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ECE 5233 Satellite Communications Prepared by: Dr. Ivica Kostanic Lecture 2: Orbital Mechanics (Section 2.1) Spring 2014

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Florida Institute of technologies Page 2 Kepler’s laws of planetary / satellite motion Equation of satellite orbits Describing the orbit of a satellite Locating the satellite in the orbit Examples Outline Important note: Slides present summary of the results. Detailed derivations are given in notes.

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Florida Institute of technologies Kepler’s laws of planetary motion Johannes Kepler published laws of planetary motion in solar system in early 17 th century Laws explained extensive astronomical planetary measurements performed by Tycho Brahe Kepler’s laws were proved by Newton’s theory of gravity in mid 18 th century Kepler’s laws approximate motion of satellites around Earth Page 3 Kepler’s laws (as applicable to satellite motion) 1.The orbit of a satellite is an ellipse with the Earth at one of the two foci 2.A line joining a satellite and the Earth’s center sweeps out equal areas during equal intervals of time 3.The square of the orbital period of a satellite is directly proportional to the cube of the semi-major axis of its orbit. Illustration of Kepler’s law

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Florida Institute of technologies Derivation of satellite orbit (1) Based on Newton’s theory of gravity and laws of motion Satellite moves in a plane that contains Earth’s origin Acting force is gravity Mass of Earth is much larger than the mass of a satellite Page 4 Satellite in Earth’s orbit Gravitational force on the satellite Newton’s 2 nd law Combining the two Constants Differential equation that determines the orbit

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Florida Institute of technologies Derivation of satellite orbit (2) Note: Detailed derivations of the satellite trajectory are given in the notes Page 5 Solution of the motion differential equation gives trajectory in the form of an ellipse Coordinate system – rotated so that the satellite plane is the same as (X 0,Y 0 ) plane Not all values for eccentricity give stable orbits Eccentricity in interval (0,1) gives stable elliptical orbit Eccentricity of 0 gives circular orbit Eccentricity = 1, parabolic orbit, the satellite escapes the gravitational pull of the Earth Eccentricity > 1, hyperbolic orbit, the satellite escapes gravitational pull of the Earth p = 1; e = 0.2 fi = 0:0.01:2*pi; r = p./(1+cos(fi)); polar(fi,r)

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Florida Institute of technologies Describing the orbit of a satellite (1) E and F are focal points of the ellipse Earth is one of the focal points (say E) a – major semi axis b – minor semi axis Perigee – point when the satellite is closest to Earth Apogee – point when the satellite is furthest from Earth The parameters of the orbit are related Five important results: 1.Relationship between a and p 2.Relationship between b and p 3.Relationship between eccentricity, perigee and apogee distances 4.2 nd Kepler’s law 5.3 rd Kepler’s law Page 6 Elliptic trajectory – cylindrical coordinates Basic relationship of ellipse

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Florida Institute of technologies Describing the orbit of a satellite (2) Page 7 1. Relationship between a and p 2. Relationship between b and p Consider point P:FP+EP=2 a Since FP=EP, EP= a From triangle CEP 3. Relationship between eccentricity, perigee and apogee distances

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Florida Institute of technologies Describing the orbit of a satellite (3) Page 8 4. 2 nd Kepler’s law The area swept by radius vector 5. 3 nd Kepler’s law Integrating both sides

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Florida Institute of technologies Locating the satellite in the orbit (1) Known: time at the perigee t p Determine: location of the satellite at arbitrary time t>t p Page 9 Definitions: S – satellite O – center of the Earth C – center of the ellipse and corresponding circle - distance between satellite and center of the Earth - “true anomaly” - “eccentric anomaly” A circle is drawn so that it encompasses the satellite’s elliptical trajectory - average angular velocity - mean anomaly

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Florida Institute of technologies Locating the satellite in the orbit (2) Algorithm summary: 1.Calculate average angular velocity: 2.Calculate mean anomaly: 3.Solver for eccentric anomaly: 4.Find polar coordinates: 5.Find rectangular coordinates Page 10 Notes: Detailed derivations provided in the notes In 3, solution is determined numerically In 4, equation for true anomaly gives two values. One of them needs to be eliminated

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Florida Institute of technologies Examples Example 2.1.1.Geostationary orbit radius Example 2.1.2 Low earth orbit Example 2.1.3 Elliptical orbit Example C1.Location of satellite in the orbit Page 11 Note: Examples are worked out in notes

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