X-Git-Url: https://melusine.eu.org/syracuse/G/git/?p=pst-eqdf.git;a=blobdiff_plain;f=gravitation%2FKepler16.tex;fp=gravitation%2FKepler16.tex;h=a55e126e793032756b30a9178a56408f5e9140f1;hp=0000000000000000000000000000000000000000;hb=0747313425c353e89ef3195c3b04ab229fadd780;hpb=dc2142392ad1b3095ceb0172db23f1ca85045ffa

diff --git a/gravitation/Kepler16.tex b/gravitation/Kepler16.tex
new file mode 100644
index 0000000..a55e126
--- /dev/null
+++ b/gravitation/Kepler16.tex
@@ -0,0 +1,594 @@
+
+\documentclass{article}
+\usepackage[a4paper,margin=2cm]{geometry}
+\usepackage[T1]{fontenc}
+\usepackage[latin1]{inputenc}%
+\usepackage[garamond]{mathdesign}
+\usepackage{pst-eqdf,pst-node,pst-tools}
+\usepackage{array,amsmath}
+\usepackage{animate}
+\usepackage{wasysym}
+\newpsstyle{vecteurA}{arrowinset=0.05,arrowsize=0.1,linecolor={[rgb]{1 0.5 0}}}
+\newpsstyle{vecteurB}{arrowinset=0.05,arrowsize=0.1,linecolor={[rgb]{0 0.5 1}}}
+\newpsstyle{vecteurC}{arrowinset=0.1,arrowsize=0.2,linecolor={[rgb]{1 0 0}}}
+\makeatletter
+%% adapté de \psRandom du package pstricks-add
+%% pour rendre aléatoire la taille des étoiles
+%% Manuel Luque
+\newdimen\pssizeStar
+\def\psset@sizeStar#1{\pssetlength\pssizeStar{#1}}
+\psset@sizeStar{1pt}
+\define@key[psset]{pst-eqd}{randomPoints}[1000]{\def\psk@randomPoints{#1}}
+\psset[pst-eqd]{randomPoints=1000}
+\define@boolkey[psset]{pst-eqd}[Pst@]{color}[true]{}
+\psset[pst-eqd]{color=false}
+\def\psRandomStar{\pst@object{psRandomStar}}%
+\def\psRandomStar@i{\@ifnextchar({\psRandomStar@ii}{\psRandomStar@iii(0,0)(1,1)}}
+\def\psRandomStar@ii(#1){\@ifnextchar({\psRandomStar@iii(#1)}{\psRandomStar@iii(0,0)(#1)}}
+\def\psRandomStar@iii(#1)(#2)#3{%
+  \def\pst@tempA{#3}%
+  \ifx\pst@tempA\pst@empty\psclip{\psframe(#2)}\else\psclip{#3}\fi
+  \pst@getcoor{#1}\pst@tempA
+  \pst@getcoor{#2}\pst@tempB
+  \begin@SpecialObj
+  \addto@pscode{
+    \pst@tempA\space /yMin exch def
+    /xMin exch def
+    \pst@tempB\space /yMax exch def
+    /xMax exch def
+    /dy yMax yMin sub def
+    /dx xMax xMin sub def
+    rrand srand                 % initializes the random generator
+    /getRandReal { rand 2147483647 div } def
+     \psk@randomPoints {
+    /DS \pst@number\pssizeStar\space getRandReal mul def
+    \@nameuse{psds@\psk@dotstyle}
+     \ifPst@color getRandReal 1 1 sethsbcolor \fi
+     getRandReal dx mul xMin add
+     getRandReal dy mul yMin add
+     Dot
+     \ifx\psk@fillstyle\psfs@solid fill \fi stroke
+    } repeat
+  }%
+  \end@SpecialObj
+  \endpsclip
+  \ignorespaces
+}
+\makeatother
+%%%%%%%%%%%%%%%%%%
+%timeline
+\begin{filecontents}{kepler16.dat}
+::0x0
+::1
+::2
+::3
+::4
+::5
+::6
+::7
+::8
+::9
+::10
+::11
+::12
+::13
+::14
+::15
+::16
+::17
+::18
+::19
+::20
+::21
+::22
+::23
+::24
+::25
+::26
+::27
+::28
+::29
+::30
+::31
+::32
+::33
+::34
+::35
+::36
+::37
+::38
+::39
+::40
+::41
+::42
+::43
+::44
+::45
+::46
+::47
+::48
+::49
+::50
+::51
+::52
+::53
+::54
+::55
+::56
+::57
+::58
+::59
+::60
+::61
+::62
+::63
+::64
+::65
+::66
+::67
+::68
+::69
+::70
+::71
+::72
+::73
+::74
+::75
+::76
+::77
+::78
+::79
+::80
+::81
+::82
+::83
+::84
+::85
+::86
+::87
+::88
+::89
+::90
+::91
+::92
+::93
+::94
+::95
+::96
+::97
+::98
+::99
+::100
+::101
+::102
+::103
+::104
+::105
+::106
+::107
+::108
+::109
+::110
+::111
+::112
+::113
+::114
+::115
+::116
+::117
+::118
+::119
+::120
+::121
+::122
+::123
+::124
+::125
+::126
+::127
+::128
+::129
+::130
+::131
+::132
+::133
+::134
+::135
+::136
+::137
+::138
+::139
+::140
+::141
+::142
+::143
+::144
+::145
+::146
+::147
+::148
+::149
+::150
+::151
+::152
+::153
+::154
+::155
+::156
+::157
+::158
+::159
+::160
+::161
+::162
+::163
+::164
+::165
+::166
+::167
+::168
+::169
+::170
+::171
+::172
+::173
+::174
+::175
+::176
+::177
+::178
+::179
+::180
+::181
+::182
+::183
+::184
+::185
+::186
+::187
+::188
+::189
+::190
+::191
+::192
+::193
+::194
+::195
+::196
+::197
+::198
+::199
+::200
+\end{filecontents}
+%%%%%%%%%%%%%%%%%
+\title{Gravitation : une planète à deux soleils comme Tatooine celle de Luke Skywalker dans \textit{La Guerre des étoiles}}
+\date{10 juillet 2\,012}
+\begin{document}
+\maketitle
+\section{La représentation avec les données de la NASA}
+Ceci est une tentative pour essayer de schématiser une planète orbitant autour d'une étoile binaire, comme Kepler-16b. On parle dans ce cas-là d'une planète \textit{circumbinaire}.
+Le site de la NASA dédié à cette planète\footnote{http://kepler.nasa.gov/Mission/discoveries/kepler16b/}, fournit un grand nombre de renseignements sur les étoiles et la planète, ce qui permet de reconstituer leurs trajectoires respectives. Voici les caractéristiques utiles pour le schéma et l'animation.
+\begin{center}
+\begin{tabular}{|ll|}
+\hline
+\textbf{ Paramètres}&\textbf{Valeurs}\\ \hline
+  \textit{Étoile A}&\\ \hline
+  Masse, $M_A(M_{\astrosun})$&0.6897\\
+  Rayon, $R_A(R_{\astrosun})$&0.6489\\ \hline
+  \textit{Étoile B}&\\ \hline
+  Masse, $M_B(M_{\astrosun})$&0.20255\\
+  Rayon, $R_B(R_{\astrosun})$&0.22623\\ \hline
+  \textit{Planète b}&\\ \hline
+  Masse, $M_b(M_{\jupiter})$&0.333\\
+  Rayon, $R_b(R_{\jupiter})$&0.7538\\ \hline
+  \textit{Orbite de l'étoile binaire}&\\ \hline
+  Période(jours)&41.076\\
+  Demi-grand axe(ua) $\mathrm{a_1}$&0.22431\\
+  Excentricité $\mathrm{e_1}$&0.15944\\
+  Argument du périastre(deg) $\omega_1$&263.464\\ \hline
+  \textit{Orbite de la planète circumbinaire} &\\ \hline
+   Période(jours)&228.776\\
+  Demi-grand axe(ua)$\mathrm{a_2}$&0.7048\\
+  Excentricité $\mathrm{e_2}$&0.0069\\
+  Argument du périastre(deg) $\omega_2$&318\\ \hline
+\end{tabular}
+\end{center}
+
+\begin{center}
+\begin{pspicture}(-5,-5)(5,5)
+\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt]%
+\pstVerb{% pour le système binaire d'étoiles
+        /a1 0.22431 def % semi-major axis
+        /e1 0.15944 def % eccentricity
+        /w1 263.464 def % argument of periaspe
+        /MA 0.6897 def  % masse of star A
+        /MB 0.20255 def % masse of star B
+        /Mt MA MB add def % masse totale
+        /P1 41.076 def  % period (day)
+        % pour la planète
+        /Mb 0.3333 def  % masse of planet b (M_Jupiter)
+        /a2 0.7048 def  % semi-major axis
+        /e2 0.0069 def  % eccentricity
+        /w2 318 def  % argument of periaspe
+        /P2 228.776 def % period (day)
+% on en déduit le paramètre de chacune des ellipses
+        /p1 a1 1 e1 dup mul add mul def
+        /p2 a2 1 e2 dup mul add mul def
+        }%
+\pnode(!/radius p1 1 e1 w1 w1 sub cos mul add div def
+        /radius1 radius MB Mt div mul neg def
+         radius1 w1 cos mul 5 mul
+         radius1 w1 sin mul 5 mul){MA0}
+\parametricplot[plotpoints=360,unit=5,linecolor=red]{0}{360}{
+    /radius p1 1 e1 t w1 sub cos mul add div def
+    /radius1 radius MB Mt div mul neg def
+     radius1 t cos mul
+     radius1 t sin mul}
+\pscircle[fillstyle=solid,fillcolor=yellow!50](MA0){0.3}
+\parametricplot[plotpoints=360,unit=5,linecolor=green]{0}{360}{
+    /radius p1 1 e1 t w1 sub cos mul add div def
+    /radius2 radius MA Mt div mul def
+     radius2 t cos mul
+     radius2 t sin mul}
+\pnode(!/radius p1 1 e1 w1 w1 sub cos mul add div def
+        /radius1 radius MA Mt div mul def
+        radius1 w1 cos mul 5 mul
+        radius1 w1 sin mul 5 mul){MB0}
+\pscircle[fillstyle=solid,fillcolor=red](MB0){0.15}
+\parametricplot[plotpoints=360,unit=5,linecolor=blue]{0}{360}{
+    /radius p2 1 e2 t w2 sub cos mul add div def
+     radius t cos mul
+     radius t sin mul}
+\pnode(!/radius p2 1 e2 w2 w2 sub cos mul add div def
+        radius w2 cos mul 5 mul
+        radius w2 sin mul 5 mul){Mb0}
+\pscircle[fillstyle=solid,fillcolor=blue](Mb0){0.07}
+\end{pspicture}
+\end{center}
+\section{La simulation avec PSTricks}
+\begin{center}
+\begin{pspicture}(-3,-1)(6,7)
+\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt]%
+\pstVerb{/x01 -3 def /y01 0 def
+         /x02  5 def /y02 0 def
+         /xr012 x02 x01 sub def
+         /yr012 y02 y01 sub def
+         /x0 2 def /y0 5 def
+         /xr002 x02 x0 sub def
+         /yr002 y02 y0 sub def
+         /xr001 x01 x0 sub def
+         /yr001 y01 y0 sub def
+         /M0 0.2 def
+         /M1 0.07 def
+         /M2 3.2e-4 def
+         /Mt M1 M2 add def
+         /xG012 x01 M1 mul x02 M2 mul add Mt div def
+         /yG012 y01 M1 mul y02 M2 mul add Mt div def
+         }%
+\pnode(0,0){O}
+\pnode(!x01 y01){M1}
+\pnode(!x02 y02){M2}
+\pnode(!x0 y0){M0}
+\pnode(!xG012 yG012){C12} % centre de masse de M1 et M2
+\pscircle[fillstyle=solid,fillcolor=yellow!50](M2){0.21}
+\pscircle[fillstyle=solid,fillcolor=red!50](M1){0.5}
+\pscircle[fillstyle=solid,fillcolor=blue!50](M0){0.07}
+\psline[linestyle=dotted](M1)(M2)
+\psline[linestyle=dashed](M2)(M0)(M1)
+\psline{<->}(6,0)(0,0)(0,7)
+\uput[u](M0){$M_0$}
+\uput[r](0,6.8){$y$}
+\uput[u](5.8,0){$x$}
+\uput[l](0,6.8){$\mathcal{R}$}
+\uput{0.6}[d](M1){$M_1$}
+\uput{0.25}[d](M2){$M_2$}
+\rput(M1){\psline[style=vecteurC]{->}(!xr012 3 div yr012 3 div)\uput[d](!xr012 3 div yr012 3 div){$\overrightarrow{F}_{2/1}$}
+          \psline[style=vecteurC]{->}(!xr001 5 div neg yr001 5 div neg)\uput[r](!xr001 5 div neg yr001 5 div neg){$\overrightarrow{F}_{0/1}$}}
+\rput(M2){\psline[style=vecteurC]{->}(!xr012 3 div neg yr012 3 div neg)\uput[d](!xr012 3 div neg yr012 3 div neg){$\overrightarrow{F}_{1/2}$}
+             \psline[style=vecteurC]{->}(!xr002 3 div neg yr002 3 div neg)\uput[r](!xr002 3 div neg yr002 3 div neg){$\overrightarrow{F}_{0/2}$}}
+\rput(M0){\psline[style=vecteurC]{->}(!xr002 3 div yr002 3 div)\uput[r](!xr002 3 div yr002 3 div){$\overrightarrow{F}_{2/0}$}}
+\rput(M0){\psline[style=vecteurC]{->}(!xr001 5 div yr001 5 div)\uput[r](!xr001 5 div yr001 5 div){$\overrightarrow{F}_{1/0}$}}
+\end{pspicture}
+\end{center}
+Pour cela, on considère un système de trois corps en interaction gravitationnelle : l'étoile $M_1$ de masse $m_1$, l'étoile $M_2$ de masse $m_2$ constituant l'étoile binaire et une planète $P$ notée $M_0$, de masse $m$ orbitant autour de cette étoile double.
+
+Avec : $\overrightarrow{r_{01}}=\overrightarrow{M_0M_1}$, $\overrightarrow{r_{02}}=\overrightarrow{M_0M_2}$ et $\overrightarrow{r_{12}}=\overrightarrow{M_1M_2}$ on pose :
+\begin{itemize}
+  \item $\overrightarrow{r}_{01}=\overrightarrow{r}_1-\overrightarrow{r}_0$
+  \item $\overrightarrow{r}_{02}=\overrightarrow{r}_2-\overrightarrow{r}_0$
+  \item $\overrightarrow{r}_{12}=\overrightarrow{r}_2-\overrightarrow{r}_1$
+\end{itemize}
+Chacune des forces s'exprime par :
+\[
+\left\{
+\begin{array}[m]{l}
+\overrightarrow{F}_{1/0}=\mathcal{G}\dfrac{m_1m_0}{r_{10}^3}\overrightarrow{r_{01}}=-\overrightarrow{F}_{0/1}\\[1em]
+\overrightarrow{F}_{2/0}=\mathcal{G}\dfrac{m_2m_0}{r_{20}^3}\overrightarrow{r_{02}}=-\overrightarrow{F}_{0/2}\\[1em]
+\overrightarrow{F}_{2/1}=\mathcal{G}\dfrac{m_1m_2}{r_{12}^3}\overrightarrow{r_{12}}=-\overrightarrow{F}_{1/2}
+\end{array}
+\right.
+\]
+L'application de la loi de Newton à chacun des corps donne :
+\[
+\left\{
+\begin{array}[m]{l}
+m_0\dfrac{\mathrm{d^2}\overrightarrow{r_0}}{\mathrm{d}t^2}=\hphantom{-}\mathcal{G}\dfrac{m_1m_0}{r_{10}^3}\overrightarrow{r_{01}}+
+                                          \mathcal{G}\dfrac{m_2m_0}{r_{20}^3}\overrightarrow{r_{02}}\\[1em]
+m_1\dfrac{\mathrm{d^2}\overrightarrow{r_1}}{\mathrm{d}t^2}=-\mathcal{G}\dfrac{m_0m_1}{r_{10}^3}\overrightarrow{r_{01}}+
+                                          \mathcal{G}\dfrac{m_1m_2}{r_{12}^3}\overrightarrow{r_{12}}\\[1em]
+m_2\dfrac{\mathrm{d^2}\overrightarrow{r_2}}{\mathrm{d}t^2}=-\mathcal{G}\dfrac{m_0m_2}{r_{20}^3}\overrightarrow{r_{02}}-
+                                          \mathcal{G}\dfrac{m_1m_2}{r_{12}^3}\overrightarrow{r_{12}}
+\end{array}
+\right.
+\]
+Ce qui conduit à un système de 6 équations différentielles :
+\[
+\left\{
+\begin{array}[m]{l}
+\ddot{x_0}=\hphantom{-}\mathcal{G}\dfrac{m_1}{r_{10}^3}(x_1-x_0)+\mathcal{G}\dfrac{m_2}{r_{20}^3}(x_2-x_0)\\[1em]
+\ddot{y_0}=\hphantom{-}\mathcal{G}\dfrac{m_1}{r_{10}^3}(y_1-y_0)+\mathcal{G}\dfrac{m_2}{r_{20}^3}(y_2-y_0)\\[1em]
+\ddot{x_1}=-\mathcal{G}\dfrac{m_0}{r_{10}^3}(x_1-x_0)+\mathcal{G}\dfrac{m_2}{r_{12}^3}(x_2-x_1)\\[1em]
+\ddot{y_1}=-\mathcal{G}\dfrac{m_0}{r_{10}^3}(y_1-y_0)+\mathcal{G}\dfrac{m_2}{r_{12}^3}(y_2-y_1)\\[1em]
+\ddot{x_2}=-\mathcal{G}\dfrac{m_0}{r_{10}^3}(x_2-x_0)-\mathcal{G}\dfrac{m_1}{r_{12}^3}(x_2-x_1)\\[1em]
+\ddot{y_2}=-\mathcal{G}\dfrac{m_0}{r_{02}^3}(y_2-y_0)-\mathcal{G}\dfrac{m_1}{r_{12}^3}(y_2-y_1)
+\end{array}
+\right.
+\]
+
+\begin{verbatim}
+%   0    1    2    3    4    5    6    7    8   9     10    11
+% y[0] y[1] y[2] y[3] y[4] y[5] y[6] y[7] y[8] y[9] y[10] y[11]
+%  x0   y0  x'0  y'0   x1   y1  x'1  y'1   x2   y2   x'2   y'2
+\def\GravAlgIIIcorps{%
+  y[2]|y[3]|%
+  M1*(y[4]-y[0])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5+M2*(y[8]-y[0])/((y[8]-y[0])^2+(y[9]-y[1])^2)^1.5|%
+  M1*(y[5]-y[1])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5+M2*(y[9]-y[1])/((y[8]-y[0])^2+(y[9]-y[1])^2)^1.5|%
+  y[6]|y[7]|%
+  -M0*(y[4]-y[0])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5+M2*(y[8]-y[4])/((y[8]-y[4])^2+(y[9]-y[5])^2)^1.5|%
+  -M0*(y[5]-y[1])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5+M2*(y[9]-y[5])/((y[8]-y[4])^2+(y[9]-y[5])^2)^1.5|%
+  y[10]|y[11]|%
+  -M0*(y[8]-y[0])/((y[8]-y[0])^2+(y[9]-y[1])^2)^1.5-M1*(y[8]-y[4])/((y[8]-y[4])^2+(y[9]-y[5])^2)^1.5|%
+  -M0*(y[9]-y[1])/((y[8]-y[0])^2+(y[9]-y[1])^2)^1.5-M1*(y[9]-y[5])/((y[8]-y[4])^2+(y[9]-y[5])^2)^1.5}
+\end{verbatim}
+
+
+\newpage
+% y[0] y[1] y[2] y[3] y[4] y[5] y[6] y[7] y[8] y[9] y[10] y[11]
+%  x0   y0  x'0  y'0   x1   y1  x'1  y'1   x2   y2   x'2   y'2
+\def\GravAlgIIIcorps{%
+  y[2]|y[3]|%
+  M1*(y[4]-y[0])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5+M2*(y[8]-y[0])/((y[8]-y[0])^2+(y[9]-y[1])^2)^1.5|%
+  M1*(y[5]-y[1])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5+M2*(y[9]-y[1])/((y[8]-y[0])^2+(y[9]-y[1])^2)^1.5|%
+  y[6]|y[7]|%
+  -M0*(y[4]-y[0])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5+M2*(y[8]-y[4])/((y[8]-y[4])^2+(y[9]-y[5])^2)^1.5|%
+  -M0*(y[5]-y[1])/((y[4]-y[0])^2+(y[5]-y[1])^2)^1.5+M2*(y[9]-y[5])/((y[8]-y[4])^2+(y[9]-y[5])^2)^1.5|%
+  y[10]|y[11]|%
+  -M0*(y[8]-y[0])/((y[8]-y[0])^2+(y[9]-y[1])^2)^1.5-M1*(y[8]-y[4])/((y[8]-y[4])^2+(y[9]-y[5])^2)^1.5|%
+  -M0*(y[9]-y[1])/((y[8]-y[0])^2+(y[9]-y[1])^2)^1.5-M1*(y[9]-y[5])/((y[8]-y[4])^2+(y[9]-y[5])^2)^1.5}
+%
+
+\begin{center}
+\def\InitCond{x00 y00 v0x0 v0y0 x01 y01 v0x1 v0y1 x02 y02 v0x2 v0y2}
+\psset{method=rk4}
+\begin{pspicture}(-8,-8)(8,8)
+\psframe*[linecolor={[cmyk]{1 1 0 0.7}}](-8,-8)(8,8)
+\pstVerb{
+/arccos {
+   dup
+   dup mul neg 1 add sqrt
+   exch
+   atan
+} def
+        /G 1 def
+        /w2 55 def
+        /x00 7 w2 cos mul def /y00 7 w2 sin mul def
+        /x01 0.4962 def
+        /y01 0 def
+        /v0x1 0 def
+        /v0y1 .135 def
+        /x02 -1.69 def
+        /y02 0 def
+        /v0x2 0 def
+        /v0y2 -.46 def
+        /v0x0 0.36 w2 sin mul neg def
+        /v0y0 0.36 w2 cos mul def
+         /M1 0.7 def
+         /M2 0.2 def
+         /M0 3e-4 def
+         /Mt M1 M2 add def
+         /xG012 x01 M1 mul x02 M2 mul add Mt div def
+         /yG012 y01 M1 mul y02 M2 mul add Mt div def
+         }%
+\psgrid[subgriddiv=0,gridcolor=white,griddots=10,gridlabels=0pt](-8,-8)(8,8)%
+\psequadiff[plotpoints=1000,algebraic,
+            plotfuncx=y 4 get
+                      y 4 get M1 mul
+                      y 8 get M2 mul add
+                      M1 M2 add div
+                      sub,
+            plotfuncy=5 get
+                      dup M1 mul
+                      y 9 get M2 mul add
+                      M1 M2 add div
+                      sub,
+            tabname=XAYA,
+            saveData,filename=XAYA.dat
+%            ]{0}{18}{\InitCond}{\GravAlgIIIcorps}
+            ]{0}{120}{\InitCond}{\GravAlgIIIcorps}
+\listplot[unit=1,linecolor=yellow]{XAYA aload pop}
+\psequadiff[plotpoints=1000,algebraic,
+            plotfuncx=y 8 get
+                      y 4 get M1 mul
+                      y 8 get M2 mul add
+                      M1 M2 add div
+                      sub,
+            plotfuncy=9 get
+                      dup M2 mul
+                      y 5 get M1 mul add
+                      M1 M2 add div
+                      sub,
+            tabname=XBYB,
+            saveData,filename=XBYB.dat
+%            ]{0}{18}{\InitCond}{\GravAlgIIIcorps}
+            ]{0}{120}{\InitCond}{\GravAlgIIIcorps}
+\listplot[unit=1,linecolor=red]{XBYB aload pop}
+\psequadiff[plotpoints=1000,algebraic,
+            plotfuncx=y 0 get
+                      y 4 get M1 mul
+                      y 8 get M2 mul add
+                      M1 M2 add div
+                      sub,
+            plotfuncy=1 get
+                      y 5 get M1 mul
+                      y 9 get M2 mul add
+                      M1 M2 add div
+                      sub,
+            tabname=XbYb,
+            saveData,filename=XPYP.dat
+            ]{0}{120}{\InitCond}{\GravAlgIIIcorps}
+\listplot[unit=1,linecolor=blue]{XbYb aload pop}
+\pnode(!x01 xG012 sub y01 yG012 sub){M01}
+\pnode(!x02 xG012 sub y02 yG012 sub){M02}
+\pnode(!x00  xG012 sub y00  yG012 sub){M00}
+\pscircle*[linecolor=yellow](M01){0.4}
+\pscircle*[linecolor=red](M02){0.16}
+\pscircle*[linecolor=white](M00){0.07}
+%\rput(M01){\psline[unit=2,linecolor=red]{->}(!v0x1 v0y1)}
+%\rput(M02){\psline[unit=2,linecolor=blue]{->}(!v0x2 v0y2)}
+\psdot(!xG012 yG012)
+\end{pspicture}
+\end{center}
+\section{Animation avec pst-eqdf et animate}
+\begin{center}
+\def\nFrames{200}% 200 images
+\begin{animateinline}[controls,timeline=kepler16.dat,loop,%
+                     begin={\begin{pspicture}(-8,-8)(8,8)},
+                     end={\end{pspicture}}]{5}% 5 images/s
+\pstVerb{/XY1 [(XAYA.dat) run] def
+         /XY2 [(XBYB.dat) run] def
+         /XY3 [(XPYP.dat) run] def
+         }%
+\psframe*[linecolor={[cmyk]{1 1 0 0.7}}](-8,-8)(8,8)
+\psRandomStar[linecolor={[rgb]{1,1,0.5}},
+    randomPoints=1000,sizeStar=1pt](-8,-8)(8,8){\psframe[linestyle=none](-8,-8)(8,8)}
+%\listplot[linecolor=gray]{XY1 aload pop}
+%\listplot[linecolor=gray]{XY2 aload pop}
+%\listplot[linecolor=gray]{XY3 aload pop}
+\newframe
+\multiframe{\nFrames}{i=0+10}{% 1 point sur 10
+\pstVerb{/X1 XY1 \i\space get def
+         /Y1 XY1 \i\space 1 add get def
+         /X2 XY2 \i\space get def
+         /Y2 XY2 \i\space 1 add get def
+         /X3 XY3 \i\space get def
+         /Y3 XY3 \i\space 1 add get def
+        }%
+%\psdot(!X1 Y1)
+\pscircle*[linecolor=yellow](!X1 Y1){0.5}
+\pscircle*[linecolor=red](!X2 Y2){0.2}
+\pscircle*[linecolor=white](!X3 Y3){0.07}
+}
+\end{animateinline}
+\end{center}
+\end{document}
+