R_ECI

The Earth Center Inertial (ECI) Cartesian coordinates of the observation location will need to be determined for all three observation times using the location's geodetic longitude, latitude, altitude above sea level and the sidereal times.

To do this, we first need to determine the distance from the geocentric center of the Earth to the observation location. The equation is already known from Step 13 of the Orbit Propagation section of this website:

R_obs = { [ cosl / R_E ]² + [ sinl / R_P ]² }^-1/2 + A

where: R_E = the Equatorial Radius of the Earth = 6378.137 km; and
            R_P = the Polar Radius of the Earth = 6356.752 km;

R_obs = 6367.8243 km

Next, we need to determine the sidereal time for all three observation times. In order to accomplish this, we can choose a convenient time to set a benchmark sidereal time and then use the time elapsed between the benchmark time and each of the three observation times to find our individual sidereal times.

For this example, the benchmark time (t₀) will be 23:00:00.00 UTC January 3, 2012, which exactly corresponds to t₁. The corresponding benchmark sidereal time can be found using any planetarium software. Make sure that you have your longitude, latitude and altitude correctly entered in your software of choice before generating the benchmark sidereal time. For the geodetic coordinates in this example, the benchmark sidereal time was found to be:

a_s0 = 0^h.7291666

The second and third sidereal times that correspond to the second and third observation times can be found by using the following equation:

a_si = 15^o/hr [a_s0 + 1.0027379 (t_i - t_o) / 3600 ]

where i = the observation number (1, 2, or 3)

t₁ - t₀ = 0 (since t₁ = t₀ in this example)
t₂ - t₀ = 328.22 seconds
t₃ - t₀ = 670.00 seconds

a_s1 = 10^o.937499
a_s2 = 12^o.308827
a_s3 = 13^o.736809

Now the ECI Matrix can be determined using the following equations:

R_{ECI1i =}  R_obs coslcosa_si
R_{ECI2i =}  R_obs coslsina_si
R_{ECI3i =}  R_obs sinl

where i = the observation number and also the column number of the ECI Matrix (1, 2, or 3)

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