ISSN (e): 2250 – 3005 || Volume, 06 || Issue, 01 ||January – 2016 || International Journal of Computational Engineering Research (IJCER)
Existence and Stability of the Equilibrium Points in the Photogravitational Magnetic Binaries Problem When the Both Primaries Are Oblate Spheriods Mohd Arif Department of mathematics, Zakir Husain Delhi college New Delhi 110002, India Abstract : In this article we have discussed the equilibrium points in the photogravitational magnetic binaries problem when the bigger primary is a oblate spheroid and source of radiation and the small primary is a oblate body and have investigated the stability of motion around these points.
Key words : magnetic binaries problem, source of radiation, stability, equilibrium points, restricted three body problem, oblate body.
I. Introduction In (1950) Radzievskii have studied the Sun-planet-particle as photogravitational restricted problem, which arise from the classical restricted three body problem when one of the primary is an intense emitter of radiation. In (1970) Chernikov and in (1980) Schuerman have studied the existence of equilibrium points of the third particle under the influence of gravitation and the radiation forces. The stability of these points was studied in the solar problem by Perezhogin in (1982). The lagrangian point and there stability in the case of photogravitational restricted problem have been studied by K.B.Bhatnagar and J. M. Chawla in (1979). Mavraganis. A. (1979) have studied the stationary solutions and their stability in the magnetic-binary problem when the primaries are oblate spheroids. Khasan (1996) studied libration solution to the photogravitational restricted three body problem by considering both of the primaries are radiating. He also investigated the stability of collinear and triangular points. In (2011) Shankaran, J.P.Sharma and B.Ishwar have been generalized the photogravitational non-planar restricted three body problem by considering the smaller primary as an oblate spheroid. The existence and stability of collinear equilibrium points in, the planar elliptical restricted three body problem under the eect of the photogravitational and oblateness primaries have been discussed by C. Ramesh kumar and A. Narayan in (2012). In (2015) Arif. Mohd. have discussed the equilibrium points and there stability in the magnetic binaries problem when the bigger primary is a source of radiation. In this article we have discussed the equilibrium points and there stability in the photogravitational magnetic binaries problem when the bigger primary is a oblate spheroid and source of radiation and the small primary is a oblate body.
II. Equation of motion Two oblate bodies (the primaries), in which the bigger primary is a source of radiation with magnetic fields move under the influence of gravitational force and a charged particle P of charge q and mass m moves in the vicinity of these bodies. The equation of motion and the integral of relative energy in the rotating coordinate system including the effect of the gravitational forces of the primaries on the charged particle P fig(1) written as:
z
đ?‘€1 đ?‘&#x;1 đ?‘&#x;
đ?‘€2 đ?‘&#x;2
y
Fig(1)
đ?‘Ľ − đ?‘Ś Ć’= đ?‘ˆđ?‘Ľ đ?‘Ś + đ?‘Ľ Ć’= đ?‘ˆđ?‘Ś đ?‘Ľ 2 + đ?‘Ś 2 = 2U − C
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(1) (2) (3)
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Existence and Stability of the Equilibrium Points in the‌ Where đ?‘ž Ć’ =2 đ?œ” – ( 31 + đ?‘ˆ=
đ?œ”2
đ?‘ž 1 đ??ź1
+
2(1âˆ’Âľ)r 51 đ?œ” đ?‘ž1 2
r1
2
đ?œ† đ?‘&#x;23
r1
+
(1âˆ’Âľ) đ?‘ž 1 đ??ź1 Âľ 2r 31
+
r2
đ?œ†đ??ź2 2Âľr 52
2
(đ?‘Ľ + đ?‘Ś ) +
2 đ?‘ž 1 1âˆ’Âľ
+
) ,đ?‘ˆđ?‘Ľ =
đ?‘Ľ + đ?‘Ś +
2
đ?œ•đ?‘ˆ
and đ?‘ˆđ?‘Ś =
đ?œ•đ?‘Ľ
1âˆ’Âľ
− Âľđ?‘Ľ
+
r 31
đ??ź2
đ?œ•đ?‘ˆ
đ?œ•đ?‘Ś đ??ź1
2r 51
+
đ?œ”đ?œ† Âľ
đ?‘Ľ 2 + đ?‘Ś 2 + (1 − Âľ)đ?‘Ľ
Âľ
+
r 32
đ??ź2 2r 52
+
(4)
2r 32
đ??źđ?‘– , (đ?‘– = 1,2) The moments of inertia of the primaries given as the difference of the axial and equatorial moments of inertia. đ?‘ž1 is the source of radiation of bigger primary. 3đ??ź1 3đ??ź2 đ?œ” =1+ + 2(1 − Âľ) 2Âľ Here we assumed 1. Primaries participate in the circular motion around their centre of mass 2. Position vector of P at any time t be đ?‘&#x; = (đ?‘Ľđ?‘– + đ?‘Śđ?‘— + đ?‘§đ?‘˜) referred to a rotating frame of reference O(đ?‘Ľ, đ?‘Ś, đ?‘§) which is rotating with the same angular velocity đ?œ” = (0, 0, đ?œ”) as those the primaries. 3. Initially the primaries lie on the đ?‘Ľ-axis. 4. The distance between the primaries as the unit of distance and the coordinate of one primary is (Âľ, 0, 0) then the other is (Âľâˆ’1, 0, 0). 5. The sum of their masses as the unit of mass. If mass of the one primaries Âľ then the mass of the other is (1âˆ’Âľ). 6. The unit of time in such a way that the gravitational constant G has the value unity and q= mc where c is the velocity of light. đ?‘€ đ?‘&#x;12 = (đ?‘Ľ − Âľ)2 +đ?‘Ś 2 , đ?‘&#x;22 = (đ?‘Ľ + 1 − Âľ)2 + đ?‘Ś 2 , đ?œ† = 2 (đ?‘€1 , đ?‘€2 are the magnetic moments of the primaries đ?‘€1
which lies perpendicular to the plane of the motion). đ?‘ž1 is the source of radiation of bigger primary. Therefore, instead of dealing with the full equations of planar magnetic-binaries problem, it makes more sense to work with a system of equations that describe the motion of the charged particle in the vicinity of the secondary mass, this type of system was derived by Hill in 1878. By making some assumptions and transferring the origin of the coordinate system to the second mass fig(2) the equations of motion (1) and (2), become đ?œ − đ?œˆ đ?‘“1 = đ?‘ˆđ?œ đ?œˆ + đ?œ đ?‘“1 = đ?‘ˆđ?œˆ Where 1 đ?‘“1 =2 đ?œ” – ( 3 + đ?‘ˆ=
đ?œ”2 2
r1
(5) (6) đ??ź1 2(1âˆ’Âľ)r 51
( đ?œ +đ?œ‡âˆ’1
2
+
đ?œ† đ?‘&#x;23 2
+
+đ?œˆ )+
đ?œ†đ??ź2 2Âľr 52 đ?œ” đ?‘ž1
(1âˆ’Âľ)
) , đ?‘ˆđ?œ =
đ?œ•đ?‘ˆ đ?œ•đ?œ
and đ?‘ˆđ?œˆ =
đ?œ +đ?œ‡âˆ’1
2
+ đ?œˆ
2
đ?œ•đ?‘ˆ đ?œ•đ?œˆ 1âˆ’Âľ
âˆ’Âľ đ?œ +đ?œ‡âˆ’1
r 31
+
đ??ź1 2r 51
đ?œˆ2+(1âˆ’Âľ)đ?œ +đ?œ‡âˆ’1Âľr23 + đ??ź22r25+đ?‘ž11âˆ’Âľr1+đ?‘ž1đ??ź12r13+Âľr2+đ??ź22r23
+
đ?œ”đ?œ† Âľ
đ?œ +đ?œ‡âˆ’1
2
(7)
đ?‘&#x;12 = (đ?œ − 1)2 +đ?œˆ 2 , đ?‘&#x;22 = đ?œ 2 + đ?œˆ 2 And new Jacobi constant, Cn , is given by đ?œ 2 + đ?œˆ 2 = 2U − Cn
(8)
đ?‘Ś
P
đ?‘&#x;2
đ?‘&#x;1
đ?œˆ đ?‘Ľ
đ?‘€2
đ?‘€1
đ?œ Fig(2) www.ijceronline.com
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+
Existence and Stability of the Equilibrium Points in the‌ III. Equilibrium points The location of equilibrium points in general is given by đ?‘ˆđ?œ = 0 (9) đ?‘ˆđ?œˆ = 0 (10) The solution of equations (9) and (10) results the equilibrium points on the đ?‘Ľ-axis (đ?œˆ = 0), called the collinear equilibrium points and other are on đ?‘Ľđ?‘Ś-plane (đ?œˆ ≠0) called the non-collinear equilibrium points(ncep). Here we have solved these equations numerically by use the software mathematica. The tables (1----12) shows that for đ?œ† > 0 and for different values of đ?‘ž1 there exist four collinear equilibrium points which we denoted by đ??ż1 , đ??ż2 , đ??ż3 and đ??ż4 but in some values of đ?‘ž1 there exist only two equilibrium points đ??ż1 and đ??ż4 . In fig 3 and fig 4 we give the position of the points đ??ż1 and đ??ż4 respectively for various values of Âľ. In these figs the curve đ?‘ž1 = 1 correspond to the case when radiation pressure is not taken into consideration and other curves corresponds to đ?‘ž1 = .3, đ?‘ž1 = .5 and đ?‘ž1 = .9 . Here we observed that due to the radiation pressure the both points đ??ż1 and đ??ż4 shifted towards the origin. It may be also observed that the deviation of the location of the point đ??ż4 from đ?‘ž1 = 1 decreases as Âľ increases and this becomes zero when Âľ ≈ .35 and again this deviation increases. Âľ = 3.37608 Ă— 10−4 đ?‘ž1 = 1 đ?œ† đ?œ† đ?œ† đ?œ†
=1 =3 =5 =7
Âľ = 3.37608 Ă— 10−4 đ?‘ž1 = .5 đ?œ† đ?œ† đ?œ† đ?œ†
=1 =3 =5 =7
Âľ = 3.37608 Ă— 10−4 đ?‘ž1 = .9 đ?œ† đ?œ† đ?œ† đ?œ†
=1 =3 =5 =7
Âľ = .132679 đ?‘ž1 = 1 đ?œ† đ?œ† đ?œ† đ?œ†
=1 =3 =5 =7
Âľ = .132679 đ?‘ž1 = .5 đ?œ† đ?œ† đ?œ† đ?œ†
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=1 =3 =5 =7
đ??ż1
đ??ż2
đ??ż3
đ??ż4
2.39051 2.71276 3.10013 3.51476 Table (1) đ??ż1 đ??ż2
-
−1.74821 −2.84499 −3.58096 −4.17329
đ??ż3
đ??ż4
-
−1.78185 −2.86665 −3.59742 −4.18691
đ??ż3
đ??ż4
-
−1.75494 −2.84931 −3.580426 −4.17611
đ??ż3
đ??ż4
2.45144 .662328 2.72307 .715733 3.06977 .742964 3.46219 .7617777 Table(4) đ??ż1 đ??ż2
.870220 .869666 .868395 .866981
−1.78160 −2.87033 −3.60486 −4.14701
đ??ż3
đ??ż4
2.19628 .69686 2.49117 .753724 2.88831 .783278 3.33211 .805181 Table(5)
.870099 .867533 .864287 .851632
−1.80221 −2.88535 −3.61684 −4.20709
2.13876 2.49977 2.94143 3.40119 Table(2) đ??ż1 đ??ż2 2.34821 2.67596 3.07147 3.49348 Table(3) đ??ż1 đ??ż2
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Existence and Stability of the Equilibrium Points in the… µ = .132679 𝑞1 = .9 𝜆 𝜆 𝜆 𝜆
=1 =3 =5 =7
µ = .230437 𝑞1 = 1 𝜆 𝜆 𝜆 𝜆
=1 =3 =5 =7
µ = .230437 𝑞1 = .5 𝜆=1 𝜆=3 𝜆=5 𝜆=7
µ = .230437 𝑞1 = .9 𝜆=1 𝜆=3 𝜆=5 𝜆=7
µ = .458505 𝑞1 = 1 𝜆 𝜆 𝜆 𝜆
=1 =3 =5 =7
µ = .458505 𝑞1 = .5 𝜆=1 𝜆=3 𝜆=5 𝜆=7
µ = .458505 𝑞1 = .9 𝜆=1 𝜆=3 𝜆=5 𝜆=7
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𝐿1
𝐿2
2.40893 .667415 2.68390 .721353 3.03785 .748851 3.43826 .767909 Table(6) 𝐿1
𝐿4
.779097 .772205 .760817 -
−1.80690 −2.88902 −3.62218 −4.213880
𝐿3
𝐿4
.775622 .745650 -
−1.81767 −2.89914 −3.63080 −4.22138
𝐿3
𝐿4
.778739 .770714 .755144 -
−1.80904 −2.89104 −3.6239 −4.21538
𝐿3
𝐿4
.585582 -
−1.86992 −2.93591 −3.66579 −4.25648
𝐿3
𝐿4
.552828 -
−1.85712 −2.93435 −3.66641 −4.25784
𝐿3
𝐿4
.583822 -
−1.86739 −2.9356 −3.66591 −4.25675
𝐿2
2.31037 .533495 2.50977 2.80384 3.18660 Table(11) 𝐿1
𝐿3
𝐿2
2.57508 .438243 2.77151 3.03536 3.36433 Table(10) 𝐿1
−1.78571 −2.87334 −3.60726 −4.19903
𝐿2
2.44904 .571296 2.69438 .658573 3.01936 .710819 3.40138 Table(9) 𝐿1
.790781 .869445 .867999 .866363
𝐿2
2.23395 .618412 2.49202 .726268 2.85571 3.71960 Table(8) 𝐿1
𝐿4
𝐿2
2.49180 .562939 2.73493 .649373 3.05355 .697523 3.42741 Table(7) 𝐿1
𝐿3
𝐿2
2.53151 .449693 2.72855 2.99667 3.33327 Table(12)
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Existence and Stability of the Equilibrium Points in the‌ q1  1
q1  .3
q1  .5
q1  .9
q1  1
q1  .3
q1  .5
q1  .9
 
0.4 0.4
0.3 0.3
0.2
0.2
0.1
0.1
2.1
2.2
2.3
2.4
L1
2.5
1.86
1.84
Fig(3)
1.82
1.80
1.78
1.76
L4
Fig(4)
In Figs (5) , (6) and (7) we give the position of the non collinear equilibrium points đ??ż5 and đ??ż6 for Âľ = 3.37608 Ă— 10−4 , Âľ = .132679 and Âľ = .230437 respectively. In these figs black dots denote the position of đ??ż5 and đ??ż6 when đ?‘ž1 = 1 (no radiation) and blue, orange and purple dots denote the position when đ?‘ž1 = .9, đ?‘ž1 = .5 and đ?‘ž1 = .3 respectively Here we observed that both đ??ż5 and đ??ż6 moves towards the small primary from left to right when Âľ = 3.37608 Ă— 10−4 and come up to down when Âľ = .132679 but when Âľ = .230437 this shifting is deferent .
Fig(5)
Fig(6)
Fig(7)
IV. Stability of motion near equilibrium points Let (đ?‘Ľ0 , đ?‘Ś0 ) be the coordinate of any one of the equilibrium point and let đ?œ‰, đ?œ‚ denote small displacement from the equilibrium point. Therefore we have đ?œ‰ = đ?œ − đ?‘Ľ0 , đ?œ‚ = đ?œˆ − đ?‘Ś0 , Put this value of đ?œ and đ?œˆ in equation (5) and (6), we have the variation equation as: 0 0 đ?œ‰ − đ?œ‚ đ?‘“0 = đ?œ‰ đ?‘ˆđ?œ đ?œ + đ?œ‚ đ?‘ˆđ?œ đ?œˆ (11) 0
đ?œ‚ + đ?œ‰ đ?‘“0 = đ?œ‰ đ?‘ˆđ?œ đ?œˆ + đ?œ‚ đ?‘ˆđ?œˆđ?œˆ 0 (12) Retaining only linear terms in đ?œ‰ and đ?œ‚. Here superscript indicates that these partial derivative of đ?‘ˆare to be evaluated at the equilibrium point (đ?‘Ľ0 , đ?‘Ś0 ) . So the characteristic equation at the equilibrium points are 0
0
0
02
đ?œ†1 4 + đ?œ†1 2 đ?‘“ 2 − đ?‘ˆđ?œ đ?œ − đ?‘ˆđ?œ đ?œˆ + đ?‘ˆđ?œ đ?œ đ?‘ˆđ?œˆđ?œˆ 0 − đ?‘ˆđ?œ đ?œˆ =0 (13) The equilibrium point (đ?‘Ľ0 , đ?‘Ś0 ) is said to be stable if all the four roots of equation (13) are either negative real numbers or pure imaginary. In tables 13 to 22 we have given these roots for the collinear equilibrium points and in 23 to 25 for non-collinear equilibrium points.
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Existence and Stability of the Equilibrium Points in the… µ = 3.37608 × 10−4 𝑞1 = 1 𝜆=1 𝜆=3
2.39051 2.71276
±0.240282 −0.75647 ± 0.75567ⅈ
±1.66982𝑖 0.75647 ± 0.75567ⅈ
𝜆=5
3.10013
−1.22735 ± 0.602771ⅈ
1.22735 ± 0.60277ⅈ
µ = .132679 𝑞1 = 1 𝜆=1 𝜆=3
2.45144 2.72307
±0.0172224 −0.88978 ± 0.692155ⅈ
±1.51261𝑖 0.88978 ± 0.692155ⅈ
𝜆=5
3.06977
−1.34709 ± 0.43637ⅈ Table(13)
1.34709 ± 0.43637ⅈ
µ = 3.37608 × 10−4 𝑞1 = .5 𝜆=1 𝜆=3
𝐿1
𝐿1
𝜆1
𝜆1
1,2
1,2
𝜆1
𝜆1
3,4
3,4
2.13876 2.49977
±0.454719 −0.374651 ± 0.756143ⅈ
±1.75661𝑖 0.374651 ± 0.756143ⅈ
𝜆=5 µ = .132679 𝑞1 = .5 𝜆=1 𝜆=3
2.94143
−0.890746 ± 0.78258ⅈ
0.890746 ± 0.78258ⅈ
2.19628 2.49117
±0.436948 −0.350631 ± 0.752211ⅈ
±1.82384𝑖 0.350631 ± 0.752211ⅈ
𝜆=5
2.88831
−0.914426 ± 0.791328ⅈ Table(14)
0.914426 ± 0.791328ⅈ
µ = 3.37608 × 10−4 𝑞1 = .9 𝜆=1 𝜆=3
𝐿1 2.34820 2.67596
±0.290048 −0.68180 ± 0.771888ⅈ
±1.7208𝑖 0.68180 ± 0.771888ⅈ
𝜆=5
3.07147
−1.16170 ± 0.656217ⅈ
1.16170 ± 0.656217ⅈ
µ = .132679 𝑞1 = .9 𝜆=1 𝜆=3
2.40893 2.68390
±0.218523 −0.800298 ± 0.727709ⅈ
±1.62633𝑖 0.800298 ± 0.727709ⅈ
𝜆=5
3.03785
−1.27069 ± 0.535704ⅈ Table(15)
1.27069 ± 0.535704ⅈ
µ = 3.37608 × 10−4 𝑞1 = 1 𝜆=1 𝜆=3
𝐿4 −1.74821 −2.84499
±2.39908 ±3.54145
±0.791718 ±0.94908
𝜆=5
−3.58096
±4.3150
±0.967011
µ = .132679 𝑞1 = 1 𝜆=1 𝜆=3 𝜆=5
−1.78160 −2.87033 −3.60486
±2.49069 ±3.72008 ±4.54733 Table(16)
±0.84689 ±0.95589 ±0.96894
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𝜆1
𝜆1
1,2
1,2
Open Access Journal
𝜆1
𝜆1
3,4
3,4
Page 6
Existence and Stability of the Equilibrium Points in the… µ = 3.37608 × 10−4 𝑞1 = .5 𝜆=1
𝐿4
𝜆1
1,2
𝜆1
3,4
−1.78185
−1.24438 ± 0.30131ⅈ
−1.24438 ± 0.30131ⅈ
𝜆=3 𝜆=5 µ = .132679 𝑞1 = .5 𝜆=1
−2.86665 −3.59792
±2.20970 ±2.82656
±1.14979 ±1.08871
−1.80221
−1.28605 ± 0.293247ⅈ
1.28605 ± 0.293247ⅈ
𝜆=3 𝜆=5
−2.88535 −1.78160
±2.35634 ±2.94064 Table(17)
µ = 3.37608 × 10−4 𝑞1 = .9 𝜆=1 𝜆=3 𝜆=5 µ = .132679 𝑞1 = .9 𝜆=1 𝜆=3 𝜆=5
𝐿4
µ = .132679 𝑞1 = 1 𝜆=1 𝜆=3 𝜆=5 µ = .132679 𝑞1 = .5 𝜆=1 𝜆=3 𝜆=5 µ = .132679 𝑞1 = .9 𝜆=1 𝜆=3 𝜆=5
𝜆1
1,2
±1.13038 ±1.14106
𝜆1
3,4
−1.75494 −2.84931 −3.58426
±2.23837 ±3.32748 ±4.06590
±0.83362 ±0.96769 ±0.979341
−1.78571 −2.87733 −3.60726
±2.32260 ±3.49758 ±4.28720 Fig(18)
±0.88322 ±0.97246𝑖 ±0.97991𝑖
𝐿2
𝜆1
1,2
𝜆1
3,4
0.662328 0.715735 0.742964
±1.71992 ±0.55122𝑖 ±1.45132𝑖
±23.4554𝑖 ±44.8950𝑖 ±63.7128𝑖
0.69686 0.753724 0.783278
±0.470315 ±0.917093𝑖 ±1.02355𝑖
±34.5159𝑖 ±69.0878𝑖 ±99.1660𝑖
.667415 .721353 .748851
±1.46455 ±0.73834𝑖 ±1.45377𝑖 Fig(19)
±24.8856𝑖 ±47.7345𝑖 ±68.0377𝑖
µ = .230427 𝑞1 = 1
𝐿2
𝜆1
𝜆=1 𝜆=3 𝜆=5 µ = .230427 𝑞1 = .5 𝜆=1 𝜆=3 µ = .230427 𝑞1 = .9 𝜆=1 𝜆=3 𝜆=5
.562939 .649373 .697523
±2.51635𝑖 ±2.96704𝑖 ±3.42845𝑖
±12.4741𝑖 ±27.999𝑖 ±43.6595𝑖
.618412 .726268
±1.6177𝑖 ±2.1378𝑖
±18.0079𝑖 ±51.1495𝑖
.571296 .658573 .710819
±2.36742𝑖 ±2.81635𝑖 ±3.32662𝑖
±13.0801𝑖 ±29.6179𝑖 ±48.0887𝑖
1,2
𝜆1
3,4
Table(20)
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Existence and Stability of the Equilibrium Points in the… µ = .132679 𝑞1 = 1 𝜆=1 𝜆=3 𝜆=5 µ = .132679 𝑞1 = .5 𝜆=1 𝜆=3 𝜆=5 µ = .132679 𝑞1 = .9 𝜆=1 𝜆=3 𝜆=5
µ = .230427 𝑞1 = 1 𝜆=1 𝜆=3 𝜆=5 µ = .230427 𝑞1 = .5 𝜆=1 𝜆=3 µ = .230427 𝑞1 = .9 𝜆=1 𝜆=3 𝜆=5 µ = 3.37608 × 10−4 𝜆=1 𝑞1 = 1
𝐿3
𝜆1
𝜆1
1,2
3,4
.870822 .869666 .868395
±5.99808𝑖 ±5.90060𝑖 ±5.79689𝑖
±457.911𝑖 ±448.533𝑖 ±438.453𝑖
.870099 .867533 .864287
±2.93265𝑖 ±2.84998𝑖 ±2.73754𝑖
±457.573𝑖 ±433.943𝑖 ±403.094𝑖
.790781 .869445 .867999
±1.07964𝑖 ±5.28311𝑖 ±5.18223𝑖 Table(21)
±101.901𝑖 ±447.697𝑖 ±435.918𝑖
𝐿3
𝜆1
𝜆1
1,2
3,4
.779097 .772205 .760817
±5.62696𝑖 ±5.25535𝑖 ±4.77456𝑖
±82.8032𝑖 ±79.8394𝑖 ±74.2119𝑖
.775622 .745650
±2.63302𝑖 ±2.29635𝑖
±84.0107𝑖 ±62.1799𝑖
.778739 .770714 .755144
±4.9964𝑖 ±4.6565𝑖 ±4.15761𝑖 Table(22)
±83.4792𝑖 ±79.3113𝑖 ±70.649𝑖
𝐿5 , 𝐿6 𝜉 𝜂 . 435766 , ± .680940
±3.5603
±3.53295ⅈ
𝑞1 = .3
. 953254 ,
± .722112
±2.60299
±1.396864ⅈ
𝑞1 = .5
. 704880,
± .748658
±3.26453
±2.295976ⅈ
𝑞1 = .9
. 46976,
± .695638
±3.69993
±4.721654ⅈ
𝜆1
1,2
𝜆1
3,4
Table(23) µ = .132679 𝜆=1 𝑞1 = 1
𝐿5 , 𝐿6 𝜉 𝜂 1.13455 , ±1.23929
±2.33559
±0.78432ⅈ
𝑞1 = .3
1.06213 ,
± .780120
±2.62362
±1.3502ⅈ
𝑞1 = .5
1.11706,
± .958210
±2.52099
±1.18674ⅈ
𝑞1 = .9
1.14413,
± 1.19237
±2.34015
±0.88217𝑖
𝜆1
1,2
𝜆1
3,4
Table(24)
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Existence and Stability of the Equilibrium Points in the‌ Âľ = .230437 đ?œ†=1 đ?‘ž1 = 1
đ??ż5 , đ??ż6 đ?œ‰ đ?œ‚ 0. 820584 , Âą1.22540
Âą2.76584
Âą1.49003â…ˆ
đ?‘ž1 = .3
0.922591 ,
Âą .770110
Âą3.02968
Âą1.97359â…ˆ
đ?‘ž1 = .5
0.868804,
Âą .930507
Âą2.52099
Âą1.18674â…ˆ
đ?‘ž1 = .9
0.840901,
Âą 1.80790
Âą2.34015
Âą0.88217đ?‘–
đ?œ†1
1,2
đ?œ†1
3,4
Table(25)
V.
Conclusion
In this article we have seen that for đ?œ† > 0 and for different values of đ?‘ž1 and mass parameter Âľ = 3.37608 Ă— 10−4 (mars-Jupiter), Âľ = .132679 (Saturn-Uranus), Âľ = .230437 (Jupiter-Saturn), Âľ = .458505 (UranusNeptune), there exists four or two collinear equilibrium points. Here we observed that due to the radiation pressure the points đ??ż1 and đ??ż4 shifted towards the origin. It may be also observed that the deviation of the location of the point đ??ż4 from đ?‘ž1 = 1 decreases as Âľ increases and this becomes zero when Âľ ≈ .35 and again this deviation increases. We have also seen that the points đ??ż1 , đ??ż2 and đ??ż3 move away from the centre of mass and đ??ż4 moves toward the centre of mass as đ?œ† increases. Here we have also observed that the non-collinear equilibrium points đ??ż5 and đ??ż6 moves towards the small primary from left to right when Âľ = 3.37608 Ă— 10−4 and come up to down when Âľ = .132679 but when Âľ = .230437 this shifting is deferent. We have observed that the points đ??ż1 , đ??ż4 , đ??ż5 and đ??ż6 are unstable while the point đ??ż3 are stable for the given values of Âľ, đ?‘ž1 and đ?œ†. We have also seen that the point đ??ż2 is unstable for Âľ = .132679 and for đ?‘ž1 = 1, đ?‘ž1 = .5 , đ?‘ž1 = .9 when đ?œ† = 1 and for other given values this point đ??ż2 is stable.
References [1] [2] [3] [4] [5]
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
Arif. Mohd. (2010). Motion of a charged particle when the primaries are oblate spheroids international journal of applied math and Mechanics. 6(4), pp.94-106. Arif. Mohd. (2015). A study of the equilibrium points in the photogravitational magnetic binaries problem. International journal of modern sciencees and engineering technology, Volume 2, Issue 11, pp.65-70. Bhatnagar. K.B, J.M.Chawla.(1979) A study of the lagrangian points in the photogravitational restricted three body problem Indian J. pure appl. Math., 10(11): 1443-1451. Bhatnagar. K.B, Z. A.Taqvi, Arif. Mohd.(1993) Motion of a charged particle in the field of two dipoles with their carrier stars. Indian J. pure appl. Math., 24(7&8): 489-502. C. Ramesh Kumar, A. Narayan (2012) Existence and stability of collinear equilibrium points in elliptical restricted three body problem under the effects of photogravitational and oblateness primaries, International ournal of Pure and Applied Mathematics Volume 80 No. 4, 477-494. Chernikov,Yu. A. , (1970). The photogravitational restricted three body problem. Soviet Astronomy AJ. vol.14, No.1, pp.176181. Mavraganis A (1978). Motion of a charged particle in the region of a magnetic-binary system. Astroph. and spa. Sci. 54, pp. 305-313. Mavraganis A (1978). The areas of the motion in the planar magnetic-binary problem. Astroph. and spa. Sci. 57, pp. 3-7. Mavraganis A (1979). Stationary solutions and their stability in the magnetic -binary problem when the primaries are oblate spheroids. Astron Astrophys. 80, pp. 130-133. Mavraganis A and Pangalos. C. A. (1983) Parametric variations of the magnetic -binary problem Indian J. pure appl, Math. 14(3), pp. 297- 306. Perezhogin, A.A.( 1982). The stability of libration points in the restricted photogravitational circular three- body problem. Academy of Science, U.S.S.R. 20, Moscow. Radzievsky, V.V.(1950). The restricted problem of three bodies taking account of light pressure, Astron Zh, vol.27, pp.250. Ragos, O., Zagouras, C. (1988a). The zero velocity surfaces in the photogravitational restricted three body problem. Earth, Moon and Planets, Vol.41, pp. 257-278. Schuerman, D.W. (1980). The restricted three body problem including radiation pressure. Astrophys. J., Vol.238, pp.337. Shankaran et al. (2011),International Journal of Engineering, Science and Technology, Vol. 3, No. 2, pp. 6367.
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