Mohd Arif / International Journal of Modern Sciences and Engineering Technology (IJMSET)
ISSN 2349-3755; Available at https://www.ijmset.com
Volume 2, Issue 11, 2015, pp.65-70
A Study of the Equilibrium Points in the Photogravitational Magnetic
Binaries Problem
Mohd Arif
Department of mathematics,
Zakir Husain Delhi college,
New Delhi 110002, India.
Email:[email protected]
Abstract
In this article we have discussed the equilibrium points in the photogravitational magnetic binaries
problem when bigger primary is a source of radiation and have investigated the stability of motion
around these points. It is seen that there are five equilibrium points, three collinear and two non collinear. We have observed that all points are unstable.
Key words: magnetic binaries problem, source of radiation, stability
1. 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). In (2011) Shankaran, J.P.Sharma and B.Ishwar have been generalised the
photogravitational non-planar restricted three body problem by considering the smaller primary as an
oblate spheroid. In this article I have discussed the equilibrium points and there stability in the magnetic
binaries problem when the bigger primary is a source of radiation.
2. EQUATION OF MOTION:
Two 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 (small body) of charge q and
mass m moves in the vicinity of these bodies. The question of the magnetic-binaries problem is to
describe the motion of this particle fig(1). 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 small
body written as:
z
𝑀1
P
𝑟1
𝑟
𝑀2
y
𝑟2
Fig.(1)
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Mohd Arif / International Journal of Modern Sciences and Engineering Technology (IJMSET)
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Volume 2, Issue 11, 2015, pp.65-70
𝑥 − 𝑦 ƒ= 𝑈𝑥
𝑦 + 𝑥 ƒ= 𝑈𝑦
𝑥 2 + 𝑦 2 = 2U − C
where
𝑞
𝜆
𝜕𝑈
𝜕𝑈
ƒ =2 – ( 31 + 3 ) , 𝑈𝑥 =
and 𝑈𝑦 =
r1
𝑟2
𝑈 = (𝑥 2 + 𝑦 2 )
𝜕𝑥
1
2
+
𝑞1
r 31
+
(1)
(2)
(3)
𝜕𝑦
𝜆
𝑟23
𝑞 1µ
+𝑥
r 31
−
𝜆 1−µ
𝑟23
+
𝑞 1 1−µ
r1
+
µ
(4)
r2
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 𝑟 = (𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘) referred to a rotating frame of reference
O(𝑥, 𝑦, 𝑧) which is rotating with the same angular velocity 𝜔 = (0, 0, 1 ) as those the primaries.
3.Initially the primaries lie on the 𝑥-axis.
4.The distance between the primaries as the unit of distance and the coordinate of one primary is (µ, 0, 0)
then the other is (µ−1, 0, 0).
5.The sum of their masses as the unit of mass. If mass of the one primaries µ then the mass of the other is
(1− µ).
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.
𝑀
𝑟12= (𝑥 − µ) 2+𝑦 2 , 𝑟22 = (𝑥 + 1 − µ) 2+ 𝑦 2 , 𝜆 = 2
(𝑀1 , 𝑀2 are the magnetic moments of the
𝑀1
primaries which lies perpendicular to the plane of the motion). 𝑞1 is the source of radiation of bigger
primary.
3. EQUILIBRIUM POINTS:
To find the locations of the equilibrium points, we must solve the following equations
𝑈𝑥 = 0
(5)
𝑈𝑦 = 0
(6)
The solution of equations (5) and (6) results the equilibrium points, three on the 𝑥-axis (𝑦 = 0), called the
collinear equilibrium points and other are on 𝑥𝑦-plane (𝑦 ≠ 0) called the non-collinear equilibrium
points(ncep). For 𝜆 > 0 we found that there exist three collinear equilibrium points, i.e. one within the
interval µ, +∞ , µ, 1 + µ , (µ − 1, −∞), which we denote by 𝐿1 , 𝐿2 , 𝐿3 respectively. In figs 2, 3 and 4
we give the positions of the point 𝐿1 in figs 5,6,7 the position of 𝐿2 and in figs 8,9,10 of 𝐿3 for 𝜆 = 1, 𝜆 =
2 and 𝜆 = 3 respectively for various values of µ. In these figs the line 𝑞1 = 1correspond to the case when
radiation pressure is not taken into consideration . Here we observed that due to radiation pressure the
point 𝐿1 shifted towards the origin whereas the point 𝐿2 and 𝐿3 go away from the origin.
q1 1
q1 .5
q1 .9
1 .8
q1 1
q1 .5
q1 .9
1 .8
1 .6
1 .6
1 .4
1 .4
1 .2
1 .0
1 .2
0 .8
1 .0
0 .6
0 .0 5
0 .1 0
0 .1 5
µ
Fig.(2)
0 .2 0
0 .2 5
0 .3 0
0 .0 5
0 .1 0
0 .1 5
0 .2 0
0 .2 5
0 .3 0
µ
Fig.(3)
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Mohd Arif / International Journal of Modern Sciences and Engineering Technology (IJMSET)
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Volume 2, Issue 11, 2015, pp.65-70
q1 1
q1 .5
q1 .9
q1 1
1 .8
q1 .5
q1 .9
1 .7
1 .6
1 .8
1 .4
1 .9
1 .2
2 .0
1 .0
0 .0 5
{
0 .1 0
q1 1
0 .1 5
0 .2 0
q1 .5
0 .2 5
0 .3 0
0 .0 5
0 .1 0
0 .1 5
0 .2 0
q1 .9
µ
0 .3 0
µ
Fig.(4)
q1 1
0 .2 5
Fig.(5)
q1 .5
q1 .9
q1 1
q1 .5
q1 .9
2 .2
2 .0
2 .3
2 .1
2 .4
2 .2
2 .5
2 .3
0 .0 5
0 .1 0
0 .1 5
0 .2 0
0 .2 5
0 .0 5 0 .1 0 0 .1 5 0 .2 0 0 .2 5 0 .3 0
0 .3 0
µ
µ
Fig.(6)
q1 1
Fig.(7)
q1 .5
q1 .9
q1 1
q1 .5
q1 .9
0 .0 0 6
0 .0 0 4
0 .0 0 2
0 .0 0 2
0 .0 5
0 .1 0
0 .1 5
0 .0 0 4
0 .2 0
0 .2 5
0 .3 0
0 .0 1 0
0 .0 0 8
0 .0 0 6
0 .0 0 4
0 .0 0 2
0 .0 0 2
0 .0 0 6
0 .0 5 0 .1 0 0 .1 5 0 .2 0 0 .2 5 0 .3 0
0 .0 0 4
µ
Fig.(8)
µ
Fig.(9)
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Mohd Arif / International Journal of Modern Sciences and Engineering Technology (IJMSET)
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Volume 2, Issue 11, 2015, pp.65-70
q1 1
q1 .5
q1 .9
2
0 .0 1 5
1
0 .0 1 0
1.0
0.5
0 .0 0 5
0.5
1.0
1.5
2.0
1
0 .0 5 0 .1 0 0 .1 5 0 .2 0 0 .2 5
2
0 .3 0
µ
Fig.(10)
Fig.(11)
In Fig (11) we give the position of the points 𝐿4 and 𝐿5 for various values of µ and 𝜆 = 1. In this fig red
dots denote the position of 𝐿4 and 𝐿5 when 𝑞1 = 1 (no radiation) and green dots when 𝑞1 = .5. Here
we observed that due to the radiation pressure both 𝐿4 and 𝐿5 moves towards the origin, we also
observed that for some values of µ these points shifted towards the small primary.
4. STABILITY OF MOTION NEAR EQUILIBRIUM POINTS:
Let (𝑥 0 , 𝑦0 ) be the coordinate of any one of the equilibrium point and let 𝜉, 𝜂 denote small displacement
from the equilibrium point. Therefore we have
𝜉 = 𝑥 − 𝑥0 ,
𝜂 = 𝑦 − 𝑦0 ,
Put this value of 𝑥 and 𝑦 in equation (1) and (2), we have the variation equation as:
0
𝜉 − 𝜂 𝑓0 = 𝜉 𝑈𝑥𝑥 0 + 𝜂 𝑈𝑥𝑦
(7)
0
0
𝜂 + 𝜉 𝑓0 = 𝜉 𝑈𝑥𝑦 + 𝜂 𝑈𝑦𝑦
(8)
Retaining only linear terms in 𝜉 𝑎𝑛𝑑 𝜂. Here superscript indicates that these partial derivative of 𝑈 are to
be evaluated at the equilibrium point (𝑥 0 , 𝑦0 ) . So the characteristic equation at the equilibrium points
are
0
0
02
𝜆1 4 + 𝜆1 2 𝑓 2 − 𝑈𝑥𝑥 0 − 𝑈𝑥𝑦
+ 𝑈𝑥𝑥 0 𝑈𝑦𝑦 − 𝑈𝑥𝑦
=0
(9)
The equilibrium point (𝑥 0 , 𝑦0 ) is said to be stable if all the four roots of equation (9) are either negative
real numbers or pure imaginary.
From tables 1 ,2, 3,4 and 5 it is clear that all the equilibrium point in these tables are unstable for 𝜆 = 1
and 𝑞1 = .5.
Table (1)
µ
01
𝐿1
.923304
𝜆1 1,2
±1.8044
𝜆1 3,4
±.86334i
.05
.1
.12
.15
.6
0.96328
1.013272
1.033266
1.06326
1.513166
±1.8455
±1.8955
±1.9152
±1.9443
±2.3377
±0.8861i
±.91399 i
±.92476i
±0.9407i
±1.1542i
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Mohd Arif / International Journal of Modern Sciences and Engineering Technology (IJMSET)
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Volume 2, Issue 11, 2015, pp.65-70
Table (2)
µ
.01
𝐿1
−0.0000481
.05
.1
.12
.15
.6
−0.00098
−0.002448
−0.00256
−0.001666
0.18252
𝜆1 1,2
±0.05291
±7547.07
±0.2651
±0.31923
±0.4094
±1.9070
𝜆1 3,4
±985.68i
±754.71i
929.87i
±543.16i
±286.71i
±17.443i
Table (3)
µ
.01
𝐿1
−2.02498
𝜆1 1,2
±2.5789
𝜆1 3,4
±1.8005
.05
.1
.12
.15
.6
−1.9783
−1.9195
−1.8958
−1.86008
−1.2803
±2.5918
±2.6212
±2.6316
±2.6481
±3.0884
±1.7921
±1.7818
±.1.7777
±1.7719
±1.7481
Table (4)
µ
𝐿4,5
𝑥
𝐿4,5
𝑦
𝜆1
1,2
𝜆1
3,4
.01
0.98139
±0.016751
±1.6053
±0.82842i
.05
.1
.12
.15
.6
−0.2363
.21761
−0.09823
1.16761
−0.18099
±0.97848
±.96514
±0.98263
±0.16317
±1.0835
±2.7613
±2.435
±2.7673
±1.4909
±2.1115
±2.0897i
±1.5387i
±2.0814i
±0.6969i
±1.3889i
5. CONCLUSIONS:
In this article we have seen that there exist five equilibrium points, three collinear and two non-collinear
we observed that due to radiation pressure the point 𝐿1 moves toward the origin whereas the point 𝐿2 and
𝐿3 go away from the origin for various values of µ and for 𝜆 = 1, 𝜆 = 2 and 𝜆 = 3. We also observed
that for 𝜆 = 1 the points 𝐿4 and 𝐿5 also shifted towards the origin when solar radiation pressure taken into
consideration. It may also noted that for some values of µ these points 𝐿4 and 𝐿5 shifted towards the
small primary. We have observed that all points given in tables 1,2,3 and 4 are unstable.
6. REFERENCES:
[1].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.
[2].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.
[3].Bhatnagar. K.B, Z. A.Taqvi, Arif. Mohd.(1993) Motion of a charged particle in the field of two dipoles with
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[4].Chernikov,Yu. A. , (1970). The photogravitational restricted three body problem. Soviet Astronomy AJ. vol.14,
No.1, pp.176-181.
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Mohd Arif / International Journal of Modern Sciences and Engineering Technology (IJMSET)
ISSN 2349-3755; Available at https://www.ijmset.com
Volume 2, Issue 11, 2015, pp.65-70
[5].Mavraganis A (1978). Motion of a charged particle in the region of a magnetic -binary system. Astroph. and
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[6].Mavraganis A (1978). The areas of the motion in the planar magnetic-binary problem. Astroph. and spa. Sci.
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[10].Rad zievsky, V.V.(1950). The restricted problem of three bodies taking account of light pressure, Astron Zh,
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[11].Ragos, O., Zagouras, C. (1988a). The zero velocity surfaces in the photogravitational restricted three body
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[12].Schuerman, D.W. (1980). The restricted three body problem including radiation pressure. Astrophys. J.,
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AUTHOR’S BRIEF B IOGRAPHY
Dr. Mohd. Arif: He is an Associate Professor in Department of Mathematics of Zakir
Husain Delhi College (Delh i University) Since 1995. He have Published three book and
several research papers in international journals.
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