File:LagrangianPoints.gif
Summary
| Description |
English: Dynamics of test masses places near the 5 Lagrangian points, showing that L4 and L5 are stable, while the other aren't (the mass ratio between the two big masses is 40).
Shown in both the inertial and the rotating frame. |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1256941485360140290 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
Mathematica 12.0 code
sunMass = 40`20;
planetMass = 1`20;
totalMass = sunMass + planetMass;
\[CapitalDelta]r = 6`20;
planetPos = {sunMass/totalMass \[CapitalDelta]r, 0};
sunPos = {-(planetMass/totalMass) \[CapitalDelta]r, 0};
\[Omega] = Sqrt[totalMass/\[CapitalDelta]r^3];
eq = FullSimplify[
sunMass (sunPos - {x, y})/Norm[sunPos - {x, y} ]^3 + planetMass (planetPos - {x, y})/Norm[planetPos - {x, y}]^3 + \[Omega]^2 {x, y}, {x \[Element] Reals, y \[Element] Reals}];
Lp = {x, y} /. NSolve[eq == {0, 0}, {x, y}]
tmax = 500;
eq2 = FullSimplify[
sunMass (sunPos - {x[t], y[t]})/Norm[sunPos - {x[t], y[t]} ]^3 + planetMass (planetPos - {x[t], y[t]})/Norm[planetPos - {x[t], y[t]}]^3 + \[Omega]^2 {x[t], y[t]} - 2 Cross[{0, 0, \[Omega]}, Join[D[{x[t], y[t]}, {t, 1}], {0}]][[1 ;; 2]], {x[t] \[Element] Reals, y[t] \[Element] Reals}];
nLp = Dimensions[Lp][[1]];
extrasat =
Partition[Flatten[Table[(# + Lp[[j]]) & /@ RandomReal[{-0.0025, 0.0025}, {20, 2}], {j, 1, nLp}]], 2];
nextra = Dimensions[extrasat][[1]]
sol = Table[
NDSolve[{D[{x[t], y[t]}, {t, 2}] == eq2, x[0] == extrasat[[j, 1]], y[0] == extrasat[[j, 2]], x'[0] == 0, y'[0] == 0}, {x, y}, {t, 0,tmax}, WorkingPrecision -> 19, MaxSteps -> 10^6], {j, 1, nextra}];
p0 = Table[
GraphicsRow[{
Graphics[{
Yellow, Disk[RotationMatrix[\[Omega] t].sunPos, 0.5], Disk[RotationMatrix[\[Omega] t].planetPos, 0.3],
Red,
Table[
Disk[RotationMatrix[\[Omega] t].Lp[[j]], 0.05], {j, 1, 5}],
Text[Style["L4", Bold], RotationMatrix[\[Omega] t].Lp[[1]] + {0.4, 0.4} ],
Text[Style["L2", Bold], RotationMatrix[\[Omega] t].Lp[[2]] + {0.4, 0.4} ],
Text[Style["L3", Bold], RotationMatrix[\[Omega] t].Lp[[3]] + {0.4, 0.4} ],
Text[Style["L5", Bold], RotationMatrix[\[Omega] t].Lp[[4]] + {0.4, 0.4} ],
Text[Style["L1", Bold], RotationMatrix[\[Omega] t].Lp[[5]] + {0.4, 0.4} ],
Opacity[0.5], LightGray,
Table[
Disk[RotationMatrix[\[Omega] t].Evaluate[{x[t], y[t]} /. sol[[j]][[1]] ], 0.1], {j, 1, nextra}]
}, Background -> Black, PlotRange -> {{-10, 10}, {-10, 10}},
PlotLabel -> "Inertial frame", LabelStyle -> {Bold, White}]
,
Graphics[{
Yellow, Disk[sunPos, 0.5], Disk[planetPos, 0.3],
Red,
Table[Disk[Lp[[j]], 0.05], {j, 1, 5}],
Text[Style["L4", Bold], Lp[[1]] + {0.4, 0.4} ],
Text[Style["L2", Bold], Lp[[2]] + {0.4, 0.4} ],
Text[Style["L3", Bold], Lp[[3]] + {0.4, 0.4} ],
Text[Style["L5", Bold], Lp[[4]] + {0.4, 0.4} ],
Text[Style["L1", Bold], Lp[[5]] + {0.4, 0.4} ],
Opacity[0.5], LightGray,
Table[
Disk[Evaluate[{x[t], y[t]} /. sol[[j]][[1]] ], 0.1], {j, 1, nextra}]
}, Background -> Black, PlotRange -> {{-10, 10}, {-10, 10}},
PlotLabel -> "Rotating frame", LabelStyle -> {Bold, White}]
}]
, {t, 0, 100, 0.1}];
ListAnimate[p0]
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
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This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. |
| The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.
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