Mathematical Applications in Modern Science

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1 Effect of Perturbatons n the Corols Centrfugal Forces on the Locaton Stablty of the Equlbrum Solutons n Robe s Restrcted Problem of + Bodes BHAVNEET KAUR Lady Shr Ram College for Women Department of Mathematcs Unversty of Delh INDIA bhavneet.lsr@gmal.com RAJIV AGGARWAL Sr Aurobndo College Department of Mathematcs Unversty of Delh INDIA rajv agg97@yahoo.com Abstract: In ths problem, one of the prmares of mass m s a Roche Ellpsod flled wth a homogeneous ncompressble flud of densty ρ. The smaller prmary of mass m s a pont mass outsde the Ellpsod. The thrd the fourth bodes of mass m m 4 respectvely are small sold spheres of densty ρ ρ 4 respectvely nsde the Ellpsod, wth the assumpton that the mass the radus of the thrd the fourth body are nfntesmal. We assume that m s descrbng a crcle around m. The masses m m 4 mutually attract each other, do not nfluence the motons of m m but are nfluenced by them. In ths paper, we have studed the effect of perturbatons n the corols centrfugal forces on the locaton stablty of the equlbrum ponts n Robe s restrcted problem of + bodes wth the bgger prmary a Roche ellpsod Chrashekhar,987. We have taken nto consderaton all the three components of the pressure feld n dervng the expresson for the buoyancy force vz due to the own gravtatonal feld of the flud that orgnatng n the attracton of m that arsng from the centrfugal force. In ths paper, equlbrum solutons of m m 4 ther lnear stablty are analyzed. We have proved that there exst only sx equlbrum solutons of the system, provded they le wthn the Roche Ellpsod. In a system the prmares are consdered as earth-moon m, m 4 as cebergs, the equlbrum solutons of m m 4 respectvely when the dsplacement s gven n the drecton of x axs or x axs are unstable. Robe s Restrcted Problem; Roche Ellpsod; Equlbrum Soluton; Stablty; Buoyancy Force. Introducton Robe 977 has nvestgated a new knd of restrcted three-body problem n whch one of the prmares of mass m s a rgd sphercal shell flled wth a homogeneous ncompressble flud of densty ρ. The smaller prmary s a mass pont m outsde the shell. The thrd body of mass m, supposed movng nsde the shell, s a small sold sphere of densty ρ, wth the assumpton that the mass the radus of the thrd body are nfntesmal. He further assumed that the mass m descrbes a Kepleran orbt around the mass m. He has proved that the centre of the frst prmary, s the only equlbrum soluton for all values of the densty parameter K, mass parameter µ, eccentrcty parameter e. Further, he has dscussed the lnear stablty of ths equlbrum soluton. He has explctly dscussed two cases. In the frst case, the orbt of m around m s crcular n the second case, the orbt s ellptc, but the shell s empty.e. no flud nsde t or the denstes of m m are equal. In each case, the doman of stablty has been nvestgated for the whole range of parameters occurrng n the problem. In the above problem, the exstence of only one equlbrum soluton namely, centre of the frst prmary s dscussed. Hallan & Rana 00 studed the exstence the lnear stablty of all the equlbrum solutons n the Robe s restrcted three-body problem. They have proved that besdes the centre, there are other equlbrum solutons whch exst only when K 0 the second prmary moves around the frst n a crcular orbt. In addton to the exstng collnear equlbrum soluton, they have shown the exstence of one more whch s stable only under certan restrctons on K µ. There are two trangular equlbrum solutons whch are always unstable. In addton, there are an nfnte number of equlbrum solutons lyng on a crcle centre at the second prmary, provded the solutons are nsde the sphercal shell. The crcular equlbrum solutons are always unstable. ISBN:

2 Szebehely 967 consdered the effect of small perturbatons n the corols force on the stablty of equlbrum ponts n the classcal restrcted problem, keepng the centrfugal force constant proved that the collnear ponts reman unstable. The range of stablty of the trangular ponts ncreases or decreases dependng upon whether the change ɛ n the corols force s postve or negatve thereby establshng that the corols force s a stablzng force. In the same problem, Bhatnagar & Hallan 978 studed the effect of the changes ɛ ɛ n the corols centrfugal forces respectvely on the stablty of equlbrum ponts concluded that the collnear ponts reman unstable for the trangular ponts, the range of stablty ncreases or decreases dependng upon whether the pont ɛ, ɛ le n one or the other of the two parts n whch the lne 6ɛ 9ɛ = 0 dvdes the ɛ, ɛ plane. Shrvastava Garan 99 studed the effect of small perturbatons n the corols centrfugal forces on the locaton of equlbrum ponts n the Robe s crcular restrcted three body problem when the denstes ρ ρ are equal evaluated the concomtant shft n the locaton of the equlbrum pont. In dervng the expresson for the buoyancy force Robe assumed that the pressure feld of the flud ρ has sphercal symmetry about the centre of the shell. He has taken nto account one of the three components of the pressure feld, that s, due to the own gravtatonal feld of the flud. Plastno & Plastno 995 nvestgated the effect of the remanng two components that orgnatng n the attracton of m, that arsng from the centrfugal force. They ncorporated these two components of pressure feld nto the dynamcs of the Robe model by consderng that the flud m adopts the shape of a Roche ellpsod. They have shown that the effect of the buoyancy force mght be thought as beng equvalent to a perturbaton of the corols force that the equlbrum pont s always stable,when the densty of the thrd body s greater than that of the surroundng medum. Gordano 997 studed the effect of drag force on the stablty of equlbrum ponts, both n the Robe s problem the problem studed by Plastno. Hallan & Rana 00 studed the effect of small perturbatons ɛ ɛ n the corols centrfugal forces respectvely on the locaton lnear stablty of the equlbrum ponts n the Robe s crcular problem wth denstes ρ ρ equal to zero vz, K equal to zero. They proved that there s only one equlbrum pont whch les on the lne jonng the centre of the shell the second prmary les to the rght or left of centre of the shell accordng as ɛ s postve or negatve. They have further studed the lnear stablty of the equlbrum pont. Hallan & Rana 00 later studed the effect of perturbatons n the corols centrfugal forces on the locaton ans stablty of the equlbrum ponts n the Robe s crcular restrcted problem wth densty parameter havng arbtrary value. Many authors have worked on problem of + bodes. Sgnfcant work s that of Whpple 984. He studed equlbrum solutons of the restrcted problem of + bodes. He further studed the lnear stablty of all the equlbrum solutons. Bhavneet & Rajv 0 extended the Robe s restrcted threebody problem to + bodes. Bhavneet & Rajv 0 studed the Robe s restrcted problem of + bodes when the bgger prmary s a Roche Ellpsod. They studed the equlbrum solutons of the nfntesmal masses analysed ther lnear stablty. In ths paper, we study the effect of perturbatons n the corols centrfugal forces on the locaton stablty of the equlbrum ponts n Robe s restrcted problem of + Bodes. Statement of the Problem Equatons of Moton In ths paper, one of the prmares of mass m s a Roche Ellpsod flled wth homogeneous ncompressble flud of densty ρ. The second prmary s a mass pont m m > m outsde the Ellpsod. The thrd the fourth body of mass m m 4 respectvely are small sold spheres of densty ρ ρ 4 respectvely nsde the Ellpsod, wth the assumpton that the mass radus of the thrd the fourth body are nfntesmal. Let R be the dstance between the centres of mass of m m. We assume that m descrbes a crcular orbt of radus R around m wth constant angular velocty ω. The masses m m 4 mutually attract each other, do not nfluence the motons of m m but are nfluenced by them. As n the case of classcal restrcted problem Szebehely 967,we adopt a unformly rotatng coordnate system Ox x x wth orgn of the coordnate system at the centre of the bgger prmary,ox pontng towards m Ox x beng the orbtal plane of m around m. The coordnate system Ox x x s as shown n the Fgure. Let the synodc system of coordnates ntally concdent wth the nertal system rotate wth angular velocty ω. Ths s the same as the angular velocty of m whch s descrbng a crcle around m. The equatons of moton of m m 4 n the Robe s crcular restrcted problem of + bodes n dmensonless cartesan coordnates Bhavneet & Rajv ISBN:

3 M x, x, x x O 0,0,0 M 4 x 4, x 4, x 4 R x Fgure : Geometry of the Robe s restrcted problem of + bodes when the bgger prmary m s a Roche Ellpsod the smaller prmary a pont mass 0are U = [ µj + x x + B + µ R j + µ { ρ [B + ρ { + µ = U, = U, = U } ] } + { m j µ j =, D = ρ m + m ρ B = πgρ I A A } ], 4, j =, 4; j. 5 A To acheve them we fx the unts such that m +m =, R =. We choose t n such a way that G =. The quantty µ becomes numercally equal to the rato m m + m. In the new unts ω =. The perturbatons n the corols centrfugal forces are expressed wth the help of parameters α, β, the unperturbed value of each beng unty. We take α β as α = + ɛ, ɛ, β = + ɛ, ɛ. Consequently, the equatons of moton of m m 4 can be wrtten as or α + α βx βx = U, 6 = U, 7 = U α = V + α, 8, 9 = V, 0 = V V =U + β x µ β + x = µ j [ + D B + { + R j { +µ ]. + + µ β + ɛ + ɛ = µ j } } The Equatons of moton of m m 4 can be rewrtten as xj = µ j R j + [ D + µ C + ɛ ], + + ɛ = µ j xj R j + [ D µ C + ɛ ], xj R j + D µ C 4, j =, 4; j. 5 C l = πgρ A l 6 µ = µ A l l =,,. 7 ISBN:

4 Equlbrum Solutons µ = µ πgρ. 8 The equlbrum solutons of m m 4 are gven by.e., µ j µ j µ j wth µ 4 V = V xj R j xj R j xj R j = V = 0 =, 4. + { D + µ C + ɛ } = 0, 9 + { D µ C + ɛ } = 0, 0 + D µ C = 0., j =, 4; j. Equlbrum Solutons lyng on x axs The Equatons 0, are satsfed wth, x ; =, 4 equal to zero. Hence we determne ; =, 4 from the Equaton 9 by puttng, x ; =, 4 equal to zero, we get x x 4 µ x x 4 + x 4 x x 4 x + { D + µ C + ɛ } x = 0. { D4 + µ C + ɛ } x 4 = 0. Multplyng the Equaton by µ by µ 4 addng, we get x 4 = λx 4 λ = D µ 5 D 4 µ 4 Substtutng n the Equaton,we get x = ± [ ] µ 4, {D + µ C + ɛ } + λ Mathematcal Applcatons n Modern Scence 6 [ x 4 = λ µ 4 {D + µ C + ɛ } + λ ]. 7 Hence x, 0, 0 x 4, 0, 0 are the equlbrum solutons for m m 4 respectvely provded they le wthn the Roche s Ellpsod. Equlbrum Solutons lyng on x axs If x = x = 0, then [ x = ± x 4 = D [ µ D 4 µ 4 µ 4 {D µ C + ɛ } + λ ], 8 µ 4 {D µ C + ɛ } + λ ]. 9 Hence 0, x, 0 0, x 4, 0 are the equlbrum solutons for m m 4, respectvely, provded they le wthn the Roche s Ellpsod. Equlbrum Solutons lyng on x axs If x = x = 0, then [ x = ± µ 4 {D µ C + ɛ } + λ ], 0 x 4 = D [ ] µ µ 4. D 4 µ 4 {D µ C + ɛ } + λ Hence 0, 0, x 0, 0, x 4 are the equlbrum solutons for m m 4 respectvely provded they le wthn the Roche s Ellpsod. There exst two equlbrum solutons of the system lyng each on x axs or x axs or x axs. Hence, there exst sx equlbrum solutons of the system, provded they le wthn Roche Ellpsod. We observe that there are no other equlbrum solutons except these sx. The equlbrum solutons of m are shown n Fgure.We observe that the equlbrum solutons donot exst when ρ = ρ ; ρ = ρ 4 or D = 0, =, 4 or when ρ = ρ ; ρ ρ 4. 4 Stablty of Equlbrum Solutons Let m m 4 be dsplaced to x j,, ; j =, 4 α j = X j +α j, = are very small X j, ISBN:

5 x x R Fgure : Locaton of Equlbrum Solutons n the Robe s Restrcted Problem of + Bodes when the Bgger Prmary s a Roche Ellpsod the Smaller Prmary a pont mass when the perturbatons n the corols centrfugal forces are taken nto consderaton. Crcles denote the postons of m trangles denote the postons of m 4. X j, Xj ; j =, 4 are the equlbrum soluton of m m 4 respectvely. The varatonal equatons of m m 4 are α α α + α α =α α =α V x + α V + α α =α + α V x x V V x x V x o + α x V x o o, o + α V x o o, o + α V o, =, 4 x o 4 V = µ j x Rj + µ V j xj = x µ j xj R 5 j µ V j xj = x V = µ j x Rj + R 5 j µ V j xj = x V = µ j x Rj + R 5 j xj xj µ j xj R 5 j R 5 j µ j xj, j =, 4; j. R 5 j + D + µ C + ɛ. xj + D µ C + ɛ. + D µ C Here the upper suffx o denotes the evaluaton of the dervatves at the equlbrum solutons under consderaton. Stablty of Equlbrum Solutons lyng on x axs Let the equlbrum soluton x 4, 0, 0 x, 0, 0 of m m 4 be ds- x + α, α, α placed to x 4 + α 4, α4, α4. The varatonal equatons of m m 4 are α α α = α α + α α = α p α = α p + λ + p λ = D µ D 4 µ 4, + λ + λ p + λ + p 5 6 =, 4 7 p = D + µ C + ɛ = p + ɛ p = D µ C + ɛ = p + ɛ p = D µ C. ISBN:

6 From the Equaton 7, we ascertan that the moton of m m 4 parallel to x axs s stable when D + µ C + ɛ D µ C D + µ C + ɛ D µ C > + D µ D 4 µ 4 =, 4 n case ρ > ρ, ρ > ρ 4 8 < + D µ D 4 µ 4 =, 4 n case ρ < ρ, ρ < ρ 4 9 The remanng Equatons 5 6 admt solutons of the form α j = A j e Ljt, =,, ; j =, 4. The characterstc equatons of m m 4 are gven by L 4 + L 4 + ɛ Q + Q + Q Q = 0 Q = p Q = p + λ + λ p + λ. =, 4 40 Let Λ = L, we obtan Λ + Λ 4 + ɛ Q + Q + Q Q = 0 =, 4. 4 we must have S + S < 0, S S > 0, > 0. Case I:ρ < ρ, ρ < ρ 4. In ths case Q < 0. Now, S S > 0 f Q < 0.e. f, D + µ C + ɛ D µ C + ɛ < + D µ D 4 µ 4 =, 4 4 Ths mples that S + S < 0, > 0. Hence, the equlbrum solutons of m m 4 are stable f the Equaton 4 holds, else unstable. Case II:ρ > ρ, ρ > ρ 4. Thus Q > 0. Now, S S > 0 f Q > 0. The equlbrum solutons of m m 4 are stable f D µ C D + µ C λ + ɛ > 0, + λ + λ 4 + λ + λ D + µ C + D µ C + ɛ + λ + λ < 4 + ɛ, 6 + 4ɛ + p 4 + λ + λ 6ɛ p 4 + λ p + λ + λ 8 + ɛ p + λ + λ p + p + λ 8ɛ > 0 + λ The shaded regon n Fgure shows the regon n whch the equlbrum solutons of m are stable for λ = 0. ɛ 0 8, ɛ Let the roots of the Equaton 4 be S, S. Then, S + S = Q + Q 4 + ɛ, S S = Q Q. Dscrmnant of the Equaton 4 s = 6 + 4ɛ + Q Q 8 + ɛ Q + Q p The equlbrum solutons x, 0, 0 x 4, 0, 0 of m m 4 respectvely when the dsplacement s gven n the drecton of x axs or x axs are stable f S S are real negatve or p Fgure : The shaded regon corresponds to the regon of stablty for the equlbrum solutons of m lyng on x axs ISBN:

7 Stablty of Equlbrum Solutons lyng on x axs Let the equlbrum solutons 0, x, 0 0, x 4, 0 of m m 4 be dsplaced to α, x + α, α α 4, x4 + α 4, α4. The varatonal equatons of m m 4 are α α α = α α + α α = α α = α p p + λ + p + λ + λ p + λ + p =, The moton of m m 4 parallel to x axs s stable when p p > + λ =, 4 n case ρ > ρ, ρ > ρ 4 47 p p < + λ =, 4 n case ρ < ρ, ρ < ρ 4 48 The characterstc equatons of m m 4 are gven by L 4 + L 4 + ɛ Q + Q + Q Q = 0 Q = p Q = p + λ + λ p + λ. =, 4 49 Case I:ρ < ρ, ρ < ρ 4. In ths case D > 0, λ > 0, p < 0, p < 0 =, 4. Thus Q < 0. The equlbrum solutons x 4, 0, 0 x, 0, 0 of m m 4 respectvely when the dsplacement s gven n the drecton of x axs or x axs are stable f Q < 0,.e. f D µ C + ɛ D + µ C + ɛ < + λ =, 4 50 as then S + S < 0, > 0. Case II:ρ > ρ, ρ > ρ 4. In ths case D < 0, λ > 0, p > 0, p > 0 =, 4. Thus Q > 0. Now, S S > 0, f Q > 0. The regon that s common to Q > 0, Q + Q < 4 + ɛ > 0 s the stable regon for the equlbrum solutons x, 0, 0 x 4, 0, 0 of m m 4 respectvely when the dsplacement s gven n the drecton of x axs or x axs. The shaded regon n Fgure 4 shows the regon n whch equlbrum solutons of m are stable for λ = 0. ɛ 0 8, ɛ 0 9 Q + Q = p =6 + 4ɛ + + 6ɛ + λ 8 + ɛ p + λ + p + λ + ɛ + λ + λ 4 + λ + λ 4 + λ p p + λ + λ + λ p p p + λ 8ɛ + λ 5 Stablty of Equlbrum Solutons lyng on x axs Let the equlbrum soluton 0, 0, x 0, 0, x 4 of m m 4 be dsplaced to α, α, x + α α 4, α4, x4 + α 4. The varatonal equatons of m m 4 are α α α α + α = α α = α α = α p p + λ + p p + λ + p + λ + λ 5 5 =, 4 54 From the Equaton 54, we ascertan that the moton of m m 4 parallel to x axs s stable f p < 0 ISBN:

8 Mathematcal Applcatons n Modern Scence when the dsplacement s gven n the drecton of x axs or x axs are stable. If Q > 0 Q > 0, that s common the regon 40 0 p X\ to Q > 0, Q > 0, Q + Q < 4 + > 0 s the stable regon for the equlbrum solutons of m m4 when the dsplacement s gven n the drecton of x axs or x axs. Case II: Smlar analyss follows when ρ > ρ, ρ > ρ4. The shaded regon n Fgure 5 shows the regon n whch equlbrum solutons of m are stable for λ = , λ Q +Q = p +p p + 0 +λ +λ 40 p X\ Fgure 4: The shaded regon corresponds to the regon of stablty for the equlbrum solutons of m lyng on x axs. =6 + 4 p p p + p + or equvalently D > 0 =, 4.e, ρ < ρ ρ < ρ4. The characterstc equatons of m m4 from the Equaton 5 5 are gven by L4 + L 4 + Q + Q + Q Q = 0 0 p +λ!! p X\ =, 4 55 p X\ Q = p 0 p +λ Q = p p. +λ p X\ or equvalently Λ + Λ 4 + Q + Q + Q Q = 0 5 Fgure 5: The shaded regon corresponds to the regon of stablty for the equlbrum solutons of m lyng on x axs. =, wth Λ = L Case I:ρ < ρ, ρ < ρ4. In ths case D > 0, λ > 0, p < 0, p < 0, p < 0 =, 4. As p < 0, the moton of m m4 parallel to x axs s always stable. Now, S S > 0 f Q > 0 Q > 0 or f Q < 0 Q < 0. If Q < 0 Q < 0, then S +S < 0, > 0. In ths case, the equlbrum solutons of m m4 ISBN: Applcatons Applcaton Moton of ceberg n the Earth-Moon system. We take ρ = 07 kg/m, ρ = 804 kg/m, ρ4 = 80 kg/m We have assumed a = Equatoral radus of earth = km. a = km. 86 λ +λ

9 a = km. Roche Lmt 0 Equatoral radus of moon = 78. km. Polar radus of moon = 76.0 km. Unts of Mass: In dmensonless unts, m + m =. Mass of Earth m = kg. Mass of Moon m = kg. Thus, kg = new unt Therefore, 4 µ = m m + m = 0.0. m µ = = 0 8. m + m m 4 µ 4 = = 0 7. m + m Unts of Dstance: Dstance between Earth Moon = km = new unt Thus, metre = new unt. Hence, ρ = 9645 new unts. ρ = 7550 new unts. ρ 4 = 7607 new unts. D = ρ = 0.77 ρ D 4 = ρ = 0.67 ρ 4 a = a = a = C = C = C = We take ɛ 0 8, ɛ 0 9. Table gves the equlbrum solutons for m m 4 usng the above data. The numercal solutons are correct n all dgts prnted. Fgure 6 plots the equlbrum solutons of m lyng on x axs vs µ. The graph s plotted for the postve value of x. For the earth moon system, we have µ = 0.0. The hghlghted pont represents the equlbrum soluton of m lyng on x axs for ths value of µ. Fgure 7 plots the correspondng equlbrum soluton of m 4 lyng on x axs vs µ when the equlbrum soluton of m m 4 are related by the Equaton 4. Stablty of Equlbrum Solutons lyng on x axs x axs The moton of m m 4 parallel to x axs s stable. The equlbrum solutons of m m 4 respectvely when the dsplacement s gven n the drecton of x axs or x axs are unstable as they donot satsfy all the condtons of stablty. Stablty of Equlbrum Solutons lyng on x axs The moton of m m 4 parallel to x axs s unstable. The equlbrum solutons of m m 4 respectvely when the dsplacement s gven n the drecton of x axs or x axs are unstable. 5. Applcaton Moton of submarnes n the Earth-Moon System We take ρ = 07 kg/m, ρ = 00 kg/m, ρ 4 = 00 kg/m In new unts ρ = 9645 new unts. ρ = 00 new unts. ρ 4 = 69 new unts. D = ρ = ρ D 4 = ρ = 0.44 ρ 4 a = a = a = C = C = C = We observe that n ths case the equlbrum solutons of m m 4 donot exst. 6 Dscusson In ths paper, we have studed the effect of perturbatons n the corols centrfugal forces on the locaton stablty of the equlbrum ponts n Robe s restrcted problem of + bodes. Plastno & Plastno 995 revsed the Robe s restrcted three-body problem under the assumpton that the flud body assumes ISBN:

10 the shape of a Roche Ellpsod. They took nto consderaton all the three components of the pressure feld n dervng the expresson for the buoyancy force vz, due to the own gravtatonal feld of the flud, that orgnatng n the attracton of m that arsng from the centrfugal force. Bhavneet & Rajv 0 studed the equlbrum solutons lnear stablty of the mnor bodes m m 4 n the Robe s restrcted problem of + bodes when the bgger prmary s a Roche ellpsod.? studed the equlbrum solutons lnear stablty of the mnor bodes m m 4 n the Robe s restrcted problem of + bodes when the bgger prmary s a Roche ellpsod smaller prmary an oblate body. In the paper Bhavneet & Rajv 0, we had neglected the tdal deformaton of the flud body m due to m. Also the components of buoyancy force arsng from the attracton of m centrfugal forces were gnored. We had studed the moton of the system n two dmenson as here by extendng to three dmenson we have determned the equatons of moton of m m 4 n the x drecton as well. All these have been ncorporated n the present study. The equatons of moton, equlbrum solutons of m m 4 ther stablty are analyzed takng nto account the above consderatons, perturbatons n the corols centrfugal forces takng m a Roche Ellpsod. Our results are n contrast to those of Plastno & Plastno 995. In ther study, when ρ = ρ, every pont wthn the Roche Ellpsod s an equlbrum soluton n general 0, 0, 0 s the only equlbrum soluton. We have taken two nfntesmal masses wthn the Roche Ellpsod as Plastno & Plastno 995 has taken only one nfntesmal mass wthn the Roche Ellpsod. When there are small perturbatons ɛ ɛ n the corols centrfugal forces respectvely, n the Robe s restrcted problem of + bodes, the number of equlbrum solutons are the same as n the problem wth no perturbaton Bhavneet & Rajv 0, but postons of the equlbrum solutons have changed. The change n the corols force does not affect the postons of the equlbrum solutons. There exst two equlbrum solutons of the system lyng each on x axs or x axs or x axs. Hence, there exst sx equlbrum solutons of the system, provded they le wthn Roche Ellpsod. We observe that there are no other equlbrum solutons except these sx It may also be observed that the problem of + bodes can be extended to + n bodes takng n nfntesmal masses wthn the ellpsod. Ths problem has applcaton n the moton of submarnes n the earth moon setup. The problem can also be appled to the moton of ceberg. Earth s the only celestal body we know of whch has flud the test partcles wthn the flud are consdered as the submarnes. In the solar or the extra solar system, we donot know of any other celestal body havng lqud nsde t. Our entre work s applcable n solar or extra solar system when f such a system s dscovered. 7 Concluson In ths paper, we have studed the moton of two nfntesmal masses m m 4 supposed movng nsde m n three dmensons, takng m as a pont mass m a Roche Ellpsod. We have studed the effect of perturbatons n the corols centrfugal forces on the locaton stablty of the equlbrum ponts n Robe s restrcted problem of + bodes. We have taken nto consderaton all the three components of the pressure feld n dervng the expresson for the buoyancy force vz, due to the own gravtatonal feld of the flud, that orgnatng n the attracton of m that arsng from the centrfugal force.we have proved that there exst only sx equlbrum solutons of the system, provded they le wthn the Roche Ellpsod. In a system the prmares are consdered as earth-moon m, m 4 as ceberg, the equlbrum solutons of m m 4 respectvely when the dsplacement s gven n the drecton of x axs or x axs are unstable. Table : Equlbrum Solutons x, x 4 =,, of m m 4 lyng on x, x, x axs respectvely. x x 4 x axs ± x axs ± x axs ± ISBN:

11 x Μ Fgure 6: Equlbrum soluton of m postve root lyng on x axs vs µ. The hghlghted pont represents the equlbrum soluton of m for µ = 0.0 x Μ Fgure 7: Equlbrum soluton of m 4 lyng on x axs vs µ. The hghlghted pont represents the equlbrum soluton of m 4 for µ = 0.0 ISBN:

12 References: Bhatnagar, K. B. Hallan, P.P.:978, Effect of Perturbatons n Corols Centrfugal Forces on the Stablty of Lbraton Ponts n the Restrcted Problem, Celestal Mechancs 8,05-. Bhavneet Kaur, Rajv Aggarwal, Robe s restrcted problem of + bodes when the bgger prmary s a Roche ellpsod, Acta Astronautca, Volume 89, AugustSeptember 0, Pages -7, ISSN , Szebehely, V.S.:967 Theory of orbts, Academc Press, New York. Szebehely V, Peters CF 967 Complete soluton of a general problem of three bodes, Astronomcal Journal 7:876-88, DOI 0.086/055. Wkpeda:0, Roche Lmt, Wkpeda. Whpple, A.L.:984, Equlbrum Solutons of the Restrcted Problem of + Bodes, Celestal Mechancs,7-94. Chrashekhar, S.:987 Ellpsodal Fgures of Equlbrum Chapter 8, Dover Publcaton Inc, New York. Gordano, C.M., Plastno, A.R. Plastno, A.:997, Robe s Restrcted Three-Body Problem wth Drag, Celestal Mechancs Dynamcal Astronomy 6, Hallan, P.P. Rana, N.:00, The Exstence Stablty of Equlbrum Ponts n the Robe s Restrcted Three-Body Problem, Celestal Mechancs Dynamcal Astronomy 79, Hallan, P.P. Rana, N.:00, Effect of Perturbatons n Corols Centrfugal Forces on the Locaton Stablty of Equlbrum Pont n the Robe s Crcular Restrcted Three-Body Problem, Planetary Space Scence 99, Hallan, P.P. Rana, N.:00, Effect of Perturbatons n the Corols Centrfugal Forces on the Locatons Stablty of the Equlbrum Pont n the Robe s Crcular Problem wth Densty Parameter Havng Arbtrary Value, Indan J. pure appl. Math., 47, Kaur, Bhavneet Aggarwal, R.:0 Robe s Problem: Its Extenson to + Bodes, Astrophyscs Space Scence 9, Plastno, A.R. & Plastno, A. :995, Robe s Restrcted Three-Body Problem Revsted, Celestal Mechancs Dynamcal Astronomy 6, Robe, H.A.G.:977, A New Knd of Three-Body Problem, Celestal Mechancs 6, 4-5. Shrvastava, A. K. Garan, D.: 99, Effect of perturbaton on the locaton of lbraton pont n the Robe restrcted problem of three bodes, Celest. Mech. & Dyn. Astr. 5, 677. ISBN: