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## ECE 5233 Satellite Communications

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**ECE 5233 Satellite Communications**Prepared by: Dr. Ivica Kostanic Lecture 3: Orbital Elements (Sections 2.2-2.7) Spring 2014**Outline**• Orbital elements (geocentric equatorial coordinates) • Rotating coordinate system • Two Line Element (TLE) data • Mapping between coordinate systems • Examples Important note: Slides present summary of the results. Detailed derivations are given in notes.**Geocentric equatorial coordinate system (GEC)**• GEC – fixed rectangular coordinate system • GEC – moves through the space, but does not rotate • Used in astronomy to map the sky • The angles of interest • W - right ascension – angle from positive x-axis to the point where satellite comes out of the equatorial plane • i– inclination of the orbit – angle between orbital plane and equatorial plane • w – argument of perigee – angular distance between perigee and the point where the satellite comes out of the equatorial plane X axis points to “first point of Aries” – distant star All satellites have their GEC coordinates given in “Two line elements” (TLE) data**Example: Two line data for space station**TLE data – used by NORAD and NASA TLA – data is used for precise calculations of satellite positions Access: http://celestrak.com/NORAD/elements/**Rotating rectangular system**• Natural way to view space objects if you are on Earth • System is fixed to the Earth (i.e. it translates and rotates along with the Earth) • X-axis goes through (0,0) lat-lon point • In summary: 3 systems are used • Orbital systems • GEC system • Rotating system • Position of the satellite is mapped between the coordinated systems using linear transformations Angular velocity of Earths rotation (72 urad/sec) Time since last alignment between GEC and rotating system Rotating and GEC systems align once/day (at different times)**Transformation between coordinate systems**Mapping between orbital system and GEC Mapping between GEC and rotating system Mapping between orbital and and rotating system**Calculation of**Angle between GEC and rotating system t – time in min after Universal Time midnight Julian day reference point: Noon of December 31st, 1899; Start of JD 2415020 JD calculator: http://www.nr.com/julian.html**Calculation of**Example: calculate WeTe for January 15th, 2011 at 5PM EST 1. Calculate t (A:1320) 2. Determine JD (A: 2455577) 3. Use spreadsheet above Answer: ~ 85 degrees**Six orbital elements**• To specify position of a satellite one needs 6 orbital elements • Selection somewhat arbitrary • Quantities adopted by the text • Eccentricity (e) • Semi-major axis (a) • Time at the perigee (tp) • Right ascending node angle (W) • Inclination (i) • Argument of the perigee (w) • Quantities adopted by the TLE data • Eccentricity (e) • Mean motion in rev/day (Mm) • Mean anomaly (M) • Right ascending node angle (W) • Inclination (i) • Argument of the perigee (w) Note: TLE data is given for a given time reference For calculation of time at perigee For calculation of semi-major axis**Example 1.**Calculate rotating coordinates for ISS at the time when TLE data are taken TLE Data for ISS (obtained on OCT 26, 2013): 1 25544U 98067A 13298.22562148 .00015844 00000-0 27472 -3 0 8812 2 25544 51.6491184.0276 000228277.223068.9667 15.4953682854871 Note: Calculation details are given in notes. Some results are as follows: • Eccentric anomaly E = 1.204173 • Semi-major axis: a = 6783.8km • Orbital coordinates: x0 = 2430.21km; y0 = 6332.95km • Rotating coordinates: xr = -4201.9km, yr = -4428.57km, zr = 2957.01km