Lecture 2, Space Segment

Presentation / Lecture 2, Space Segment

Date Submitted: 06 June 2001

Written by RPC Telecommunications Ltd.. Website: http://www.rpctelecom.com

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This is the second in the series of general satcom tutorial lectures submitted by RPC Telecommunications.

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-Section 1
Orbit Theory: Kepler's Laws.
-Section 2
Orbit Theory: Formulae.
-Section 3
The Real World.
-Section 4
Types & Implications of Orbit.
-Section 5
Satellite Hardware.
-Section 6
Launch Vehicles.
-Section 7
Flight Plan & Maintaining Satellite In Orbit.
-Section 8
Stationkeeping & Stabilisation.

Laws Of Motion

Kepler's laws were originally developed to describe the motion of planetary bodies, but they are just as relevant for theoretical satellite orbits around the earth.

  • Each planet moves around the sun in a ellipse with the sun at one focus (i.e motion is in a plane).
  • A line from the sun to a planet (radius vector) sweeps out equal areas in equal intervals of time.
  • The ratio of the square of the orbit period (T) to the cube of the semi-major axis (a) is a constant (i.e T2/a3 = constant)

The geometry of an ellipse:

and: c2=a2-b2

Eccentricity (e) is defined as:

e2=1-(b/a)2

Newton's universal law of gravitation tells us that:

F=(G.M.m)/r2

From Newton's Second Law of Motion we get:

F=m.a

Centripetal acceleration (assume circular motion for simplicity):

a=v2/r

Taking these three expressions, and simplifying we get the following equation:

v=(G.M/r)0.5

i.e ω=(G.M/r3)0.5

Keplerian Elements
The "Keplerian Elements" are a set of 6 parameters which define the position of a satellite at a given epoch (date and time) as well as defining the shape and position of its elliptical orbit around the Earth.

The 6 parameters are outlined below.

  • Right Ascention of the Ascending Node
    The angle in the equatorial plane between the direction of the first point of Aries and the direction to the ascending node, measured counter-clockwise when viewed from the north side of the equatorial plane.
  • Argument of Perigee
    The angle in the plane of the satellite orbit between the ascending node and the perigee, measured in the direction of the satellite motion.

  • Mean Anomaly
    The angle at the Earth's centre, measured from the perigee in the direction of the satellite's motion, which the satellite would have if it were moving at a constant angular speed, i.e M = 2*π*t/T (radians), where T is the orbital period.
  • Semi Major Axis
    Half the length of the longest diameter of the ellipse.
  • Inclination
    The angle between the Earth's equatorial plane and the plane of the satellite, measured positively above the equatorial plane with reference to the ascending segment of the orbit.
  • Eccentricity
    A term defining the "circularity" of the ellipse; e=0 is a cirlce, e = 1 is a line (i.e a totally flat ellipse).

With these elements at a known epoch, it is possible to fix the position of the satellite in space and predict its theoretical future movement.

Simplifying Assumptions
  • Consider the Earth to be stationary and the origin of all coordinates is located at the Earth's centre
  • Consider the satellite and the Earth (much more significant) to be spherically symmetric (i.e treat m and M as point masses)
  • Assume that the satellite is subject only to one force - the gravitational attraction from the Earth - which varies as the inverse ratio to the square of the distance between centres of mass (this is highly unrealistic in the real world)

Note: The theoretical equations can fully determine the relative orbital motion of two bodies, but the case of three bodies has no analytical solution.

Next: Section 2 - Orbit Theory: Formulae.

 
 

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