Skip to main content

differential equations - Solar System N body Simulation


This is an N-Body simulation for the Sun and the following planetary bodies: Mercury, Venus,Earth,Mars,Jupiter,Saturn,Uranus,Neptune and Pluto.


Initial Parameters



Ecc = {0.20563069, 0.00677323, 0.01671022, 0.09341233,0.04839266, 
0.05415060, 0.04716771, 0.00858587, 0.24880766};(*eccentricity of bodies*)

a = {0.38709893, 0.7233319899999999, 1.00000011, 1.52366231,
5.2033630099999995, 9.537070319999998, 19.191263929999998,
30.06896348, 39.48168677}; (*semi major axis of bodies*)

r = a (1 - Ecc^2)/(1 + Ecc Cos[\[Psi]]); (*orbital position*)
rx={0, 0.3075, 0.718433, 0.98329, 1.38133, 4.95156, 9.02063, 18.2861, 29.8108, 29.6583} (*x component of position*)
ry={0, 0., 0., 0., 0., 0., 0., 0., 0., 0.}

v = {0, 0.03406085426835039`, 0.020363076269733636`,
0.017491554631468408`, 0.015304294697344465`,
0.007915195286690359`, 0.005880353628887382`,
0.004116410730170449`, 0.0031640275881454545`,
0.0035297581940090896`};(*initial velocity*)
T = {0, 88.0, 224.7, 365.2, 687.0, 4331, 10747, 30589, 59800, 90560};(* period of orbit in days*)

Solving the differential equations


eq = {Table[
x[i]''[t] ==

Sum[If[j == i,
0, (-\[Mu][[j]] (x[i][t] -
x[j][t]))/((x[i][t] - x[j][t])^2 + (y[i][t] -
y[j][t])^2)^(3/2)], {j, 10}], {i, 10}],
Table[y[i]''[t] ==
Sum[If[j == i,
0, (-\[Mu][[j]] (y[i][t] -
y[j][t]))/((x[i][t] - x[j][t])^2 + (y[i][t] -
y[j][t])^2)^(3/2)], {j, 10}], {i, 10}]};


var = Join[Table[x[i], {i, 10}], Table[y[i], {i, 10}]];

orb = NDSolve[{eq, Table[x[i][0] == rx[[i]], {i, 10}],
Table[y[i][0] == 0, {i, 10}], Table[x[i]'[0] == 0, {i, 10}],
Table[y[i]'[0] == v[[i]], {i, 10}]}, var, {t, 0, 90600}];

Plotting the bodies


plot2D = Show[
Table[ParametricPlot[
Evaluate[{x[i][t], y[i][t]} /. orb], {t, 0, 30000}(*,

PlotStyle\[Rule]None*), PlotRange -> 5], {i, 10, 10}]];

Animating the bodies


Animate[Show[plot2D, 
Graphics[Table[{Red, PointSize[0.02],
Point[{x[i][t], y[i][t]} /. orb]}, {i, 1, 10}]]], {t, 0, 90000},
AnimationRate -> 50, AnimationRunning -> False]

The Problem The simulation appears to work however, at around t=30000, the sun begins to drift which drags the rest of the bodies with it. Please see below for(first image) t=1000 and (second image)t=30000


As a result of this drift, pluto does not reach its aphelion of 49.3 AU.



Im aware that with N-body simulations, integration errors occur over time but could be an error in the code that might be causing this??


t=1000 t=30000




Comments

Popular posts from this blog

plotting - How to draw lines between specified dots on ListPlot?

I would like to create a plot where I have unconnected dots and some connected. So far, I have figured out how to draw the dots. My code is the following: ListPlot[{{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4,13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full] I have thought using ListLinePlot command, but I don't know how to specify to the command to draw only selected lines between the dots. Do have any suggestions/hints on how to do that? Thank you. Answer One possibility would be to use Epilog with Line : ListPlot[ {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4, 13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full, Epilog -> { Line[ ...

dynamic - How can I make a clickable ArrayPlot that returns input?

I would like to create a dynamic ArrayPlot so that the rectangles, when clicked, provide the input. Can I use ArrayPlot for this? Or is there something else I should have to use? Answer ArrayPlot is much more than just a simple array like Grid : it represents a ranged 2D dataset, and its visualization can be finetuned by options like DataReversed and DataRange . These features make it quite complicated to reproduce the same layout and order with Grid . Here I offer AnnotatedArrayPlot which comes in handy when your dataset is more than just a flat 2D array. The dynamic interface allows highlighting individual cells and possibly interacting with them. AnnotatedArrayPlot works the same way as ArrayPlot and accepts the same options plus Enabled , HighlightCoordinates , HighlightStyle and HighlightElementFunction . data = {{Missing["HasSomeMoreData"], GrayLevel[ 1], {RGBColor[0, 1, 1], RGBColor[0, 0, 1], GrayLevel[1]}, RGBColor[0, 1, 0]}, {GrayLevel[0], GrayLevel...

list manipulation - Selecting multiple columns from a matrix?

Sample data: data = { {{2013, 1, 1}, 24.13, 167.67, 231.82}, {{2013, 1, 2}, 32.15, 170.92, 225.99}, {{2013, 1, 3}, 35.43, 172.68, 221.67}, {{2013, 1, 4}, 36.73, 173.05, 218.32}, {{2013, 1, 5}, 58.19, 165.96, 197.05}, {{2013, 1, 6}, 69.99, 163.50, 187.52}, {{2013, 1, 7}, 71.37, 154.21, 175.58}, {{2013, 1, 8}, 72.51, 149.66, 163.25}}; I want a DateListPlot with three graphs, so for a matrix formed by columns 1 and 2, one for columns 1 and 3, and 1 for columns 1 and 4. At the moment I'm using this code: data2 = Transpose[{data[[All, 1]], data[[All, 2]]}]; data3 = Transpose[{data[[All, 1]], data[[All, 3]]}]; data4 = Transpose[{data[[All, 1]], data[[All, 4]]}]; DateListPlot[{data2, data3, data4}, Joined -> True, Filling -> {3 -> {1}}] but I have a hunch that this can be done more efficiently. I don't like the Transpose s in particular. Any ideas? edit (for extra credit) What if I need to multiply the second column by 2, which in my solution is simp...