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= Apsidal Precession =

The following is [[ https://en.wikiquote.org/wiki/John_Maynard_Keynes#Misattributed  | "vaguely right"]], '''for estimates only'''.  Actual orbit planning will require higher precision, inclusion of perturbations for the sun, moon, and other planets, and also constellation management to avoid collisions and facilitate delivery to the final orbital destination.

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The Earth's equatorial bulge causes orbits (perigee and apogee) to drift westward.  From AE Roy ''Orbital Motion '' 1978:

|| $ \Omega $  || longitude of the ascending node                                ||
|| $ a $       || semimajor axis                                                 ||
|| $ e $       || eccentricity                                                   ||
|| $ i $       || inclination                                                    ||
|| $ \mu $     || standard gravitational parameter, 398600.4418 km³/s² for Earth ||
|| $ J_2 $     || zonal ablateness factor, 1.08262668e-3 for Earth               ||
|| $ 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                          ||
|| $ T $       || unperturbed orbital period                                     ||
|| $ \omega $  || $\omega ~=~ { \large { 2 \pi } \over T } $ "average" orbit angular frequency (different from Roy) ||

Roy writes about $\omega$ as if it is the angle from the orbiting body perpendicular the equatorial plane ... or something.  Confusing.  We will use $\omega$ for the average angular frequency.
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$ {n_0}^2 ~=~ \mu / {a_0}^3 $  . . . unperturbed mean motion  $ ~~ n_0  ~ \approx ~ { \large { { 2 \pi } \over T } } $

$ \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]  ~\approx~ n_0 $

$ { \Large { { \partial  \Omega } \over { \partial t } } } ~=~ -{ \large { { 3 ~ J_2 R^2 } \over { 2 ~~~ p^2 ~ } } } ~ \bar{n} ~ \cos( i ) ~\approx~ -{ \large { { 3 \over 2} ~ { { J_2 R^2 } \over { a^2 ( 1 - e^2 )^2 } }  ~ { { 2 \pi } \over T } } } ~ \cos( i ) $

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=== Wikipedia ===

A similar result from Wikipedia's [[https://en.wikipedia.org/wiki/Nodal_precession | Nodal precession ]] page.

$ \omega_p ~=~ - { \Large { 3 \over 2 } ~ \left( R \over { a (1-e^2) } \right)}^2 ~ J_2 ~ \omega ~ \cos i $
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$ \omega_p \approx \Delta \omega $ is a westward perturbation for a prograde orbit with $ \cos i ~> 0 $.  We can correct the period by increasing the orbit time a little (decreasing $ \omega $ ) by increasing the perigee radius $ r_p $ (and thus $a$ ).

$ \omega ~=~ { \Large \sqrt{ \mu \over a^3 } } ~~~~~~~~~~~~~~ \Delta \omega ~=~ { \Large \sqrt{ \mu \over a^3 } } \left( -{ \Large { { 3 \over 2 } { 1 \over a } } } \Delta a \right) ~~~~~~~~~~~~~~ { \Large { { \Delta \omega } \over \omega } } ~=~ - { \Large { { 3 \over 2 } { { \Delta a } \over a } } } $

so

$ \Delta a ~\approx ~ A ~ { \Large \left( R \over { a (1-e^2) } \right)}^2 ~ J_2 ~ \cos i ~~~~~~~~~~~~~~~~ $ also $~~ \Delta T ~\approx { \Large {3 \over 2}} ~T~ { \Large \left( R \over { a (1-e^2) } \right)}^2 ~ J_2 ~ \cos i $

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=== An example ===

An 8 degree inclined orbit, $ r_p $ = 8378 km, $ r_a $ = 76000 km, period 86164.099 seconds 
 
So, $ a $ = 42164 km, e = 0.80189, p = 15051.209 km,

$ { \large { { 3 ~ J_2 R^2 } \over { 2 ~~~ p^2 ~ } } } ~= $ 2.92784e-4

The apsides will precess by about the same amount per orbit.

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