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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: orbital-mechanics
      2. Calculate the topocentric declination δ using the following equation: orbital-mechanics
      3. Calculate the topocentric right ascension α using the following equations: orbital-mechanics
      4. Calculate the direction cosine unit vector using the following equation:
    orbital-mechanics
    1. Calculate the geocentric position vector r using the following equation: orbital-mechanics
    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: orbital-mechanics

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

 

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