Lecture 2, Space Segment

 Presentation / Lecture 2, Space Segment Date Submitted: 06 June 2001 Written by RPC Telecommunications Ltd.. Website: http://www.rpctelecom.com ExcellentGoodAveragePoorVery Poor   Rate This Article This is the second in the series of general satcom tutorial lectures submitted by RPC Telecommunications. Comment On This Article
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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.