1. MAE 5410 – Astrodynamics Lecture 5 Orbit in Space Coordinate Frames and Time

2. Orienting the orbit plane So far, we’ve solved for the orbital motion in the orbital plane (PQW) which is given by the following parameters that can be calculated from a position and velocity at any epoch time Now we’ll orient the orbit plane (i.e. PQW) in space using three angles. Since the orbit is inertially fixed, we use the Earth Centered Inertial frame as a reference. ECI: The X-Y axes are the the Earth’s equatorial plane, with X pointing along the intersection of the equator and the ecliptic (vernal equinox or line of Aries) direction. Z is along the Earth spin axis. These directions change ever so slightly (Earth precession has 26,000 year period with a 18.6 year 9 arcmin nodding) so the vernal equinox direction at a particular time is used as a standard. Right now, J2000 is the standard reference. In 2025, we’ll switch to J2050.

3. Inclination, i Angle between the orbit plane and the equatorial plane Increasing the orbital inclination increases the maximum latitude of the groundtrack (in fact, the maximum latitude equals the orbit inclination)

4. Longitude of the Ascending Node,  Angle between the X-axis and the intersection of the orbit plane and equatorial plane (the nodal vector)

5. Argument of Perigee,  Angle from the nodal vector to the periapsis point (eccentricity vector, or )

6. Putting it all together

7. Some special cases

8. r(t) and v(t) in ECI In Lecture 3 we found the position and velocity in the PQW frame: In this lecture we defined orbital elements that locate the PQW frame wrt the ECI frame. To get from PQW to ECI, we perform a coordinate transformation:

9. Single Axis Rotations

10. Transformation from ECI to PQW First do a three axis rotation of , then a one axis rotation of I, then a three axis rotation of  :

11. r(t) and v(t) in ECF To get from PQW to ECI we invert the previous transformation, which turns out to just be the transpose: To get from ECI to ECF we rotate through the Greenwich mean sidereal time: ECF Greenwich meridian GST ECI

12. r(t) in SEZ To get from ECF to the topocentric-horizon frame, SEZ, we rotate through latitude,, and longitude,  and subtract off the position vector to the site on the Earth: ECF SEZ  This vector can then be used to find the azimuth and elevation of the satellite with respect to the observer on the ground