Applied Mathematical Sciences, Vol. 9, 2015, no. 32, 1551 - 1563

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2015.46431

Periodic Solutions for Rotational Motion of

an Axially Symmetric Charged Satellite

Yehia A. Abdel-Aziz

National Research Institute of Astronomy and Geophysics (NRIAG)

Cairo, Egypt, University of Hail, Department of Mathematics

P.O. Box 2440, Kingdom of Saudi Arabia

Muhammad Shoaib

University of Ha’il, Department of Mathematics

P.O. Box 2440, Kingdom of Saudi Arabia

Copyright © 2014 Yehia A. Abdel-Aziz and Muhammad Shoaib. This is an open access article

distributed under the Creative Commons Attribution License, which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The present paper is devoted to the investigation of sufficient conditions for the

existence of periodic solutions in the vicinity of stationary motion of a charged

satellite in elliptic orbit. Lorentz forces which result from the motion of a charged

satellite relative to the magnetic field of the Earth are considered. An axial

symmetry is assumed about the center of mass of the satellite. In addition to the

Lorentz force the perturbation caused by gravitational and magnetic fields of the

Earth are considered. The stationary solutions and periodic orbits close to them are

obtained using the Lyapunov theorem of holomorphic integral. Numerical results

are used to explain the periodic motion of a certain satellite. It is shown that the

charge-to-mass ratio has a significant influence on the periodic motion of satellites.

Keywords: Periodic solutions, Lorentz force, Rotational motion, charged

spacecraft, gravitational force

1. Introduction

When a spacecraft is moving in geomagnetic field in Low Earth Orbit ( LEO ) then due to the accumulation of charging on the surface of the spacecraft the Lorentz

1552 Yehia A. Abdel-Aziz and Muhammad Shoaib

force is generated. A detailed analysis of Lorentz Augmented Orbit (LAO) is

available in [1].The rotational motion of a rigid artificial satellite subject to

gravitational and magnetic torque was treated in detail in [2-6]. Chen and Liu [7]

uses the Poincare map technique to investigate the behavior of periodic and chaotic

motion of the spinning gyrostat satellite. The attitude dynamics of a satellite is

described by the sixth order Euler- Poisson equations. To understand the periodic

behavior of a rigid body Kolosoff’s [8] transformed its equations of motion to a

planar system of equations as was done by Lyapunov in [9]. Abdel-Aziz [10] found

the periodic solution of a spacecraft using an approximated model of the Earth

magnetic field model. Abdel-Aziz and Shoaib [15] studied the effects of Lorentz

force on the attitude dynamics of an electyrostatic spacecraft moving in a circular

orbit and used charge to mass ratio as semi-passive control. In this paper we use a

similar model of Lorentz force but the spacecraft is considered to be in an elliptic

orbit. In [16] Abdel-Aziz and Shoaib devolped the torque due to Lorentz force as a

function of orbital elements and investigated the effects of lorentz force on the

existance and stability of equilibrium positions in Pitch-Roll-Yaw directions.

In the present work we consider a satellite moving in an elliptic orbit under the

influence geomagnetic field, Lorentz force and gravitational field. Using the

isothermal coordinates we reduce the equations of motion of the satellite to the

planar system of equations. The existence of periodic orbits in the neighborhood of

stationary solutions is obtained with the help of the Lyapunov method. The

numerical results are used to show the significance of Lorentz force on the position

and stability of periodic orbits.

2. Equations of motion and their reduction of order

Consider an axially symmetric charged satellite under the action of Lorentz

force, gravitational and geomagnetic fields. Two Cartesian coordinate systems are

introduced. The origin of the coordinate system is taken to be at the center of mass

O of the satellite. The orbital system of coordinate is named as ''' ZYOX and the

coordinate system for the body of satellite is named as 000 ZYOX . In the orbital

system 'OX , 'OY and 'OZ are in the orbital direction, normal to the orbit and

along the radius vectors of the satellite. The attitude motion of satellite is defined by

three Euler angles namely , and . Here is the angle of precession and

is the angle of self-rotation. The angle between the Z -axis of both the coordinate

systems is taken to be . The 0Z -axis is taken to be the axis of symmetry. The

principal moments of inertia of the satellite are taken to be CBA ,, . Due to the

symmetry about 0Z -axis BA = . The unit vectors of the orbital coordinate system

are named �⃗�, 𝛽, �⃗�.

�⃗� = (𝛼1, 𝛼2, 𝛼3), 𝛽 = (𝛽1, 𝛽2, 𝛽3) and �⃗� = (𝛾1, 𝛾2, 𝛾3), (1)

where

𝛼1 = cos𝜓 cos𝜙 − sin𝜓 sin𝜙 cos𝜃,

Periodic solutions for rotational motion 1553

𝛼2 = −cos𝜓 sin𝜙, 𝛼3 = −cos𝜃 sin𝜓 cos𝜙, sin𝜃 sin𝜓), 𝛽1 = sin𝜓 cos𝜑 + cos𝜃 cos𝜓 sin𝜙, 𝛽2 = −sin𝜓 sin𝜙 + cos𝜃 cos𝜓 cos𝜙, 𝛽3 = −sin𝜃 cos𝜓, 𝛾1 = sin𝜃 sin𝜙, 𝛾2 = sin𝜃 cos𝜙, 𝛾3 = cos𝜃.

The Lagrangian of the system can be written as in Yehia [11]

𝑳 =1

2 �⃗⃗⃗�𝑇𝑰�⃗⃗⃗� − 𝑽𝟎, (2)

where MLG VVV=V 0 is the total potential of the forces acting on the

spacecraft. GV , LV ,and MV are the potentials due to the gravitation of the Earth,

Lorentz force and magnetic field respectively, I is the interia matrix of the

spacecraft.

Let �⃗⃗⃗� = (𝑝, 𝑞, 𝑟) and �⃗⃗⃗�0 = (𝑝0, 𝑞0, 𝑟0) are the angular velocities of the

satellite in the inertial and orbital reference frames respectively. As the orbital

system rotate in space with an orbital angular velocity about the axis, which is

perpendicular to the orbital plane. The relation between the angular velocities in the

two systems is given by �⃗⃗⃗� = �⃗⃗⃗�0 + Ω 𝛽. According to Abdel-Aziz [10], we can

write.

(𝑝, 𝑞, 𝑟) = (�̇�sin𝜃sin𝜑 + �̇�cos𝜑 + Ω𝛽1, �̇�sin𝜃cos𝜑 − �̇�sin𝜑 +Ω𝛽2, �̇�cos𝜃 + �̇� + Ω𝛽3) (3)

The gravitation potential of the Earth is [2]

23

=2

T

G V I (4)

2.1 Potential due to Lorentz force

The magnetic field is expressed as [12]

𝐵 =𝐵0

𝑟3 [2cos𝜙 �̂� + sin𝜙�̂� + 0𝜃], (5)

where 0B is the strength of the magnetic field in Wpm. The acceleration in inertial

coordinates is given by

�⃗� =�⃗�

𝑚= −

𝜇

𝑟3𝑟 +

𝑞

𝑚(�⃗⃗�𝑟𝑒𝑙 × �⃗⃗�). (6)

The Lorentz force (per unit mass) can be written as

1554 Yehia A. Abdel-Aziz and Muhammad Shoaib

�⃗�𝐿 =𝑞

𝑚(�⃗⃗�𝑟𝑒𝑙 × �⃗⃗�), (7)

�⃗⃗�𝑟𝑒𝑙 = �⃗⃗� − �⃗⃗⃗�𝑒 × 𝑟, (8)

where, �⃗⃗�is the inertial velocity of the spacecraft, �⃗⃗⃗�𝑒is the angular velocity vector

of the Earth and �⃗⃗�𝑟𝑒𝑙 is velcoity of the spacecraft relative to the geomagnetic field.

In agreement with [12], we have used

�⃗⃗� = �̇��̂� + 𝑟�̇��̂� + 𝑟�̇�sin𝜙 𝜃 (9)

𝑟 = 𝑟 �̂�, (10)

�⃗⃗⃗�𝑒 = 𝜔𝑒�̂�, (11)

.̂sinˆcos=ˆ rz (12)

Therefore, the acceleration due to Lorentz force in inertial coordinates is given by

�⃗�𝐿 =𝑞𝐵0

𝑚 𝑟2 [−(�̇� − 𝜔𝑒)(sin2𝜙 �̂� + sin2𝜙 �̂�) + (�̇�

𝑟sin𝜙 − 2�̇�cos𝜙) 𝜃]. (13)

As stated in [16], the Lorentz acceleration experienced by the geomagnetic field is

decomposed to the three components R, , T N (radial, trnsversal and normal)

respectively as functions of orbital elements as follows:

𝑅 =𝑞𝐵0

𝑚𝑟2 ( 2322 cos 1 cos/)sinsin(1 feipfie ), (14)

𝑇 =𝑞𝐵0

𝑚√𝜇𝑝(

23 cos 1 cos/ feipr

r

fipe sinsin/ 2 23 2cos 1cos fef (15)

𝑁 =𝑞𝐵0

𝑚√𝜇𝑝(

232 22 [1 ] 2 / cos 1 cossin sine i f p i e f

23/ cos 1 cosp i e f +

3

2 2

cos /

1 sin sin

r ip

r i f

3 22 4

3 2 2

sinsin cos1 cos 2 1 cos

1 sin cos

i f fe f e f

p i f

) (16)

Periodic solutions for rotational motion 1555

where, i , , af , and e are the inclination of the orbit on the equator, argument

of the perigee, true anomaly , the semi-major axis and the eccentricity of the

satellite orbit respectively.

The final form of the potential due to Lorentz force can be written as below.

𝑽𝐿 = �⃗�0 𝑨𝑻(𝑅, 𝑇, 𝑁)𝑇 , (17)

where, �⃗�0 = (𝑥0, 𝑦0, 𝑧0)is the radius vector of the spacecraft relative to the center

of mass of the spacecraft, 1 2 3

0 0 0

.

0 0 0

T

A

2.2 Potential of the magnetic field of the Earth

Let a dipole magnetic field �⃗⃗� = (𝐵1, 𝐵2, 𝐵3), and the magnetic moment M⃗⃗⃗⃗ =(0,0, 𝑚3), of the satellite. Therefore the potential of the geomagnetic field is

= T

M M BV (18)

As in Wertz [13] we can write geomagnetic field and the total magnetic moment of

the orbital system directed to the tangent of the orbital plane, normal to the orbit,

and in the direction of the radius respectively as the following:

],cos)(2cos[3sin2

=3

0

3

1 mm

'

m fr

MaB (19)

,cos2

=3

0

3

2

'

mr

MaB (20)

].sin)(2sin[3sin2

=3

0

3

3 mm

'

m fr

MaB (21)

,cos=3

'

mgmm (22)

where, 15

0

3 107.943= Ma , 168.6='

m is co-elevation of the dipole, 𝛼𝑚 =

109. 3∘, and f is the true anomaly measured from ascending node and gm is the

magnitude of the total magnetic moment. Hence, we can write the potential due to

the Earth magnetic field as follows

3 3= .MV m B (23)

Therefore, the final form of the potential of the problem is

2 22

0 3 3 0

3 1= ( ) sin cos ( )sin cos cos [ cos sin ]cos

2 2V C A A m B f z R T N

(24)

1556 Yehia A. Abdel-Aziz and Muhammad Shoaib

It is clear that in addition to the first term of gravitational effects, the main

parameters of the potential of the satellite motion is the charge-to-mass ratio and 0z

.The Lagrangian can be written as below.

.2

1)(

2

1= 0

222 VCrqpAL (25)

It is clear that is a cyclic variable, therefore we can use *== fCrL

(arbitrary

constant).Using equation (24) and the cyclic integral, we can construct the

Routhian as follows

,== 210

* RRRt

fLR

(26)

where

2 22

2 = [ ],sin2

AR

*

1 = sin [ sin cos cos cos ] ,R A A f

2 2 *2

0 3 3 0

3 1= ( ) sin cos ( )sin cos cos [ cos sin ]cos

2 2R C A A m B f z R T N

(27)

2.3 Reduction of the equations of motion

Using Lyapunov method of the holomorphic integral [9], the system of

equation of motion are reduced to another simpler form. Let

,=sin ux (28)

,1)(1

=220 vmv

vd

C

Ay

v

(29)

,= ddt (30)

,arcsin= u (31)

),1

/(arccos=

2*vm

CAv

(32)

.]1

1[=,=

2*

2*

vm

vA

C

CAm

(33)

In the new coordinates [14], the Routhian function can now be written as below.

,))()((2

1= 21

22 Uxyyx

R

(34)

Periodic solutions for rotational motion 1557

where

,]/[1)(1= 1/221/22*

1 vvmu (35)

𝛿2 = (1 − 𝑢2)[Ω √𝐴/ 𝐶 + 𝑓∗

√𝐴 𝐶

(1+𝑚∗ 𝑣2)1/2

(1−𝑣2)1/2 (1−𝑢2)1/2]𝑣/[1 − 𝑣2]1/2 (36)

𝑈 = 𝜇 (−3

2Ω2(𝐶 − 𝐴)

𝑣2

𝐶(1+𝑚∗𝑣2)−

1

2Ω2(1 − 𝑢2 +

1

𝐴[𝑚3𝐵3 − Ω *f ] [1 −

𝑢2]1

2 [1−𝑣2

1+𝑚∗ 𝑣2]

1

2) + 𝑧0 [1 − 𝑢2]1

2 [−𝑅 − 𝑇 𝑣√

𝐴

𝐶

1+𝑚 𝑣2+ 𝑁 [

1−𝑣2

1+𝑚∗ 𝑣2]

1

2 ]). (37)

The equation derived from the Routhian can be written as below. 𝑑

𝑑𝜏(

∂ 𝑅

∂ 𝑥′) −∂ 𝑅

∂ 𝑥= 0,

𝑑

𝑑𝜏(

∂ 𝑅

∂ 𝑦′) −∂ 𝑅

∂ 𝑦= 0. (38)

The above equations are transformed to the following planar system of equations.

∂2𝑥

∂𝜏2 = −Γ ∂𝑦

∂𝜏+

∂ 𝑈

∂ 𝑥,

∂2𝑦

∂𝜏2 = Γ ∂𝑥

∂𝜏+

∂ 𝑈

∂ 𝑦, (39)

Where

Γ =∂

∂ 𝑥(𝛿1) −

∂

∂ 𝑦(𝛿2) =

cos2𝑥

(1−𝑣2)12

(−Ω v1 cos−1 𝑥) − [Ω 𝑚∗ −

(𝑓∗

𝐴) cos−3𝑥] (𝑣2 − 𝑣4)v1 − [Ω (1 − 𝑣2)

1

2v1 + (*f

𝐴) cos−3𝑥 v1

2] (1 + 2𝑣 − 𝑣2),

(40)

v1 = (1 + 𝑚∗ 𝑣2)1

2.

The Jacobi integral of the new system is

.2)()(= 22 Uyx

h

(41)

The region of possible motions can be determined by the inequality

𝑈(𝑥, 𝑦, ℎ, 𝑓∗) ≥ 0.

3. The periodic solutions and numerical results The potential function in isothermal coordinates [9, 14] can be written as below.

𝑈(𝑥, 𝑦) = 𝜇 (ℎ + 𝑤), (42)

𝑤 = 3

2Ω2(𝐶 − 𝐴)

𝑣2(𝑥,𝑦)

𝐶v12 −

Ω2

2(1−𝑢2(𝑥,𝑦))+ 1/A(𝑚3𝐵3 − Ω𝑓∗)(1 −

𝑢2(𝑥, 𝑦))1

2v1−1 (1 − 𝑣2(𝑥, 𝑦))

1

2 + 𝑧0 (1 − 𝑢2(𝑥, 𝑦))1

2 (−𝑅 −

𝑇v1−2𝑣2(𝑥, 𝑦)√

𝐴

𝐶+ 𝑁v1

−1 (1 − 𝑣2(𝑥, 𝑦))1

2). (43)

1558 Yehia A. Abdel-Aziz and Muhammad Shoaib

The stationary solutuons of the new system of as given in Yehia [14] are

= 0, = 0, = 0.U U U

x y z

(44)

Here Γ, 𝑢, 𝑣 are functions of x, and y; U is the force function. The system (44) can

be written as

0.=0,=0,= why

w

x

w

(45)

We assume that *( , ) = ( ), =1,2,3,...i i i iP x y P f i are the stationary solutions and

)(= *fhh ii is the corresponding energy constant. Equations 0=

x

w

and 0=

y

w

will give the values of ix and iy and 0=wh will determine the energy

constant.

To find periodic solutions around the stationary points for some values of *f , let

.=,= 00 yyyxxx (46)

The system of equations given by (40) becomes

0.=);,(2)()(

0,=);,(),(

0,=);,(),(

00

22

00002

2

00002

2

hyyxxUyx

hyyxxy

Uyyyxx

y

hyyxxx

Uyyyxx

x

(47)

With the aid of Lyapunov theorem [9] we find periodic solutions in the following

way.

.=,=,=1=

0

)(

1=

)(

1=

s

s

s

ss

s

ss

s

hchhycyxcx

(48)

Here sc are parameters, sh are constants and

)(=)(),(=)( )()()( TyTxx sss . The period T can also be written as

)(12

=2

=1=0

s

s

s

cTT

(49)

To simplify the system, transform the variable .u

.= 0u (50)

In the same way as in [14] we approximate )(sx and )(sy as follows.

Periodic solutions for rotational motion 1559

).sincos(=

),sincos(=

)(

2

)(

2

1=

2

)(

)(

1

)(

1

1=

1

)(

urburaay

urburaax

r

s

r

s

s

r

s

s

r

s

r

s

s

r

s

s

(51)

For 1=s , equation (46) takes the following form

,=,= (1)

0

(1)

0 ycyyxcxx (52)

The time t and the coordinates satisfy the following relationship.

.))(),((=0

0

dyxtt (53)

Therefore equations (41) reduces to:

0,=

0,=

(1)

002

(1)22

0

(1)

002

(1)22

0

ud

xd

ud

yd

ud

yd

ud

xd

(54)

The zero subscript in the above equations corresponds to the value ,=,= 00 yyxx

.=,=,=

,=,=,=

,=,=,=

00

2

0

002

2

0

002

2

0

yyxxyx

w

yyxxy

w

yyxxx

w

(55)

Equation (54) has the following characteristic equation.

.) 4()(2

1= 222

0

2

00 (56)

Lets consider the two possible cases.

1) 0> 2 ( w has extremum at 0P ). In this case two different frequencies

are possible if the following condition is satisfied.

. 2> 22

0 (57)

The above inequality holds at all points where w has a maximum.

2) 0< 2 ( w has saddle point at 0P ). The only possible frequency in this

case is when the positive sign is taken in equation (56).

1560 Yehia A. Abdel-Aziz and Muhammad Shoaib

In the case of non-commensurable frequencies the values of sh , )(sc , and )(sT

can easily be obtained. For a bounded c the series will absolutely converge. The

following value of (1)(1) , yx and h are obtained from equations (54).

0.=,cossin= ),(= 100

(1)2

0

(1) huuyx (58)

The system of equations given below is the first approximation of the solution of

equation (41).

.=

),cossin(=

,sin)(=

0

000

2

00

hh

uucyy

ucxx

(59)

The periodic solution in equations (59) could be transformed again as a

solution in and . To aid the understanding of the existence of periodic orbits

some numerical examples are given in figures (1-3) for various charge to mass

ratios. In figure (1a), the charge to mass ratio is taken to be 0.001. It can be seen

here that two types of orbits exist in this special case. One type is the usual elliptical

orbits and the other is a figure 8 type of orbit. Figure 8 type of orbits are not very

common. The numerical example in figures (1b, 1c) shows the dependence of the

position of the periodic orbit around the stationary positions on the charge-to-mass

ratio. These figures explains that the Lorentz force have significant influence on the

positions of the periodic solutions.

For different values of charge-to- mass ratio figures (1-3) show the oscillation

of the energy integral h, and the maximum and saddle points of the energy integral

as in equation (69).

Figure 1. The phase plane of at a) 0.001=/mq b) 0.02=/mq c)

0.3=/mq .

Periodic solutions for rotational motion 1561

Figure 2. The maximum and saddle points of the energy integral h at

0.001,0.1=/mq .

Figure 3. The maximum and saddle points of the energy integral h at

0.5=/0.2,=/ mqmq .

4. Conclusions

This paper presented the sufficient conditions for the existence of periodic

solutions close the stationary motion of an axially symmetric charged satellite in an

elliptic orbit. We investigated the rotational motion of the satellite under the action

of the Lorentz forces, gravitational and magnetic fields of the Earth. The Routhian

function of the satellite motion is constructed. The equations of motion of the

satellite are reduced to the planar equations of motion. The existence of periodic solutions are obtained using Lyapunov method of the holomorphic integral. Numerical

1562 Yehia A. Abdel-Aziz and Muhammad Shoaib

results have shown the effects of Lorentz force, in particularly the charge to mass

ratio on the position of periodic solutions and the maximum and saddle points of the

energy integral. In addition to the elliptical orbits some very exotic figure 8 orbits

have also been identified in the vicinity of stationary motion.

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Received: June 15, 2015; Published: February 27, 2015