UNIT-1

Introduction to Satellite Communications

Satellite Communication combines the missile

and microwave technologies

The space era started in 1957 with the

launching of the first artificial satellite

(sputnik)

Satellite Communications

• Satellite-based antenna(e) in stable orbit above earth.

• Two or more (earth) stations communicate via one or more satellites serving as relay(s) in space.

• Uplink: earth->satellite.

• Downlink: satellite->earth.

• Transponder: satellite electronics converting uplink signal to downlink.

Satellite Communications

• Satellite Transponder is a microwave device consisting of receiver, repeater and regenerator in orbit

• Satellite transmission involves sending signals to satellite that receive, amplify, and transmit back to earth

Capabilities of Satellite Transmission

• Point-to-point transmission

– To transfer large volume of data

– Voice, data, etc communication

– Video conference

• Point-to-multipoint transmission

– Data communication

– Internet

– Video conference

• Broadcast services such as television

Satellite Network Configurations

Advantages of Satellite Communication

• LARGE CAPACITY

- One satellite = 10 transponders = 10x120 Mbit/s. Total transmission capacity = 1 Gbit/s

• COVERAGE

- A single geostationary satellite can provide communications coverage for some 42.4% of the Earth’s surface, using much less power and much less infrastructure than would be required for a terrestrial system with similar coverage.

• WIDEBAND SERVICE

- allows for transmission of:

- TV

- high bit date rate

• High availability

- There are very few sources of disruption to the Earth-satellite propagation path that cannot be factored into the original link budget, which means that satellite communications have a very high availability.

- Good quality

- Again, since the variations in the satellite path are few and well-characterized, the link budget for a particular path can be determined to guarantee a desired level of quality of service.

History of satellite communication

• 1945 Arthur C. Clarke publishes an essay about „Extra

Terrestrial Relays“

• 1957 first satellite SPUTNIK

• 1960 first reflecting communication satellite ECHO

• 1963 first geostationary satellite SYNCOM

• 1965 first commercial geostationary satellite Satellit „Early Bird“

(INTELSAT I): 240 duplex telephone channels or 1 TV

channel, 1.5 years lifetime

• 1976 three MARISAT satellites for maritime communication

• 1982 first mobile satellite telephone system INMARSAT-A

• 1988 first satellite system for mobile phones and data

communication INMARSAT-C

• 1993 first digital satellite telephone system

• 1998 global satellite systems for small mobile phones

Sputnik 1

• Launched October 14, 1957 – from the Baikonur

Cosmodrome in Kazakhstan

• 184 pounds

• Orbital period 90 minutes

• Broadcast “beep beep” – 20 and 40 MHz

• Shocked the US into action – Started space race

Now: Boeing 702 DBS Satellite

• 134.5 feet long

• 2645 lbs payload

• 11,464 lbs takeoff weight

• Over 100 high-power transponders (94 active/24 spare)

• Up to 25 kW power

• Xenon-Ion Propulsion System

• Built for direct broadcast and point to point services.

Intercontinental telephone,

data, and video relay

• Initially satellite links were only:

– One-way video and data traffic

– Backup to undersea telephone cables

• Because:

– Nominal 1-2 second time delay for a round-trip voice message.

VSAT - Private Networks

• VSAT – Very Small Aperture

Terminal

• Replaces wireline data connections to businesses – Convenience stores,

malls, restaurants, gas stations

• Common uses – Muzak background music

– Credit card transactions

– Corporate communications

• 64kbps to 2Mbps

Mobile Satellite Services

• Inmarsat - communications to ships at sea.

• Expanded – Aircraft

– Trucks

– Rail locomotives.

– Suitcase sized terminals • Used extensively in disaster situations and remote

exploration.

• Not suitable for handheld equipment – Antennas and terminals required

• Analog and digital services are used.

In the Future ?

Internet backbone services

• Teledesic

– Internet in the sky

– 120 Mb uplink

– 720 Mb downlink.

– Ka band

• LEO constellation

– Inter-satellite links

– Scalable

• Viability in question

– Iridium debacle

• System scaled back

– From 240 satellites

– To only 30 satellites

– Nothing launched yet

Major Organisations

• INTELSAT (1964), global (about 140 countries),

FSS and BSS systems

• EUTELSAT (1977) 47 countries (Europe and former

USSR countries), FSS and BSS systems

• INMARSAT (1979) global, mobile systems

• SES Astra (1988) private, DTH-TV

Satellite Services

• FSS Fixed Satellite Services (VSAT

networks,..)

• MSS Mobile Satellite Services (Inmarsat

systems,...)

• BSS Broadcasting Satellite Services ( TV,

DVB..)

• RDSS Radiodetermination Satellite Services

(GPS)

Satellite Orbits

Satellite Orbits GEO

advantages:

- the satellite appears to be fixed (immovable) when viewed from the Earth, no tracking required for earth station antennas

- about. 40% of the earth`s surface is in view from the satellite

disadvantages:

- high attenuation level (power loss) (200dB) on the path

- large signal delay (238-284ms)

- polar regions (latitudes > 81 deg.) are not covered

LEO

advantages:

- much smaller attenuation compare GEO satellites

- low signal delay

disadvantages:

- short period satellite visibility (through earth station), many times during the day

- Doppler effect

- many satellites are required for establishing continuous transmission

Satellite period and orbits

10 20 30 40 x106 m

24

20

16

12

8

4

radius

satellite

period [h] velocity [ x1000 km/h]

synchronous distance

35,786 km

• Four different types of satellite orbits can be identified

depending on the shape and diameter of the orbit:

• GEO: geostationary orbit, ca. 36000 km above earth

surface

• LEO (Low Earth Orbit): ca. 500 - 1500 km

• MEO (Medium Earth Orbit) or ICO (Intermediate

Circular Orbit): ca. 6000 - 20000 km

• HEO (Highly Elliptical Orbit) elliptical orbits

Orbits I

Orbits II

earth

km

35768

10000

1000

LEO

(Globalstar,

Irdium)

HEO

inner and outer Van

Allen belts

MEO (ICO)

GEO (Inmarsat)

Van-Allen-Belts:

ionized particels

2000 - 6000 km and

15000 - 30000 km

above earth surface

Space Weather

Geostationary satellites

• Orbit 35.786 km distance to earth surface, orbit in equatorial plane

(inclination 0°)

complete rotation exactly one day, satellite is synchronous to earth rotation

• fix antenna positions, no adjusting necessary

• satellites typically have a large footprint (up to 34% of earth surface!),

therefore difficult to reuse frequencies

• bad elevations in areas with latitude above 60° due to fixed position above

the equator

• high transmit power needed

• high latency due to long distance (ca. 275 ms)

not useful for global coverage for small mobile phones and data

transmission, typically used for radio and TV transmission

LEO systems

• Orbit ca. 500 - 1500 km above earth surface

• visibility of a satellite ca. 10 - 40 minutes

• global radio coverage possible

• latency comparable with terrestrial long distance connections, ca. 5 - 10

ms

• smaller footprints, better frequency reuse

• but now handover necessary from one satellite to another

• many satellites necessary for global coverage

• more complex systems due to moving satellites

• Examples:

• Iridium (start 1998, 66 satellites)

• Globalstar (start 1999, 48 satellites)

MEO systems

• Orbit ca. 5000 - 12000 km above earth surface

• comparison with LEO systems:

• slower moving satellites

• less satellites needed

• simpler system design

• for many connections no hand-over needed

• higher latency, ca. 70 - 80 ms

• higher sending power needed

• special antennas for small footprints needed

• Example:

• ICO (Intermediate Circular Orbit, Inmarsat) start ca. 2000

GEO vs LEO

GEO

• advantages:

- the satellite appears to be fixed (immovable) when viewed from the Earth, no tracking required for earth station antennas

- about. 40% of the earth`s surface is in view from the satellite

• disadvantages:

- high attenuation level (power loss) (200dB) on the path

- large signal delay (238-284ms)

- polar regions (latitudes > 81 deg.) are not covered

LEO

• advantages:

- much smaller attenuation compare GEO satellites

- low signal delay

• disadvantages:

- short period satellite visibility (through earth station),

- many times during the day

- Doppler effect

- many satellites are required for establishing continuous transmission

Communication Satellites

Overview of LEO/MEO systems

Iridium Globalstar ICO Teledesic

# satellites 66 + 6 48 + 4 10 + 2 288

altitude(km)

780 1414 10390 ca. 700

coverage global 70° latitude global global

min.elevation

8° 20° 20° 40°

frequencies[GHz(circa)]

1.6 MS29.2

19.5

23.3 ISL

1.6 MS

2.5 MS

5.1

6.9

2 MS

2.2 MS

5.2

7

19

28.8

62 ISL

accessmethod

FDMA/TDMA CDMA FDMA/TDMA FDMA/TDMA

ISL yes no no yes

bit rate 2.4 kbit/s 9.6 kbit/s 4.8 kbit/s 64 Mbit/s

2/64 Mbit/s

# channels 4000 2700 4500 2500

Lifetime[years]

5-8 7.5 12 10

costestimation

4.4 B$ 2.9 B$ 4.5 B$ 9 B$

Spectrum Allocation

Frequency Spectrum concepts:

• Frequency: Rate at which an electromagnetic wave reverts its polarity (oscillates) in cycles per second or Hertz (Hz).

• Wavelength: distance between wavefronts in space. Given in meters as:

λ = c/f

Where: c = speed of light (3x108 m/s in vacuum) f = frequency in Hertz

• Frequency band: range of frequencies.

• Bandwidth: Size or “width” (in Hertz) or a frequency band.

• Electromagnetic Spectrum: full extent of all frequencies from zero to infinity.

Radio Frequencies (RF)

• RF Frequencies: Part of the electromagnetic spectrum

ranging between 300 MHz and 300 GHz. Interesting properties:

– Efficient generation of signal power

– Radiates into free space

– Efficient reception at a different point.

Differences depending on the RF frequency used:

- Signal Bandwidth

- Propagation effects (diffraction, noise, fading)

- Antenna Sizes

Microwave Frequencies

• Sub-range of the RF frequencies approximately from

1GHz to 30GHz. Main properties:

- Line of sight propagation (space and atmosphere).

- Blockage by dense media (hills, buildings, rain)

- Wide bandwidths compared to lower frequency bands.

- Compact antennas, directionality possible.

- Reduced efficiency of power amplification as frequency grows:

Radio Frequency Power OUT

Direct Current Power IN

Radio Frequency Spectrum

Commonly Used Bands

Frequency Bands

Frequency allocation

Band Downlink Bands, GHz Uplink Bands, GHz

Uhf – Military0.25 - 0.27 (Approximately) 0.292 - 0.312 (Approximately)

S Band

L Band

C Band - Commercial 3.7 - 4.2 5.925 - 6.425

X Band - Military 7.25 - 7.75 7.9 - 8.4

Ku Band - Commercial 11.7 - 12.2 14.0 - 14.5

Ka Band - Commercial 17.7 - 21.2 27.5 - 30.0

Ka Band - Military 20.2 - 21.2 43.5 - 45.5

Q/V Geostationary 37.5 - 40.5 47.2 - 50.2

Q/V Non-geostationary 37.5 - 38.5 48.2 - 49.2

W Band 66.0 - 67.0 71.0 - 72.0

C-Band Ku-band

Antennas

Insights on Frequency Selection: (Part 1: Lower frequencies, stronger links)

• LEO satellites need lower RF frequencies:

– Omni-directional antennas on handsets have low gain - typically

G = 0 db = 1

– Flux density F in W/m2 at the earth’s surface in any beam is

independent of frequency

– Received power is F x A watts , where A is effective area of

antenna in square meters

– For an omni-directional antenna A = G λ2/ 4 π = λ2/ 4 π

– At 450 MHz, A = 353 cm2, at 20 GHz, A = 0.18 cm2

– Difference is 33 dB - so don’t use 20 GHz with an omni!

Insights on Frequency Selection: (Part 2: Higher frequencies, higher capacity)

• GEO satellites need more RF frequencies

– High speed data links on GEO satellites need about 0.8 Hz of RF bandwidth per bit/sec.

– A 155 Mbps data link requires 125 MHz bandwidth

– Available RF bandwidth:

C band 500 MHz (All GEO slots occupied)

Ku band 750 MHz (Most GEO slots occupied)

Ka band 2000 MHz (proliferating)

Q/V band ?

Satellite Link Performance Factors

• Distance between earth station antenna and satellite

antenna

• For downlink, terrestrial distance between earth

station antenna and “aim point” of satellite

– Displayed as a satellite footprint

• Atmospheric attenuation

– Affected by oxygen, water, angle of elevation, and higher

frequencies

Elevation

Elevation:

angle between center of satellite beam

and surface

minimal elevation:

elevation needed at least

to communicate with the satellite

Atmospheric attenuation Example: satellite systems at 4-6 GHz

elevation of the satellite

5° 10° 20° 30° 40° 50°

Attenuation of

the signal in %

10

20

30

40

50

rain absorption

fog absorption

atmospheric

absorption

Applications

• Traditionally

– weather satellites

– radio and TV broadcast satellites

– military satellites

– satellites for navigation and localization (e.g., GPS)

• Telecommunication

– global telephone connections

– backbone for global networks

– connections for communication in remote places or

underdeveloped areas

– global mobile communication

replaced by fiber optics

Initial application of GEO Satellites:

Telephony

• 1965 Early Bird 34 kg 240 telephone ccts.

• 1968 Intelsat III 152 kg 1 500 circuits

• 1986 Intelsat VI 1,800 kg 33,000 circuits

• 2000 Large GEO 3000 kg 8 - 15 kW power

1,200 kg payload

Current GEO Satellite Applications:

• Broadcasting - mainly TV at present

– DirecTV, PrimeStar, etc.

• Point to Multi-point communications

– VSAT, Video distribution for Cable TV

• Mobile Services

– Motient (former American Mobile Satellite),

INMARSAT, etc.

Satellite Navigation:

GPS and GLONASS

• GPS is a medium earth orbit (MEO) satellite system

– GPS satellites broadcast pulse trains with very accurate time signals

– A receiver able to “see” four GPS satellites can calculate its position within 30 m anywhere in world

– 24 satellites in clusters of four, 12 hour orbital period

• “You never need be lost again”

– Every automobile and cellular phone will eventually have a GPS location read-out

base station

or gateway

Typical satellite systems

Inter Satellite Link

(ISL) Mobile User

Link (MUL) Gateway Link

(GWL)

footprint

small cells

(spotbeams)

User data

PSTN ISDN GSM

GWL

MUL

PSTN: Public Switched

Telephone Network

Space Segment

Satellite

TT&C Ground Station

Satellite System Elements

Ground Segment

Earth

Stations

Coverage Region

SCC

Space Segment

– Satellite Launching Phase

– Transfer Orbit Phase

– Deployment

– Operation

– TT&C - Tracking Telemetry and Command Station: Establishes an control and monitoring link with satellite. Tracks orbit distortions and allows correction planning. Distortions caused by irregular gravitational forces from non-spherical Earth and due to the influence of Sun and Moon forces.

– SSC - Satellite Control Center, a.k.a.:

– OCC - Operations Control Center

– SCF - Satellite Control Facility

Provides link signal monitoring for Link Maintenance and Interference monitoring.

– Retirement Phase

Space segment

Satellite Transponder

Ground Segment

Earth Station = Satellite Communication Station (air, ground or sea, fixed or mobile).

FSS – Fixed Satellite Service MSS – Mobile Satellite Service

Collection of facilities, users and applications.

System Design Considerations

Basic Principles

Signals

• Signals:

– Carried by wires as voltage or current

– Transmitted through space as electromagnetic waves.

– Analog: Voltage or Current proportional to signal. E.g. Telephone.

– Digital: Generated by computers.

Ex. Binary = 1 or 0 corresponding to +1V or –1V.

Separating Signals

• Up and Down:

– FDD: Frequency Division Duplexing.

f1 = Uplink

f2 = Downlink

– TDD: Time Division Duplexing.

t1=Up, t2=Down, t3=Up, t4=Down,….

– Polarization

V & H linear polarization

RH & LH circular polarizations

Separating Signals (so that many transmitters can use the same transponder

simultaneously)

• Between Users or “Channels” (Multiple Access):

– FDMA: Frequency Division Multiple Access; assigns

each transmitter its own carrier frequency f1 = User 1; f2 = User 2; f3 = User 3, …

– TDMA: Time Division Multiple Access; each

transmitter is given its own time slot t1=User_1, t2=User_2, t3=User_3, t4 = User_1, ...

– CDMA: Code Division Multiple Access; each

transmitter transmits simultaneously and at the same

frequency and each transmission is modulated by its

own pseudo randomly coded bit stream Code 1 = User 1; Code 2 = User 2; Code 3 = User 3

Digital Communication System

Current Trends in Satellite

Communications

• Bigger, heavier, GEO satellites with multiple roles

• More direct broadcast TV and Radio satellites

• Expansion into Ka, Q, V bands (20/30, 40/50 GHz)

• Massive growth in data services fueled by Internet

• Mobile services:

– May be broadcast services rather than point to point

– Make mobile services a successful business?

The Future for Satellite

Communications

• Growth requires new frequency bands

• Propagation through rain and clouds becomes a problem as RF frequency is increased

– C-band (6/4 GHz) Rain has little impact 99.99% availability is possible

– Ku-band (10-12 GHz) Link margin of 3 dB needed for 99.8% availability

– Ka-band (20 - 30 GHz) Link margin of 6 dB needed for 99.6% availability

The Future for Satellite

Communications

• Low cost phased array antennas for mobiles are needed

– Mobile systems are limited by use of omni-directional antennas

– A self-phasing, self-steering phased array antenna with 6 dB gain can quadruple the capacity of a system

– Directional antennas allow frequency re-use

The Future for Satellite

Communications • Expected revenues from all Satellite

Communications services should reach $75 billion by

2005

• Satellite Direct-to-Home (DTH) Video and Internet

services appear to be the major drivers

Orbital Mechanics

Part 1

Kinematics & Newton’s Law

• s = ut + (1/2)at2

• v2 = u2 + 2at

• v = u + at

• F = ma

s = Distance traveled in time, t

u = Initial Velocity at t = 0

v = Final Velocity at time = t

a = Acceleration

F = Force acting on the object

Newton’s

Second Law

FORCE ON A SATELLITE

• Force = Mass

Acceleration

• Unit of Force is a Newton

• A Newton is the force required to accelerate

1 kg by 1 m/s2

• Underlying units of a Newton are therefore

(kg)

(m/s2)

ACCELERATION FORMULA

• a = acceleration due to gravity = / r2 km/s2

• r = radius from center of earth

• = universal gravitational constant G multiplied by the mass of the earth ME

• is Kepler’s constant and = 3.9861352 105 km3/s2

• G = 6.672 10-11 Nm2/kg2 or 6.672 10-20 km3/kg s2 in the older units

FORCE ON A SATELLITE : 2

Inward (i.e. centripetal force)

Since Force = Mass Acceleration

If the Force inwards due to gravity = FIN then

FIN = m ( / r2)

= m (GME / r2)

F1 (Gravitational

Force)

v (velocity)

Why do satellites stay moving and

in orbit?

F2 (Inertial-Centrifugal

Force)

Orbital Velocities and Periods

Satellite Orbital Orbital Orbital

System Height (km) Velocity (km/s) Period

h min s

INTELSAT 35,786.43 3.0747 23 56 4.091

ICO-Global 10,255 4.8954 5 55 48.4

Skybridge 1,469 7.1272 1 55 17.8

Iridium 780 7.4624 1 40 27.0

FORCE ON A SATELLITE

Forces acting on a satellite in

a stable orbit around the earth.

Gravitational force is inversely

proportional to the square of

the distance between the

centers of gravity of the

satellite and the planet the

satellite is orbiting, in this case

the earth. The gravitational

force inward (FIN, the

centripetal force) is directed

toward the center of gravity of

the earth. The kinetic energy

of the satellite (FOUT, the

centrifugal force) is directed

diametrically opposite to the

gravitational force. Kinetic

energy is proportional to the

square of the velocity of the

satellite. When these inward

and outward forces are

balanced, the satellite moves

around the earth in a “free fall”

trajectory: the satellite’s orbit.

If FOUT = FIN

the object is in

FREE FALL

FREE FALL???

ORBIT LIMITS

Geographical Coordinates Earth Centric Coordinate System

The earth is at the center

of the coordinate system

Reference planes coincide

with the equator and the

polar axis

Orbital Plane Coordinates

The earth is at the

center of the coordinate

system but ………

Reference is the plane

of the satellite’s orbit

Balancing the Forces

Inward Force

r

mGME

F 3

rEquation (2.7)

F

G = Gravitational constant = 6.672 10-11 Nm2/kg2

ME = Mass of the earth (and GME = = Kepler’s constant)

m = mass of satellite

r = satellite orbit radius from center of earth

r= unit vector in the r direction (positive r is away from earth)

Balancing the Forces

Outward Force F

2

2

dt

dmF

rEquation (2.8)

Equating inward and outward forces we find

2

2

3 dt

d

r

rr Equation (2.9), or we can write

032

2

rdt

d rr Equation (2.10)

Second order differential

equation with six unknowns:

the orbital elements

• We have a second order differential equation

• See text for a way to find a solution

• If we re-define our co-ordinate system into polar

coordinates (see Fig.) we can re-write equation

as two second order differential equations in

terms of r0 and 0

THE ORBIT

Polar Coordinates

In the plane of the

orbit

Polar coordinate system in the plane of the satellite’s orbit. The plane of the orbit

coincides with the plane of the paper. The axis z0 is straight out of the paper from the

center of the earth, and is normal to the plane of the satellite’s orbit. The satellite’s

position is described in terms of the radius from the center of the earth r0 and the angle

this radius makes with the x0 axis, Φo.

THE ORBIT

• We have a second order differential equation

• If we re-define our coordinate system into polar coordinates

(see Fig. 2.3) we can re-write equation (2.5) as two second

order differential equations in terms of r0 and 0.

and

THE ORBIT

• Solving the two differential equations leads to six constants

(the orbital constants) which define the orbit, and three

laws of orbits (Kepler’s Laws of Planetary Motion)

• Johaness Kepler (1571 - 1630) a German Astronomer and

Scientist

KEPLER’S THREE LAWS

• Orbit is an ellipse with the larger body (earth) at

one focus

• The satellite sweeps out equal arcs in equal time

(NOTE: for an ellipse, this means that the orbital

velocity varies around the orbit)

• The square of the period of revolution equals a

CONSTANT ´ the THIRD POWER of SEMIMAJOR

AXIS of the ellipse

Review: Ellipse analysis

• Points (-c,0) and (c,0) are the foci.

•Points (-a,0) and (a,0) are the vertices.

• Line between vertices is the major axis.

• a is the length of the semimajor axis.

• Line between (0,b) and (0,-b) is the minor axis.

• b is the length of the semiminor axis.

12

2

2

2

b

y

a

x

222 cba

Standard Equation:

y

V(-a,0)

P(x,y)

F(c,0) F(-c,0) V(a,0)

(0,b)

x

(0,-b)

abA

Area of ellipse:

The orbit as it appears in the orbital plane, The point O is the center of the earth and the

point C is the center of the ellipse. The two centers do not coincide unless the

eccentricity, e, of the ellipse is zero (i.e., the ellipse becomes a circle and a = b). The

dimensions a and b are the semimajor and semiminor axes of the orbital ellipse,

respectively.

KEPLER 1: Elliptical Orbits

e = ellipse’s eccentricity

O = center of the earth (one

focus of the ellipse)

C = center of the ellipse

a = (Apogee + Perigee)/2

KEPLER 1: Elliptical Orbits (cont.)

Equation 2.17 in text:

(describes a conic section,

which is an ellipse if e < 1)

)cos(*1 0

0e

pr

e = eccentricity

e<1 ellipse

e = 0 circle

r0 = distance of a point in the orbit to the

center of the earth

p = geometrical constant (width of the

conic section at the focus)

p=a(1-e2)

0 = angle between r0 and the perigee

p

KEPLER 2: Equal Arc-Sweeps

Figure 2.5

Law 2

If t2 - t1 = t4 - t3

then A12 = A34

Velocity of satellite is

SLOWEST at APOGEE;

FASTEST at PERIGEE

Kepler’s Laws – 1 & 2

KEPLER 3: Orbital Period

Orbital period and the Ellipse are related by

T2 = (4 2 a3) / (Equation 2.21)

That is the square of the period of revolution is equal to a

constant the cube of the semi-major axis.

IMPORTANT: Period of revolution is referenced to inertial space, i.e., to

the galactic background, NOT to an observer on the surface of one of the

bodies (earth).

= Kepler’s Constant = GME

Kepler’s 3rd Law: T² = (4π²a³)/μ

μ = 3.986004418 × 105 km/s²

Numerical Example 1

The Geostationary Orbit:

Sidereal Day = 23 hrs 56 min 4.1 sec

Calculate radius and height of GEO orbit: • T2 = (4 2 a3) / (eq. 2.21)

• Rearrange to a3 = T2 /(4 2)

• T = 86,164.1 sec

• a3 = (86,164.1) 2 x 3.986004418 x 105 /(4 2)

• a = 42,164.172 km = orbit radius

• h = orbit radius – earth radius = 42,164.172 – 6378.14

= 35,786.03 km

Solar vs. Sidereal Day

• A sidereal day is the time between consecutive crossings of any

particular longitude on the earth by any star other than the sun.

• A solar say is the time between consecutive crossings of any

particular longitude of the earth by the sun-earth axis.

– Solar day = EXACTLY 24 hrs

– Sidereal day = 23 h 56 min. 4.091 s

• Why the difference?

– By the time the Earth completes a full rotation with respect to an

external point (not the sun), it has already moved its center

position with respect to the sun. The extra time it takes to cross

the sun-earth axis, averaged over 4 full years (because every 4

years one has 366 deays) is of about 3.93 minutes per day.

LOCATING THE SATELLITE IN

ORBIT: 1

Start with Fig. 2.6 in Text o is the True

Anomaly

See eq. (2.22)

C is the

center of the

orbit ellipse

O is the

center of the

earth

NOTE: Perigee and Apogee are on opposite sides of the orbit

LOCATING THE SATELLITE

IN ORBIT

• Need to develop a procedure that will allow the

average angular velocity to be used

• If the orbit is not circular, the procedure is to use a

Circumscribed Circle

• A circumscribed circle is a circle that has a radius

equal to the semi-major axis length of the ellipse and

also has the same center

Locate Satellite in Orbit

= Average angular velocity

E = Eccentric Anomaly

M = Mean Anomaly

M = arc length (in radians) that the

satellite would have traversed since

perigee passage if it were moving

around the circumscribed circle

with a mean angular velocity

ORBIT CHARACTERISTICS

Semi-Axis Lengths of the Orbit

21 e

pa

where

2hp

and h is the magnitude of

the angular momentum

See eq. (2.18)

and (2.16)

2/121 eab where Ch

e2

See eqn.

(2.19)

and e is the eccentricity of the orbit

ORBIT ECCENTRICITY

• If a = semi-major axis,

b = semi-minor axis, and

e = eccentricity of the orbit ellipse,

then

ba

bae

NOTE: For a circular orbit, a = b and e = 0

Time reference

• tp Time of Perigee = Time of closest

approach to the earth, at the same time, time

the satellite is crossing the x0 axis, according to

the reference used.

• t- tp = time elapsed since satellite last passed

the perigee.

ORBIT DETERMINATION 1:

Procedure:

Given the time of perigee tp, the eccentricity e and the length of the semimajor axis a:

• Average Angular Velocity (eqn. 2.25)

• M Mean Anomaly (eqn. 2.30)

• E Eccentric Anomaly (solve eqn. 2.30)

• ro Radius from orbit center (eqn. 2.27)

• o True Anomaly (solve eq. 2.22)

• x0 and y0 (using eqn. 2.23 and 2.24)