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

2. 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.

3. 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

4. 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:

5. 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)

6. Transformation between coordinate systems
Mapping between orbital system and GEC
Mapping between GEC and rotating system
Mapping between orbital and and rotating system

7. 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
JD calculator:

8. 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: )
3. Use spreadsheet above
Answer: ~ 85 degrees

9. 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

10. Example 1.
Calculate rotating coordinates for ISS at the time when TLE data are taken
TLE Data for ISS (obtained on OCT 26, 2013):
U 98067A
Note: Calculation details are given in notes.
Some results are as follows:
Eccentric anomaly E =
Semi-major axis: a = km
Orbital coordinates: x0 = km; y0 = km
Rotating coordinates: xr = km, yr = km, zr = km