Differences between revisions 30 and 31
 ⇤ ← Revision 30 as of 2018-04-07 21:16:10 → Size: 3144 Editor: KeithLofstrom Comment: ← Revision 31 as of 2018-04-07 22:01:05 → ⇥ Size: 3752 Editor: KeithLofstrom Comment: Deletions are marked like this. Additions are marked like this. Line 4: Line 4: 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. Line 27: Line 30: ${ \Large { { \partial \Omega } \over { \partial t } } } ~=~ -{ \large { { 3 ~ J_2 R^2 } \over { 2 ~~~ p^2 ~ } } } ~ \bar{n} ~ \cos( i ) ~\approx ~ -{ \large { { 3 \over 2 } ~ { { J_2 R^2 } \over { a ( 1 - e^2 )^2 } } ~ { { 2 \pi } \over T } } } ~ \cos( i )$ ${ \Large { { \partial \Omega } \over { \partial t } } } ~=~ -{ \large { { 3 ~ J_2 R^2 } \over { 2 ~~~ p^2 ~ } } } ~ \bar{n} ~ \cos( i ) ~\approx~ -{ \large { { 3 \over 2} ~ { { J_2 R^2 } \over { a^2 ( 1 - e^2 )^2 } } ~ { { 2 \pi } \over T } } } ~ \cos( i )$ Line 34: Line 37: $\omega_p ~=~ - { \Large { 3 \over 2 } ~ { { R^2 } \over { a (1-e^2) )^2 } } } ~ J_2 ~ \omega ~ \cos i$ $\omega_p ~=~ - { \Large { 3 \over 2 } ~ \left( R \over { a (1-e^2) } \right)}^2 ~ J_2 ~ \omega ~ \cos i$ Line 36: Line 39: $\omega_p \approx \Delta \omega$ 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 $a$ ). $\omega_p \approx \Delta \omega$ 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 $a$ ). Line 38: Line 41: $\omega ~=~ { \Large \sqrt{ \mu \over a^3 } } ~~~~~~~~~~~~~~ \Delta \omega ~=~ { \Large \sqrt{ \mu \over a^3 } } \left( -{ \Large { { 3 \over 2 } { 1 \over a } } } \Delta a \right) ~~~~~~~~~~~~~~ { \Large { { \Delta \omega } \over \omega } } ~=~ - { \Large { { 3 \over 2 } { { \delta a } \over a } } }$ $\omega ~=~ { \Large \sqrt{ \mu \over a^3 } } ~~~~~~~~~~~~~~ \Delta \omega ~=~ { \Large \sqrt{ \mu \over a^3 } } \left( -{ \Large { { 3 \over 2 } { 1 \over a } } } \Delta a \right) ~~~~~~~~~~~~~~ { \Large { { \Delta \omega } \over \omega } } ~=~ - { \Large { { 3 \over 2 } { { \Delta a } \over a } } }$so$\Delta a ~\approx ~ A ~ { \Large \left( R \over { a (1-e^2) } \right)}^2 ~ J_2 ~ \cos i ~~~~~~~~~~~~~~~~$ also $~~ \Delta T ~\approx { \Large {3 \over 2}} ~T~ { \Large \left( R \over { a (1-e^2) } \right)}^2 ~ J_2 ~ \cos i$

# Apsidal Precession

The following is "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.

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^2 ( 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 } ~ \left( R \over { a (1-e^2) } \right)}^2 ~ J_2 ~ \omega ~ \cos i

\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

### 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.

ApsidalPrecession (last edited 2018-04-07 22:01:05 by KeithLofstrom)