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Now it's finally time to add the spaceship, the fourth body to all the three body problem, and I think

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this is really an exciting project and a really important project because we are actually planning here

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a moon mission.

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And I want to show you what we can do to reach the orbit of the Moon from Earth using a spaceship.

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Of course, there are some effects that we are disregarding here, like the atmosphere, which gives

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a lot of friction to the spaceship.

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So that's actually a pretty important aspect.

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But what we will do here is we will start from a rocket that orbits the Earth and then we tell the rocket,

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What do we have to do to reach the Moon?

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All right.

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But before we do this, let's consider the spaceship and let me tell you how we can add the spaceship

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to our differential equations.

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And the first important thing is that I want to assume here that the mass of the spaceship is extremely

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small compared to the other massive masses of the Sun, the Earth and even the Moon.

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So, of course, you could take the all three eight differential equations and add now a new term to

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all three of those tests would be actually very easy, and you would just have to add a new term.

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Basically, here I have copied the old stuff, the old f already function.

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You would just have to add to your new term with all four.

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However, we don't even have to do this here because we can just assume that the mass of the spaceship

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is so small that the motion of the Sun, the Earth and the Moon is not affected.

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And I mean, it's hardly imaginable that the Sun will move just because the spaceship is flying by.

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But in reality, this is really happening, but it's just by a tiny amount.

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I don't know how much, but it's really a tiny amount so we can neglect.

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So we say in this approximation, which is an excellent approximation, the three differential equations

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remain and we will add a fourth one just describing the spaceship.

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And of course, we have three terms and this will be the gravitational forces and gravity with the Sun,

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the Earth and the Moon, and then we can order these terms.

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These are not six terms actually we owed them by one or two or three or four.

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So we have no three terms here, three terms here, and we have to program this now for the f OTV.

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So to do this?

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First of all, let me introduce here four.

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This is our going from nine to 12.

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This one would not be correct.

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Any more would be.

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One is now from 12 to 15.

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So let's just delete it.

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We don't need it.

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And then we need three more distances.

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One four is A. P. Linear algebra dot norm and then one to four.

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We could also write four minus one.

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Wouldn't matter.

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It would give the same thing.

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And then we add to other relations are two four, three four.

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And this would be our two minus R four and four.

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And is our three like this?

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OK, and now we can go ahead and just program this equation here.

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So first, we have three terms like this, which would be m one divided by our one for power of three

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times one.

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Then another one with M two are two four.

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And then here are two, and the next one and three are three, four or three.

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And then we take basically this one here.

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But then we have three terms.

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First one is M one divided by R one for power three plus and two divided by our two for power three.

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Another one, which is three divided by R three for power three times are for.

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And then here we have now the velocities going from 12 to 24, and I think that's already it.

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So we run, it's no error, which is great, and we have to also tweak the starting conditions.

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So we have to act here now.

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The space ship and the space ship is basically, let me think, basically similar to the Moon.

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So in a sense, the spaceship is supposed to orbit the Earth like the Moon orbits the Earth, and it

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will orbit the Earth with a position of our Sun, Earth and then our orbits.

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And the velocity will be minus V orbits and the Earth, and now we just need to include here our orbits

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and V orbits.

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And to do this, I ask your new cell and I've prepared this before we have in meter four times 10 to

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the power of seven and meter, which is a standard stationary orbit in space of satellites, for example.

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And the corresponding velocity would be this and meters per second.

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Of course.

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And we can run this now.

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And now we can go ahead and define the starting values for a spaceship.

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So here, if we've got to change this off or before, and of course, we must include it here to specter

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for the starting conditions for start.

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And we for start.

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OK, so you see, it's pretty similar to the Moon, just that the radius and velocity are smaller than

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for the Moon because, you know, it's much closer to the Earth.

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OK, so now I think we are almost finished.

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I think we can just see what happens.

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And let me out here another plot with our spaceship.

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I choose to color purple.

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And now here we must change it to nine and 10.

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So let's see.

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OK, it's running, it's running for 10 years.

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That's probably too long, I guess.

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Maybe I should have gone for just one year.

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Yeah.

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Let me let me rerun this.

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Let me colonel restart in front of.

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It's not a whole notebook will be run again.

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And.

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Now we are already back at our simulation of the spaceship because the rest of the notebook is very,

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very fast.

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And though it's finished.

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OK, and it seems like this is here on the scale of Sun and Earth.

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Then we have the moon on top and now we also have our spaceship on top.

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So great, but we can't see anything.

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Of course we have to analyze now the trajectory was back to Earth and Moon.

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And for this, I scroll up where we did the same thing already for the Moon.

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So this was this one here.

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And let me copy this, I a. just write X with respect to Moon, sorry with respect to Earth, because

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we will now plot the Moon and the spaceship and now we will plot basically this one we leave.

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This is green.

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This will be the moon and knowledge and another one which will be purple.

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This will be nine minus three and 10 minus four, which will be spaceship minus Earth.

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So now we see our Earth is fixed, so we have changed our coordinate system and fixed it to the position

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of the Earth.

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And you see, the Moon is rotating like this and the spaceship is rotating like this.

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So it's really on a stationary orbit quite close to the Earth and quite far from the Moon.

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So that's really, as I said, a typical trajectory of the of the satellites.

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So like from which we get our GPS and our internet.

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OK.

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So and the following lectures.

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I want to play now a bit with the starting conditions.

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You can if you want pulsed video and just try a bit for yourself, but I will explain to you four very

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interesting cases, and let's get started with the next lecture.