Variable Symbol Value
Time 1 t1 23:00:00.00
Time 2 t2 23:05:28.22
Time 3 t3 23:11:10.00
Longitude q -76.889444
Latitude l +44.121944
Altitude A 0.079 km

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:

Robs = { [ cosl / RE ]2 + [ sinl / RP ]2 }-1/2 + A

where: RE = the Equatorial Radius of the Earth = 6378.137 km; and
            RP = the Polar Radius of the Earth = 6356.752 km;

Robs = 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:

as0 = 0h.7291666

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

asi = 15o/hr [as0 + 1.0027379 (ti - to) / 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

as1 = 10o.937499
as2 = 12o.308827
as3 = 13o.736809

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

RECI1i =  Robs coslcosasi
RECI2i =  Robs coslsinasi
RECI3i =  Robs sinl

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

4488.1674 km 4466.1251 km 4440.4533 km
RECI = 867.33155 km 974.49353 km 1085.4885 km
4433.2015 km 4433.2015 km 4433.2015 km







Step 4: ECI Matrix Was Last Modified On September 23, 2013