First Order Normalization in the Generalized Photogravitational Restricted Three Body Problem with Poynting-Robertson Drag

Abstract

In this paper we have studied non-linear stability of triangular equilibrium points. We have performed first order normalization in the generalized photogravitational restricted three body problem with Poynting-Robertson drag. In this problem we have taken bigger primary as source of radiation and smaller primary is an oblate spheroid. At first we have expanded the Lagrangian function in power series of x and y, where (x, y) are the coordinates of the triangular equilibrium points. Then the relation between the roots of the characteristic equation for the linearised system is obtained. Key Words:First order normalization / Generalised Photogravitational/ RTBP.

Full text

Page 1

FIRST ORDER NORMALISATION IN THE GENERALISED
PHOTOGRAVITATIONAL RESTRICTED THREE BODY PROBLEM WITH
POYNTING-ROBERTSON DRAG

B.S. Kushvah1, J.P. Sharma2 and B. Ishwar3

1 JRF DST Project, 2 Co-PI DST Project, 3 PI. DST Project
University Department of Mathematics,
B.R.A. Bihar University Muzaffarpur – 842001, India
Email: bskush@hotmail.com

In this paper we have studied non-linear stability of triangular equilibrium points. We
have performed first order normalization in the generalized photogravitational
restricted three body problem with Poynting-Robertson drag. In this problem we have
taken bigger primary as source of radiation and smaller primary is an oblate spheroid.
At first we have expanded the Lagrangian function in power series of x and y, where
(x, y) are the co-ordinates of the triangular equilibrium points. Then the relation
between the roots of the characteristic equation for the linearised system is obtained.


Key Words: First order normalization / Generalised Photogravitational/ RTBP.

1. Introduction

The restricted three body problem describes the motion of an infinitesimal
mass moving under the gravitational effect of the two finite masses, called primaries,
which move in circular orbits around their centre of mass on account of their mutual
attraction and the infinitesimal mass not influencing the motion of the primaries. The
classical restricted three body problem is generalized to include the force of radiation
pressure, the Poynting – Robertson effect and oblate ness effect.
J. H. Poynting (1903) considered the effect of the absorption and subsequent
re-emission of sunlight by small isolated particles in the solar system. His work was
later modified by H. P. Robertson (1937) who used a precise relativistic treatments of
the first order in the ratio of he velocity of the particle to that light.
The effect of radiation pressure and P. R. drag in the restricted three body
problem has been studied by Colombo et at. (1966), Chernikov Yu. A. (1970) and
Schuerman (1980) who discussed the position as well as the stability of the

Page 2

Lagrangian equilibrium points when radiation pressure, P-R drag force are included.
Murray C. D. (1994) systemematically discussed the dynamical effect of general drag
In the planer circular restricted three body problem, Liou J. C. et al. (1995) examined
the effect of radiation pressure, P-R drag and solar wind drag in the restricted three
body problem.
Moser’s conditions (1962), Arnold’s theorem (1961) and Liapunov’s theorem
(1956) played a significant role in deciding the nonlinear stability of an equilibrium
point. Applying Arnold’s theorem (1961) Leontovic (1962) examined the nonlinear
stability of triangular points. Moser gave some modifications in Arnold’s theorem.
Then Deprit and Deprit (1967) investigated the nonlinear stability of triangular points
by applying Moser’s modified version osf Arnold’s theorem (1961).
Bhatnagar and Hallan (1983) studied the effect of perturbations on the
nonlinear stability of triangular points. Maciejewski and Gozdziewski (1991)
described the normalization algorithms of Hamiltonian near an equilibrium point.
Further, Bhatnagar examined the nonlinear stability of L4. Niedzielska (1994)
investigated the nonlinear stability of the libration points in the photogravitational
restricted three body problem. Mishra P and Bhola Ishwar (1995) studied the second
order normalization in the generalized restricted problem of three bodies, smaller
primary being an oblate spheroid.
In this paper we discuss on the first order normalization in the generalized
photo gravitational restricted three body problem with Poynting-Robertson drag.
Further we will study non-linear stability of the triangular equilibrium point.
For this we will apply Arnold’s theorem (1961) and follow the procedure as adopted
by Bhatnagar and Hallan (1983), Arnold proved that if :

(i)
(
212211
, pairs allfor 0
κκωω≠+kk
)
of rational integers and
(ii) Determinant D
, where
0
≠
21,ωω
are the basic frequencies for the
linear dynamical system,
()
, det
=
bijD

(
3, 2 , 1,
=
ji
)

() 2, 1,
0
2
=






∂∂
∂
I
=
==
ji
I
H
bij
IjIi
ji


(),2 , 1
0
33
=






∂
∂
==
==
i
I
H
bb
IjIi
i
ii

,

0
33=
b
[] .......
+
2
2
1
2
221
2
12211
+++−=
CIIBI AIIIH
ωω


2

Page 3

is the normalized Hamiltonian with
then an each energy manifold
exist invariant tori of quasi-periodic motion, which divide the manifold, and
consequently the equilibrium is stable. This is valid for a system with two degrees of
freedom, which is the case under consideration. Moser has shown that Arnold’s
theorem is true if the condition (i) of the theorem is replaced by
all pairs (
rational integers such that |
)
2
k

and
in the neighbourhood of equilibrium, there
as the action momenta coordinates,
1I
h
2I
H =
0
2221
≠+ωω
kk
for
1, k
. 4||
2
|
1
≤+ kk

2. First Order Normalization
Equations of motion are

() (
r
)
()()
5
2
2
3
2
3
1
1
2
1
2
31
1
2
r
x
A
r
x
xq
xny
&
nx
& &
Ux
−+
−
−+
−
+−
−=−=
µ
µ
µµ
µµ


()(
[
)
]






−+++
+
r
1
−
ny x
&
y yx
&
x
x
r
W
&
µ
µ
22
1
1
… (1)

()
5
2
2
3
2
3
1
1
2
2
3
1
2
r
yA
r
y
r
yq
ynx
&
nyUy
µ
µ
µ
−−
−
−=+=
& &


()
[]
(






+++++−
µµ
xny
&
y yx
&
x
r
y
r
W
&
2
1
2
1
1
) … (2)
where
()(
Cd
)
,1,
2
3
1,
11
2
2
1
1






−=+=
−−
=
Fg
Fp
qAn
q
µ
W


()()
,1,,
106 . 5
1
2
2
2
2
2
2
2
1
5
yxryxr
as
q
+−+=++=
×
−=
−
µµχ


2
1
21
2
≤
+
=
mm
m
µ
.
Where
are masses of the primaries. The perturbed mean motion of the
primaries is n and force function is
21, mm
()
()
3
2
2
21
1
22
2
2
1
1
2r
A
rr
q
yx
n
U
µ
µ
µ
++
−
++=


()
r












+
−
++
+
µ
µ
2
x
y
arcn
y yx
2
1
x
W tan
1
&&
… (3)
The Lagrangian function of the problem can be written as

3

Page 4

()
(
y x
)
()
()
21
1
22
2
22
1
22
1
rr
q
yx
n
y x
&
ny
&
x
&
L
µ
µ
+
−
+++−++=
&


()
r












+
−
++
+
µ
µ
2
µ
2
x
y
arcn
y yx
&
x
W
r
A
tan
2
1
1
3
2
2
&
+
, … (4)
and the Hamiltonian H is

()()
()
21
1
22
2
22
1
22
1
rr
q
yx
n
y
&
x
&
y
&
px
&
pLH
yx
µ
µ
−
−
−+−+=++−=








+
+−
µ
µ
2
x
y
arcnW
r
A
tan
1
3
2
2
, … (5)
where Px, Py are the momenta coordinates given by
∂
=
nyx
x
&

()






++=
∂
∂
=
++−=
∂
y
r
W
2
nxy
&
y
&
L
p
x
r
W
2
L
p
y
x
2
1
1
2
1
1
&
µ
… (6)
The coordinates of the triangular equilibrium points are


−=
0*
13
µ
()
()











−
−








−
1
µ
+



−
1
−
0
2
2
00
2
2
21
2
222
5
1
1
x
A
xy
A
AnW
xx
δ
µ
δ
µ



()
()
21
2
0
2
2
2
3
0
2
2
2
2
1
0*
2
1
13
2
17
22
−
3
112
1


















−
−




−+





µ
−
µ
−−
−=
y
A
y
A
AnW
yy
δ
δ
µ
µ
δ
µδ


where
31
21
2
0
2
0
,
4
1,
2
qyx
=






−±=−=
δ
δ
δµ
δ
… (7)
We shift the origin to L4 for that, we change

xxx
+→
*
,
The Lagrangian function L becomes
(
({
*
2
()
+++
r
rr
21
2
,
yyy
+→
*
)
)()}()()
{
(
}
2
*
2
*
2
*
22
2
&
1
yyxx
n
yyx
&
(
y
&
xxny
&
x
&
L
+++++−+++=

))()
µ
()





++
+
x
−
++
2
1
++
+
−
µµ
µ
µ
x
yy
arcn
r
y
&
yyxxx
W
Aq
*
***
1
3
2
21
tan
2
1


4

Page 5

… (8)
Where

()()
[
[
[
[
]
(
yxfbyaxr,
1
21
22
1
1
=
]
+++=
−
) {say}

()()() yxfb
]
yaxr,1
2
21
22
1
2
=++−+=
−
−


()()() yxfbyaxr,
3
1
22
2
1
=+++=
−
−



()()
]
() yxfbyaxr,1
4
2/3
22
3
2
=++−+=
−
−

() yxf
ax
by
arc,tan
5
=






+
+


**
and ,
µ
ybxa
=+=

Expanding L in power series of x and y, we get
()




++++−=+++++=
+++++=
y
&
px
&
pLLLHHHHHH
LLLLLL
yx
..........
or
.........................
21043210
43210

… (9)
where
() (
+
) (
q
)
()
[]
21
2
2
21
22
1
22
2
0
11
2
[
−
−
+−++−+=
bababa
n
L
µµ


()
]
a
b
arc
1
nWba
A
2
tan1
23
2
2
2
−+−
−
+µ

()()() (
q
)
()
{
3
2
3
11
2
1
11faafxby axnbx
&
y
&
anL
−+−++++−=
µµ

()
[]
}
() [
q
][]
{
bfb.fybfnWaf
A
2
3
2
3
1131
5
2
2
113
−+−−++−−+µµ
µ

[]
}
()
3131
5
2
2
3
2
fy
&
bx
& +
aWafnW bf
A
++− +µ

()
(
y x
)
()
() (
q
)
[
{
3
1
22
11
222
2
22
2
131
2|
1
22
1
fafxyx
n
y x
&
ny
&
x
&
L
−++++−++=
µ
&

()
[]
()
[
15
]
[]]
ab f nWfaf
A
2
faf
2
31
5
2
2
7
2
2
3
2
2
3
2
2451113
−−−+−−+
µ
µ


5

Page 6

() [
q
]
()
[][
()
[
15
] b 1af
A
2
bafabf xy16612
7
2
2
5
2
5
11
−+−+−+
µ
µµ

[]]
() [
q
][][
3
2
25
2
3
1
25
11
222
3
22
3
1
33122
2
fbffbfyafbf
nW
−+−−+−−µµ

[
15
]]}

+
2
++−+
31
2
31
5
2
27
2
2
245
2
f
y yx x
Wabf nWfbf
A
&&
µ

()
[
y x
]



+−+−
2
3
3
3
22
f abx yabfy ybx xa
&&&&

() (
q
)
[
()()
[]
{
191159151
1
L
5
2
3
7
2
5
1
37
11
3
3
3
−+−−++−−=
afafafafxL
µµ

()()
[]
[]]
] b
bfbafnWfaaf
A
2
2
3
223
31
7
2
3
9
2
2
281 451 105
+−−−+−−+µ

() (
q
)
[
()
[
fafbf
]
bafy
[
x
5
2
2
7
2
5
1
27
11
2
311531513
+−−++−−+µµ

()
[]

−−−+−−+
afabf
nW
3
)
a
bfbaf
A
2
2
3
23
3
1
7
2
2
9
2
2
28151105
µ

() (
q
(
f
2
()()
[
11531513
µ
5
2
27
2
5
1
27
11
2
−+−+
]
+
(
a
−−+
afbffabfxy
A
2
µµ
))
[[]

+
bf
2
3
2
+−−−+−−+
bafbf
nW
3
fba
23
3
23
3
1
7
2
29
2
1681151105

() (
q
)()
[
bfbfbfbfy
5
2
27
2
5
1
37
11
3
9 159 151
+−++−−+µµ

[][]]

−−+−+
afab fnW bfbf
A
2
2
3
23
31
7
2
39
2
2
2845105
µ

()
()
[][]








−++−
+
2
+
2
3
22
33
2
3
2
1
216 28
2 |
1
fyb sfabxyfafx
y
&
bx
&
a
W
(
105
)()
[
()()
[
105
µ
]
{
5
2
2
7
2
4
9
21
5
1
27
1
49
1
4
4
4
91 9011990
1
L
fafafqfafafxL
+−−−+−+−=µ

()()
[]
[]]
] b 1
abfbafnWfafaf
A
2
3
3
34
31
7
2
2
9
2
2
11
2
2
2448 4516301945
−−+−−−+µ

() (
q
(
a
) (
q
µ
)
]
)
[
()()
[
105
µ
nW
afbafab
)
1
fbafy
[
x
A
45145
(
2
10514
µ
7
2
3
9
2
7
1
39
11
3
−−−+−−+
µ
)
[]

−
2
3
23
3
24fa
++−−−−−+
224
3
44
3
1
9
3
11
2
2
4814448
4
[
105
µ
3151945
[
y
2
x
fbafafbafbf
(()
]
5
2
2
2
9
2
5
1
229
11
22
1211210516fbaffbaf
−−+−−+


6

Page 7

()()
[][]
7
2
2
2
9
2
2
2
11
2
2
1511051945
2
fbafbaf
A
+−−−+µ

[
144
]

−−−
abfbafabf
nW
4
3
3
34
3
34
3
1
48144
() (
q
)
[
()()
[
105
µ
] b1afbafab fabfxy4514510514
7
2
39
2
7
1
39
11
3
−−−+−−+µ

()()
[]
[]
2
3
224
3
23
3
44
3
1
9
2
3 11
2
) (
q
2
2414448 48
[
µ
4
)
13151945
2
4
fbafbfbf
nW
bafbaf
A
+−−−−−−+
]
µ
(
[]
5
2
27
2
49
2
5
1
27
1
49
11
9901059901051fbfbffbfbfy
+−++−−+µ


[]
[
]}}
abfabf
nW
4
fbfbf
A
2
3
3
34
3
1
7
2
29
2
411
2
2
24 48 45 630945
+−−+−+µ

()(){
[]
ybx abf axaf
y
&
bx
3 | . 2
a
W
2
3
32
31
8 4832448
+−++−
+
+
&

+

[]
[] }
b
3
3
32
3
2
3
24488483fbf axyaf
+−++−
The second order part
of the Hamiltonian H corresponding to Lagrangian L given
in (8) takes the form
2
H

()
()
GxyFy Exxp ypnppH
yxyx
+++−++=
2222
2
2
1
… (10)
where
(){
()
()
[]
5
2
2
2
5
1
22
1
1221
2
1
fbafbaqE
−−+−−−=µµ


()
[]
}
ab f nWfba
A
2
31
7
2
22
2
23) 1(2
2
15
++−−
µ
… (11)

(){
()
[]
5
2
225
1
22
1
) 1
−
(221
2
1
fabfabqF
−+−−−=µµ


[]
}
abfnWfab
A
2
31
7
2
22
2
2 ) 1
−
(2
2
(
15
++−
µ
… (12)

){()b
) }
3
f
af
(
abfqG1616
5
2
5
11
−+−−=µµ
(
a


)
222
1
7
22
1 15ab nWbfA
−−−+µ
… (13)
3. Perturbed Basic Frequencies
Now we follow the method of Whittaker (1965) to find the canonical
transformation from the phase space (
yx,,
)
yxp,p
in to the phase space product of

7

Page 8

the angle co-ordinates (
in
2
I
, so that the second order part of the Hamiltonian is normalized. In order
to examine the stability of the motion, we consider the following set of linear
equations of the variables x, y :

21,
)
φφ
and the action momenta ()
21, II
and of the first order
2121
1, I
Ex+= 2
Fy+= 2
ny
+
nx
−
n
1
F
0
λ
λ
)
λ
0
=
()
{}
FE +
4
2+ EF
−
2
44
yx
npGy
x
H
∂
p
−
∂
=−
2
λ

xy
npGx
y
H
∂
p
+
∂
=−
2
λ

p
p
H
∂
x
x
x
=
∂
=
2
λ


p
p
H
∂
y
y
y
=
∂
=
2
λ

i.e., … (14)
0
=
AX
where,















=












−−
−
−
λ
=
y
x
p
p
y
x
X
n
n
G
nG
2
E
A and
1
0
2
λ
since
0||0
(
=⇒≠
AX.
i.e.,
()
22
24224
+−+−+++
FEnnGnFE
λ
… (15)
This is the characteristic equation whose discriminant is
(
nFED
++=
4
)
nnGEF
−+−
2422
2

Stability is possible only when
.We have seen, the Lagrangian function L and
the Hamiltonian are the functions of q, A2 and W1. Hence they are affected by
radiation pressure force, oblateness and P–R drag. So we conclude that these forces
affect stability of triangular equilibrium points.
0
>
D

Acknowledgement
We are thankful to D.S.T, Govt. of India, New Delhi for sanctioning a project
DST/MS/140/2K on this topic.


8

Page 9

9
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