Chapter 2.

Orbits and Launching Methods

Satellite Communication
Lecture 3

Chapter 2. Orbits and Launching Methods

2.1 Introduction
2.2 Kepler’s First Law
2.3 Kepler’s Second Law
2.4 Kepler’s Third Law
2.5 Definitions of Terms for Earth-Orbiting Satellites
2.6 Orbital Elements
2.7 Apogee and Perigee Heights 37
2.8 Orbit Perturbations
2.8.1 Effects of a nonspherical earth
2.8.2 Atmospheric drag

 Motion of Space Objects

1571 – 1630 Johannes Kepler

Discovered orbital path to be elliptical around focus
point Keplers 3 laws of planetary motion

1642 – 1727 Sir Isaac Newton

Physical Principals – Universal law of Gravitation

 Origin of planetary laws

Derived 3 laws
based
upon his
observations
Sir. Johannes
Kepler of planetary
motion. 4
Kepler’s Laws (Three laws of Planetary Motion)
LAW 1: Law of orbit
The orbit of a planet about the Sun is an ellipse with the
Sun's center of mass at one focus
LAW 2: Law of areas
A line joining a planet and the Sun sweeps out equal
areas in equal intervals of time
LAW 3: Law of Time period
The squares of the periods of the planets are proportional
to the cubes of their semi-major axes

Kepler’s First Law

LAW 1: The orbit of a planet about the Sun is an

ellipse with the Sun's center of mass at one
focus.

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 1- Law of orbit: The path followed by a satellite around the primary
will be an ellipse.

The eccentricity e is given by

2 2
a −b
e=
a
Major Minor
axis b
axis
F2 F1
Center of
ellipse
b

a a

➢ The eccentricity e and the semimajor axis a are two of the orbital
parameters specified for satellites orbiting the earth.
➢ For an elliptical orbit 0 < e < 1. when e = 0 the orbit becomes circular.
7

Cont….

Earth’s orbit has an eccentricity of 0.017 (nearly circular)

Pluto’s orbit has an eccentricity of 0.248 (the largest in our solar

system)

Satellites also follow Kepler’s 1st Law But Earth can replace
sun at Focus

Kepler’s Second Law

LAW 2: A line joining a planet and the Sun sweeps out equal areas in
equal intervals of time

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 Kepler’s Second Law
The line from the sun to a planet sweeps
out equal areas in equal time intervals.
t1

t2 areaA

t4

aphelion
(slowest) perihelion
(fastest)
areaA = areaB if t2-t1 = t4-t3
areaB

t3
So… Satellites go faster at Perigee than at Apogee
Reason: conservation of specific mechanical energy
10

 Kepler’s Third Law LAW 3:

The period of an orbit depends on the altitude of the orbit
OR
The square of the period is proportional to the cube of its
mean distance from primary focus

3- Law of Time Period: The periodic time of orbit is

proportional to the cube of the mean distance
between the two bodies (which is the semimagor
axis). The orbital period is given in seconds by:

3 2 2π n is the mean motion

a ∝P P is the orbital
Period in sec
P=
n
of the satellite in
radians per second.

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As n is proportional to the orbital period, Kepler’s third low can be written as

3 µ µ Is the earth’s gravitational constant

a = 2
n µ = 3.986005 × 10 14 3
m /s 2

➢ The above equation applies only to the ideal situation of a satellite

orbiting a perfectly spherical earth of uniform mass.

➢ The importance of Kepler’s third law is that it shows there is a fixed

relationship between period and semimajor axis.

➢ One very important orbit in particular, known as the geostationary

orbit, is determined by the rotational period of the earth.

➢ In anticipation of this, the approximate radius of the geostationary

orbit is determined in the following example. 12

Example 2.1

Kepler’s Third Law

 2π P = 24 * 60 * 60
P=
n

Kepler’s Third Law

3 µ
a = 2
n

14
 Apogee
Apogee
Inclination
Inclination
Definitions & orbital parameters:
V

Perigee Right
Perigee Right Ascension
Ascension
Apogee: Farthest from earth

Perigee: Closest approach to earth

Line of apsides: Joining perigee & apogee through center of the earth

Ascending node: Point where the orbit crosses the equatorial plane going from south to north

Descending node: Point where the orbit crosses the equatorial plane going from north to south

Line of nodes: Line joining the ascending and descending node through the center of the earth

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Subsatellite path: The path traced out on the earth's surface

directly below the satellite.

Apogee: The point farthest from earth, ha is the apogee height.

Perigee: The point of closest approach to earth, hp is the apogee height

ha

hp
Dr. Hesham A. Mohamed 16

 Line of apsides: The line joining the perigee and apogee through the
center of the earth.
Ascending node: The point where the orbit crosses the equatorial
plane going from south to north.
Descending node: The point where the orbit crosses the equatorial
plane going from north to south.
Line of nodes: The line joining the ascending and descending nodes
through
the center of the earth (The intersection of the equatorial
plane
Inclination. The and
angle the orbital
between plane) plane and the earth’s equatorial plane
the orbital
It is measured at the ascending node from the equator to the orbit,
going from east to north. The greatest latitude, north or south,
reached by the subsatellite path is equal to the inclination.

Line of apsides

1
17
7

Prograde orbit. An orbit in which the satellite moves in the same

direction as the earth's rotation.

Inclination of a prograde orbit always lies between 0° and 90°.

Retrograde orbit. An orbit in which the satellite moves in a direction counter

to the earth’s rotation (Most satellites are launched in a prograde orbit)

Inclination of a retrograde orbit always lies between 90° and 180°.

Retrograde orbit.
Prograde orbit.
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Argument of perigee:

The angle from ascending node to perigee, measured in the

orbital plane at the earth's center, in the direction of satellite

Motion (0 – 360 degrees). The argument of perigee is shown

as ω in the figure below.

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Orbital Elements
Earth-orbiting artificial satellites are defined by six orbital elements

referred to as the keplerian element set.

• Semi-major axis (a)

▪ Eccentricity (e)

• Mean anomaly (M)

▪ Argument of perigee (w)

▪ Inclination (i)

▪ Right ascension of the ascending Node (Ω)

Equatorial bulge and other perturbing forces causes slow variations

in w and Ω. The values are specified for the reference time or epoch.

These elements values are given in certain epoch and tabulated as

Example
NASA 2.2 Calculate
two-line elementsthe semimajor axis for the satellite parameters given in
Table 2.1. 20
NASA: National Aeronautics and Space Administration.

 Apogee and Perigee Heights

▪ The apogee and perigee heights are two parameters required when defining the satellite orbit
in space.
▪ From the geometry of the ellipse, the length of the radius vectors at apogee and perigee of the
satellite orbit can be obtained as (DRAW THE ELLIPSE IN THE APPENDIX)

ra = a(1+ e)
rp = a(1- e)
ha

In order to find the apogee and perigee heights,

the radius of the earth must be subtracted from
the radii lengths, as shown in the following
example.
hp

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 Orbital Elements
NASA: National Aeronautics and Space Administration.

Example 2.2 Calculate the semimajor axis for the satellite parameters
−
given that n0=0.00104

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𝑺
𝟏
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 Apogee and Perigee Heights
▪ The apogee and perigee heights are two parameters required when defining the satellite orbit in space.
▪ From the geometry of the ellipse, the length of the radius vectors at apogee and perigee of the satellite
orbit can be obtained as (DRAW THE ELLIPSE IN THE APPENDIX)

▪ ra = a(1+e)
▪ rp = a(1- e)
ha

In order to find the apogee and perigee heights, the radius of

the earth must be subtracted from the radii lengths, as shown
in the following example.

■ Apogee Height: hp

❑ ra = a (1 + e)
❑ ha = ra - R
■ Perigee Height:
❑ rp = a (1 – e)
❑ hp =rp – R

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 Apogee and Perigee Heights

▪ The apogee and perigee heights are two parameters required when

defining the satellite orbit in space.

▪ From the geometry of the ellipse, the length of the radius vectors at

apogee, ra, and perigee, rp, of the satellite orbit can be obtained as

ra = a (1+ e)

Where rpthe
ra is the distance from = acenter
(1- e) of the earth to the apogee

point,

rp is the distance from the center of the earth to the perigee

point, and a is the semi-major axis.

❑ The apogee height: is the distance from the Earth’s surface to the apogee.

❑ The perigee height: is the distance from the Earth’s surface to the perigee.

➢ The In order to find the apogee and perigee heights, the radius of the

earth (R) must be subtracted from the radii lengths.

The apogee height (ha) = ra – R

The Perigee height (hp) = ra – R

ha

hp

Apogee and Perigee Heights

ha

■ Apogee Height:
❑ Ra = a (1 + e)
❑ Ha = Ra - R
■ Perigee Height: hp

❑ Rp = a (1 – e)
❑ Hp = Rp – R
■ Given e=0.0011501, a=7192.3 km, and earth radius, R=6371 km.
❑ Ra = 7192.3 (1 + 0.0011501) = 7200.6 km
❑ Ha = 7200.6 – 6371 = 829.6 km
❑ Rp = 7192.3 (1 - 0.0011501) = 7184.1 km
❑ Hp = 7184.1 – 6371 = 813.1 km

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 Orbital Elements
▪ A satellite's position at a specific time can be determined using six

distinct orbit characteristics named the "Keplerian orbit elements".

▪ Orbital elements defines the set of parameters needed to uniquely specify

the location of an orbiting satellite.

These orbital elements are:
1- Semi-major axis (a)

2- Eccentricity (e)

The semi-major axis and the eccentricity give the shape of the ellipse.

3- Mean anomaly(M0): gives the position of a satellite in its orbit at a reference

time known as the epoch.

4- Argument of perigee (w): gives the rotation of the orbit’s perigee point

relative to the orbit’s nodes in the earth’s equatorial plane.

5- Inclination (i)

6- Right ascension of the ascending Node (Ω)

 Orbital Elements
NASA: National Aeronautics and Space Administration.

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