Lagrange Points

circular orbit approximation

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

\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

spreadsheet

Earth-Sun Lagrange points

• Earth-Sun Lagrange at 235 Re original

• magnetotail at 200 Re original

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