File:Tidallock.gif
Tidallock.gif (360 × 360 pixels, file size: 3.67 MB, MIME type: image/gif, looped, 1,801 frames)
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Summary[edit]
DescriptionTidallock.gif |
English: Numerical simulation showing a satellite entering a situation of tidal locking with the (much more massive) main body. As perfectly spherical satellites won't show any tidal locking (and to simplify both calculation and visualization) I used a satellite where the mass is concentrated in only two points (the orange and purple spots).
Reaching the tidal lock from a randomly chosen initial condition takes ages, so only the last part of the evolution is shown. |
Date | |
Source | https://twitter.com/j_bertolotti/status/1067762417092972544 |
Author | Jacopo Bertolotti |
Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
Mathematica 11.0 code[edit]
(*To make the calculation simpler the satellite has its mass \ concentrated in only two points (a spherically symmetric satellite \ won't show any tidal locking)*) d = 1.; (*half-distance between the two points with mass*) m0 = 10; (*mass of the mai n body*) mp = 1; (*mass of each massive \ point on the satellite*) p1 = r[t] {Cos[\[Theta][t]], Sin[\[Theta][t]]} + d {Cos[\[Phi][t]], Sin[\[Phi][t]]}; p2 = r[t] {Cos[\[Theta][t]], Sin[\[Theta][t]]} - d {Cos[\[Phi][t]], Sin[\[Phi][t]]}; T1 = mp/2 FullSimplify[Norm[D[p1, t] ]^2, Assumptions -> {r[t] > 0, r'[t] \[Element] Reals, \[Theta][t] \[Element] Reals, \[Phi][t] \[Element] Reals, \[Theta]'[t] \[Element] Reals, \[Phi]'[t] \[Element] Reals}] T2 = mp/2 FullSimplify[Norm[D[p2, t] ]^2, Assumptions -> {r[t] > 0, r'[t] \[Element] Reals, \[Theta][t] \[Element] Reals, \[Phi][t] \[Element] Reals, \[Theta]'[t] \[Element] Reals, \[Phi]'[t] \[Element] Reals}] V1 = (-m0 mp)/ FullSimplify[Norm[p1], Assumptions -> {r > 0, t \[Element] Reals, f \[Element] Reals}] V2 = (-m0 mp)/ FullSimplify[Norm[p2], Assumptions -> {r > 0, t \[Element] Reals, f \[Element] Reals}] L = FullSimplify[T1 + T2 - V1 - V2, Assumptions -> {r[t] > 0, r'[t] \[Element] Reals, \[Theta][t] \[Element] Reals, \[Phi][t] \[Element] Reals, \[Theta]'[t] \[Element] Reals, \[Phi]'[t] \[Element] Reals}] (*Euler-Lagrange equations*) e1 = FullSimplify[D[D[L, \[Theta]'[t]], t] - D[L, \[Theta][t]]]; e2 = FullSimplify[D[D[L, \[Phi]'[t]], t] - D[L, \[Phi][t]]]; e3 = FullSimplify[D[D[L, r'[t]], t] - D[L, r[t]]] tmax = 5000; r0 = 6; (*starting distance of the satellite*) s1 = NDSolve[{e1 == 0, e2 == 0, e3 == 0, \[Theta][0] == 0, \[Phi][0] == 0.8, r[0] == r0, \[Theta]'[0] == 0.25, \[Phi]'[0] == 0.2, r'[0] == 0}, {\[Theta], \[Phi], r}, {t, 0, tmax}, MaxSteps -> Infinity] plots = Table[ Graphics[{Disk[{0, 0}, 2], Thick, Circle[({r[t] Cos[\[Theta][t]], r[t] Sin[\[Theta][t]]} /. s1)[[ 1]], d], Orange, Disk[(p1 /. s1)[[1]], 0.5], Purple, Disk[(p2 /. s1)[[1]], 0.5] }, PlotRange -> {{-15, 15}, {-15, 15}}] , {t, 3900, 4800, 0.5}]; ListAnimate[plots]
Licensing[edit]
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.
http://creativecommons.org/publicdomain/zero/1.0/deed.enCC0Creative Commons Zero, Public Domain Dedicationfalsefalse |
This file, which was originally posted to
https://twitter.com/j_bertolotti/status/1030470604418428929, was reviewed on 29 November 2018 by reviewer Ronhjones, who confirmed that it was available there under the stated license on that date.
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