Apsidal Precession

The Earth's equatorial bulge causes orbits to drift, with westward orbit perigee and apogee drifting westward. From AE Roy Orbital Motion 1978:

\Omega

longitude of the ascending node

a_0

semimajor axis

e

eccentricity

i

inclination

\mu

standard gravitational parameter, 398600.4418 km³/s² for Earth

J_2

zonal ablateness factor

p

p ~=~ a ( 1 - e^2 ) = r_p r_a / a

n_0

unperturbed mean motion

\bar{n}

perturbed mean motion

R

Earth Equatorial Radius = 6378.137 km

{n_0}^2 ~=~ \mu / {a_0}^3 . . . unperturbed mean motion

\bar{n} ~=~ n_0 \left[ 1 + { \large { { 3 ~ J_2 R^2 } \over { 2 ~~~ p^2 ~ } } } \left( 1 - {\large { 3 \over 2 } } \sin( i )^2 \right) (1-e^2)^{1/2} \right]

{ \Large { { \partial \Omega } \over { \partial t } } } ~=~ -{ \large { { 3 ~ J_2 R^2 } \over { 2 ~~~ p^2 ~ } } } ~ \bar{n} ~ \cos( i )