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ECE 5233 Satellite Communications Prepared by: Dr. Ivica Kostanic Lecture 3: Orbital Elements (Sections 2.2-2.7) Spring 2014

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Florida Institute of technologies Page 2 Orbital elements (geocentric equatorial coordinates) Rotating coordinate system Two Line Element (TLE) data Mapping between coordinate systems Examples Outline Important note: Slides present summary of the results. Detailed derivations are given in notes.

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Florida Institute of technologies 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 - right ascension – angle from positive x-axis to the point where satellite comes out of the equatorial plane oi – inclination of the orbit – angle between orbital plane and equatorial plane – argument of perigee – angular distance between perigee and the point where the satellite comes out of the equatorial plane Page 3 X axis points to “first point of Aries” – distant star All satellites have their GEC coordinates given in “Two line elements” (TLE) data

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Florida Institute of technologies Example: Two line data for space station Page 4 TLE data – used by NORAD and NASA TLA – data is used for precise calculations of satellite positions Access: http://celestrak.com/NORAD/elements/

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Florida Institute of technologies 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 Page 5 Rotating and GEC systems align once/day (at different times) In summary: 3 systems are used o Orbital systems o GEC system o Rotating system o 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

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Florida Institute of technologies Transformation between coordinate systems Page 6 Mapping between orbital system and GEC Mapping between GEC and rotating system Mapping between orbital and and rotating system

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Florida Institute of technologies Calculation of Page 7 Angle between GEC and rotating system t – time in min after Universal Time midnight Julian day reference point: Noon of December 31 st, 1899; Start of JD 2415020 JD calculator: http://www.nr.com/julian.html

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Florida Institute of technologies Calculation of Page 8 Example: calculate e T e for January 15 th, 2011 at 5PM EST 1. Calculate t (A:1320) 2. Determine JD (A: 2455577) 3. Use spreadsheet above Answer: ~ 85 degrees

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Florida Institute of technologies Six orbital elements To specify position of a satellite one needs 6 orbital elements Selection somewhat arbitrary Page 9 Quantities adopted by the text o Eccentricity ( e ) o Semi-major axis ( a ) o Time at the perigee ( t p ) o Right ascending node angle ( ) o Inclination ( i ) o Argument of the perigee ( ) Quantities adopted by the TLE data o Eccentricity ( e ) o Mean motion in rev/day ( M m ) o Mean anomaly ( M ) o Right ascending node angle ( ) o Inclination ( i ) o Argument of the perigee ( ) Note: TLE data is given for a given time reference For calculation of time at perigee For calculation of semi-major axis

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Florida Institute of technologies 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 51.6491 184.0276 0002282 77.2230 68.9667 15.4953682854871 Note: Calculation details are given in notes. Some results are as follows: o Eccentric anomaly E = 1.204173 o Semi-major axis: a = 6783.8km o Orbital coordinates: x0 = 2430.21km; y0 = 6332.95km o Rotating coordinates: xr = -4201.9km, yr = -4428.57km, zr = 2957.01km Page 10

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