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| $ \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 } } } $ | $ \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 } } } $ |
Apsidal Precession
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.
{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 ( 1 - e^2 )^2 } } ~ { { 2 \pi } \over T } } } ~ \cos( i )
Wikipedia
A similar result from Wikipedia's Nodal precession page.
\omega_p ~=~ - { \Large { 3 \over 2 } ~ { { R^2 } \over { a (1-e^2) )^2 } } } ~ J_2 ~ \omega ~ \cos i
This 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 } } }
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.