Orbital Aspects of Satellite Communications

GMU - TCOM 507 - Spring 2001 Class: Jan-25-2001

(C) Leila Z. Ribeiro, 2001 1

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ORBITAL MECHANICS

A Compilation by: M.LENIN BABU,M.Tech.,

Lecturer,Dept. of ECE, Bapatla Engineering College

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Topics covered according to

syllabus

• Kepler’s laws of motion

• Locating the satellite in the orbit

• Locating the satellite w.r.t earth.

• Orbital elements

• Look angle determination



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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

4

FORCE ON A SATELLITE : 1

• 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)

• In Imperial Units 1 Newton = 0.2248 ft lb.



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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

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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)



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Reference Coordinate Axes 1:

Earth Centric Coordinate

System

Fig. 2.2 in text

The earth is at the

center of the coordinate

system

Reference planes

coincide with the

equator and the polar

axis

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Reference Coordinate Axes

2: Satellite Coordinate

System

Fig. 2.3 in text

The earth is at the

center of the

coordinate system and

reference is the plane

of the satellite’s orbit



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Balancing the Forces - 2

Inward Force

r

mGME

F 3

r

Equation (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)

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Balancing the Forces - 3

Outward Force F

2

2

dt

dmF

r

Equation (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



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• We have a second order differential

equation

• See text p.21 for a way to find a solution

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

polar coordinates (see Fig. 2.4) we can re-

write equation (2.11) as two second order

differential equations in terms of r0 and 0

THE ORBIT - 1

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THE ORBIT - 2

• 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



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KEPLER’S THREE LAWS

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

• The satellite sweeps out equal arcs (area) 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 SEMI-MAJOR AXIS of the ellipse

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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.

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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:



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KEPLER 1: Elliptical Orbits

Figure 2.6 in text

Law 1

The orbit is an

ellipse

e = ellipse’s eccentricity

O = center of the earth

(one focus of the ellipse)

C = center of the ellipse

a = (Apogee + Perigee)/2

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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



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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

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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



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Numerical Example 1

The Geostationary Orbit: Sidereal Day = 23 hrs 56 min 4.1 sec

Calculate radius and height of GEO orbit:

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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



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IN ORBIT: 2 • 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

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IN ORBIT: 3 Fig. 2.7 in the text

= 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



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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

Che

2

See eqn.

(2.19)

and e is the eccentricity of the

orbit

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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



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It is related to the radius ro by

ro = a(1-ecosE)

Thus

a - ro = aecosE

We can develop an expression that relates eccentric anomaly E to the average angular velocity η, which yields

η dt = (1-ecosE) dE

Let tp be the time of perigee. This is simultaneously the time of closest approach to the earth; the time when the satellite is crossing the xo axis; and the time when E is zero. Integrating both sides of the equation we obtain

η (t – tp) = E – e sin E

The left side of the equation is called the mean anomaly, M. Thus

M =η (t – tp) = E – e sin E

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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.



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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 (from eqn. 2.30)

• ro Radius from orbit center (eqn. 2.27)

• o True Anomaly ( eq. 2.22)

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

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ORBIT DETERMINATION 2:

• Orbital Constants allow you to determine coordinates (ro, o) and (xo, yo) in the orbital plane

• Now need to locate the orbital plane with respect to the earth

• More specifically: need to locate the orbital location with respect to a point on the surface of the earth



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WITH RESPECT TO THE EARTH

• The orbital constants define the orbit of the

satellite with respect to the CENTER of the earth

• To know where to look for the satellite in space,

we must relate the orbital plane and time of

perigee to the earth’s axis

NOTE: Need a Time Reference to locate the satellite. The

time reference most often used is the Time of Perigee, tp

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GEOCENTRIC EQUATORIAL

COORDINATES - 1

• zi axis Earth’s rotational axis (N-S poles

with N as positive z)

• xi axis In equatorial plane towards FIRST

POINT OF ARIES

• yi axis Orthogonal to zi and xi

NOTE: The First Point of Aries is a line from the

center of the earth through the center of the sun at

the vernal equinox (spring) in the northern

hemisphere



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GEOCENTRIC EQUATORIAL

COORDINATES - 2

Fig. 2.8 in text

To First Point of Aries

RA = Right Ascension

(in the xi,yi plane)

= Declination (the

angle from the xi,yi plane

to the satellite radius)

NOTE: Direction to First Point of Aries does NOT rotate

with earth’s motion around; the direction only translates

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LOCATING THE SATELLITE - 1

• Find the Ascending Node

–Point where the satellite crosses

the equatorial plane from South to

North

–Define and i

–Define

Inclination

Right Ascension of the Ascending

Node (= RA from Fig. 2.6 in text)

See next slide



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DEFINING PARAMETERS

Orbit passes through

equatorial plane here

First Point

of Aries

Fig. 2.9 in text

Center of earth

Argument of Perigee

Right Ascension Inclination

of orbit

Equatorial plane

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DEFINING PARAMETERS 2



35

36


• and i together locate the Orbital

plane with respect to the

Equatorial plane.

• locates the Orbital coordinate

system with respect to the

Equatorial coordinate system.



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• Astronomers use Julian Days or Julian Dates

• Space Operations are in Universal Time

Constant (UTC) taken from Greenwich Meridian

(This time is sometimes referred to as “Zulu”)

• To find exact position of an orbiting satellite at a

given instant, we need the Orbital Elements

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ORBITAL ELEMENTS (P. 29)

• Right Ascension of the Ascending Node

• i Inclination of the orbit

• Argument of Perigee (See Figures 2.6 &

2.7 in the text)

• tp Time of Perigee

• e Eccentricity of the elliptical orbit

• a Semi-major axis of the orbit ellipse (See

Fig. 2.4 in the text)



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Numerical Example 2: Given a Space Shuttle Circular orbit (height = h = 250

km). Use earth radius = 6378 km. Determine:

a. Period = ?

b. Linear velocity = ?

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Numerical Example 3: Elliptical Orbit: Perigee = 1,000 km, Apogee = 4,000 km

a. Period = ?

b. Eccentricity = ?

Orbital Aspects of Satellite Communications

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