Step 10 - Finding the Geocentric Cartesian Coordinates

Element	_Symbol	Value
Epoch	t_o	05:54:21.171 UTC December 16, 2007
Inclination	i	51^o.9970
R.A. of the Ascending Node	a_W₀	251^o.0219
Eccentricity	e	0.0001492
Argument of Perigee	w₀	33^o.8641
Mean Anomaly at TLE Epoch	M_o	326^o.2322
Mean Motion	n	12.62256095 orbits / solar day
Propagation Time	Dt	1.7677141 solar days
Mean Anomaly at Time t	M(t)	1.37777389 radians (78^o.940629)
True Anomaly at Time t	n(t)	78^o.95065818
Semi-major Axis	a	7791.787473 km
Perigee Distance	P	7790.624938 km
Geocentric Distance	r(t)	7791.564499 km
Precessed R.A. of Asc. Node	a_W(t)	245^o.6400244
Precessed Arg. of Perigee	w(t)	37^o.77767416
Argument of Latitude	m(t)	116^o.7283323
R.A. Difference	Da	129^o.278845
Geocentric R.A.	a_g	14^o.9188694
Geocentric Declination	d_g	+44^o.73125163

The geocentric equatorial coordinates (R.A. and Dec.) of the satellite are known, but they are useless to observers on the surface of the Earth. We need to translate these coordinates to a location on the surface of the Earth that we wish to observe the satellite from.

First, we need to convert the satellite polar coordinates (r(t), a_g, d_g) into their equivalent Geocentric Cartesian Coordinates (x_g, y_g, z_g) using the following equations:

x_g = r(t)cosa_gcosd_g
y_g = r(t)sina_gcosd_g
z_g = r(t)sind_g

x_g = 5348.663965 km
y_g = 1425.05581 km
z_g = 5483.565179 km

In order to confirm that the Cartesian coordinates have been calculated correctly, add the sum of the squares of the answers and take the number's square root. The number you finally arrive at should be the original geocentric satellite distance, r(t):

x_g² + y_g² + z_g² should = r(t)²

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Step 10: Geocentric Cartesian Was Last Modified On April 01, 2014