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| Deletions are marked like this. | Additions are marked like this. |
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| || $ \omega $ || $\omega ~=~ { \large { 2 \pi } \over T } $ "average" orbit angular frequency (different from Roy) || | |
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| $ {n_0}^2 ~=~ \mu / {a_0}^3 $ . . . unperturbed mean motion | 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 ~ \approx ~ { \large { { 2 \pi } \over T } } $ | $ {n_0}^2 ~=~ \mu / {a_0}^3 $ . . . unperturbed mean motion $ ~~ n_0 ~ \approx ~ { \large { { 2 \pi } \over T } } $ |
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| Roy writes about $\omega$ as if it is the angle from the orbiting body perpendicular the equatorial plane ... or something. Confusing. | ---- === Wikipedia === A similar 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 $ ---- 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 } } } $ |
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| === An example === | |
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| === An example === |
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| $ \bar{n} ~= n_0 \times 1.000171899 $ which suggests the orbital period is reduced by 0.000171869 × 86164.099 or 14.809 seconds |
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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 } ~ { { 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:
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longitude of the ascending node |
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semimajor axis |
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eccentricity |
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inclination |
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standard gravitational parameter, 398600.4418 km³/s² for Earth |
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zonal ablateness factor, 1.08262668e-3 for Earth |
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unperturbed mean motion |
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perturbed mean motion |
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Earth Equatorial Radius = 6378.137 km |
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unperturbed orbital period |
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Roy writes about 
a03 
T2
= n0
1+3 J2R22 p2 
1−23sin(i)2
(1−e2)1
2
n0
t
= −3 J2R22 p2 n cos(i) −23 J2R2a(1−e2)2 T2 cos(i)
Wikipedia
A similar result from Wikipedia's Nodal precession page.
p = −23 R2a(1−e2))2 J2 cosi
This is a westward perturbation for a prograde orbit with 
0
\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.