Final answer:
This question involves finding various vectors and orbital elements for a satellite observed from a ground station. It requires conversion between different coordinate systems and calculations based on the given data.
Step-by-step explanation:
(a) To find the range vector and the relative velocity vector in the topocentric horizon basis, we need to convert the given data into Cartesian coordinates. The range vector can be found by multiplying the range magnitude by the unit vector in the direction of the satellite's azimuth and elevation angles. The relative velocity vector can be found by multiplying the range rate magnitude by the unit vector in the direction of the satellite's azimuth and elevation rates.
(b) To find the Local Sidereal Time (LST) of the tracking station, we need to calculate the Greenwich Sidereal Time (GST) at the given time and longitude. LST can be found by adding the East longitude to the GST.
(c) To find the radius and velocity vectors in the geocentric equatorial frame in the topocentric horizon basis, we need to convert the range vector and relative velocity vector into the geocentric equatorial frame. This can be done using the rotation matrix.
(d) To find the radius and velocity vectors in the geocentric equatorial frame in the geocentric equatorial basis, we need to convert the range vector and relative velocity vector from the topocentric horizon basis to the geocentric equatorial basis using the rotation matrix.
(e) The orbital elements can be determined using the given data, such as the range, range rate, azimuth, and azimuth rate. The orbital elements include the semi-major axis, eccentricity, inclination, right ascension of the ascending node, argument of periapsis, and true anomaly.