Lagrange Points

circular orbit approximation


Period s

Radius m

Gravitational Parameter m3/s2

Sun

\mu_s = 1.3271244002e20

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

spreadsheet


Earth-Sun Lagrange points

Lagrange (last edited 2017-07-09 19:27:50 by KeithLofstrom)