Quesion :

Determine the orbit of an object that you have observed. You will determine the object state vector at a time of the first observation and create a simulations of the orbital trajectory.

Solution :

The steps followed to solve the problems are as follows:

1. Determine the UTC (Coordinated Universal Time) for the three observations to calculate the Local Sidereal Time for each observation. Nairobi is located at UTC + 3 so the time in UTC is the time given – 3.
2. Calculate the Local Sidereal Time for each observation

   Date	Local Time*	Azimuth Angle (deg.)	Elevation Angle (deg.)	Range (km)	Local Sidereal Time	Universal Time
   17-05-20	12:06 AM	272.4822	21.9305	39364	229.322	21:06
   17-05-20	1:00 AM	272.4401	22.0888	39348	242.859	22:00
   17-05-20	1:30 AM	272.3864	22.5174	39307	250.38	22:30

3. Calculate the three geocentric position vectors corresponding to the three sets of azimuth, elevation and slant range data at each time given the earth observation location:
   1. Using the altitude H, latitude φ and local sidereal time θ of the site, calculate its geocentric position vector R from the following equation:
   2. Calculate the topocentric declination δ using the following equation:
   3. Calculate the topocentric right ascension α using the following equations:
   4. Calculate the direction cosine unit vector using the following equation:
   
   1. Calculate the geocentric position vector r using the following equation:
   The results obtained by MATLAB were as the following:
4. Applying Gibbs’ Method to the three obtained position vectors to get the velocity vector corresponding to the second position vector: - The result was as following: We take the initial r and v to be r2 and v2:
5. Knowing r and v the 6 COE could be obtained easily using the below flow graph [1] and the results were:

Since the eccentricity is between 0 and 1, the orbit is elliptical.