et al yy p I, (1) 8 and Here xy 6 1 0p I. (15) xx, yy, xy are the values f xx, yy, xy respetively at the libratin pints. If, then the redued harateristi Equatin (1) an be written as P Q 0, (16) where I P n xx yy 1 p, (17) 71 9 Q xx yy xy p I. (18) Fr mmensurability, let (16). Then k be the rati f the rts f the redued harateristi Equatin P P Q 1 P P Q 1, where k is a psitive integer. k That is, P k k Q 1 0. (19) The mbinatin f the Equatins (17), (18) and (19) yields k p I k p I k p I (0) The rts f Equatin (0) are alled ritial mass dented by and given by k p I k p I (1) Putting p 0, we get k p I

7 AAM: Intern. J., Vl 1, Issue 1 (June 017) 1 k I k I k I [Same as Hassan et al. (013)] and by putting pi 0, we get 1 16k k 1. [Same as Szebehely (1981) and Tukness (1995)] 5. Numerial Integratin Figure 1. Maximum velity and Maximum displaement Envelpes Fr numerial integratin, let us intrdue x x1, y x, x x3, y x and reduing tw send rder differential equatin f mtin given in Equatin (1) t the fur first rder differential equatins as dx1 dx x3, x, dt dt dx3 dx nx, nx3, dt x1 dt x () where p n 1 1 3I 1 r r, (3) 1 3 r1 r r

8 M. R. Hassan et al. with r x x and r x 1 x are the mmenta rrespnding t the -rdinates, If p1and p mass, then p1 x1 nxand p x nx1. x x f the infinitesimal 1 The Hamiltn-Jabi equatins f mtin in annial frm, are () H H x1, x, p1 p H H p1, p, x1 x with the rrespnding Hamiltnian p n 1 1 3I H p p p x p x C. (5) r1 r r T slve Equatin () numerially, we apply Runge-Kutta methd within the limits given by Tukness (1995). On nstrutin f maximum displaement and maximum velity envelpes f the third bdy, a large initial velity is given t the third bdy in the diretin f 110 and integrating the trajetry fr a spaeraft and amunt f time and heking t shw if the third bdy rsses the x axis. If the third bdy rsses the x axis befre the time limit, the initial velity was dereased by dimensinless time units and the whle predure was again started and ntinued this press until a maximum velity was fund that allwed the third bdy t librate arund L fr the full length f time limit withut rssing the x axis. This predure will ntinue fr full 360 surrunding L. Speifying the maximum initial velity allwed in a ertain diretin nstitutes a velity vetr with magnitude and diretin. By nneting the end pints f all suh velity vetrs, we will get maximum velity envelpe. In a similar way if the third bdy starts with zer initial velity frm L and within speified amunt f time, the third bdy will be at a displaement frm L alng a ertain diretin withut rssing the x axis and then the end pint f this displaement nstitutes a displaement vetr. This predure will ntinue fr all 360 surrunding L. By nneting the end pints, all suh displaement vetrs will nstitute a maximum displaement envelpe. The maximum velity envelpe and maximum displaement envelpe are defined in the hpe f shwing that the third bdy will remain within a speified area arund L fr an infinitely lng perid f time with given speifi initial nditins.

9 AAM: Intern. J., Vl 1, Issue 1 (June 017) 3 6. Cmparisn by Pinare Surfae f setin In this setin, the maximum velity and maximum displaement envelpes bundary values are investigated using time limits f Tukness (1995) fr numerial integratin and Pinare surfae f setin. The slutin f the Hamiltnian equatin f mtin in Equatin () an be represented as a trajetry in a fur dimensinal phase spae. Beause f the existene f the integral f mtin in Equatin (5), the trajetry lays a three-dimensinal subspae H whih is equal t a nstant f the phase spae. The suessive intersetin f this three-dimensinal trajetry with a tw dimensinal surfae is alled Pinare surfae f setin. The intersetin f 3D trajetry with the surfae 3 p 3 y 1 I 3 3 in the psitive diretin is the Pinare surfae f setin in ur ase. In Figure, the Pinare surfae f setin has n lear visibility, s in Figure 3 (Enlarged view f Figure ), the phase spae an be divided int tw regins. One regin ntains disnneted islands and the send regin nsists f primarily regular trajetries whih are islated frm the islands. It means the hati belt f the Pinare setin has been redued due t the intrdutin f phtgravitatin and blateness. The maximum velity and maximum displaements were investigated using the Pinare surfae f setins fr varius values f. Initial velities and psitins frm the maximum velity and maximum displaement envelpes were numerially integrated using the Hamiltnian equatins f mtin and the intersetin with the surfae 3 p 3 y 1 I 3 3 were pltted fr 500 t 000 rbits fr y 0 nly using Runge-Kutta 7/8 integratr. Figure 3 shws the Pinare surfae f setin fr 0.001, 108 and values f initial dimensinless velities 0.0, 0.5, 0.30, 0.35, 0.0, 0. fr the third bdy rbiting at L, whih are the same as Hassan et al. (013). Therefre, pht-gravitatin has n effet n the Pinare surfae f setin rrespnding t maximum velity and maximum displaement envelpes. Using stability riteria established in this study, 0. is the value f the maximum velity in the diretin f 108. The velity greater than 0. represents unstable mtin and velity less than 0. represents quasi-peridi mtin.

10 M. R. Hassan et al. Figure. Surfae f Pinare Setin fr Figure 3. Enlarged view f Figure 0.001, 108 Figure depits the Pinare surfae f setin fr 0.001, 0 and the maximum initial displaement allwed, whih an als be explred in enlarged view f Figure. Fllwing Tukness (1995), the maximum velity and maximum displaement envelpes an be used t detet when hati mtin f the third bdy is abut t take plae. Figure. Surfae f Pinare Setin fr Figure 5. Enlarged view f Figure 0.001, 0 7. Stability f the Infinitesimal Mass in the neighburhd f L As in Tukness (1995), we have hsen final time 1000 f t and bundary x axis, s size f the maximum velity and maximum displaement envelpes vary arding t the values

11 AAM: Intern. J., Vl 1, Issue 1 (June 017) 5 f, I, pand t f under the defined bundary. Sine the final time t f 1000 and bundary is the x axis, hene they are assumed t be fixed and thus different envelpes vary arding t, Iand p. Fr the Sun-Jupiter system, we take μ as a fixed value beause we have taken Pinare setins nly fr tw different values f θ i.e., fr θ 0 and θ 108 as in Tukness (1995) and Hassan et al. (013). Fllwing Hassan et al. (013), we have alulated the areas f displaement envelpes and 1 velity envelpes by nsidering the frmula rd with the prper limits f fr different envelpes. Beause f the estimated area f the envelpes is a measure f stability, a mparisn an be made n hw stability varies arding t the value f, Iand p. In Figure 6, it is visible that the area f velity envelpes beame narrwer in mparisn t the lassial ase (fr I 0 ) with the inrease f blateness parameter I. Thus, the stability perentage dereases with the inrease f I as in the Figure 6, the velity envelpe fr I 0.01 bemes the interir part f that fr I 0.1. In Figure 7, the displaement envelpes beame narrwer with very little effet f blateness I 0.1and We have nt nfirmed that, whether I have sme signifiant effet r nt n the displaement envelpes. Figure 6. Maximum velity envelpes Figure 7. Maximum displaement envelpes fr µ = and I = 0, 0.01, 0.1 fr µ = and I = 0, 0.01, 0.1 Figure 8 shws the psitin f L in the mmn area f the velity envelpe and displaement envelpe in unperturbed mtin in the Sun-Jupiter system. Figure 9 depits that the area f velity envelpes and displaement envelpes beme maximum rrespnding t the same value f i.e., rrespnding t this value f mtin f the third bdy is mst stable whih means fr this value f, the third bdy will librate in the neighburhd f L fr maximum t f 1000 (nn-dimensinal time) withut rssing the x axis. Als, frm this graph, we an nlude that fr 0. 07, bth the area f velity envelpe and displaement envelpes beme zer, s we an say that this value f is fr

12 6 M. R. Hassan et al. the mst unstable mtin i.e., the third bdy will esape frm the neighburhd f L fr Figure 8. Maximum displaement and Figure 9. Area f maximum velity and velity envelpes fr 0.001&0.011 displaement in Sun-Jupiter system in unperturbed mtin 8. Order f Cmmensurability k Table 1. Order f Cmmensurability (Fr I 0) k p 0 5 p 10 p p 10 3 p 10.5 p 10 p

13 AAM: Intern. J., Vl 1, Issue 1 (June 017) 7 Table. Order f Cmmensurability (Fr I 10 ) k p 0 5 p 10 p p 10 3 p 10.5 p 10 p In the Table 1, ritial mass has been alulated fr p 0,10,10, keeping I 0. It is seen in the table that varies frm t fr k 1 i.e., dereases with the inrease f p. Similarly in every rw f the Table 1, dereases with the inrease f p fr k,3,...10 i.e., inreases with the inrease f mmensurability k. In eah lumn f the Table 1, inreases with the inrease f p frm 0t10..5 In Figure 10, graphs shw the variatin f with p 0,10 and 10 rrespnding t eah value f k 1,, It is seen that inreases rapidly frm k 1,,3 whereas inreases very slwly fr k, 5, That is, runs almst thrugh a hrizntal straight line after k 3. Similar ase happened in graphs f Figure 11 and Figure 1 fr 1 I 10 and I 10, respetively with a visible differene, whih an be seen in Figure In Figure 13 the graphs f versus p fr I 0,10 and10 are drawn frm the first rw f Tables 1, and 3. It is seen that is almst nstant fr all values f pand k 1. Similarly ther graphs an be seen fr k, 3, All graphs almst will be hrizntal lines. Therefre, phtgravitatin has insignifiant effet n fr mmensurability 5 k 1,, In Figure 1, fr the fixed values f p 10, the graphs f versus I fr k 1,, and fr all values f, I, are almst same, in whih dereases when I inreases.

14 8 M. R. Hassan et al. 1 Table 3. Order f Cmmensurability (Fr I 10 ) k p 0 5 p 10 p p 10 3 p 10.5 p 10 p k Table. Cmbined Effet f p 0, p 5 10, p pand I n Critial Mass 10, p , p 3 10, p.5 10, p 10, I 0 3 I 10.5 I 10 I I 10 1 I I In Table, ritial mass has been alulated by the frmulae given in Equatin (1) when bth f pand I p 10 5, I 10 3, p 10, I 10.5 are allwed t vary. Fr p , I 10, p 10 3, I , p 10.5, I 10 1 and p , I 10, has been alulated fr k 1,, and graphs were pltted in Figure 15. In this figure, all pairs f pand I represent almst the same graph with a slight differene. With the inrease f bth the parameter pand I tgether, dereases. Thus, by the redutin f ritial mass, stability range f the triangular libratin pints is redued arding t the riterin under nsideratin.

15 AAM: Intern. J., Vl 1, Issue 1 (June 017) p=0 p=10^-.5 p=10^ k p=0 p=10^-.5 p=10^ k Figure 10. Pltting f vs. k fr I 0 Figure 11. Pltting f vs. k fr I p=0 p=10^-.5 p=10^ k I=0 I=10^- I=10^ p Figure 1. Pltting vs. k fr I 1 10 Figure 13. Pltting vs. p fr k k=1 1 k= 0.98 k= k= k=5 0.9 k= I Cmbined Effet k=1 k=3 k=5 k=7 k=10 p, I Figure 1. Pltting vs. I fr p 5 10 Figure 15. Cmbined Effet f pand I n

16 30 M. R. Hassan et al. 9. Cnlusin The present study aims t hek the jint effet f slar radiatin f the Sun and blateness f the Jupiter n the stability f the triangular libratin pints L 5, in the Sun-Jupiter system. In setins, 3 and f the present wrk, all mathematial derivatins are given, whih are required t disuss the stability f the triangular libratin pints. In setin 5, fr numerial integratin we have applied furth-rder Runge-Kutta methd with a seventh-rder autmati step-size ntrl in the FORTRAN prgramming fr slving the fur first-rder differential equatins f the system (). In setin 6, with the tehniques given by Tukness (1995), we have prdued data f x 1,x,x 1,x,C, then within the time limit f Tukness, Pinare surfae f setin are pltted in Figures 3, and 5. In lassial ase f Tukness (1995), the regins f mtin in Pinare surfae f setin an be divided int three regins. In the first regin, all trajetries are regular and peridi, in the send regin trajetries are quazi-peridi and in the third regin hati belt is seen in whih islands are sattered thrughut the regin f mtins but in ur ase, there is a signifiant hange in the hati regin fr bth the ase f 0, 108. In the hati regin f ur Pinare surfae f setin, the sattered islands have been signifiantly redued. Thus, the hati belt has been redued due t the intrdutin f the phtgravitatin f the Sun and blateness f Jupiter. In setin 7, we have mpared the stability riterin f the third bdy in the viinity f the triangular libratin pints with that f the lassial ase given by Tukness (1995). In Figure 6, the area f velity envelpes is maximum fr I 0 but fr I 0. 01, the area f velity envelpes has been redued and fr I 01., the area f velity envelpes is further redued; i.e., by inreasing the blateness parameter I, the area f velity envelpes dereases. In ther wrds, we an say that due t blateness f Jupiter, the perentage f stability f triangular libratin pints redued; i.e., blateness f the Jupiter has a signifiant effet n the stability f triangular libratin pints f the Sun-Jupiter system but n signifiant effet f phtgravitatin is seen in the area f velity envelpes. As far as the area f displaement envelpes is nerned, n effet f phtgravitatin and blateness is fund, whih is visible in Figure 7 and 8; i.e., due t phtgravitatin f the Sun and blateness f the Jupiter, the area f the displaement envelpes is nt redued. Frm all the setins disussed abve and the graphs drawn in Figures 10, 11, 1, 13, 1 and 15, we nlude that a great effet f blateness and a very little effet f phtgravitatin have redued the range f stability f the triangular libratin pints in mparisn f the lassial ase. Aknwledgement We are very thankful t the UGC (University Grants Cmmissin), ERO, Klkata and New Delhi, India fr santining the Minr Researh Prjet and supprting the exeutin f suh imprtant researh.

17 AAM: Intern. J., Vl 1, Issue 1 (June 017) 31 REFERENCES Alfriend, K.T. (1970). The Stability f the Triangular Lagrangian Pint s Cmmensurability f Order Tw, Celestial Mehanis and Dynamial Astrnmy, 1(3), Bhatnagar, K.B. and Chawla, J.M. (1977). The Effet f Oblateness f the bigger Primary n Cllinear Libratin Pints in the restrited prblem f three bdies, Celestial Mehanis and Dynamial Astrnmy, 16(), Breakwell, J. and Pringle, R. (1965). Resnanes affeting Mtin near the Earth-Mn Equilateral Libratin Pints. Prgress in Astrnautis and Aernautis, Aademi Press, New Yrk, 17, Deprit, A. and Deprit-Barthlme, A. (1967). Stability f the Triangular Lagrangian Pints, The Astrnmial Jurnal, 7(), Deprit, A. and Palmre, J. (1966). Analytial ntinuatin and first-rder Stability f the shrt-perid Orbits at L in the Sun-Jupiter system, The Astrnmial Jurnal, 71(), Deprit, A., Henrard, J., Palmre, J. and Prie, J.R. (1967). The Trjan manifld in the system Earth-Mn, Mnthly Nties f the Ryal Astrnmial Siety, vl. 137, pp Henrard, J. (1970). Cnerning the genealgy f lng perid families at L, Astrnmy and Astrphysis, 5, Hassan, M.R., Antia, H.M. and Bhatnagar, K.B. (013). Psitin and velity sensitiveness at the Triangular Libratin Pints in the Restrited Prblem f three bdies when the bigger Primary is an Oblate bdy, Astrphysis and Spae Siene, 36(1), Lentvih, A.M. (196). On the Stability f the Lagrange Peridi Slutins fr the Restrited Prblem f three bdies, Sviet Mathematis Dklady, 3, 5-8 Markeev, A.P. (1969). On the Stability f the Triangular Libratin Pints in the Cirular bunded three-bdy Prblem, Jurnal f Applied Mathematis and Mehanis, 33(1), Markeev, A. (1973). On the Stability prblem fr the Lagrange slutins f the restrited three-bdy prblem, Jurnal f Applied Mathematis and Mehanis, 37(), Markells, V.V., Papadakis, K.E. and Perdis, E.A. (1996). Nn-linear Stability znes arund Triangular Equilibria in the Plane Cirular Restrited three-bdy Prblem with Oblateness, Astrphysis and Spae Siene, 5(1), MKenzie, R. and Szebehely, V. (1981). Nn-linear Stability arund the Triangular Libratin Pints, Celestial Mehanis and Dynamial Astrnmy, 3(3), Nayfeh, A.H. (1971). Tw-t-ne Resnanes near the Equilateral Libratin Pints, Amerian Institute f Aernautis and Astrnautis Jurnal, 9(1), Papadakis, K.E. (1998). Effet f Radiatin n the Nn-linear Stability znes f the Lagrangian Equilibrium Pints, Celestial Mehanis and Dynamial Astrnmy, 7(), 35-

18 3 M. R. Hassan et al. Pedersen, P. (1933). On the Peridi Orbits in the neighburhd f the Triangular Equilibrium Pints in the Restrited prblem f three bdies, Mnthly Nties f the Ryal Astrnmial Siety, 9, Sklski, A.G. (1975). Stability f the Lagrange slutins f the restrited three-bdy Prblem fr the Critial rati f the Masses, Jurnal f Applied Mathematis and Mehanis, 39(), Szebehely, V. (1967). Thery f Orbits, Aademi Press, New Yrk Tukness, D.G. (1995). Psitin and velity sensitiveness at the Triangular Libratin Pints, Celestial Mehanis and Dynamial Astrnmy, 61(1), Zslt, S., Baliant, E. and Christ, E. (000). The Phase Spae Struture arund L in the Restrited three-bdy Prblem, Celestial Mehanis and Dynamial Astrnmy, 78(1),