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| The Earth's equatorial bulge causes orbits to drift, with westward orbit perigee and apogee drifting westward. From AE Roy ''Orbital Motion '' 1978: | The Earth's equatorial bulge causes orbits (perigee and apogee) to drift westward. From AE Roy ''Orbital Motion '' 1978: |
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| || $ a_0 $ || semimajor axis || | || $ a $ || semimajor axis || |
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| || $ J_2 $ || zonal ablateness factor || | || $ J_2 $ || zonal ablateness factor, 1.08262668e-3 for Earth || |
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| || $ T $ || unperturbed orbital period || | |
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| $ \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] $ | $ n_0 ~ \approx \large ( { 2 \pi } \over T $ |
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| $ { \Large { { \partial \Omega } \over { \partial t } } } ~=~ -{ \large { { 3 ~ J_2 R^2 } \over { 2 ~~~ p^2 ~ } } } ~ \bar{n} ~ \cos( i ) $ | $ \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 ) ~=~ -{ \large { { 3 ~ J_2 R^2 } \over { 2 ~~~ a ( 1 - e^2 ) } } ~ { { 2 \pi } \over T } ) ~ \cos( i ) $ Roy writes about $\omega$ as if it is the angle from the orbiting body perpendicular the equatorial plane ... or something. Confusing. ---- === 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 $ \bar{n} ~= n_0 \times 1.000171899 $ which suggests the orbital period is reduced by 0.000171869 × 86164.099 or 14.809 seconds The apsides will precess by about the same amount per orbit. ---- === Wikipedia differs? === A different result from Wikipedia's [[https://en.wikipedia.org/wiki/Nodal_precession | Nodal precession ]] page. $ \omega_p ~=~ - { \Large { 3 \over 2 } ~ { { R^2 } \over { a (1-e^2) )^2 } } } ~ J_2 ~ \omega ~ \cos i $ |
Apsidal Precession
The Earth's equatorial bulge causes orbits (perigee and apogee) to drift westward. From AE Roy Orbital Motion 1978:
|
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 |
{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 ) ~=~ -{ \large { { 3 ~ J_2 R^2 } \over { 2 ~~~ a ( 1 - e^2 ) } } ~ { { 2 \pi } \over T } ) ~ \cos( i )
Roy writes about \omega as if it is the angle from the orbiting body perpendicular the equatorial plane ... or something. Confusing.
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
\bar{n} ~= n_0 \times 1.000171899 which suggests the orbital period is reduced by 0.000171869 × 86164.099 or 14.809 seconds
The apsides will precess by about the same amount per orbit.
Wikipedia differs?
A different 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