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ECE 5233 Satellite Communications

ECE 5233 Satellite Communications. Prepared by: Dr . Ivica Kostanic Lecture 2: Orbital Mechanics (Section 2.1). Spring 2014. Outline . Kepler’s laws of planetary / satellite motion Equation of satellite orbits Describing the orbit of a satellite Locating the satellite in the orbit

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ECE 5233 Satellite Communications

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

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

  3. Kepler’s laws of planetary motion • Johannes Kepler published laws of planetary motion in solar system in early 17th century • Laws explained extensive astronomical planetary measurements performed by Tycho Brahe • Kepler’s laws were proved by Newton’s theory of gravity in mid 18th century • Kepler’s laws approximate motion of satellites around Earth • Kepler’s laws (as applicable to satellite motion) • The orbit of a satellite is an ellipse with the Earth at one of the two foci • A line joining a satellite and the Earth’s center sweeps out equal areas during equal intervals of time • 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

  4. 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 Gravitational force on the satellite Newton’s 2nd law Constants Combining the two Differential equation that determines the orbit Satellite in Earth’s orbit

  5. Derivation of satellite orbit (2) Solution of the motion differential equation gives trajectory in the form of an ellipse Note: Detailed derivations of the satellite trajectory are given in the notes • Coordinate system – rotated so that the satellite plane is the same as (X0,Y0) 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)

  6. 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: • Relationship between a and p • Relationship between b and p • Relationship between eccentricity, perigee and apogee distances • 2ndKepler’s law • 3rdKepler’s law Elliptic trajectory – cylindrical coordinates Basic relationship of ellipse

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

  8. Describing the orbit of a satellite (3) 4. 2ndKepler’s law The area swept by radius vector 5. 3ndKepler’s law Integrating both sides

  9. Locating the satellite in the orbit (1) • Known: time at the perigee tp • Determine: location of the satellite at arbitrary time t>tp 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

  10. Locating the satellite in the orbit (2) • Algorithm summary: • Calculate average angular velocity: • Calculate mean anomaly: • Solver for eccentric anomaly: • Find polar coordinates: • Find rectangular coordinates 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

  11. 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 Note: Examples are worked out in notes

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