Satellite-Communication-Systems.ppt

SATELLITE
COMMUNICATION SYSTEMS

Communication satellites bring the world
to you anywhere and any time…..

What exactly is a satellite?
• The word satellite originated from the Latin word “Satellit”- meaning
an attendant, one who is constantly hovering around & attending to a
“master” or big man.
• For our own purposes however a satellite is simply any body that
moves around another (usually much larger) one in a mathematically
predictable path called an orbit.
• A communication satellite is a microwave repeater staion in space that
is used for tele communcation , radio and television signals.
• The first man made satellite with radio transmitter was in 1957.
. There are about 750 satellite in the space, most of them are used for
communication.

How do Satellites Work?
* Two Stations on Earth want to communicate through radio
broadcast but are too far away to use conventional means.
The two stations can use a satellite as a relay station for their
communication.
* One Earth Station transmits the signals to the satellite. Up link
frequency is the frequency at which Ground Station is
communicating with Satellite.
* The satellite Transponder converts the signal and sends it down
to the second earth station. This frequency is called a Downlink.

Consider the light bulb example:

Advantages of satellite over terrestrial communication :
* The coverage area of a satellite greatly exceeds that of a
terrestrial system.
* Transmission cost of a satellite is independent of the distance
from the center of the coverage area.
* Satellite to Satellite communication is very precise.
* Higher Bandwidths are available for use.
Disadvantages of satellites:
* Launching satellites into orbit is costly.
* Satellite bandwidth is gradually becoming used up.
* There is a larger propagation delay in satellite communication
than in terrestrial communication.

How does a satellite stay in it’s orbit?

How do we escape gravity & place an
object in orbit?
• If an object is fired
fast enough it should
escape the earths pull.
• This is done through
the use of Rocket
Launchers

Multi-stage Rockets
• Stage 1: Raises the
payload e.g. a satellite to
an elevation of about 50
miles.
• Stage 2: Satellite 100
miles and the third stage
places it into the transfer
orbit.
• Stage 3: The satellite is
placed in its final geo-
synchronous orbital slot
by the AKM, a type of
rocket used to move the
satellite.

Major problems for satellites
• Positioning in orbit
• Stability
• Power
• Communications
• Harsh environment

Positioning
• This can be achieved by several methods
• One method is to use small rocket motors
• These use fuel - over half of the weight of
most satellites is made up of fuel
• Often it is the fuel availability which
determines the lifetime of a satellite
• Commercial life of a satellite typically 10-
15 years

Stability
• It is vital that satellites are stabilised
- to ensure that solar panels are aligned properly,
communication antennae are aligned properly
• Early satellites used spin stabilisation
- either this requires an inefficient omni-directional
aerial Or antennae were precisely counter-rotated in
order to provide stable communications.
* Modern satellites use reaction wheel
stabilisation - a form of gyroscopic stabilisation.

Power
• Modern satellites use a variety of power means
• Solar panels are now quite efficient, so solar
power is used to generate electricity
• Batteries are needed as sometimes the satellites
are behind the earth - this happens about half the
time for a LEO satellite
• Nuclear power has been used - but not
recommended

Satellite - satellite communication
• It is also possible for
satellites to
communicate with
other satellites
• Communication can
be by microwave or
by optical laser
1.
2.
1.
2.
1.
2.
Point-Point System Crosslink System Hybrid System

Harsh Environment
• Satellite components need to be specially
“hardened”
• Circuits which work on the ground will fail very
rapidly in space
• Temperature is also a problem - so satellites use
electric heaters to keep circuits and other vital
parts warmed up - they also need to control the
temperature carefully

Early satellites
• Telstar
– Allowed live transmission across the Atlantic
• Syncom 2
– First Geosynchronous satellite
TELSTAR SYNCOM 2

Satellite orbits
Classification of orbits:

* Circular orbits are simplest
* Inclined orbits are useful for coverage of
equatorial regions
* Elliptical orbits can be used to give quasi
stationary behavior viewed from earth using 3 or
4 satellites
* Orbit changes can be used to extend the life of
satellites

Classification of orbits:
Satellite orbits are also classified based on their
heights above the earth:
– GEO
– LEO
– MEO
– Molniya Orbit
– HAPs

Geostationary Earth Orbit (GEO)
• These satellites are in orbit 35,786 km above the earth’s
surface along the equator.
• Objects in Geostationary orbit revolve around the earth
at the same speed as the earth rotates. This means GEO
satellites remain in the same position relative to the
surface of earth.

GEO contd.
• Advantages
– A GEO satellite’s distance from earth gives it a large coverage
area, almost a fourth of the earth’s surface.
– GEO satellites have a 24 hour view of a particular area.
– These factors make it ideal for satellite broadcast and other
multipoint applications
– Minimal doppler shift
• Disadvantages
– A GEO satellite’s distance also cause it to have both a
comparatively weak signal and a time delay in the signal,
which is bad for point to point communication.
– GEO satellites, centered above the equator, have difficulty for
broadcasting signals to near polar regions
– Launching of satellites to orbit are complex and expensive.

Low Earth Orbit (LEO)
• LEO satellites are much closer to the earth than GEO satellites,
ranging from 500 to 1,500 km above the surface.
• LEO satellites don’t stay in fixed position relative to the surface,
and are only visible for 15 to 20 minutes each pass.
• A network of LEO satellites is necessary for LEO satellites to be
useful

The Iridium system has 66 satellites in six LEO orbits,
each at an altitude of 750 km.
Iridium is designed to provide direct worldwide voice and data
communication using handheld terminals, a service similar to
cellular telephony but on a global scale

LEO Contd.
• Advantages
 A LEO satellite’s proximity to earth compared to a GEO
satellite gives it a better signal strength and less of a time
delay, which makes it better for point to point communication.
 A LEO satellite’s smaller area of coverage is less of a waste
of bandwidth.
• Disadvantages
 A network of LEO satellites is needed, which can be costly
 LEO satellites have to compensate for Doppler shifts cause by
their relative movement.
 Atmospheric drag effects LEO satellites, causing gradual
orbital deterioration.

Medium Earth Orbit (MEO)
• A MEO satellite is in orbit somewhere between 8,000
km and 18,000 km above the earth’s surface.
• MEO satellites are similar to LEO satellites in
functionality.
• MEO satellites are visible for much longer periods of
time than LEO satellites, usually between 2 to 8 hours.
• MEO satellites have a larger coverage area than LEO
satellites.

MEO contd.
• Advantage
 A MEO satellite’s longer duration of visibility and
wider footprint means fewer satellites are needed in a
MEO network than a LEO network.
• Disadvantage
 A MEO satellite’s distance gives it a longer time
delay and weaker signal than a LEO satellite, though
not as bad as a GEO satellite.

MEO satellites
Glonass (Russian)
The GPS constellation
calls for 24 satellites to be
distributed equally among
six circular orbital planes

Molniya Orbit
 Used by Russia for decades.
 Molniya Orbit is an elliptical orbit. The satellite remains in a
nearly fixed position relative to earth for eight hours.
 A series of three Molniya satellites can act like a GEO satellite.
 Useful in near polar regions.

High Altitude Platform (HAP)
One of the newest ideas in satellite communication.
 A blimp or plane around 20 km above the earth’s surface is used
as a satellite.
 HAPs would have very small coverage area, but would have a
comparatively strong signal.
 Cheaper to put in position, but would require a lot of them in a
network.

Satellite frequency band
Band
Downlink,
GHz
Uplink, GHz
Bandwidth,
MHz
L 1.5 1.6 15
S 1.9 2.2 70
C 4 6 500
Ku 11 14 500
Ka 20 30 3500

Solar day and Sidereal day
• A day is defined as the time that it takes the
Earth to rotate on its axis.
• However, there is more than one way to define
a day:
– A sidereal day is the time that it takes for the Earth
to rotate with respect to the distant stars.
– A solar day is the time that it takes to rotate with
respect to the Sun.

The Length of the Day
• A solar day is slightly longer than a sidereal day.
– A sidereal day is 23h 56m 4.091s.
• We set our watches according to the solar day.
• Astronomers use sidereal time because we are
mostly interested in distant celestial objects.

• A solar day is measured using the passage of the
Sun across the sky—it lasts 24 hours
• A sidereal day (from the Latin word meaning
star) is measured with respect to fixed stars—it
lasts a little less than 24 hours.
• Each solar day the Earth rotates 360 degrees with
respect to the Sun
• Each sidereal day the Earth rotates 360 degrees
with respect to the background stars
• During each solar day the motion of the Earth
around the Sun means the Earth rotates 361
degrees with respect to the background stars

• The actual length of a sidereal day on Earth is 23 hours 56
minutes 4 seconds
• This means that the Earth has to rotate slightly more than one
turn with respect to a fixed star to reach the same Earth-Sun
orientation (solar day)

The difference between solar days and sidereal days
means that a given star will rise earlier each day
These 3 photos show how Orion reaches the same
position in the sky 4 minutes earlier on each
consecutive day.

Apparent Solar Time
• Apparent solar time is the time measured with
respect to the actual position of the Sun.
– At noon, the Sun would be exactly on the meridian.
– 1 P.M. would be exactly one hour after the Sun was on
the meridian.
– 9 A.M. would be exactly 3 hours before the Sun was on
the meridian.
– The apparent solar time depends on your longitude.

Origin of planetary laws
Sir. Johannes Keppler
 Derived 3 laws based
upon his observations
of planetary motion.
Sir.Tycho Brahe
• Introduced precision into
astronomical measurements.
• Mentor to Johannes Keppler

Kepler’s 1st Law: Law of Ellipses
The orbits of the planets are ellipses with
the sun at one focus

Kepler’s 2nd Law: Law of Equal Areas
The line joining the planet to the center of the sun
sweeps out equal areas in equal times
T6
T5
T4 T3
T2
T1
A2
A3
A4
A5
A6
A1

Kepler’s 3rd Law: Law of Harmonics
The squares of the periods of
two planets’ orbits are
proportional to each other as
the cubes of their semi-
major axes:
T1
2/T2
2 = a1
3/a2
3
In English:
Orbits with the same semi-
major axis will have the
same period

Newton’s Laws
• Kepler’s laws only describe the planetary motion
without attempting to suggest any explanation as
to why the motion takes place in that manner.
Sir .Issac Newton
• Derived three laws of motion.
• Derived the Law of Universal
Gravitation.
• Explained why Kepler’s laws
worked.

Newton’s 1st Law: Law of Inertia
• Every body continues in a state of uniform
motion unless it is compelled to change that
state by a force imposed upon it

Newton’s 2nd Law: Law of Momentum
• Change in momentum is proportional to and
in the direction of the force applied
• Momentum equals mass x velocity
• Change in momentum gives: F = ma
F
F

Newton’s 3rd Law: Action - Reaction
• For every action, there is an equal and
opposite reaction
• Hints at conservation of momentum

Newton’s Law of Universal Gravitation
Between any two objects there exists a force of
attraction that is proportional to the product
of their masses and inversely proportional to
the square of the distance between them
Fg = G( )
M1m2
r2

Apogee and Perigee
• In astronomy, an apsis is the point of greatest or least distance
of the elliptical orbit of an astronomical object from its center of
attraction, which is generally the center of mass of the system.
• The point of closest approach is called the periapsis (Perigee)
or pericentre and the point of farthest excursion is called the
apoapsis (apogee)
• A straight line drawn through the perigee and apogee is the line
of apsides. This is the major axis of the ellipse.
Ascending & Descending nodes
• These are the 2 points at which the orbit of a satellite penetrates
the equatorial plane.

Classical orbital elements
• Six independent quantities are sufficient to
describe the size, shape and orientation of an
orbit.
These are
– a, the semi-major axis
– , the eccentricity
– i, the inclination
– , the right ascension of the ascending node
– , the argument of perigee
– tp, mean anamoly

• The semi-major axis describes the size of the orbit. It
connects the geometric center of the orbital ellipse with
the periapsis, passing through the focal point where the
center of mass resides.
• The eccentricity shows the ellipticity of the orbit.
• The inclination is the angle between the plane of the
orbit and the equatorial plane measured at the ascending
node in the northward direction.
• The right ascension of an ascending node is the angle
between the x axis and the ascending node.
• The argument of periapsis (perihelion) is the angle in the
orbital plane between the line of nodes and the perigee of
the orbit.
• The mean anomaly is the time elapsed since the satellite
passed the perigee.

Major parameters of an elliptical orbit
• Satellite trajectory
• Satellite period
• Satellite velocity
• Satellite position

Satellite Trajectory
The path of a satellite in space may be obtained
under the following assumptions:
1.The satellite and earth are symmetric spherically
and may be treated as point masses.
2.There are no other forces acting on the system
besides the gravitational forces.
3.The mass of the earth is much greater than
satellite.
These assumptions lead to the two body problem.

Applying Newton's laws to such systems,
..
∑F = m r (second law) …………(1)
F = -GMm. r (third law) ……………..(2)
r2 r
Substituting (1) in (2) we get,
.. ..
r + GM .r = 0 (or) r + μ .r = 0
r3 r3
..
Where r = vector acceleration in the given coordinate system
r = vector from M (mass of earth) to m (mass of satellite)
r = distance between M and m , μ = GM (gravitational parameter)
A partial system is easy to obtain and is adequate for illustrating the size and shape of
an orbit.
The resulting trajectory equation has a general form of conic section:
r = P ; p = a geometric constant called parameter of conic
1+e cos θ = (r v cos ф)2 / μ
e = the eccentricity which determines type of conic section
=√(1-P/a)
θ = angle between r and the point on the conic nearest the focus
ф = flight elevation angle , v = satellite velocity
a = semi-major axis = (ra+rb)/2

Satellite period
The period T of a satellite is given as:
T2= 4 П2 .a3 (period depends only on semi major axis,a)
μ
For a satellite in circular orbit around earth-
T2= 4 П2 .(R+h)3
μ
Where , R= radius of earth,
h= satellite altitude

Satellite velocity
Total specific mechanical energy ε of a satellite is the sum of
kinetic energy/unit mass and potential energy/unit mass,
but there is an interchange between these energies.
Thus a satellite slows down when it moves up and gains
speed as it loses height.
The velocity of a satellite in an elliptic orbit is :
V2= μ(2/r -1/a)
also ε = V2/2 - μ /r and ε = –μ /2a
For circular orbit the equation reduces to:
V2 = μ /r

Satellite position with time
The origin O is the geocentre.
The satellite at any instant tp is assumed to be at S.
The circle is drawn from centre C of the ellipse with a radius equal to
the semi major axis and a perpendicular BM is drawn passing through
the point S.
Angle E is called eccentric anomaly and angle θ is the true anomaly.

Satellite position
For an elliptic orbit, the time tp elapsed from a perigee pass is defined as-
tp = T/2Π (E-e sin E)
= (T/2Π)M ; where M = E-e sin E
Eccentric anomaly is defined as
E = arccos[ (e + cosθ)/(1+ e cosθ)]
where θ = true anomaly
= 2tan
-1 {[( 1+e)/(1-e)]1/2 tanE/2}
When θ=0 ,the mean and true anomalies are equal.
Hence distance between satellite and geocentre is
r = a(1-e2)/(1-ecosθ)

GEOSYNCHRONOUS
AND
GEOSTATIONARY ORBITS

GEOSYNCHRONOUS ORBITS
• A geosynchronous orbit is the one with an orbital
period (the time needed to orbit once around the
Earth) that matches the rotation rate of the Earth.
This is a sidereal day, which is 23 hours 56
minutes and 4 seconds in length.
• A geosynchronous earth orbit is sometimes
referred to as the Clarke orbit or Clarke belt, after
Arthur C. Clarke, who first suggested its‘ existence
in 1945 and proposed its use for communications
satellites

Clarke Orbit
• The Clarke orbit meets the
concise set of specifications
for geosynchronous satellite
orbits:
– (1) be located directly above
the equator
– (2) travel in the same
direction as Earth's rotation at
6840 mph
– (3) have an altitude of 22,300
miles above Earth
– (4) complete one revolution in
24 hours

Geo synchronous Satellites
• There is only one geosynchronous earth orbit.
• It is occupied by a large number of satellites. In fact, the
geosynchronous orbit is the most widely used earth orbit for the
obvious reason.
• An international agreement initially mandated that all satellites
placed in the Clarke orbit must be separated by at least 1833 miles.
• This stipulation equates to an angular separation of 4° or more,
which limits the number of satellite vehicles in a geosynchronous
earth orbit to less than 100.
• Today, however, international agreements allow satellites to be
placed much closer together.

Geo stationary orbit
• A geostationary orbit is a special case of a
geosynchronous orbit.
• A satellite is in a geostationary orbit when it appears
stationary from the point of view of an observer on the
Earth's surface.
This can only occur when:
• The orbit is geosynchronous
• The orbit is a circle
• The orbit lies in the plane of the Earth's equator
• Thus, a geosynchronous satellite will be geostationary
only with the additional restrictions of it being in a
circular orbit situated over the equator.

Geostationary Vs. Polar Orbiting
http://cimss.ssec.wisc.edu/satmet/modules/sat_basics/images/orbits.jpg

Geostationary Satellites
The satellite velocity in this
orbit is 3075 m/s.
• Operate in the 2.0 GHz to
18 GHz range
• When the inclination and
eccentricity of the orbit is
zero, the satellite appears to
be stationary to an observer
from ground.

Geostationary Satellites in Orbit

Geostationary Satellite Coverage
http://www.ssec.wisc.edu/mcidas

Geo-stationary satellites
• The geometric considerations like satellite elevation/look
angle etc are very vital for reliable communication
satellite system design.
Applications:
Telecommunication systems
Radio
Data Transmission systems

Satellite elevation:
The elevation of a satellite,η is the angle which a satellite makes with
the tangent at the specified point on the earth.
η = arc tan [(cosψ-σ)/ sin ψ]
Where, coverage angle ψ = arc cos (cosθc cosφcs )
φcs = φc - φs and σ =R /(R+h) = 0.151
In terms of elevation angle:
ψ = 900 – η-sin-1(cos η / 6.63235)
In terms of tilt angle : ψ = sin -1(6.6235 sinγ- γ) where θc = latitude
of earth station, φc = the longitude, φs = longitude of sub satellite
point, R=radius of earth, h=satellite height above equator
Tilt angle γ = arc tan [sin ψ / (6.6235-cos ψ)

Azimuth:
The azimuth ξ is the angle which the satellite direction makes
with the direction of true north measured in the clockwise direction.
The azimuth ξ = arc tan [tan φcs /sinθc]
in northern hemisphere:
ξ =1800 + A0;when the satellite is to the west of earth station
ξ =1800 - A0;when the satellite is to the east of earth station
in southern hemisphere:
ξ =3600 - A0;when the satellite is to the west of earth station
ξ =A0;when the satellite is to the east of earth station

Range:
The range d of a geostationary satellite is given by,
d = 35786[1+0.4199{1-cos ψ}1/2
In terms of radius of earth (ie, der = d/r)
der= [13.47(1-cosβ+31.624)1/2 also der = 6 .6235 sin ψ/cos η
• The angle , is the angle between the solar vector and the orbit
plane. If the solar vector is in the orbit plane,  = 0. Beta can go
to  90. The general convention is that  is positive when the sun
is on the same side of the orbit plane as the positive orbit normal
(right hand rule).

Launching of geostationary satellite:
• Initially place spacecraft with the final rocket
stage into LEO.
• After a couple of orbits, during which the orbital
parameters are measured, the final stage is
reignited and the spacecraft is launched into a
geostationary transfer orbit(GTO).
• Perigee of GTO is that of LEO altitude and
apogee that of GEO altitude.
• After a few orbits in GTO, while the orbital
parameters are measured, a rocket motor (AKM)
is ignited at apogee and GTO is raised until it is
circular geostationary orbit.
• AKM (Apogee Kick Motor) is used to circularize
the orbit at GEO and to remove any inclination
error so that the final orbit is very close to
geostationary.

Launching of geostationary satellite

Geostationary Transfer
Orbit
• BUT, if we fire a rocket motor when the satellite's at apogee,
and speed it up to the required circular orbit speed, it will stay at
that altitude in circular orbit. Firing a rocket motor at apogee is
called "apogee kick", and the motor is called the "apogee kick
motor".
• If we speed the satellite
up while it's in low
circular earth orbit it will
go into elliptical orbit,
heading up to apogee.
• If we do nothing else, it
will stay in this elliptical
orbit, going from apogee
to perigee and back again.

Phase I and II of launching spacecraft
Phase I ↑
Phase II
Few Geostationary satellites: EDUSAT, INTELSAT ,
INSAT , PAKSAT, AMERICOM …….

ORBITAL MANEUVERS
Hohmann Transfer
– Can be used to raise or lower
altitude
– Most efficient method
– At minimum, requires
completion of half revolution
of transfer orbit

Hohmann transfer
• Most satellites launched today are initially placed into an
low earth orbit.
In the next phase the satellite is injected into an elliptical
transfer orbit which has an apogee at the height of GEO and
its apsides (line joining perigee-apogee) in the equatorial
plane.
• Finally satellite is injected into GEO by imparting a
velocity increment at the apogee equal to the difference
between satellite velocity at GTO and velocity in GEO.
• A transfer between two coplanar circular orbits via
elliptical transfer orbit requires the least velocity increment
(and hence fuel). This principle was recognized by Hohmann
in 1925 and is referred as Hohmann transfer.

• A Hohmann transfer is a fuel efficient way to transfer
from one circular orbit to another circular orbit that is in
the same plane (same inclination), but a different altitude.
• To change from a lower orbit (A) to a higher orbit (C),
an engine is first fired in the opposite direction from the
direction the vehicle is traveling.
• This will add velocity to the vehicle causing its trajectory
to become an elliptic orbit (B). This elliptic orbit is
carefully designed to reach the desired final altitude of
the higher orbit (C).
• In this way the elliptic orbit or transfer orbit is tangent to
both the original orbit (A) and the final orbit (C). This is
why a Hohmann transfer is fuel efficient.
• When the target altitude is reached the engine is fired in
the same manner as before but this time the added
velocity is planned such that the elliptic transfer orbit is
circularized at the new altitude of orbit (C).

Hohmann Transfer
Target Orbit
Initial
Orbit
Transfer Orbit
The orbital inclination is given by,
cos i= sinξ1 cos θ1
where i=inclination
ξ1 =azimuth of launch
θ1 =latitude of launching site

PERTURBATIONS
• Perturbation is a term used in astronomy to
describe alterations to an object's orbit caused by
gravitational interactions with other bodies.
Major sources are:
• Effect of earth
• Third Body Effects
• Atmospheric Drag
• Solar radiation pressure
• Electro-Magnetic effect

Space Weather Effects
Space Weather Effects

Effect of earth on satellites
• The effect of gravitational force is
non uniform because of the non
uniform distribution of earth’s mass -
a slight bulge at the equator, with a
difference of 21 km between polar
and the equator radius.
• This deviation from spherical shape
causes additional forces on the
satellite.
• The effect of earth’s gravitational
pull may be expressed as the
harmonic series of the field. The
first term represents the principal
gravitational law and the higher
order terms in the series as the
perturbations.

The main effects of perturbations are:
1. The component of perturbations in the orbital plane causes the
perigee to rotate in the orbital plane.
2. Another effect of perturbations is that the orbital plane rotates around
the earth’s north-south axis.
3. The perturbating force along the orbital plane imparts a force vector
on a satellite
1. The component of perturbations in the orbital plane
causes the perigee to rotate in the orbital plane.
The rate of change of argument of perigee is
ω = 4.97[R/a]3.5 (5cos2i-1)/(1-e2)2 deg/day
where R= mean equatorial radius , a=semi major axis
i = inclination, e=eccentricity
• when i=63.40 , ω reduces to zero, implying that perigee remains
fixed in space.

2. The orbital plane rotates around the earth’s north-south axis.
The rate of change of rotation of ascending node is
Ω = 9.95[r/a]3.5 cos i /(1-e2)2 deg/day
Where r = satellite-geo centre distance
The rotation is in a direction opposite to the satellite motion. For a
geostationary orbit magnitude is 4.90/year ,implying the ascending
node rotates around the earth in 73 years.
3. The perturbating force along the orbital plane imparts a
force vector on a satellite.
For most orbits such components cancel out as the satellite position
changes continuously.
In the geostationary orbits, resultant perturbating component do not
cancel but cause a satellite to drift towards one of the two nearest
stable points on the orbit.
Stable points are approximately on the minor axis, showing that the
elliptical approx. of earth is not precisely accurate.

Third Body Effects (heavenly bodies)
• Gravitational pull of other massive bodies,
i.e. Sun, moon
• Mainly noticeable in deep space orbits

Gravitational effects from heavenly bodies:
• In LEO satellites, the influence of gravitational forces
from sun and moon are small when compared to the
gravitational force of earth.
• The order of magnitude of gravitational force of moon
and sun are main sources of perturbations in GEO
satellites.
• When nearer to heavenly bodies, the gravitational pull is
stronger and hence causes a gravity gradient. main
effect of such gradient is to change the inclination of the
orbit.
• The combined effect of sun and moon is to cause a
change in inclination of GEO satellites between 0.750
and 0.940

• The inclination of orbital plane caused by moon changes
cyclically between 0.480 and 0.670 with a period of 18.6
years. Maximum inclination change occurred in year
1987 and minimum in Feb 1997
• The change in inclination due to sun is 0.270 /year.
Note: Among the three forces affecting the inclination (gravity pull,
sun and non spherical nature of earth) the later force has a component
in the direction opposite to the former two forces.
Hence these forces cancel out at an inclination angle of about 7.50
Thus the inclination of satellite when left uncorrected oscillates around
the stable inclination with the period of about 53 years reaching a
maximum of 150 and a minimum of 00

Atmospheric Drag
• Satellites below 2000 kilometres, are actually travelling
through the Earth’s atmosphere. Collisions with air
particles, even at these high altitudes slowly act to
circularise the orbit and slow down the spacecraft
causing it to drop to lower altitudes , this effect is
known as atmospheric drag
• Emissions from the Sun cause the upper atmosphere to
heat and expand.
• These energetic solar outputs increase dramatically
during periods of high solar activity, and may result in
Earth-orbiting satellites experiencing an increase in
atmospheric drag

A satellite orbiting the Earth would continue to orbit
forever if gravity were the only force acting on it.
Perigee remains same, Apogee decreases

• Reduces satellite’s energy
• Changes the size (semi-major axis) and shape
(eccentricity)
• The effect of drag is more severe at about 180km and
causes excess heat on satellite .Unless such LEO satellites
are routinely boosted to higher orbits, they slowly fall,
and eventually burn up
• Orbital life time of satellite at 400km circular earth orbit is
typically few months, where as the life time is several
decades if they are at 800km altitude
• In the former case, functional life time depends on orbital
life time and for latter the life time of satellite equipments
is the deciding factor.
• However, for GEO satellites the governing factors are
equipment life time and fuel capacity of the satellite
(typically 10-15 years).

Solar radiation pressure
• Solar radiation pressure is the force exerted by solar
radiation on objects within its reach
• The effect of solar radiation pressure increases as the
surface area of the satellite projected in the
direction of sun increases.
• The net effect is the increase in the orbital eccentricity
and also introduces disturbing torque that effects the
north-south axis of the satellite.

• Solar wind causes radiation pressure on the satellite
• The solar wind is a stream of charged particles (a plasma) that are
ejected from the upper atmosphere of the sun. It consists mostly of
electrons and protons with energies of about 1 keV. These particles
are able to escape the sun's gravity because of the high temperature
of the corona, and also because of high kinetic energy
• These perturbations are corrected periodically

Effects of Solar radiation pressure
1. HUMAN HEALTH
• Intense solar flares produce very high energy particles
that can be as harmful to people as low-energy
radiation from nuclear blasts. Earth’s atmosphere and
magnetosphere provide protection for people on the
ground, but astronauts in space are subject to
potentially lethal doses of radiation. The penetration of
high energy particles into living cells leads to
chromosome damage and, potentially, cancer.
• Airline pilots and flight crews, as well as frequent
fliers, also receive increased doses of radiation from
solar flares. If you were travelling in an aircraft at high
altitudes during a major solar flare, the amount of
radiation you would be exposed to can be equivalent to
getting a chest x-ray.

2. COMMUNICATIONS
• Stormy space weather can damage Earth-orbiting
satellites such as those carrying TV and mobile
phone signals.
• During high levels of solar activity, satellites are
bombarded with high energy particles. If the
deeply penetrating electrons build up faster than
the charges are able to dissipate out of the satellite
material, a discharge can result that is capable of
damaging the satellite electronics.
• These processes can result loss of control and even
satellite failure.

3. NAVIGATION
• A Global Positioning System (GPS) receiver uses radio signals from
several orbiting satellites to determine the range, or distance, from
each satellite, and determines from these ranges the actual position of
the receiver.
• The radio signals must pass through the ionosphere, the uppermost
part of the Earth’s atmosphere, and in doing so are subjected to
variations in the electron density structure of the ionosphere.
• Changes in the electron density due to space weather activity can
change the speed at which the radio waves travel introducing a
“propagation delay” in the GPS signal. Changing propagation delays
cause errors in the determination of the range.
• An increase in space weather activity may cause widespread
disruption to aircraft and ship navigation and emergency location
systems that rely heavily on satellite navigation data.

Electro-Magnetic effect
• Interaction between the Earth’s magnetic
field and the satellite’s electro-magnetic
field results in magnetic drag

Magnetic storm
A geomagnetic storm is a temporary disturbance of the earth’s
magnetosphere caused by a disturbance in space weather. A geomagnetic
storm is caused by a solar wind shock wave. This only happens if the
shock wave travels in a direction toward Earth.
The solar wind pressure on the magnetosphere will increase or decrease
depending on the Sun's activity. These solar wind pressure changes
modify the electric currents in the ionosphere.
Magnetic storms usually last 24 to 48 hours, but some may last for many
days.

Non geostationary constellations
• The design of constellations can be categorized according
to inclination, altitude and eccentricity.
On the basis of inclination, two types of constellations are
designed.
Type I constellations are those having their orbital planes
with a common intersection point.
Eg: Polar constellations
• Type II constellations have optimized inclined orbit
constellations and distribute satellites uniformly.
• Eg: inclined constellations
• Depending on altitudes, constellations may be
LEO,MEO etc.
• A hybrid of orbital altitudes are also possible within a
system (A LEO satellite can be used together with a
geostationary orbit satellite)

Advantages of Non-geostationary constellations
1. Since these orbits are closer to earth, the free space loss is
lower and hence it is possible to use hand held terminals.
Path loss at 1.5 GHz for LEO=152.87 dB, MEO=175.96
dB and GEO=187.10dB.
2. LEO and MEO reduces the propagation delay which
reduces or eliminates delay related problems.
3. These orbits offer a higher frequency reuse.
Maximum distance between two points which view a
satellite at an elevation angle 100 is
LEO (at altitude 700km) = 3885 km
MEO (10000km) = 12790 km
GEO (36000km) = 15914 km
Hence note that 4 LEO satellites would cover the same
geographical distance as a single GEO. Thus LEO has 4
times frequency reuse than GEO.

4 . Distributed architecture of LEO and MEO orbits make
them more resistant to satellite failures and hence more
reliable.
5. Competitions between operators has triggered a feverish
technical, regulatory and financial activity in the
industry.
Current non geostationary proposals are-
• MEO system -Offers a real time services. Medium/high
bit rates communication facility. eg: ICO system
• Little LEO -Offers a low bit rate non-real time services
such as messaging (bit rate< 4kbps). eg. ORBCOM
• Big LEO -Offering medium bit rate interactive services
such as voice (bit rate 1-10 kbps) eg: Iridium.
• Broad band LEO -Offers broadband services such as
internet high speed file download
(bit rate=16kbps to 1 Gbps) eg: Teledesic.

Polar constellations
• A polar orbit is an orbit in which a satellite passes above or
nearly above both poles of the body (usually a planet such
as the earth) being orbited on each revolution. It therefore
has an inclination of (or very close to) 900 to the equator.
• Polar orbits are often used for earth-mapping, earth
observation, as well as some weather satellites
• The disadvantage to this orbit is that no one spot on the
Earth's surface can be sensed continuously from a satellite
in a polar orbit.
• Polar satellites include: Defense meteorological satellite
program (DMSP), Landsat, SPOT and NOAA. Landsat and
SPOT are Commercial polar orbiters and are intended for
geophysical remote sensing
• To achieve a polar orbit requires more energy, thus more
propellant is needed than an orbit of low inclination

Eg. of the positions of a sun-synchronous satellite in 12 hour intervals
Sun synchronous satellites pass over any given latitude at almost the
same local time during each orbital pass

Polar constellations
Here : Ψ = coverage circle
m = number of orbital planes
n = satellites /plane
Δ = cos-1[cos Ψ/cos Π/m]

Single coverage:
• Satellites in adjacent planes move in same direction, shifted with
respect to each other by half intra-orbit satellite separation (Π/m),
where m = number of planes.
The separation between adjacent planes is (Ψ+Δ) and the relative
geometry remains constant because they move in phase.
• Satellites are separated by 2Δ,when the satellites move in opposite
directions and the relative geometry is not constant.
The total number of satellites,
N = 4 /(1-cos Ψ) ; 1.3n < m < 2.2n
In the cases of non integer, next highest integer satisfying the
inequality can be taken.
If the N is much large, then the condition (n-1) Ψ + (n+1) Δ = П

• When the coverage is required beyond a latitude λ,the equations are –
(n-1) Ψ+(n+1) Δ = П cos λ and N = 4cos λ/(1-cos Ψ)
• The coverage efficiency of the constellations is given by NΩ/4П
Where NΩ = total solid angle
Ω = solid angle bounded by a single satellite=2П(1-cos Ψ)
Triple coverage:
• The constellation geometry is similar to single coverage case, with at
least three satellites must be visible at all points.
• The coverage angle is adjusted such that at least 3 satellites lie within
angle Ψ of each point of set.
• The resulting relationship for providing triple coverage from pole up
to latitude λ is
N = 11cos λ/(1-cos λ) ; 1.4n < mcos λ < 2.4n

Inclined orbit
• A satellite is said to occupy an inclined orbit around the earth if
the orbit exhibits an angle other than zero degrees with the
equatorial plane
• They have an inclination between 0 degrees (equatorial orbit)
and 90 degrees
• This family of satellites provides unbiased worldwide coverage
by deploying satellites in circular orbits of same period and
inclination, distributed uniformly on the sphere.
• The orbital altitude of these satellites is generally on the order of a
few hundred km, so the orbital period is on the order of a few
hours. These satellites are not sun-synchronous, however, so they
will view a place on Earth at varying times.

Adjacent orbital planes are separated equally around a reference
plane (equatorial).
Within each orbit ,neighboring satellites have equal angular
separation.

Inclined constellations
αi = right ascension angle of ith orbital plane =2Пi / P
βi = inclination angle of ith orbit
γi = initial phase angle of ith satellite = m αi
m = (0 to N-1)/Q ; N = PQ (P,Q are integers)
Q = number of satellites per plane

Hybrid constellations
• This combines the various types orbits for full earth
coverage
• These orbits have different orbital period
• Eg: using circular orbits for covering equatorial regions
and elliptical orbits for higher altitude regions
Eg: using GEO for covering equatorial regions and
inclined orbits for polar regions

Regional coverage:
In some cases it is necessary to cover only a part of the world. This is
made possible by number of spot beams. Here it is necessary to
ensure that all the satellites pass over the same service area.
Eg: equatorial regions may be covered by deploying satellites in
equatorial planes
Using elliptical orbits inclined at 63.40 can be used for covering high
altitude because satellites in these orbits dwell over high altitudes
over a considerable time.
Thuraya allows to create more
than 200 spot beams and handle
13,750 simultaneous phone calls.
Telecommunications Services
offered are:
• Voice
• Fax at 9.6 Kbps
• Data at 9.6 Kbps

Constellations for store and forward system:
• Systems which do not require real time coverage (messaging/paging)
is less stringent because gaps in coverage are allowed, provided at
least one satellite is visible within (ta-td) where ta=specified end to
end delay , td=delay in message transfer
• Eg :A single satellite in polar orbit can cover every regions of earth
within a time dependent on the orbital period
ORBCOMM provides low cost,
reliable, two-way data communications
services around the world through a
global network of 29 low-earth orbit
(LEO) satellites and accompanying
ground infrastructure. The system can
send and receive short messages,
between six bytes and several kilobytes

Random and phased constellations
• Constellations can be categorized on the basis of phase relationship
between satellites with respect of each other.
• In random constellation all or some constellation parameters such
as altitude inclination, inter orbital plane separation and inter satellite
phase are chosen at random.
• In phased constellation all these parameters are well defined.
Random constellations are simpler to maintain but are inefficient in
terms of coverage property and tend to randomly crowd the celestial
sphere around the chosen altitude.

Design considerations of a non GEO satellite systems
1.Traffic distribution and coverage:
• A constellation design depends on service area and geographical
distribution of traffic within that area.
• A worldwide coverage is essential for an operator interested in global
operations, but regional operator is interested in only a specific
region. Hence the constellations are completely different.
• Good RF visibility ensures adequate signal strength before a
connection is established and the increase in spectrum reuse.
• Complexity in coverage design is the dynamic variation in position
of the footprint of each satellite and the constellation as a whole,
making the geometric relationship time dependent.
• Hence an estimate can be made on the basis of known growth trends
from existing systems, population density, per capita income,
existing infrastructure, market segmentation due to competition and
prevailing economic /political condition of the target market.

2. Satellite capacity:
• The capacity required per satellite increases as the orbit altitude
increases because a satellites field of view and captured traffic
increases with the altitude
• Total constellation capacity is the sum of capacities of satellites
• At higher altitudes satellite capacities are better shared
• Hence as the altitude increases, total constellation capacity reduces
and more efficient constellation capacity is utilized
3. State of spacecraft technology:
• Antenna size and complexity -as the altitude increases , larger
antennas are required to meet link quality objective and maintain
frequency reusability
• Spacecraft DC power -DC power determines the capacity of the
satellite
• Inter satellite link -satellites with inter links influence the network
routing scheme

4.Terminal characteristic and communication requirement-
• The size of terminals and their communication capability
influence a satellites power and sensitivity requirements
• RF power of a handset is limited by radiation safety considerations,
battery size /capacity and the target terminal cost
• If the satellites are brought closer , power required can be reduced
but number of satellites in the constellations increases.
5.Quality of service:
• Quality of service refers to RF link reliability, propagation delay
and signal quality measured as bit error rate
• Higher link reliability requires higher elevation angle
• Propagation conditions improve as the elevation angle increases
because number of obstructions reduces
• Propagation delay is related to the altitude of orbit
• Hence for interactive applications lower orbits are best and non real
time applications are insensitive to altitude
• Signal quality is related to link conditions and issues such as carrier
to noise power density/modulation/coding schemes

6. Spectrum availability:
• Frequency reusability can be increased by spatial/polarization
diversity
• This is achieved by using spot and shaped beams
• For a given spot beam size lower altitude constellations can give
increased reusability
• Additional measures like modulation, coding and multiple access
schemes can maximize radio resource
• 7. Orbital considerations:
• Space environment affects the orbit selection
• Atmospheric drag, eclipses, ionization
8. Launch considerations
• Important practical consideration is the launch cost, feasibility of
launching the satellites in the acceptable time frame
• Probability of launch failure and in-orbit satellite failure increases as
the number of satellites in the constellation increases.

Assignment 01
Explain about the effect of
a) eclipse due to earth
b) eclipse due to moon
c) solar interference
on geo-stationary satellites.
Submit before:18.08.2008

Problems
Q1. Find out the radius of a geostationary
satellite orbit.
Given:
T = 23Hr 56Min 4.1Sec
G = 6.672 X10-11 m3/kg/s2
M = 5974 X1024 kg
r = 6378.1414 km
take μ = √GM

Answer:
T2 = 4 П2 .(R)3
μ
R = [T√GM]2/3
-------------
2П
= [23x60x60+56x60+4.1√ 6.672 X10-11x 5974 X1024 ]
2П
= 42164.17 km
Altitude, h = R-r
= 42164.17-6378.1414
= 35786.02 km

Q2. A satellite orbiting in equatorial plane has
a period from perigee-perigee of 12 Hrs.
Given that the eccentricity is 0.002.
Calculate semi-major axis.
Given:
G = 6.672 X10-11 m3/kg/s2
M = 5974 X1024 kg
r = 6378.1414 km

Answer:
Eccentricity is 0.002 (0<e<1), hence orbit is
elliptical.
For an elliptical orbit,
T2 = 4 П2 .(a)3
μ
12x60x60 =2 П √ a3/6.672x10-11 x 5974x1024
a3 = 1.886x1025
a = 266183.1516 km

Q3. Calculate the apogee and perigee heights
for the given orbital parameters.
e=0.0011501 and
a= 7192.3 km
Given:
r = 6378.1414 km

Answer:
ra=a(1+e) = 7200.57 km
rp= a(1-e) = 7184.03 km
Apogee height, ha = ra – r = 822.14 km
Perigee height, hp = rp – r = 805.89 km

Q4. A satellite is in elliptical orbit with a
perigee of 1000 km and an apogee of 4000
km. Using a mean earth radius of 6378.14 km
, find the period of the orbit in hours ,minutes
and seconds.
Also find the ‘e’ of the orbit.

Answer:
ra = ha+R =4000+6378.14=10378.14 km
rp =hp +R=1000+6378.14=7378.14 km
a= (ra + rp)
T=√4П2a3/GM
=8320.94 sec = 2hr 19 mts 8 sec
ra = a(1+e)
e= (ra/a) -1
= 0.169

Communication satellites (comsat)
• Satellite is a RF repeater in orbit.
• The design of a satellite is governed by the communication capacity,
physical environment in which it is operated and state of
technology.
Main considerations of a comm. satellite are:-
i) Type of service to be provided (eg: mobile communication, DTH)
ii) communication capacity (transponder BW and satellite EIRP)
iii) coverage area
iv) Technological limitations
• Basic specifications are laid out for satellite depending on the
communication requirement.
• A domestic fixed satellite service it is the EIRP per carrier, number
of carriers and coverage area.
• A direct broadcast satellite it is the number of television channels and
coverage area.

First TV image of weather (1960)

First complete view of world’s weather, photographed by
TIROS 9 (13/2/1965).
Image assembled from 450 individual photographs

Environmental Conditions:
• A spacecraft must be reliable in all types of environments beginning from
launch to the in-orbit deployment and throughout its operation phase.
Most important stresses are-
a) Zero gravity :-
• At GEO, gravitational force is negligible giving rise to zero gravity effects.
• Major effect is on liquid fuel flow and hence external means are to be
provided for liquid flow.
• The absence of gravity facilitates operation of the deployment mechanisms
used for stowing antennas and solar panels during launch.
b) Atmospheric pressure and temperature :-
• At high altitudes, atmospheric pressure is extremely low (10-7 torr).
• This makes thermal conduction negligible and increase friction between
surfaces.
• Hence special materials are used for lubrication of moving parts.
• However, pressure inside the spacecraft is higher because of out gassing of
electronic components.
• The temperature of a spacecraft is mainly affected by heat from sun and
various spacecraft subsystems.
• The excursion in the external temperature varies from 330-350K during
sunlight and 95-120K during eclipses.

c) Space Particles :-
• Various types of particles like cosmic rays, protons, electrons, meteoroids,
manmade debris etc exist in space.
• Main effect of bombardment of particles on a satellite is the degradation of
solar cells and certain solid state components within the satellite.
• Effect of meteoroids is negligible in GEO satellites.
d) Magnetic fields :-
• Magnitude of earth’s magnetic field is very weak at GEO (1/300 of earths
surface).
• The effect of magnetic field can be compensated by the use of large coil.
• While Satellites passing through Van Allen belt ,deflected charged particles
that are trapped in this region affect electronic components
• Hence special manufacturing mechanisms are used to harden the
components against radiations.
e) Other Considerations :-
• Due to the variation of distance of earth from sun, a variation in DC generation
capability must be taken into account in design of satellite power system.
• Also satellites must be prepared for loss of power during eclipses and may result in
gradual degradation of solar cell efficiency.
• There are several perturbations affecting the satellites due to movement of
mechanical parts and fuel within it.
• There may be a small drift in position of antennas.

Life time and reliability
• Lifetime of a geostationary satellite is determined by the maximum
acceptable deviation in inclination and orbital location.
• Satellite is maintained in its orbital location by firing thrusters
regularly, using stored fuel
• Hence the operational lifetime of a satellite is determined by-
a) Increasing fuel capacity
b) Saving fuel by accepting orbital deviation to the maximum extent
possible.
• However there is a practical limit to a satellites fuel storage capacity.
Hence satellite lifetime is between 12-15 years.

Reliability
• The overall reliability of a satellite is governed by its critical
components.
• Reliability is improved by employing redundancy in the critical sub
systems and in components such as TWT amplifiers.
• Reliability is defined as the probability that a given component/system
performs its function within a specified time t.
• R= where λ= failure rate of a component
• Unit of λ is specified as FIT, the number of failures in 109 Hr.
t
0
dt
e
 


• Three regions can be identified
An early high failure rate region attributed to manufacturing faults, defects in
materials etc
A region of low failure attributed to random component failures
A region of high failure rate attributed to component wear-out.
In a satellite system, early failures are eliminated to a large extent during testing and
burn-in. The main aim is to minimize the random failures which occur during the
operational phase of the satellite by using reliability engineering techniques. The
beginning of wear-out failure can best be delayed by improving the manufacturing
technique and the type of material used.

• The reliability can be expressed as
R = e-λt = e-t/m ; where m =1/ λ (mean time between failures)
• When several components or sub-systems are connected in series, the
overall reliability is
Rs=R1 R2…….Rn where Ri is the reliability of the ith component.
• In terms of the failure rate :
Rs= e-(λ1+ λ2+… λn)t
• Parallel redundancy is useful when the reliability of an individual
subsystem is high.
• If Qi is the unreliability of the ith parallel element, the probability that
all units will fail is the product of the individual unreliabilities
Qs=Q1 Q2…Qi
• When the unreliabilities of all elements are equal, this expression
reduces to Qs = Qi ;Where Q is the unreliability of each element.
Therefore the reliability is
R = 1-Qs
= 1- Qi =1- (1- R)i =1- (1- e-λt)i

A Typical reliability model of a Geostationary Satellite:
• All the major sub-systems are shown in series.
Simplified reliability model
• Applying the equation for series and parallel combination, the
reliability of the communication system is obtained as
Rs =RRXRTX [1-(1-RT)2]
• When RT=0.9,reliability of transponder increases to 0.99
• Figure of merit, Fγ = r/M ;where r = R’/R
R’= reliability with redundancy employed
R= reliability without employing redundancy
M= increase in mass due to added redundancy
• The addition of redundant equipment increases the cost of the
transponder

----Introduction to Solar System Dynamics----
2.a
a: semimajor axis
e: eccentricity
v: true anomaly (0…360 deg)
rp ra
Basic orbital elements (ellipse)
rp: Radius of periapsis
(perihelion)
ra: Radius of apoapsis
(aphelion)
)
1
(
)
1
(
e
a
r
e
a
r
a
p




e=0: circle
e<1: ellipse
e=1: parabola
e>1: hyperbola
v
e
e
a
r
cos
1
)
1
( 2



v
r

----Introduction to Solar System Dynamics----
Useful orbital parameters (elliptical orbit)
1) Velocity:
2) Period:
3) Energy:
4) Angular
momentum:








a
r
GM
u
1
2
GM
a
T
3
2

a
GMm
E
2


)
1
(
,
2
e
a
M
G
m
L
u
r
m
L










M: mass of central body
m: mass of orbiting body
r: distance of m from M
(M>>m)
(Constant!)
(Constant!)

Spin stabilization
• With spin stabilization, the entire spacecraft rotates around its own
vertical axis, spinning like a top. This keeps the spacecraft's orientation
in space under control.
• The spinning spacecraft resists perturbing forces.
• Designers of early satellites used spin-stabilization for their satellites,
which most often have a cylinder shape and rotate at one revolution
every second.
• Spin stabilization was used for NASA's Pioneer 10 and 11 spacecraft,
the Lunar Prospector, and the Galileo Jupiter orbiter.

• The advantage of spin stabilization is that it is a very simple way to
keep the spacecraft pointed in a certain direction.
• A disadvantage of this stabilization is that the satellite cannot use
large solar arrays to obtain power from the Sun. Thus, it requires
large amounts of battery power.
• Another disadvantage of spin stabilization is that the instruments or
antennas also must perform “despin” maneuvers so that antennas or
optical instruments point at their desired targets.

Reaction wheel stabilisation
• With three-axis stabilization, satellites have small spinning wheels,
called reaction wheels or momentum wheels, that rotate so as to
keep the satellite in the desired orientation in relation to the Earth
and the Sun.
• If satellite sensors detect that the satellite is moving away from the
proper orientation, the spinning wheels speed up or slow down to
return the satellite to its correct position.
• Some spacecraft may also use small propulsion-system thrusters to
continually nudge the spacecraft back and forth to keep it within a
range of allowed positions.
• Voyagers 1 and 2 stay in position using 3-axis stabilization.
• An advantage of 3-axis stabilization is that optical instruments
and antennas can point at desired targets without having to perform
“despin” maneuvers

Alignment
• There are a number of components which need
alignment
– Solar panels
– Antennae
• These have to point at different parts of the sky at
different times, so the problem is not trivial

Elliptical Orbit Geometry &
Nomenclature
Periapsis
Apoapsis
Line of Apsides

R
a c
V
Rp
b
• Line of Apsides connects Apoapsis, central body & Periapsis
• Apogee~ Apoapsis; Perigee~ Periapsis (earth nomenclature)
S/C position defined by R & ,
 is called true anomaly
R = [Rp (1+e)]/[1+ e cos()]

ORBITAL ELEMENTS
Keplerian Elements: True Anomaly
i





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