#format jsmath
= Plane Crossing Velocity =

Launch loops will be located slightly south of the equator, so they will launch into latitude-inclined transfer orbits, crossing other circular equatorial orbits.  What is the relative velocity compared to an object in a circular equatorial orbit?

The launch orbit has a perigee of $ r_p $ and an apogee of $ r_a $.  Earth's standard gravitational parameter $ \mu $ = 398600.44 km³/s².  The orbital radius and velocity are:

|| $ r = { \Large { { a ( 1 - e² ) } \over { 1 + e \cos( \theta ) } } } $ || $ a = { \Large { { r_a + r_p } \over 2 } }  ~ ~ ~ ~  e = { \Large { { r_a - r_p } \over  { r_a + r_p } } } ~ ~ ~ ~ ~ ~ \phi $ is inclination $ ~ ~ ~ ~ ~ ~ \theta $ is orbital angle ||
|| tangential $ v_t = v_0 ( 1 + e \cos( \theta ) ) \cos( \phi ) $ || $ v_0 = \Large { \sqrt{ { \mu \over 2 } \left( { 1 \over r_a } + { 1 \over r_p } \right) } } $ ||
|| radial $ v_r = e v_0 \sin( \theta ) $ || north/south $ v_z = v_0 ( 1 + e \cos( \theta ) ) \sin( \phi ) $ ||

The transfer orbit crosses the equatorial plane at $ \theta = \pi/2 = $ 90 degrees, the semi latus rectum, so the equations simplify to:

|| $ r =  a ( 1 - e² ) $ || $ \Delta v_t = v_0 ( 1 - \cos( \phi ) ) $ || $ v_z = v_0 \sin( \phi ) $ || $ v_r = e v_0 $ ||

These simplify to:

|| $ r = { \Large { { 2 r_a r_p } \over { r_a + r_p } } } $ || $ v_0 = \sqrt{ \Large { \mu \over r } } $ || $ \Delta v_t = v_0 ( 1 - \cos( \phi ) ) $ ||  $ v_z = v_0 \sin( \phi ) $ || $ v_r = e v_0 $ ||

Note that $ v_0 $ is the circular orbit velocity at radius $ r $.

Assume a launch loop at 5° south latitude with a altitude of 100 km and a perigee radius of 6478 km.  What are the equator crossing velocities relative to circular orbits?

||<-2> Destination ||<-5> plane crossing $\Delta$V compared to circular  || [[attachment:planecross.ods | libreoffice spreadsheet ]] ||
||       || radius || $ r_0 $ || $ v_0 $ || $\Delta v_t $ || $ v_z $ || $ v_r $ ||
||       || km     ||  km     ||   km/s  ||   km/s  ||   km/s  ||   km/s  ||
|| ISS   ||   6800 ||    6635 ||  7.7508 ||  0.0295 ||  0.6755 ||  0.1880 ||
|| M288  ||  12770 ||    8596 ||  6.8097 ||  0.0259 ||  0.5935 ||  2.2260 ||
|| M320	 ||  13532 ||    8762 ||  6.7449 ||  0.0257 ||  0.5879 ||  2.3777 ||
|| O3B	 ||  14420 ||    8940 ||  6.6773 ||  0.0254 ||  0.5820 ||  2.5376 ||
|| GPS	 ||  26538 ||   10414 ||  6.1867 ||  0.0235 ||  0.5392 ||  3.7590 ||
|| GEO	 ||  42164 ||   11231 ||  5.9576 ||  0.0227 ||  0.5192 ||  4.3707 ||
|| Moon  || 384400 ||   12741 ||  5.5932 ||  0.0213 ||  0.4875 ||  5.4078 ||

Note that lunar $ r_0 $ comes pretty close to M288, the original server sky orbit.  It may be better to change the server sky orbit to M320 (R=2.122 Re), which moves it farther from the proton belt, LAGEOS, and the lunar crossing, and provides more sunshine.  That makes the orbital repeat cycle 9 overhead passes in two days, rather than 10.

Note also that the numbers differ from 12789 km for M288 and 14441 km for O3B/M360.  Brain fail, but this may be the J₂ oblateness effect.  If not, I've got a lot of pages to correct.