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So let's continue, we have just programmed the differential equations, and we have programmed to starting

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conditions, and now we can go ahead and solve the problem.

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So as always, we will store our solution in the variable and we will call integrate dots.

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Solve underscore, I repeat, I think by now, you should know.

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And first, we must provide, as always, a function and we must provide some time.

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So t start to T and we will to find these in a second.

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So t start and T and.

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And then we had the final restarting condition, which will be our zero, and this will be our 18 component

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factor, and we will use once again the methods are AK 45 and we will output a equidistant array of

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time points with the corresponding solutions.

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So we will use to out argument, which is optional, and we will create here an array from start to

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end with a certain number of steps.

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And I think we will use quite a lot of those.

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So let's start with 100k and then an extra one so that we have to start point as well.

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So the start times easier, just zero.

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And then for the end time, we have to think a bit because if we now write 100 here, then this will

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be 100 seconds.

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And what are 100 seconds in terms of the motion of planets or the Moon?

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It's nothing.

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So we must use a pretty large number here.

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And I want to start you with a year and this is, of course, 60 times 60 times 24 and then times 365.

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And if you want to be precise, a quarter seconds.

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So this this is a year in astronomical terms.

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And so our value will be here like this.

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And if we, for example, want to use two years, then we write two times 2.0, OK, and now we can

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test if it works.

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You see, the simulation was very fast, didn't even take a blink of an eye.

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It was finished instantly.

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So let's go ahead and see if we can plot something nice here.

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So, all right, Paul, dot plot.

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And then first of all, I want to plot the trajectory.

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So a solution to why zero and solution thought, why one?

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So I want to plot the X versus two Y components.

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And the first thing that we plot is the trajectory of the Sun and then we go ahead and add to other

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plots.

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So these will be then the Earth.

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So this three and four and we choose a different color here.

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Blue and the Moon will be six and seven and the color, for example, green and we see we get a nice

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output.

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So now let's make the plot a bit nicer.

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So let me copy here from my other notebook that I've prepared the aspect ratio that's set one and the

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labels, which are X and Y coordinates of the individual objects.

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So you see, this is now our output.

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And maybe you are wondering now.

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So first of all, we see the right trajectory.

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That's the motion of the Sun.

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And it looks pretty good.

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It's just positioned in the center and doesn't move at all.

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This is what we want.

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This is what we expect.

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And then we have the trajectory of the Moon and the Earth.

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And it doesn't look so good, to be honest, because they are moving on a spiral shaped trajectory.

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And it turns out that the problem is here the accuracy of the solver.

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The accuracy is not high enough to get a correct solution here.

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And I think this is understandable because all of our units, and especially our time is very, very

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large.

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So the standard error that is programmed is not sufficient.

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So let me show you what the standard error is.

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We have a correlative tolerance.

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So this is what is acceptable for the solver.

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This is 10 to the power of minus three, and we have an absolute tolerance, which will be added.

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So you can read if you want to know the details you can read out by just Googling solve on the Scott

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IVP, and then these things are explained.

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But these are just absolute and relative tolerances for the error.

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And this one is, what, 10 to the minus six.

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And if I run this now, we get the exact same solution.

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So yeah, these are really the standard values, and now we can test what happens if we decrease these

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values.

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For example, I multiply here by one over a thousand and I multiply here by the same number.

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And you see now the spiral, that directory is gone.

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And just to be on the safe side, let's just add a few zeros here because you see the calculation was

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still.

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Every very fast, and I will add here three more zeros and here as well one, two three.

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And you see, still the calculation is fast and we have now a nice trajectory.

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So now the green object, which is the moon, moves on a circle around the sun.

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And actually we have here blue and the green trajectory right on top of each other because the Earth

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and the Moon are so close to each other compared to the Sun.

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So we can't even see here Ervand individually.

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So let's think about how we can basically analyze or discussed this motion in more detail.

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And now I will go ahead and analyze the motion of all three objects individually and in more detail.