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 t – time in min after Universal Time midnight 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

Example: calculate WeTe for January 15th, 2011 at 5PM EST 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 51.6491 184.0276 0002282 77.2230 68.9667 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