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| $ \large f ~ = ~ 1 - f { \Large \sqrt{ k \over { 1 - f^3 } } } $ . . . iterate until convergence . . .$ r_1 ~ = ~ f * r_e $ | $ \large f ~ = ~ 1 - f { \Large \sqrt{ k \over { 1 - f^3 } } } $ . . . iterate until convergence . . . $ r_1 ~ = ~ f * r_e $ |
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| $ \large f ~ = ~ 1 + f { \Large \sqrt{ k \over { f^3 - 1 } } } $ . . . iterate until convergence . . .$ r_2 ~ = ~ f * r_e $ | $ \large f ~ = ~ 1 + f { \Large \sqrt{ k \over { f^3 - 1 } } } $ . . . iterate until convergence . . . $ r_2 ~ = ~ f * r_e $ |
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| $ \large f ~ = ~ \left( 1 + k \left( { \LARGE f \over { f + 1 } } \right)^2 \right)^{ 1 \over 3 } $ . . . iterate until convergence . . .$ r_3 ~ = ~ f * r_e $ | $ \large f ~ = ~ \left( 1 + k \left( { \LARGE f \over { f + 1 } } \right)^2 \right)^{ 1 \over 3 } $ . . . iterate until convergence . . . $ r_3 ~ = ~ f * r_e $ |
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------ === Earth-Sun Lagrange points === . [[ attachment:lagrange.pdf | Earth-Sun Lagrange ]] at 235 Re [[ http://map.gsfc.nasa.gov/ContentMedia/lagrange.pdf | original ]] . [[ attachment:bowshock.pdf | magnetotail at 200 Re ]] [[ https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20140009185.pdf | original ]] |
Lagrange Points
circular orbit approximation
|
Period s |
Radius m |
Gravitational Parameter m3/s2 |
Sun |
|
|
|
Earth |
t_e = 31558149.76 |
r_e = 1.49598023e11 |
\mu_e = 3.98600442e14 |
Moon |
t_m = 2360591.51 |
r_m = 3.84399e8 |
\mu_m = 4.904869e12 |
colinear L1, L2, L3
Test orbital radius
deviations perhaps due to eccentricity and tidal effects
\large \omega_e ~ = ~ 2 \pi / t_e ~ = ~ = 1.9909865e-7 radians/second
{ \large r_e } ~ = ~ { \LARGE \left( { \mu_s + \mu_e } \over { \omega_e }^2 \right) ^ {1 \over 3} } ~ = ~ 1.4959787e11 ~ m ~ \approx ~ { \LARGE \left( { \mu_s } \over { \omega_e }^2 \right) ^ {1 \over 3} } ~ = ~ 1.4959772e11 m (ignoring barycenter)
\large \omega_m ~ = ~ 2 \pi / t_m ~ = ~ = 2.6616995e-6 radians/second
{ \large r_m } ~ = ~ { \LARGE \left( { \mu_e + \mu_m } \over { \omega_m }^2 \right) ^ {1 \over 3} } ~ = ~ 3.8474861e8 ~ m ~ \approx ~ { \LARGE \left( { \mu_s } \over { \omega_e }^2 \right) ^ {1 \over 3} } ~ = ~ 3.831833e08 m (ignoring barycenter)
Earth-Sun Lagrange points
Circular orbits, ignoring barycenter
L1 |
r_1 |
Lagrange point between Earth and Sun |
0.990029594 r_e |
1,491,550 km in front of earth |
L2 |
r_2 |
Lagrange point behind Earth from Sun |
1.0100371234 r_e |
1,501,530 km behind earth |
L3 |
r_3 |
Lagrange point behind Sun from Earth |
1.0001286404 r_e |
19,244 km behind earth orbit |
\large { k ~ ≡ \mu_e / \mu_s ~ ~ ~ ~ f ~ ≡ ~ r / r_e }
Earth/Sun L1
{ \large {\omega_e}^2 r_1 ~ = ~ } { \LARGE { { \mu_s r_1 } \over {r_e}^3 } } { \large ~ = ~ } { \LARGE { \mu_s \over {r_1}^2 } - { \mu_e \over ( r_e - r_1 )^2 } }
\large f ~ = ~ 1 - f { \Large \sqrt{ k \over { 1 - f^3 } } } . . . iterate until convergence . . . r_1 ~ = ~ f * r_e
Earth/Sun L2
{ \large {\omega_e}^2 r_1 ~ = ~ } { \LARGE { { \mu_s r_2 } \over {r_e}^3 } } { \large ~ = ~ } { \LARGE { \mu_s \over {r_2}^2 } + { \mu_e \over ( r_e - r_1 )^2 } }
\large f ~ = ~ 1 + f { \Large \sqrt{ k \over { f^3 - 1 } } } . . . iterate until convergence . . . r_2 ~ = ~ f * r_e
Earth/Sun L3
{ \large {\omega_e}^2 r_3 ~ = ~ } { \LARGE { { \mu_s r_3 } \over {r_e}^3 } } { \large ~ = ~ } { \LARGE { \mu_s \over {r_3}^2 } - { \mu_e \over ( r_e + r_1 )^2 } }
\large f ~ = ~ \left( 1 + k \left( { \LARGE f \over { f + 1 } } \right)^2 \right)^{ 1 \over 3 } . . . iterate until convergence . . . r_3 ~ = ~ f * r_e
Moon-Earth Lagrange points
Circular orbits, ignoring barycenter
L1 |
r_1 |
Lagrange point between Moon and Earth |
0.848883087 r_m |
57,905 km in front of moon |
L2 |
r_2 |
Lagrange point behind Moon from Earth |
1.1681346921 r_m |
64,426 km behind moon |
L3 |
r_3 |
Lagrange point behind Earth from Moon |
1.001025435 r_m |
393 km behind moon orbit |