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ARTIFICIAL SATELLITES APPLIED PHYSICS GRAVITATION
ARTIFICIAL SATELLITES
*

EMILIA DANA SELECHI
PhD student - University of Bucharest, Department of Atomic and Nuclear Physics, Mgurele,
P.O. Box, MG-11, RO-077125, Bucharest, ROMANIA, e-mail: ed_seletchi@yahoo.com
Received December 21, 2004

Presenting modelling as a modern method of scientific investigation, by
combining different types of pattern: icons (images) or ideals (logical-mathematical),
this paper presents an overview on satellite system. By using the algorithm as a way
of studying objects and phenomena or as a method of solving practical and theoretical
problems, we intend to describe the optimal transfer problem of a spacecraft between
two given orbits and we have chosen to discuss the conceptual basis founded on
relativity in the Global Positioning System.
Key words: geophysics, orbits, satellites, communication.

1. ORBIT DESCRIPTIONS

Experience with satellite communications has demonstrated that satellite
systems can satisfy many military requirements. They are reliable, survivable,
secure, and a cost effective method of telecommunications. Satellite links are
unaffected by the propagation variations that interfere with hf radio. They are also
free from the high attenuation of wire or cable facilities and are capable of
spanning long distances.

Orbits generally are described according to the physical shape and the angle
of the inclination. The shape of an orbit is established by the initial launch
parameters and the later deployment techniques used.

Angle of inclination is the angle between the equatorial plane of the Earth
and the orbital plane of the satellite. An orbit with an angle of inclination of 90
0
or
near 90
0
is called a polar orbit. The satellite can survey the whole of the Earths
surface, including the poles in a few days. An equatorial orbit has an inclination
angle of
0
0
.

*

Paper presented at the 5th International Balkan Workshop on Applied Physics, 57 July 2004,
Constana, Romania.
Rom. Journ. Phys., Vol. 51, Nos. 12, P. 121130, Bucharest, 2006

Emilia Dana Selechi
2
122

Low Earth Orbit satellites are placed into a circular inclined orbit at an
altitude of roughly 700 to 1600 km above the Earths surface. The advantages of
this type of orbit are very low costs per satellite, modest demands on antenna,
receiver and transmitter performance and very low propagation latency times. The
disadvantages of this model are the limited footprint on the Earths surface, so
these orbits require many satellites to provide global coverage.

A Geosynchronous Orbit is a circular, low inclination orbit about Earth
having a period of 23 hours, 56 minutes, 4.09 seconds. The altitude is about 35 680
km. These orbits allow the satellite to observe almost a full hemisphere of the Earth
and are therefore ideal for TV and Radio Broadcasting and international
communications. These satellites have poor resolution and geosynchronous orbits
require more energy and rocket fuel.
Table 1
Typical satellite orbits
Types of satellite orbits
Altitude
The orbital period of the
satellite
LEO
(Low Earth Orbit)
Below 2 000 km
(1250 miles)
Of 90 min to 2 hours
MEO
(Medium Earth Orbit)
10 000 km
(6250 miles) 6 hours
GEO (Geosynchronous Earth Orbit)
35 680 km
(22 300 miles)
24 hours

The orbits of satellites about a central massive body can be described as
either circular or elliptical. A circular orbit is a special type of elliptical orbit. A
satellite orbiting about the Earth in circular motion is moving with a constant speed
and remains at the same height above the Earths surface. In this particular case
there is no component of force in the direction of motion. Since kinetic energy
depends on the speed of an object, the amount of kinetic energy will be constant
throughout the satellites motion. And since potential energy depends on the height
of an object, the amount of potential energy will be constant throughout the
satellites motion. So if the kinetic energy and the potential energy remain constant
the total mechanical energy of system remains constant.
1.1 CIRCULAR ORBITS
Consider a satellite with mass m orbiting a planet (the Earth).
R = the radius of orbit for the satellite (distance from center of planet) where:

h
R
R
+
=
0

(1)
6
0
6,37 10
R
m
= ,
M
=
24
5,98 10 Kg , G =
2
11
2
6,673 10
Nm
Kg
3
Artificial satellites
123
h = the height above the Earths surface, M = the Earth mass, G = universal
gravitational constant,
0
g
= acceleration of gravity at the Earth surface.
The orbital speed of the satellite is given by the following equation:

0
0
0
0
0
GM
g
v
R
R
h
R
h
=
=
+
+
(2)
The first cosmic speed and the period of the satellite are given by:

0
1
0
0 0
0
7,9
/
g
v
R
g R
Km s
R
=
= ,
0
1
0
2
R
T
g = 1h 25 min (3), (4)
The escape velocity:

2
0 0
1
2
2
11,2
v
g R
v
km s
=
=


(5)
The rotation period of a satellite:

(
)
(
)
3
3
0
0
2
0 0
2
2
R
h
R
h
T
GM
g R
+
+
=
=

(6)
The radius of orbit for the satellite:

2
2
2
0 0
3
3
2
2
4
4
g R T
GMT
R
=
=

(7)
The orbital speed of the satellite:

2
0 0
3
2 g R
v
T
=
(8)
1.2 ELLIPTICAL ORBITS
A satellite orbiting the Earth in elliptical motion will experience a component
of force in the same or the opposite direction as its motion. The speed of a satellite
in elliptical motion is constantly changing, increasing as it moves closer to the
Earth and decreasing as it moves further from the Earth. The speed of the satellite
is greatest at perigee and least at apogee, so the satellite moves from A to B thus it
loses kinetic energy and gains potential energy. As the satellite moves from B back
to A, it gains speed and loses height subsequently there is a gain in kinetic energy
and a loss of potential energy. Yet throughout the entire elliptical trajectory, the
total mechanical remains constant.
Emilia Dana Selechi
4
124
A
y
vr B
If a satellite is moving on an elliptical trajectory, (the total mechanical energy E < 0
and the eccentricity e <1, we can write:

(
)
2
2
2
2
1
2
1
1
2
e
x
x
pex
p +
+
=


2
1
2
2
2
2
2
2
2
1
1
1
1
pe
x
x
e
p
p
e
e
+
+
=

(9)














Fig. 2 The elliptic orbit of the satellite.


2
1
p
a
e
(10)

2
1
p
b
e
=
(11)


2
2
2
1
pe
c
ea
a
b
e
=
=
=

(12)

1
(1
)
1
p
r
a c
e a
e
= =
= +
(13) 5
Artificial satellites
125

(
)
2
1
1
p
r
a c
e a
e
= + = +
=
(14)
p = the parameter of the ellipse, e = the eccentricity
The equation of the ellipse can be written as:

( ) 1 cos
p
r
e
= +

(15)
Considering e = 0
2
2
2
1
2
x
x
r +
=
we obtain the equation of the circle.

1
0
1
p
r
e
= = +
(16)

2
1
p
r
e
= =


(17)

2
1
1
2
r
r
e
r
r = +
(18)

1 2
1
2
2r r
p
r
r
= +
(19)
The area A of an ellipse may be found by making the change of coordinates
'
b
x
x
a


and
'
y
y from the elliptical region R to the new region
'
R
. The
equation of the ellipse becomes:

2
'2
'
2
2
1
1
a
y
x
a
b
b + =

(20)
'2
'2
2
x
y
b
+
=
, so
'
R
is a circle o radius b.

1
1
'
'
x
x
b
a
x
a
b
x

=
=
=



(21)
The Jacobian is:
( )
( )
'
'
'
'
'
'
,
0
,
0 1
x
y
a
x y
a
x
x
b
x
y
b
x y
y
y =
=
=
(22)
Emilia Dana Selechi
6
126

( )
( )
( )
'
'
'
'
2
' '
,
R
R
R
x y
a
a
A
dx dy
dx dy
dx dy
b
ab
b
b
x y =
=
=
=
=

(23)
The equation for momentum conservation is given by:

1 1
2 2
m v r
m v r
=

(24)
The equation for energy conservation has the form:

2
2
1
2
1
2
1
1
2
2
G M m
G M m
m v
m v
r
r
+
=
+

(25)

(
)
2
2
1
1 1
2
1
1
2
2
2
(
)
GMr
r
v
r r
r
r r
r
=
=
+
+
µ

(26)

(
)
(
)
1
1
2
2
1
2
2
1
2
2
2
GMr
r
v
r r
r
r r
r
=
=
+
+
µ

(27)
where
1
v
is the speed at the perigee,
2
v
is the speed at the apogee and
µ
represents
the gravitational parameter of the central body.

G M µ

(28)
The speed for a satellite on an elliptical motion is then given by [5]:

(
) (
)
0 0
0
0
2
2
2
0
0
0
2
sin
g R h
v
R
R
h
R
h
R +
+

(29)
2. A STU