Now it's finally time to add the spaceship, the fourth body to all the three body problem, and I think
this is really an exciting project and a really important project because we are actually planning here
a moon mission.
And I want to show you what we can do to reach the orbit of the Moon from Earth using a spaceship.
Of course, there are some effects that we are disregarding here, like the atmosphere, which gives
a lot of friction to the spaceship.
So that's actually a pretty important aspect.
But what we will do here is we will start from a rocket that orbits the Earth and then we tell the rocket,
What do we have to do to reach the Moon?
All right.
But before we do this, let's consider the spaceship and let me tell you how we can add the spaceship
to our differential equations.
And the first important thing is that I want to assume here that the mass of the spaceship is extremely
small compared to the other massive masses of the Sun, the Earth and even the Moon.
So, of course, you could take the all three eight differential equations and add now a new term to
all three of those tests would be actually very easy, and you would just have to add a new term.
Basically, here I have copied the old stuff, the old f already function.
You would just have to add to your new term with all four.
However, we don't even have to do this here because we can just assume that the mass of the spaceship
is so small that the motion of the Sun, the Earth and the Moon is not affected.
And I mean, it's hardly imaginable that the Sun will move just because the spaceship is flying by.
But in reality, this is really happening, but it's just by a tiny amount.
I don't know how much, but it's really a tiny amount so we can neglect.
So we say in this approximation, which is an excellent approximation, the three differential equations
remain and we will add a fourth one just describing the spaceship.
And of course, we have three terms and this will be the gravitational forces and gravity with the Sun,
the Earth and the Moon, and then we can order these terms.
These are not six terms actually we owed them by one or two or three or four.
So we have no three terms here, three terms here, and we have to program this now for the f OTV.
So to do this?
First of all, let me introduce here four.
This is our going from nine to 12.
This one would not be correct.
Any more would be.
One is now from 12 to 15.
So let's just delete it.
We don't need it.
And then we need three more distances.
One four is A. P. Linear algebra dot norm and then one to four.
We could also write four minus one.
Wouldn't matter.
It would give the same thing.
And then we add to other relations are two four, three four.
And this would be our two minus R four and four.
And is our three like this?
OK, and now we can go ahead and just program this equation here.
So first, we have three terms like this, which would be m one divided by our one for power of three
times one.
Then another one with M two are two four.
And then here are two, and the next one and three are three, four or three.
And then we take basically this one here.
But then we have three terms.
First one is M one divided by R one for power three plus and two divided by our two for power three.
Another one, which is three divided by R three for power three times are for.
And then here we have now the velocities going from 12 to 24, and I think that's already it.
So we run, it's no error, which is great, and we have to also tweak the starting conditions.
So we have to act here now.
The space ship and the space ship is basically, let me think, basically similar to the Moon.
So in a sense, the spaceship is supposed to orbit the Earth like the Moon orbits the Earth, and it
will orbit the Earth with a position of our Sun, Earth and then our orbits.
And the velocity will be minus V orbits and the Earth, and now we just need to include here our orbits
and V orbits.
And to do this, I ask your new cell and I've prepared this before we have in meter four times 10 to
the power of seven and meter, which is a standard stationary orbit in space of satellites, for example.
And the corresponding velocity would be this and meters per second.
Of course.
And we can run this now.
And now we can go ahead and define the starting values for a spaceship.
So here, if we've got to change this off or before, and of course, we must include it here to specter
for the starting conditions for start.
And we for start.
OK, so you see, it's pretty similar to the Moon, just that the radius and velocity are smaller than
for the Moon because, you know, it's much closer to the Earth.
OK, so now I think we are almost finished.
I think we can just see what happens.
And let me out here another plot with our spaceship.
I choose to color purple.
And now here we must change it to nine and 10.
So let's see.
OK, it's running, it's running for 10 years.
That's probably too long, I guess.
Maybe I should have gone for just one year.
Yeah.
Let me let me rerun this.
Let me colonel restart in front of.
It's not a whole notebook will be run again.
And.
Now we are already back at our simulation of the spaceship because the rest of the notebook is very,
very fast.
And though it's finished.
OK, and it seems like this is here on the scale of Sun and Earth.
Then we have the moon on top and now we also have our spaceship on top.
So great, but we can't see anything.
Of course we have to analyze now the trajectory was back to Earth and Moon.
And for this, I scroll up where we did the same thing already for the Moon.
So this was this one here.
And let me copy this, I a. just write X with respect to Moon, sorry with respect to Earth, because
we will now plot the Moon and the spaceship and now we will plot basically this one we leave.
This is green.
This will be the moon and knowledge and another one which will be purple.
This will be nine minus three and 10 minus four, which will be spaceship minus Earth.
So now we see our Earth is fixed, so we have changed our coordinate system and fixed it to the position
of the Earth.
And you see, the Moon is rotating like this and the spaceship is rotating like this.
So it's really on a stationary orbit quite close to the Earth and quite far from the Moon.
So that's really, as I said, a typical trajectory of the of the satellites.
So like from which we get our GPS and our internet.
OK.
So and the following lectures.
I want to play now a bit with the starting conditions.
You can if you want pulsed video and just try a bit for yourself, but I will explain to you four very
interesting cases, and let's get started with the next lecture.