Element Symbol Value Epoch to 05:54:21.171 UTC December 16, 2007 Inclination i 51o.9970 R.A. of the Ascending Node aW0 251o.0219 Eccentricity e 0.0001492 Argument of Perigee w0 33o.8641 Mean Anomaly at TLE Epoch Mo 326o.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 (78o.940629) True Anomaly at Time t n(t) 78o.95065818 Semi-major Axis a 7791.787473 km Perigee Distance P 7790.624938 km Geocentric Distance r(t) 7791.564499 km

Some orbit elements do not remain the same after the Epoch time (t0) has passed. All satellite orbits precess in a periodic fashion.

Specifically, R.A. of Ascending Node and the Argument of Perigee will precess because of the spheroid shape of the Earth. This step is more commonly known as "J2 propagation". These two orbit elements can vary quickly over the course of a few hours. Since our chosen propagation time is over 24 hours, correcting for these critical precession terms is essential.

First, several specific constants have to be determined:

Let a1 = a / (km/Earth Radii); a1 = 7791.787473 km / (6378.135 km/Earth Radius) = 1.221640413 Earth Radii

Let J2 = 1.0826267 x 10-3 (This is the second gravitational zonal harmonic of the Earth: i.e. J2 Propagation)

Let d1 = [ 3J2RE2 (3cos2i - 1) ] / [ 4a12 (1-e2)3/2 ]: d1 = 7.468325145 x 10-5

Let a0 = -a1 [ 134d13/81 + d12 + d1/3 - 1 ]; a0 = 1.221609994 Earth Radii

Let p0 = a0 (1-e2); p0 = 1.221609967 Earth Radii

The precession of the R.A. of Ascending Node will have the following form:

DaW = 360o [ -3J2RE2nDtcosi / 2po2 ]

aW(t) = aW + DaW

aW(t) = 251o.0219 + (-5o.381875553) = 245o.6400244

The precession of the Argument of Perigee will have the following form:

Dw = 360o [ 3J2RE2nDt (5cos2i - 1) / 4po2 ]

w(t) = w + Dw

w(t) = 33o.8641 + 3o.913574162 = 37o.77767416

Special Note: The precession formulae for this section was taken directly from "The SGP Model" section of the NORAD SpaceTrack Report No. 3 technical paper. Many thanks to Felix R. Hoots, Ronald L. Roehrich and T.S. Kelso who wrote and compiled this wealth of information that made this Orbit Precession section possible.

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