Simulating the Outer Solar System

Data

The chosen units are masses relative to the sun, meaning the sun has mass $1$. We have taken $m_0 = 1.00000597682$ to take account of the inner planets. Distances are in astronomical units, times in earth days, and the gravitational constant is thus $G = 2.95912208286 \cdot 10^{-4}$.

planet	mass	initial position	initial velocity
Jupiter	$m_1 = 0.000954786104043$	[-3.5023653, -3.8169847, -1.5507963]	[0.00565429, -0.00412490, -0.00190589]
Saturn	$m_2 = 0.000285583733151$	[9.0755314, -3.0458353, -1.6483708]	[0.00168318, 0.00483525, 0.00192462]
Uranus	$m_3 = 0.0000437273164546$	[8.3101420, -16.2901086, -7.2521278]	[0.00354178, 0.00137102, 0.00055029]
Neptune	$m_4 = 0.0000517759138449$	[11.4707666, -25.7294829, -10.8169456]	[0.00288930, 0.00114527, 0.00039677]
Pluto	$m_5 = 1/(1.3 \cdot 10^8 )$	[-15.5387357, -25.2225594, -3.1902382]	[0.00276725, -0.00170702, -0.00136504]

The data is taken from the book “Geometric Numerical Integration” by E. Hairer, C. Lubich and G. Wanner.

import Plots, OrdinaryDiffEq as ODE
import ModelingToolkit as MTK
using ModelingToolkit: t_nounits as t, D_nounits as D, @mtkbuild, @variables
Plots.gr()

G = 2.95912208286e-4
M = [
    1.00000597682,
    0.000954786104043,
    0.000285583733151,
    0.0000437273164546,
    0.0000517759138449,
    1 / 1.3e8
]
planets = ["Sun", "Jupiter", "Saturn", "Uranus", "Neptune", "Pluto"]

pos = [0.0 -3.5023653 9.0755314 8.310142 11.4707666 -15.5387357
       0.0 -3.8169847 -3.0458353 -16.2901086 -25.7294829 -25.2225594
       0.0 -1.5507963 -1.6483708 -7.2521278 -10.8169456 -3.1902382]
vel = [0.0 0.00565429 0.00168318 0.00354178 0.0028893 0.00276725
       0.0 -0.0041249 0.00483525 0.00137102 0.00114527 -0.00170702
       0.0 -0.00190589 0.00192462 0.00055029 0.00039677 -0.00136504]
tspan = (0.0, 200_000.0)

(0.0, 200000.0)

The N-body problem's Hamiltonian is

\[H(p,q) = \frac{1}{2}\sum_{i=0}^{N}\frac{p_i^T p_i}{m_i} - G\sum_{i=1}^N \sum_{j=0}^{i-1}\frac{m_i m_j}{\left\lVert q_i - q_j \right\rVert}\]

where each $p_i$ and $q_i$ is a 3-dimensional vector describing the planet's position and momentum, respectively.

Here, we want to solve for the motion of the five outer planets relative to the sun, namely, Jupiter, Saturn, Uranus, Neptune, and Pluto.

const ∑ = sum
const N = 6
@variables u(t)[1:3, 1:N]
u = collect(u)
potential = -G *
            ∑(
    i -> ∑(j -> (M[i] * M[j]) / √(∑(k -> (u[k, i] - u[k, j])^2, 1:3)), 1:(i - 1)),
    2:N)

\[ \begin{equation} - 0.00029591 ~ \left( \frac{4.3728 \cdot 10^{-5}}{\sqrt{\left( - u\_{1,1}\left( t \right) + u\_{1,4}\left( t \right) \right)^{2} + \left( - u\_{2,1}\left( t \right) + u\_{2,4}\left( t \right) \right)^{2} + \left( - u\_{3,1}\left( t \right) + u\_{3,4}\left( t \right) \right)^{2}}} + \frac{0.00095479}{\sqrt{\left( - u\_{1,1}\left( t \right) + u\_{1,2}\left( t \right) \right)^{2} + \left( - u\_{2,1}\left( t \right) + u\_{2,2}\left( t \right) \right)^{2} + \left( - u\_{3,1}\left( t \right) + u\_{3,2}\left( t \right) \right)^{2}}} + \frac{1.4786 \cdot 10^{-8}}{\sqrt{\left( - u\_{1,3}\left( t \right) + u\_{1,5}\left( t \right) \right)^{2} + \left( - u\_{2,3}\left( t \right) + u\_{2,5}\left( t \right) \right)^{2} + \left( - u\_{3,3}\left( t \right) + u\_{3,5}\left( t \right) \right)^{2}}} + \frac{2.1968 \cdot 10^{-12}}{\sqrt{\left( - u\_{1,3}\left( t \right) + u\_{1,6}\left( t \right) \right)^{2} + \left( - u\_{2,3}\left( t \right) + u\_{2,6}\left( t \right) \right)^{2} + \left( - u\_{3,3}\left( t \right) + u\_{3,6}\left( t \right) \right)^{2}}} + \frac{2.264 \cdot 10^{-9}}{\sqrt{\left( - u\_{1,4}\left( t \right) + u\_{1,5}\left( t \right) \right)^{2} + \left( - u\_{2,4}\left( t \right) + u\_{2,5}\left( t \right) \right)^{2} + \left( - u\_{3,4}\left( t \right) + u\_{3,5}\left( t \right) \right)^{2}}} + \frac{3.9828 \cdot 10^{-13}}{\sqrt{\left( - u\_{1,5}\left( t \right) + u\_{1,6}\left( t \right) \right)^{2} + \left( - u\_{2,5}\left( t \right) + u\_{2,6}\left( t \right) \right)^{2} + \left( - u\_{3,5}\left( t \right) + u\_{3,6}\left( t \right) \right)^{2}}} + \frac{5.1776 \cdot 10^{-5}}{\sqrt{\left( - u\_{1,1}\left( t \right) + u\_{1,5}\left( t \right) \right)^{2} + \left( - u\_{2,1}\left( t \right) + u\_{2,5}\left( t \right) \right)^{2} + \left( - u\_{3,1}\left( t \right) + u\_{3,5}\left( t \right) \right)^{2}}} + \frac{1.2488 \cdot 10^{-8}}{\sqrt{\left( - u\_{1,3}\left( t \right) + u\_{1,4}\left( t \right) \right)^{2} + \left( - u\_{2,3}\left( t \right) + u\_{2,4}\left( t \right) \right)^{2} + \left( - u\_{3,3}\left( t \right) + u\_{3,4}\left( t \right) \right)^{2}}} + \frac{3.3636 \cdot 10^{-13}}{\sqrt{\left( - u\_{1,4}\left( t \right) + u\_{1,6}\left( t \right) \right)^{2} + \left( - u\_{2,4}\left( t \right) + u\_{2,6}\left( t \right) \right)^{2} + \left( - u\_{3,4}\left( t \right) + u\_{3,6}\left( t \right) \right)^{2}}} + \frac{7.6924 \cdot 10^{-9}}{\sqrt{\left( - u\_{1,1}\left( t \right) + u\_{1,6}\left( t \right) \right)^{2} + \left( - u\_{2,1}\left( t \right) + u\_{2,6}\left( t \right) \right)^{2} + \left( - u\_{3,1}\left( t \right) + u\_{3,6}\left( t \right) \right)^{2}}} + \frac{4.9435 \cdot 10^{-8}}{\sqrt{\left( - u\_{1,2}\left( t \right) + u\_{1,5}\left( t \right) \right)^{2} + \left( - u\_{2,2}\left( t \right) + u\_{2,5}\left( t \right) \right)^{2} + \left( - u\_{3,2}\left( t \right) + u\_{3,5}\left( t \right) \right)^{2}}} + \frac{0.00028559}{\sqrt{\left( - u\_{1,1}\left( t \right) + u\_{1,3}\left( t \right) \right)^{2} + \left( - u\_{2,1}\left( t \right) + u\_{2,3}\left( t \right) \right)^{2} + \left( - u\_{3,1}\left( t \right) + u\_{3,3}\left( t \right) \right)^{2}}} + \frac{4.175 \cdot 10^{-8}}{\sqrt{\left( - u\_{1,2}\left( t \right) + u\_{1,4}\left( t \right) \right)^{2} + \left( - u\_{2,2}\left( t \right) + u\_{2,4}\left( t \right) \right)^{2} + \left( - u\_{3,2}\left( t \right) + u\_{3,4}\left( t \right) \right)^{2}}} + \frac{7.3445 \cdot 10^{-12}}{\sqrt{\left( - u\_{1,2}\left( t \right) + u\_{1,6}\left( t \right) \right)^{2} + \left( - u\_{2,2}\left( t \right) + u\_{2,6}\left( t \right) \right)^{2} + \left( - u\_{3,2}\left( t \right) + u\_{3,6}\left( t \right) \right)^{2}}} + \frac{2.7267 \cdot 10^{-7}}{\sqrt{\left( - u\_{1,2}\left( t \right) + u\_{1,3}\left( t \right) \right)^{2} + \left( - u\_{2,2}\left( t \right) + u\_{2,3}\left( t \right) \right)^{2} + \left( - u\_{3,2}\left( t \right) + u\_{3,3}\left( t \right) \right)^{2}}} \right) \end{equation} \]

Hamiltonian System

NBodyProblem constructs a second order ODE problem under the hood. We know that a Hamiltonian system has the form of

\[\dot{p} = -\frac{\partial H}{\partial q}, \quad \dot{q} = \frac{\partial H}{\partial p}\]

For an N-body system, we can simplify this as:

\[\dot{p} = -\nabla V(q), \quad \dot{q} = M^{-1} p.\]

Thus, $\dot{q}$ is defined by the masses. We only need to define $\dot{p}$, and this is done internally by taking the gradient of $V$. Therefore, we only need to pass the potential function and the rest is taken care of.

eqs = vec(@. D(D(u))) .~ .-MTK.gradient(potential, vec(u)) ./
                         repeat(M, inner = 3)
@mtkbuild sys = MTK.System(eqs, t)
prob = ODE.ODEProblem(sys, [vec(u .=> pos); vec(D.(u) .=> vel)], tspan)
sol = ODE.solve(prob, ODE.Tsit5());

retcode: Success
Interpolation: specialized 4th order "free" interpolation
t: 862-element Vector{Float64}:
      0.0
      0.4345623854592078
     35.99898429859733
    174.61856264192386
    380.59547356844905
    611.8666224958926
    881.0171294254685
   1216.5292888230308
   1576.6203478300724
   1950.5247369459682
      ⋮
 199464.1763426005
 199535.15658191757
 199610.59925970965
 199680.2127842468
 199755.12591135743
 199821.69348971432
 199894.37470391282
 199963.8563633727
 200000.0
u: 862-element Vector{Vector{Float64}}:
 [-3.1902382, -25.2225594, -15.5387357, -10.8169456, -25.7294829, 11.4707666, -7.2521278, -16.2901086, 8.310142, -1.6483708  …  0.00354178, 0.00192462, 0.00483525, 0.00168318, -0.00190589, -0.0041249, 0.00565429, 0.0, 0.0, 0.0]
 [-3.190831391666314, -25.223301179998256, -15.537533140796315, -10.816773167663484, -25.728985182525822, 11.472022169413352, -7.251888638015288, -16.289512746418094, 8.31168109382013, -1.6475343823037847  …  0.0035416394156575887, 0.0019248511933338966, 0.004835677090231295, 0.0016819059109041002, -0.0019046284357956216, -0.004121794983806699, 0.005657137663985553, -1.2785303085738884e-9, -3.1048711440018464e-9, -2.3461051313643876e-9]
 [-3.2393549980716214, -25.28382714926436, -15.439004932934363, -10.802586698126767, -25.68807456291928, 11.574698023221423, -7.232135255028369, -16.240343014505505, 8.437431867001305, -1.5787461360440151  …  0.003530050705299505, 0.0019434002837321665, 0.004869678113502438, 0.0015771360623620687, -0.0017989198926536812, -0.003862300332365893, 0.005883789353087844, -1.0815714572233175e-7, -2.6207016211975343e-7, -1.8809265799702778e-7]
 [-3.4280420079701845, -25.516315174827085, -15.052897344777248, -10.74588663345915, -25.525275821246, 11.973377190937034, -7.151759201209805, -16.04109905292252, 8.92355578462731, -1.3047059800667917  …  0.0034833220382023663, 0.0020085086352349597, 0.004983872848475712, 0.001159925238119635, -0.0013434734330002656, -0.0027569266671721732, 0.006637565644492371, -5.641293116780171e-7, -1.355853291430368e-6, -7.856804265309118e-7]
 [-3.707045233737575, -25.851643637032474, -14.473230154989114, -10.657522537472943, -25.27360304517124, 12.561169018902879, -7.022513730851573, -15.723033045969192, 9.633516314698607, -0.8828569747252994  …  0.0034093846352291437, 0.0020830012488561394, 0.005097364894807607, 0.0005177706296843234, -0.0005615639064162919, -0.0008938228560599242, 0.00732295326725339, -1.3356559430885386e-6, -3.1755768738352596e-6, -1.2520419390932113e-6]
 [-4.018219555698541, -26.213593809252725, -13.814242309814366, -10.55249040041501, -24.977230530048878, 13.214270892868017, -6.863712622101781, -15.335273269499737, 10.411813081950893, -0.3946555485512191  …  0.003320134583039653, 0.0021327085030388186, 0.005140272710915692, -0.00022593519308113902, 0.0004056976190662129, 0.0013657697967930863, 0.00737629704852887, -2.2774483152877735e-6, -5.354540757180861e-6, -1.0850151360140068e-6]
 [-4.377351821263963, -26.615251742595444, -13.037031515584674, -10.4225811285836, -24.61415232483528, 13.964702392241312, -6.661187598566764, -14.844430810695425, 11.29056443664343, 0.18184560197133734  …  0.0032082766977699646, 0.0021422344170847346, 0.005071760741873903, -0.0011053612366158495, 0.001524119679033659, 0.00391913317522403, 0.006393763850831102, -3.352643743731525e-6, -7.783412619735192e-6, 1.1117424004516135e-7]
 [-4.820111737629151, -27.086169758809156, -12.053684429277054, -10.249222799506164, -24.134590564238305, 14.88478091799421, -6.383017215336513, -14.175283374934832, 12.342012873833715, 0.8921234781572502  …  0.003057381822894924, 0.002077267176953791, 0.004801498391555867, -0.00219007808374467, 0.0026373811216874425, 0.006361810268772439, 0.0036755225624795053, -4.4025951139865464e-6, -1.005113500232717e-5, 3.0255810736516264e-6]
 [-5.288698031255902, -27.55448768068402, -10.981811680435065, -10.049281927244104, -23.587206884911865, 15.8520692715453, -6.054200506543434, -13.389902886563227, 13.411790811130421, 1.6130640693535738  …  0.002882098483398166, 0.001910140608322444, 0.004282315243518839, -0.0032893506670255575, 0.0031164430292441457, 0.007245658434674586, -0.00043809969483555244, -4.818016508531506e-6, -1.0759768964913463e-5, 7.277884644905893e-6]
 [-5.767395463053767, -27.999807278078084, -9.852496757125376, -9.826742706901163, -22.983785100386633, 16.83289924854452, -5.681515563669695, -12.50527369793824, 14.453259951043403, 2.2785511623098795  …  0.0026864469850750635, 0.0016324385357110599, 0.0035049450533046413, -0.004295404494929296, 0.0026116141120660847, 0.005830469215816499, -0.004617706489447968, -4.262396980911033e-6, -9.199407080435744e-6, 1.1567784087392676e-5]
 ⋮
 [-15.12428321145426, -14.894874081334379, 35.667880311593166, 7.4315776951389605, 19.527694829589173, 21.925029902362844, 7.078852434351263, 16.078827530312687, -3.779058341746134, -0.31137881425065644  …  -0.0038077595252227576, -0.0021362445252223108, -0.005131207063513987, 0.00033960493122794493, 0.0057634393409658315, 0.013411117084152619, -0.000724469903621151, -6.144047199049043e-6, -1.3831359131893112e-5, 7.062596806457794e-6]
 [-15.114710858659436, -14.74747139770558, 35.789181791619086, 7.493906799414991, 19.670140847882386, 21.763222135660016, 7.045893479723227, 15.99492077658409, -4.04876070501613, -0.46291236561743043  …  -0.0037914569940202987, -0.002132891443935548, -0.0050992330755615265, 0.0005673012351435305, 0.00415292593475153, 0.009088380276305609, -0.010694180169522076, -4.605988344596329e-6, -9.710109400497054e-6, 1.6516590126110436e-5]
 [-15.104163016575788, -14.590443479633441, 35.917240726127105, 7.559675650564673, 19.820295168829617, 21.58997228355702, 7.009062917501749, 15.901670718128706, -4.334114311936603, -0.6235657813361021  …  -0.0037731551668865955, -0.0021253907795129433, -0.005056012249711443, 0.0008065985304772452, -0.00012125544499117648, -0.0011150912483529805, -0.014716418089412023, -5.2581250719376e-7, 2.293297831075599e-8, 2.028869895561162e-5]
 [-15.094090024875962, -14.445225638598082, 36.03461109434366, 7.619921572132155, 19.95769957058487, 21.42895771930952, 6.973437719918136, 15.811919443793055, -4.596163132017202, -0.7711770863362468  …  -0.003755378273250854, -0.002114927659561817, -0.005007861075214955, 0.001024577622422391, -0.004174989546655308, -0.01030655218793932, -0.009941094064845496, 3.3429289853874026e-6, 8.788061526939474e-6, 1.5667103277921365e-5]
 [-15.082887077037949, -14.2886113729952, 36.160065534762005, 7.684277012530359, 20.10432466176581, 21.254461476528693, 6.933347106815876, 15.71137646548158, -4.876744204094014, -0.9290754611443719  …  -0.0037352958226249144, -0.002099941274711382, -0.004947384959897934, 0.0012557684635037913, -0.005914457227321276, -0.013714384274526491, 0.0015911414045346097, 5.000861017418266e-6, 1.2027804229068342e-5, 4.590322135364936e-6]
 [-15.072617637514437, -14.149151810999962, 36.270803622342584, 7.741044998762698, 20.233529580763403, 21.098348757614914, 6.896202570000974, 15.61860374414486, -5.124776605861796, -1.0683309691256593  …  -0.00371662258487358, -0.002083450192965714, -0.004886307957185618, 0.0014579657718091072, -0.004127768088198349, -0.008995278383096824, 0.011298737607877113, 3.291500821785404e-6, 7.507565382821609e-6, -4.736091242059667e-6]
 [-15.06106848238177, -13.996574213275812, 36.39091653815536, 7.802573909148575, 20.37342626252894, 20.926773328706137, 6.854018931379957, 15.513639585596032, -5.394137651459188, -1.2190038670401349  …  -0.0036953459140358926, -0.0020621049845998825, -0.00481194394771111, 0.001674927483288424, 0.0002411778246242779, 0.0014023657596519635, 0.014843420826708972, -8.846206401029478e-7, -2.437961485763195e-6, -8.182542643787225e-6]
 [-15.049700635517883, -13.850415133881445, 36.504965631009156, 7.860949121362133, 20.50601071106621, 20.761661940075516, 6.81210963146918, 15.409728745894729, -5.650164631691982, -1.3614804101314608  …  -0.003674141068707149, -0.0020385077879030607, -0.004733552691032501, 0.0018783500766333657, 0.004227914291723056, 0.010411150574796236, 0.009603033468161, -4.696520395831749e-6, -1.10587272355188e-5, -3.2373546988628376e-6]
 [-15.043661226907014, -13.774270843429269, 36.563992817014835, 7.891142208622381, 20.574531427123674, 20.675354847003064, 6.789699247604722, 15.354302034897824, -5.782756769087059, -1.4349182809487384  …  -0.0036627770095602153, -0.0020250238346580327, -0.004690020656762545, 0.001982543863559828, 0.0054353243107729396, 0.01293953488877421, 0.004479354345387266, -5.852508555675117e-6, -1.3483635473863919e-5, 1.6247891406544974e-6]

plt = Plots.plot()
for i in 1:N
    Plots.plot!(plt, sol, idxs = (u[:, i]...,), lab = planets[i])
end
Plots.plot!(plt; xlab = "x", ylab = "y", zlab = "z", title = "Outer solar system")