Cataract Detection

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649 Copyright © 2011-15. Vandana Publications. All Rights Reserved.

Volume-5, Issue-3, June-2015

International Journal of Engineering and Management Research

Page Number: 649-661

A Study of Resonance in a Geocentric Satellite Including its Latitude and

Second Order Tesseral Harmonic Due to Earth’s Equatorial Ellipticity

Rajiv Aggarwal1,Sushil Yadav2, Bhavneet Kaur3

1

Sri Aurobindo College, Department of Mathematics, University of Delhi, New Delhi, INDIA

2

Maharaja Agrasen College, Department of Mathematics, University of Delhi, Delhi, INDIA

3

Lady Shri Ram College, Department of Mathematics, University of Delhi, New Delhi, INDIA

ABSTRACT

The present paper deals with the study of resonance in a geocentric satellite including its latitude and the earth’s equatorial ellipticity. We have studied the resonance between angular velocity of the satellite and earth’s equatorial ellipticity parameter

Γ

(angle between the projection of satellite in the plane of equator and the minor axis of the earth’s equatorial section). The object of this study is to determine amplitude and time period of the resonance oscillation by using a method given in Brown and Shook (1933). We have shown the effect of earth’s equatorial ellipticity parameter

Γ

on amplitude and time period including

φ

(latitude of the satellite). We observed that for increasing values of the earth’s equatorial ellipticity parameter

Γ

,

the corresponding amplitude decreases and time period also decreases.

Keywords--- Geocentric satellite. Earth’s equatorial ellipticity. Resonance

I. INTRODUCTION

Resonance plays a fundamental role in any dynamical system. It can be the source of both instability and long-term stability. Resonance occurs between two satellites or planets if there is a repetition in the geometric configuration of their position with respect to their orbits within a small period of time. This happens when the ratio of the orbital period is close to small integers. Resonance is common in the satellite system due to the effect of tides. Several authors have worked on the resonance problems in the solar system but a very less attention is given on the second order tesseral harmonic due to earth’s equatorial ellipticity.

Mohamad Radwan (2002) studied the resonance resulting from the commensurability between the mean motion of the satellite, the Moon and the Sun. A conditionally periodic solution is constructed for the motion of an Earth satellite taking into consideration the Oblateness of the Earth and the Luni-Solar attraction.

Callegari Jr. et al. (2004) analyzed the model of the dynamics of two planets near a first-order mean-motion resonance in the domain of the general three body problem. They studied the phase space of the resonance and near-resonance regions in Uranus-Neptune (2/1 resonance) system by means of surfaces of section and spectral analysis techniques.

Voyatzis et al. (2005) studied symmetric and nonsymmetrical periodic orbits in the exterior mean motion resonances 1:2, 1:3 and 1:4 with Neptune in the framework of the planar circular restricted three-body problem. They observed that in each resonance there exist two branches of symmetric elliptic periodic orbits with stable and unstable segments.

Vrbik, Jan (2013) analyzed the motion of a test particle of a planar, circular, restricted three body problem in resonance, using the Kustaanheimo-Stiefel formalism. Using 7/4 resonance as an example, he showed that a good qualitative description of the motion can be reduced to three simple equations for semi-major axis, eccentricity and resonance angle, which reveals the onset of chaos and sheds a new light on its weak nature.

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gravitation body. They showed that inclusion of the fourth

body gravitation leads the (3:1) as well as the (4:1) resonance orbits to chaos.

In our paper (2013), we have studied the resonance in a geocentric satellite due to earth’s equatorial ellipticity. We have not taken into account the latitude of the satellite. We have also given a brief description of the method of Brown and Shook (1933) which we have used to investigate the resonance In the present work, we have

investigated resonance in a geocentric satellite including its latitude and second order tesseral harmonic

J

₂(2) due to earth’s equatorial ellipticity. We have used a method given in Brown and Shook (1933) to determine the amplitude and time period of the resonance oscillation. In this study, the gravitational attraction of the sun and the moon, the sun’s radiation pressure and residual drag effects are neglected.

II. EQUATIONS OF MOTION

The equations of motion of the geocentric satellite

P r

(

, ,

θ φ

)

moving around the earth

E

in the equatorial plane are given by (Frick and Garber (1962))

(

2 2 2

)

cos

,

(1)

s

U

M

r

θ

φ

∂

−

=

∂





(

2

)

1

1 cos

,

(2)

cos

s

d

U

M

r

φ

dt

θ

φ

r

φ θ





₌

∂





_∂







1 ( )

2 2

cos sin

1 ,

(3)

cos

s

d

U

M

r

r dt

φ

θ

φ

r

φ θ

∂



₊



₌





_∂







where

2 2 2 (2) 2

2

0 0 2 0 2 0

2 2

3sin

1

3 cos

cos 2

,

(4)

2 g R

J R

J

R

U

r

φ

_φ





−





=

_

−

_

_

+

Γ

_









0

earth's gravitational potential,

gravitational aceleration on earth's surface,

radial distance between satellite and the centre of the earth,

U

g

r

=

mass of satellite,

s

M

=

2 0 (2) 2

coefficient due to earth oblateness,

radius of the earth,

coefficient due to the earth's equatorial ellipticity,

J

R

J

=

latitude of the satellite , (Fig.1)

longitude of the satellite , (Fig.1)

G

PEM

X EF

φ

θ

= ∠

=

(3)

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angle between the projection of satellite in the plane o

minor axis of the earth's equ

f equator an

atorial section, (Fig.2)

d

E

MEF

θ θ

Γ = ∠

= −

=

angular position of the minor axis of the earth's equatorial section,

angular rate of earth's rotation,

E G

E

X EF

θ

= ∠

=



,

inertial coordinate system with origin at the centre of the earth

and

plane in the earth's equatorial plane,

G G G

G G

X

Y Z

X Y

=

Substituting value of U from (4) in Equations (1), (2) and (3), we obtain

(

)

2 4

2 2 2 0 0 2 0 0 2

2 4

(2) 4 2 2 0 0

4

cos

3 3sin

1

2

9 cos

cos 2 ,

(5)

g R

J g R

r

J

g R

r

θ

φ

−

= −

+

−

Γ





(

)

(2) 4

2 2 0 0

4

1 cos

6 cos sin 2 ,

(6)

cos

J

g R

d

r

φ

dt

θ

φ

= −

r

φ

Γ



( )

4

2 2 2 0 0

4 (2) 4 2 0 0

4

1 cos sin

3 sin cos

2

6 sin cos cos 2 .

(7)

J g R

d

r

r dt

r

J

g R

r

φ

θ

φ

+

= −

−

Γ



G

X

E

(

, ,

)

P r

θ φ

r

M

F

φ

θ

Γ

E

θ

G

Y

G

Z

Fig.1 Configuration of the satellite

P

Minor Axis

Major

Axis

Γ

E

East

E

θ



Satellite

(4)

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For the unperturbed system, we have

( )2

2

0,

2

0. J

=

J

=

Equations (5), (6) and (7) become

2

2 2 2 0 0

2

cos

g R

,

(8)

r

θ

φ

−



−



= −



(

)

2

1 cos

0,

(9)

cos

d

r

φ

dt

θ

φ

=



( )

2 2

1 cos sin

0. (10)

d

r

r dt

φ

+

θ

φ

=



Using (9), we have

r

2

θ



cos

φ =

constant

=

h

(say).

(11)

Substituting the values

r

1 u

=

, ₂

cos

h

du

r

d

φ θ

= −



, 2 2 4 2

cos

hu

d u

r

d

φ θ

= −



and using (11) in Equation (5), we get

(

)

(

)

2 2 2

2 7

2 2 2

2 2 4 4 2 (2) 4 4 2

0 0 2 0 0 2 0 0

cos

3 3sin

1

9 cos

cos 2

,

2 d u

u

d

g R u

J g R u

J

g R u

φ

θ

φ

+

= −

−

×



₋

₊

_{− −}

_Γ













or

(

)

(

)

2 2 2

2 2 7 2 2

2 4 (2) 4

2 2

0 0 2 0 0 2 0 0

2 4 4

1

1 cos

3 3sin

1

9 cos

cos 2

.

2 d u

u

d

r

g R

J g R

J

g R

r

φ

θ

φ

+

= −

−

×





−

+

− −

Γ











Replacing r,

θ

 by their steady-state values

r

₀

,

θ



₀ , we get

(

)

(

)

2 2

2

2 2 7 2 2

0 0 0 0

2 4 (2) 4

2 2

0 0 2 0 0 2 0 0

2 4 4

0 0 0

1

1 cos

3 3sin

1

9 cos

cos 2

. (12)

2 d u

u

d

r

g R

J g R

J

g R

r

φ

θ

φ

+

= −

−

×





−

+

− −

Γ











We may take

0

(say),

(say).

t

nt

t

mt

θ θ

=

Γ = Γ =



(5)

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Substituting these values in Equation (12), we get

(

)

(

)

2 2

2 7 2

0 0

2 4 (2) 4

2 2

0 0 2 0 0 2 0 0

2 4 4

0 0 0

1 cos

3 3sin

1

9 cos

cos 2

.

2 d u

n

u

dt

r

g R

J g R

J

g R

t

r

φ

+

= −

−

×





−

+

− −

Γ











(13)

Since we are investigating the resonance in the satellite due to earth’s equatorial ellipticity, we are ignoring the secular terms

and terms due to the oblateness

J

₂ of the earth, we, therefore, write

(

)

(2) 4

2

2 2 2 0 0 2

2 6

0

cos

9 J

g R

cos

cos 2

.

(14)

d u

n

u

mt

dt

+

φ

=

r

φ

The solution of Equation (14) is

(2) 4 2 2 0 0

6 2 2 2

0

1 cos(

) 9

cos

cos 2

.

(15)

4 cos

J

g R

u

A

nt

B

mt

r

φ

m

n

φ





=

−

+

_

_

−

+





where

A

and

B

are constants.

We know that if any one of the denominator vanishes, the motion is indeterminate at that point. The resonance occurs at the

point where

n

cos

φ

=

2 m

.

III. RESONANCE AT THE POINT WHERE

n

cos

φ

=

2 m

To study the motion of satellite at the point where

n

cos

φ

=

2 m

, we will follow the procedure as given in

Brown and Shook

_(1933).

It is proposed to find the solution of Equation (14) when that of

(

)

2

2 2

2

cos

0,

(16)

d u

n

u

dt

+

φ

=

is periodic and is known.

The solution of Equation (16) is

cos ,

u

=

c

l

where

(

)

1

cos

,

function of ,

(17)

cos

c

l

n

t

n

c

φ

ε

φ

′

=

+

=

(6)

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(

)

( )

2

2 2

1

2

cos

cos 2

say , ,

d u

n

u

NA

mt

N

dt

+

φ

=

ψ

′

where

(2) 4

2 2 0 0

1 6

0

9 J

g R

and

cos

.

N

A

r

φ

=

1

cos 2

,

1

cos 2

,

A

mt

uA

mt

u

ψ

′ =

∂

=

ψ

=

∂

(

)

(

)

{

}

1

cos 2

.

(18)

2 c

A

mt

l

mt

l

ψ

=

+ +

−

Thus, we get

,

(19)

dc

N u

N

dt

K

l

K

l

ψ

∂

_′

∂

=

∂

(

cos

)

(

cos

)

,

(20)

dl

N u

N

n

dt

K

c

K

c

ψ

φ

∂

ψ

′

φ

∂

=

−

=

−

∂

where

(

cos

)

.

(

cos

)

22

.

,

(21)

a function of only.

u

K

n

c

l

c

φ

∂



∂



∂

=

_

_

−

∂



∂



∂

=

Since n K, are functions of

c

only, we can put (19) and (20) into canonical form with new variables c B₁, defined by

1 , (22)

dc =Kdc

(

cos

)

1

(

cos

)

.

(23)

dB

= −

n

φ

dc

= −

n

φ

Kdc

Equations (24) and (25) can be put in the form

(

)

(

)

1

,

.

dc

B

M

dt

l

dl

B

M

dt

c

ψ

∂

=

+

∂

= −

+

∂

Differentiating Equation (20) with respect to t and substituting the expressions for

dl

dt

and

dc

(7)

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(

)

₍

₎

2 2 2

2

2 2

2

cos

1 .

(24)

n

d l

N

n

dt

K

c

l

l c

c t

N

K

l c

c

c K

c

l

φ ψ

_φ

ψ

ψ ψ

ψ

∂



∂



=

_

−

_

∂

∂ ∂







∂



∂



∂



+

_

−

_

_

_

∂ ∂ ∂

∂



∂



∂





Since the last

expression of Equation (24) has the factor

N

2_{, it may, in general, be neglected in a first approximation.}

In Equation (18), we find that l andt are present in

ψ

as the sum of the periodic terms with argument l′ = −l 2mt.

The affected term, in our case, is

1

cos . (25)

2c A l

ψ

= ′

Equation (24) for

l

′

is then

(

)

(

)

₍

₎

2

2 2

1 cos

2

0. cos

2 d l

N

n

m

dt

K

c

n

m

l

ψ

φ









′

₊

₋



∂



₌







_∂

_

₋

_∂

_′

_















or

(

)

(

)

₍

₎

2

2 ₁

2 cos 2 ₂ _cos ₂ sin 0. (26)

cA

d l N

n m l

dt

φ

K c n

φ

m

  

′₋ ₋ ∂  _′₌

 

_∂ _ ₋ _

  

 

As a first approximation, we put

0, cos 0cos 0, 0(all constants).

c=c n

φ

=n

φ

K =K

Then Equation (26) can be written as

(

)

2

2 ₁

0 0

2

0 ₀

cos

2 sin

0. (27)

2 cos

2 cA

d l

N

n

m

l

dt

φ

K

c n

φ

m





′

₋

∂





_′

₌



_∂



₋













If the oscillation be small, Equation (27) can be put in the form

(

)

2

2 ₁

0 0 2

0 0

cos 2 0.

2 cos 2

cA

d l N

n m l

dt

φ

K c n

φ

m

 

′₋ ₋ ∂   _′₌

_∂  ₋ 

 

 

or

2 2

2 0 , (28)

d l p l dt

′ _′

+ =

(8)

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(

)

2 1

0

0 0

(2) 4

1 2

2 0 0 0

6 2

0 0

0

cos

,

2

9 cos

,

2 n

NA

p

c

K

c

J

g R

c

r

K

c

φ

∂





=

_

_

∂









=



_

−



_





or

( )2 2

0 2 0 1

0 3

0 0 0

3

cos , (29)

2

g J R c

p

K c

r

φ

=

and

( )

0 0

2 2

0

,

cos cos ,

K K

u u u u

n n

c φ l l φ l c

=

∂  ∂ ∂ ∂ ∂ 

=_∂ _ _∂ __∂ − _{∂ ∂} 

 

Substituting

u

=

c

cos ,

l

we get

(

)

(

)

2

(

) (

)

0 2

0

cos

,

K

n

c

l

c

l

n

c

l

c

l

c

φ

l

φ

l

c



∂



∂



∂



=

_

_

_

−

_

∂



∂



∂





Now we substitute

cos

l

=

(

n

φ

)

t

+

ε

′

and using

n

cos

c

1

c

φ

=

, we obtain

(

)

(

)

{

2

}

0 1

0

cos

,

K

=

c

n

φ

t

+

ε

′

(

)

(

)

(

)

(

)

(

)

2 2

1 0 0 0 1 0

2 2

1 0 1 0

cos

2 ,

cos

2 cos

2 .

c

n

t

c

mt

c

t

c

φ

ε

′

=

+

=

+

′

=

Γ +



==

Γ +

The solution of Equation (28) is given by

(

0

)

sin

,

(30)

l

′ =

A

pt

+

λ

where

2

2 0

,

and are constants of integration,

2 . c A

p

c

l l mt

λ =

(9)

The equation for

l

gives

(

0

)

2 sin

(31)

l

= Γ +

A

pt

+

λ

Using Equations (19), (25) and (30), the equation for

c

gives

(

)

1

0 0

0

cos , (32)

2

NA c A

c c pt

K p

λ

 

= + _{ } +

  _where

0 0 0

0 0

is determined from

cos

2 ,

since

is a known function of .

c

n

m

n

c

φ

=

The amplitude

A

and the time period

T

are given by

2

2 ,

.

c

A

T

p

π

=

where

( )2 2 0 2 0

0 3

0 0

3

1 cos

2 cos 2

g J

R

p

r

φ

c

=

Γ

Using Equation (17), we have

1 0

0 0

.

cos

c

n

φ

=

We may choose the constants of integration

1

1, 1 and 0.

2 0

c

=

c

=

ε

′

=

Hence we get

( )

3 0

3

2 2 ₂

0 2 0 0 0

2 cos 2 ,

(33)

3 cos

r

A

g J

R

n

φ

=

Γ

( )

3 0

3

2 2 ₂

0 2 0 0 0

2 2

cos 2 .

(34)

3 cos

r

T

g J

R

n

π

φ

=

Γ

IV. APPLICATION

Using the following data of a satellite, we draw Amplitude

A

versus

Γ

(Fig.3) and time period

T

versus

Γ

(Fig.4).

5 0

6392.1 10 cm,

7822.49 10 cm,

R

r

=

×

=

×

(10)

0

180 0.0009

deg/ sec,

n

π





=

_

×

_





( )2 6

2

2.32 10 ,

J

=

×

−

2 0

0

9.8 m / sec ,

0 .

g

φ

=



We make the above quantities dimensionless by taking

1unit,

E S

M

+

M

=

distance between the earth and the sun

1unit,

ES

D

=

universal gravitational constant=1unit.

G

=

where

27 33

mass of the earth=5.9742 10 gm,

mass of the sun=1.9891 10 gm,

E

S

M

=

×

=

×

13

8 3 1 2

1.4959787061 10 cm,

6.672 10 cm gm sec .

ES

D

G

− − −

=

×

=

×

(11)

2

Fig.3 Amplitude

A

versus

Γ

Fig.4 Time Period T versus

Γ

Effect of amplitude

A

and time period

T

on

φ

(latitude of the satellite) for

0



≤ Γ ≤

45

 is shown in Fig.5 and Fig.6 respectively. It is observed that amplitude

A

increases as

Γ

increases for

0



≤ ≤

φ

90

 (Fig.5) and time period T also increases as

Γ

increases for

0



≤ ≤

φ

90

 (Fig.6).

A

Γ

T

(12)

Fig.5 Effect of amplitude

A

on

φ

(latitude of satellite) for

0



≤ Γ ≤

45



Fig.6 Effect of amplitude T on

φ

(latitude of satellite) for

0



≤ Γ ≤

45



It is observed that the denominator in Equation

(15) vanishes at the point where

V. CONCLUSION

0

cos

2 ,

θ



φ

= Γ



_where

0

θ



is the steady-state orbital angular rate of the satellite,

φ

is latitude of the satellite and

Γ



is the rate of change of the parameter due to earth’s equatorial ellipticity.

Resonance occurs at the point where

θ



₀

cos

φ

= Γ

2 

. Amplitude and time period of resonance oscillation are determined by using the method as given in Brown and

φ

(13)

Shook (1933). Using the data of a satellite, we have shown

the effect of

Γ

and

φ

on amplitude and time period. We conclude that as

Γ

increases from

0

 to

45

, amplitude decreases (Fig.3) and time period also decreases (Fig.4). Finally, we have shown effect of amplitude

A

and time period

T

on

φ

for

0



≤ Γ ≤

45

 in Fig.5 and Fig.6 respectively. It is observed that amplitude

A

increases as

Γ

increases for

0



≤ ≤

φ

90

 (Fig.5) and time period

T

also increases as

Γ

increases for

0



≤ ≤

φ

90

 (Fig.6).

The present study has tremendous applications in the motion of geo-stationary satellites. These satellites are widely used for mass media and telecommunications.

REFERENCES

[1] Brown, E. W. and Shook, C. A. Planetary theory, Cambridge University Press, London and New York (1933). Reprinted by Dover, New York, 1964.

[2] Callegari Jr., N., Michtchenko, T.A. and Ferraz-Mello, S. Dynamics of two planets in the 2/1 mean-motion resonance, Celestial Mechanics and Dynamical Astronomy, 89, (2004), 201–234

[3] Frick, R. H. and Garber, T. B. Perturbations of a Synchronous Satellite, The RAND Corporation, R–399, NASA, (1962).Gallardo, T. Evaluating the signatures of the Mean Motion Resonances in the solar system, Journal of Aerospace Engineering, Sciences and Applications, (2008) vol.1, No.1.

[4] Mohamad Raddwan. Resonance caused by the Luni-Solar attractions on a satellite of the oblate earth, Astrophysics and Space Science, 282, (2002), 551–562 [5] Pourtakdoust, H. and Sayanjali, M. Fourth body gravitation effect on the resonance orbit characteristics of the restricted three-body problem, Nonlinear Dynamics, 76 (2), (2014), 955–972.

[6] Voyatzis, G. and Kotoulas, T. Planar periodic orbits in exterior resonances with Neptune, Planetary and Space Science, 53(11), (2005), 1189–1199