3
Constellation Characteristics and
Orbital Parameters
3.1 Satellite Motion
3.1.1 Historical Context
In 1543, the Polish Canon Nicolas Copernicus wrote a book called On the Revolutions of the
Heavenly Spheres, which for the first time placed the Sun, rather than the Earth, as the centre
of the Universe. According to Copernicus, the Earth and other planets rotated around the Sun
in circular orbits. This was the first significant advancement in astronomy since the Alexan-
drian astronomer Ptolemy in his publication Almagest put forward the geocentric universe
sometime during the period 100–170 AD. Ptolemy theorised that the five known planets at the
time, together with the Sun and Moon, orbited the Earth.
From more than 20 years of observational data obtained by the astronomer Tycho Brahe,
Johannes Kepler discovered a minor discrepancy between the observed position of the planet
Mars and that predicted using Copernicus’ model. Kepler went on to prove that planets orbit
the Sun in elliptical rather than circular orbits. This was summarised in Kepler’s three
planetary laws of motion. The first two of these laws were published in his book New
Astronomy in 1609 and the third law in the book Harmony of the World a decade later in 1619.
Kepler’s three laws are as follows, with their applicability to describe a satellite orbiting
around the Earth highlighted in brackets.
†
First law: the orbit of a planet (satellite) follows an elliptical trajectory, with the Sun
(gravitational centre of the Earth) at one of its foci.
†
Second law: the radius vector joining the planet (satellite) and the Sun (centre of the Earth)
sweeps out equal areas in equal periods of time.
†
Third law: the square of the orbital period of a planet (satellite) is proportional to the cube
of the semi-major axis of the ellipse.
While Kepler’s laws were based on observational records, it was sometime before these
laws would be derived mathematically. In 1687, Sir Isaac Newton published his breakthrough

1
r
2
r
r
ð3:2Þ
where F is the vector force on mass m
1
due to m
2
in the direction from m
1
to m
2
; G ¼ 6.672 £
10
211
Nm/kg
2
is the Universal Gravitational Constant; r is the distance between the two
bodies; r/r is the unit vector from m
1
to m
2
.
The Law of Universal Gravitation states that the force of attraction of any two bodies is
proportional to the product of their masses and inversely proportional to the square of the
distance between them. The solution to the two-body problem together with Newton’s Three
Laws of Motion are used to provide a first approximation of the satellite orbital motion
around the Earth and to prove the validity of Kepler’s three laws.

2
r
dt
2
¼ 2
m
r
r
3
ð3:4Þ
The above equation represents the Law of Conservation of Energy [BAT-71].
Cross multiplying equation (3.4) with r:
r £
d
2
r
dt
2
¼ 2
m
r £
r
r
3
ð3:5Þ
Mobile Satellite Communication Networks84
Since the cross product of any vector with itself is zero, i.e. r £ r ¼ 0, hence:
r £
d
2

dt

¼ 0 ð3:8Þ
Hence,
r £
dr
dt
¼ h ð3:9Þ
where h is a constant vector and is referred to as the orbital areal velocity of the satellite.
Cross multiplying equation (3.4) by h and making use of equation (3.9):
d
2
r
dt
2
£ h ¼ 2
m
r
3
r £ h ¼
m
r
3
r £ r £
dr
dt

ð3:10Þ
By making use of the rule for vector triple product: a £ (b £ c) ¼ (a·c)b 2 (a·b)c, the
rightmost term of equation (3.10) can be expressed as:

dr
dt

¼
m
d
dt
r
r

ð3:12Þ
Comparing (3.10) with (3.12) gives:
d
2
r
dt
2
£ h ¼
m
d
dt
r
r

ð3:13Þ
Integrating (3.13) with respect to t:
dr
dt
£ h ¼
m

m
r 1 rc cos
q
ð3:17Þ
where
q
is the angle between vectors c and r and is referred to as the true anomaly in the
satellite orbital plane.
By expressing:
c ¼
m
e ð3:18Þ
Hence:
r ¼
h
2
=
m
1 1 ecos
q
ð3:19Þ
Equation (3.19) is the general polar equation for a conic section with focus at the origin.
For 0 # e , 1, the equation describes an ellipse and the semi-latus rectum, p, is given by:
p ¼
h
2
m
¼ að1 2 e
2
Þð3:20Þ

1
2
r £
dr
dt
ð3:23Þ
Substituting equation (3.9) into (3.23) gives:
dA
dt
¼
h
2
ð3:24Þ
Mobile Satellite Communication Networks86
Since h is a constant vector, it follows that the satellite sweeps out equal areas in equal
periods of time. This proves Kepler’s second law.
3.1.4 The Orbital Period – Proof of Kepler’s Third Law
From equation (3.20),
h ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m
að1 2 e
2
Þ
q
ð3:25Þ
Constellation Characteristics and Orbital Parameters 87
Figure 3.1 Satellite orbital plane.
Figure 3.2 Area swept by the radius vector per unit time.
At the perigee and apogee,

where b ¼ a(1 2 e
2
)
1/2
is the semi-minor axis.
Equating (3.27) with (3.28) when t is equal to T, it follows that:
T ¼ 2
p
ffiffiffiffi
a
3
m
s
ð3:29Þ
This proves Kepler’s Third Law.
3.1.5 Satellite Velocity
Using the Law of Conservation of Energy in equation (3.4) and taking its dot product with v,
where v is the satellite velocity, gives:
d
2
r
dt
2
·
n
¼ 2
m
r
r
3

dt
¼ 2
dv
dt
·v ð3:32Þ
and
dr
2
dt
¼ 2r·
dr
dt
ð3:33Þ
Substituting (3.32) and (3.33) into (3.31) gives:
1
2
dv
2
dt
¼ 2
m
2r
3
dr
2
dt
ð3:34Þ
Mobile Satellite Communication Networks88
Integrating (3.34) with respect to t:
1

p
¼
1
2
m
að1 2 e
2
Þ
a
2
ð1 2 e
2
Þ
2
m
að1 2 eÞ
¼ 2
m
2a
ð3:36Þ
Hence
1
2
n
2
2
m
r
¼ 2
m

¼
ffiffiffiffiffiffi
m
r
a
ar
p
s
ð3:39Þ
n
a
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m
a
að1 2 eÞ
að1 1 eÞ

s
¼
ffiffiffiffiffiffi
m
r
p
ar
a
s
ð3:40Þ
where r
a

, i and
v
define the orientation of the orbital plane. They are
used to locate the satellite with respect to the rotating Earth. The latter three parameters e, a
and
n
define the orbital geometrical shape and satellite motion; they are used to locate the
satellite in the orbital plane. Figure 3.3 shows the orbital parameters with respect to the
Earth’s equatorial plane. The co-ordinate system is called the geocentric-equatorial co-ordi-
nate system, which is used to locate the satellite with respect to the Earth. In this co-ordinate
system, the centre of the Earth is the origin, O, and the xy-plane coincides with the equatorial
plane. The z-axis coincides with the Earth’s axis of rotation and points in the direction of the
North Pole, while the x-axis points to the direction of the vernal equinox. The points at which
Mobile Satellite Communication Networks90
Figure 3.3 Satellite parameters in the geocentric-equatorial co-ordinate system.
the satellite moves upward and downward through the equatorial plane are called ascending
node and descending node, respectively.
In addition to defining the location of a satellite in space, it is important to determine the
direction at which an Earth station’s antenna should point to the satellite in order to commu-
nicate with it. This direction is defined by the look angles – the elevation and azimuth angles –
in relation to the latitude and the longitude of the Earth station. The following sections discuss
the location of the satellite with respect to the different co-ordinate systems. Note: the
formulation of the satellite location outlined in the following sections assumes that the
Earth is a perfect sphere.
3.2.3 Satellite Location in the Orbital Plane
The location of a satellite in its orbit at any time t is determined by its true anomaly,
q
,as
shown in Figure 3.4. In the figure, the orbit is circumscribed by a circle of radius equal to the
semi-major axis, a, of the orbit. O is the centre of the Earth and is the origin of the co-ordinate

is given by:
d
A ¼ A
t 2 t
0
T
ð3:42Þ
Substituting (3.41) into (3.42):
d
A ¼ A
4
ðt 2 t
0
Þ
2
p
¼ A
M
2
p
ð3:43Þ
From equation (3.43)
M ¼
4
ðt 2 t
0
Þð3:44Þ
From Figure 3.4, it can be seen that the area (CPD) is equal to (a
2
E/2) and area (CDB) is

sinE ð3:48Þ
and
OB ¼ rcos
q
¼ acosE 2 ae ð3:49Þ
Adding QB
2
and OB
2
from equations (3.48) and (3.49) gives:
r ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a
2
ð1 2 ecosEÞ
2
q
ð3:50Þ
and
q
¼ 2tan
21
1 1 e
1 2 e

1=2
tan
E
2
"#

5
ð3:52Þ
3.2.4 Satellite Location with Respect to the Rotating Earth
Given the three orbital parameters,
V
, i, and
v
, as shown in Figure 3.3, the location of the
satellite in the geocentric-equatorial system in terms of the orbital plane –ordinates system is
given by [PRA-86]:
x
f
y
f
z
f
2
6
6
4
3
7
7
5
¼
cos
V
cos
v
2 sin

2cos
V
sini
sinisin
v
sinicos
v
cosi
2
6
6
4
3
7
7
5
x
0
y
0
z
0
2
6
6
4
3
7
7
5

2
6
6
4
3
7
7
5
¼
cos
4
e
T
e

sin
4
e
T
e

0
2sin
4
e
T
e

cos
4

4
e
T
e
at any time t in minutes is given by:
4
e
T
e
¼
a
g;o
1 0:25068447t ðdegreesÞð3:55Þ
where
a
g,o
¼ 99.6909833 1 36000.7689
t
1 0.00038707
t
2
(degrees);
t
¼ (JD 2 2415020)/
36525 (Julian centuries); JD ¼ 2415020 1 (Y 2 1899) £ 365 1 Int[(Y 2 1899)/4] 1 M
m
1
(D
m
2 D) 1 [(h 2 12)]/24.

m
for Julian dates calcula-
tion
Month M
m
January 334 (335 if it is a leap year)
February 306
March 275
April 245
May 214
June 184
July 153
August 122
September 92
October 61
November 31
December 0