A contour plot of the effective potential of a two-body system. (the Sun and Earth here), showing the 5 Lagrange points. An object in free-fall would trace out a contour (such as the Moon, shown).

SVG replacement for Image:Lagrange points.jpg. Not complete yet.

Diagram of w:Lagrange points for a system where the ratio of masses (primary:secondary) is 25:1. Contour lines plot effective potential in rotating frame.

Self-made in w:Gnuplot:

# Ratio of masses (primary:secondary)
R=25
left=-R*1.25
right=R*1.5
bottom=-R*1.25
top=R*1.25

set isosamples 200
unset surface
set cntrparam levels 30
set cntrparam levels incremental 1.4,0.019
set view 0,0
set contour base
set table 'lpoints.dat'
# Perturbatory force (vector)
#                         centrifugal         primary             secondary
lperturb(z)   =           z/(R+1)**2 - R*(z+1)/abs(z+1)**3 - (z-R)/abs(z-R)**3
# Effective potential (negative)
lpotential(z) = abs(z)**2/2/(R+1)**2 +       R/abs(z+1)    +     1/abs(z-R)
set parametric
splot [0:2*pi][0:right][left:right][bottom:top][1:3] v*sin(u),v*cos(u),lpotential(v*sin(u)*{1,0}+v*cos(u)*{0,1})
unset table

set term svg
set out 'lagrange.svg'

unset object
unset label
set size square
set multiplot
unset key
unset xtics
unset ytics

set parametric
# Space
set style line 6 lt 1 lw 2 lc rgb "black"
set samples 5
plot [0:4] [left:right][bottom:top] t<1?left+0.001:t<2?(t-1)*(right-left-0.002)+left+0.001:t<3?right+0.001:(t-3)*(left-right+0.002)+right-0.001,t<1?t*(top-bottom-0.002)+bottom+0.001:t<2?top-0.001:t<3?(t-2)*(bottom-top+0.002)+top-0.001:bottom linestyle 6 with filledcurves closed
set samples 100
unset parametric

# Contours
set style line 1 lt 1 lw 1 lc rgb "white"
plot [left:right][bottom:top] 'lpoints.dat' smooth bezier with lines linestyle 1

# Newton-Raphson method to calculate L1-L3
dlperturb(z) = 1.0/(R+1)**2 + 2.0*R/abs(z+1)**3 + 2.0/abs(z-R)**3
newtonraphson(x, m) = m<0 ? x : newtonraphson(x - lperturb(x)/dlperturb(x), m-1)

#set label 1 "Primary" at -1,0 offset -1,2 textcolor rgb "red"
#set label 2 "Secondary" at R,0 offset 1,1 textcolor rgb "red"
# Barycentre
set label 3 "" at 0,0 offset 1,0 point pointtype 1 pointsize 1
set label 4 "L1" at newtonraphson(R - (R+1)/(3*R)**(1.0/3), 10),0 right offset -1,-1 point pointtype 7 pointsize 1 linetype 1 lc rgb "green" textcolor rgb "green" font "Sans,14"
set label 5 "L2" at newtonraphson(R + (R+1)/(3*R)**(1.0/3), 10),0 offset 1,-1 point pointtype 7 pointsize 1 linetype 1 lc rgb "green" textcolor rgb "green" font "Sans,14"
set label 6 "L3" at newtonraphson((7.0*R+7)/(12.0*R-5)-(R+2), 10),0 offset 1,-1 point pointtype 7 pointsize 1 linetype 1 lc rgb "green" textcolor rgb "green" font "Sans,14"
set label 7 "L4" at (R-1)/2,sqrt(3)*(R+1)/2 offset 1,1 point pointtype 7 pointsize 1 linetype 1 lc rgb "green" textcolor rgb "green" font "Sans,14"
set label 8 "L5" at (R-1)/2,-sqrt(3)*(R+1)/2 offset 1,-1 point pointtype 7 pointsize 1 linetype 1 lc rgb "green" textcolor rgb "green" font "Sans,14"

set parametric
# Primary
set style line 4 lt 1 lw 1 lc rgb "white"
plot [0:2*pi] [left:right][bottom:top] 4*sin(t)-1,4*cos(t) linestyle 4 with filledcurves closed
set style line 3 lt 1 lw 1 lc rgb "blue"
plot [0:2*pi] [left:right][bottom:top] sin(t),cos(t) linestyle 3

# Secondary
plot [0:2*pi] [left:right][bottom:top] R*sin(t),R*cos(t) linestyle 3
set style line 6 lt 1 lw 1 lc rgb "black"
plot [0:2*pi] [left:right][bottom:top] 2.2*sin(t)+R,2.2*cos(t) linestyle 6 with filledcurves closed
set style line 5 lt 1 lw 1 lc rgb "#80C0FF"
plot [pi:2*pi] [left:right][bottom:top] 1.2*sin(t)+R,1.2*cos(t) linestyle 5 with filledcurves closed
set style line 8 lt 1 lw 1 lc rgb "#406080"
plot [0:pi] [left:right][bottom:top] 1.2*sin(t)+R,1.2*cos(t) linestyle 8 with filledcurves closed
# Moon
plot [0:2*pi] [left:right][bottom:top] 2.2*sin(t)+R,2.2*cos(t) linestyle 3
set style line 2 lt 1 lw 1 lc rgb "#DDDDDD"
plot [pi:2*pi] [left:right][bottom:top] 0.4*sin(t)+R+2.2*sqrt(0.5),0.4*cos(t)+2.2*sqrt(0.5) linestyle 2 with filledcurves closed
set style line 7 lt 1 lw 1 lc rgb "#222222"
plot [0:pi] [left:right][bottom:top] 0.4*sin(t)+R+2.2*sqrt(0.5),0.4*cos(t)+2.2*sqrt(0.5) linestyle 7 with filledcurves closed

unset parametric
unset multiplot

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