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So in the previous lecture, we had our own encounter, we had chosen our starting we lost teeth such

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that our lips almost hits the Moon.

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And if we then don't do anything, we will get thrown out of the trajectory around the Earth and close

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to the Moon, and we will then just circle around the sun.

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This is, of course, really nice if you want to leave Earth and want to save a bit of fuel, but it

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can also be used to break at this point and then to go on a trajectory around the Moon.

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And this is what we want to do now.

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So the task for this very special lecture is to set up a space program where we will encounter an orbit

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around the Moon and we want to plan ahead what we want to do with our engine, at which point of time.

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So.

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For this, we have to modify our equations of motion, but only a single one.

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Only the motion of the spaceship, of course.

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And also we have the same thing as before and just add a single term here.

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So I told you that I want to break and breaking in space is not really possible.

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Can I just push a brake pedal?

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Because, yeah, you have no friction.

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So what you have to do is you have to accelerate it into the opposite direction.

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You have to use an engine, some booster along the opposite direction and with opposite direction,

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I mean opposite direction with respect to the relative velocity of our spaceship compared to the Moon,

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because we want to enter an orbit around the Moon.

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So we have to break along the opposite direction of the velocity of our spaceship relative to the velocity

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of the Moon.

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Sounds difficult, but if you think about it, it's not that difficult.

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You could just say, OK, here we have our distance spaceship to the Moon.

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And at this point you look down to the moon, you see your velocity and you push the the engine and

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you accelerate along the opposite direction along which you are moving.

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OK.

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So this will be the term just some, some some prefecture times the direction.

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So the direction is really what's important here.

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And the prefecture will be, of course, time dependent because we don't want to have this false the

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whole time.

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We just want to have it when our engine is on.

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And this is really what we want to do here.

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We have times where the engine is on and we have times where the engine is off.

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So the first question would be how can you program a function which is on and off?

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Of course, we could just take our F40 and use an if statement.

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And if our time is in a specified frame, then we use the term and if not, then we don't use the term.

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But it's always better to avoid if statements and mathematicians use, in this case, the so-called

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heavy side function.

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Let me show you how this works.

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This is basically just a step function.

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So I start with Telus just for the plotting and pitot in space.

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Oops.

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Sorry, typos

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and alan space goes from, I don't know, it doesn't matter here zero to 100.

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And what we want to plot is for the x axis to T list the time and then for the wireless R plot to have

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a site function and p dot.

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Have you sites?

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We see it's an important function.

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It's implemented in NumPy, so the guy who was named heavy side.

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So that's why it's not with a y.

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What would an AI and he rewrite it rewrite T list minus some shift, for example.

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Twenty five.

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And then we have to write here.

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A number could read zero one.

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That doesn't really matter for us.

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If you want to know the details, you can look that description, but it doesn't matter.

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We should just write one here.

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This is what matters for us.

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And what happens then is it's a step function that zero for arguments that are smaller than this number,

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and it is one for arguments that are larger than this lumber.

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So now we can subtract another heavy side function with a different shift, for example.

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75.

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And so then we get such such a step, step up, step down.

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And it is one only between these two arguments and this is exactly what we want.

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We could now, of course, write this here in brackets and multiply some factor, like 100.

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And then you see, it's zero all the time and then only in this range here it's a hundreds.

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So this is what we will add to our EV ODI.

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So let me scroll back up to the notebook and copy the Odyssey and a few plots, and then I'll be back

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in a second and then we modify all of these commands and then we go to approach this moon orbit.

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So here I'm back.

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We have just discussed the equation of motion and the heavy side function.

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And now we want to add this term he with the heavy side function to our F ODI, which I have copied

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from above.

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And then also I have copied here.

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This is code that we have written before with the starting conditions and the solution by integrated

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solve underscore IVP and the plots.

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And then here, another plot spaceship tune that's all just copied from above.

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OK, so we will leave everything as it is.

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We will just use here the same starting velocity that we had before.

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One point thirty four times we orbit and then we can.

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Basically rallied as soon as we have out a Tier two additional term to the equation, so I will right

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here.

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This factor, I will call be off a break and then we have to program here this direction, which is

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basically this one.

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The velocity of the space ship with respect to the velocity of the moon and then divides by the known.

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All right.

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So, all right.

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Minus the art times before minus the three divided by and pitot taught Lynn Rock tot mom.

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And then once again, we three minus three or four or vice versa doesn't matter in this case.

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What do you have to be careful here?

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You have to write me four minus three.

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Otherwise, it will be an acceleration, which is not what you want.

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We want to really break.

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OK, so this is the new term here.

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Be our times the distance of two vectors of the velocities divided by the norm of this distance.

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And now we must, of course, program the hour before and we leave three.

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So the three is ah, and we could just take this one plus 12 plus 12.

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And B, four, nine plus 12 and 12 plus 12, you could, of course, just calculate it, but I just

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want to show you this just shift of this index by 12th because we have 12 variables.

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And yeah, the first 12 are the positions and the other 12 are the velocities.

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So now I can find the brake function.

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So this will be B art is and now we need an amplitude.

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And you have to be honest, I spend like maybe half an hour trying a bit of a parameters.

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And this is what I came up with.

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So I have 27 million so times 10 to the power of six.

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This will be Newton, 27 Newton or 27 Mega Newton.

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And we apply this force for a certain time and this we will construct by using the heavy side function,

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of course.

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So I will just copy this.

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We will, of course, use the same type of command here.

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This was our amplitude.

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Sorry, here we of course, have the exponent 10 to the power of six.

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And now we have here.

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Do you have your site function?

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OK, so we don't have to list anymore, but tea, which is important to change.

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So now for the first time, our differential equation is actually T dependent.

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But that's no problem because it was it could have been Tiede dependent all the time.

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And now we must subtract here the starting time and the end time, and we see here that we approach

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the Moon approximately after 0.11 years.

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So I will use the engine approximately at this point of time.

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So let me copy this time factor here and then multiply it by zero point one one, which will then be

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zero point one one years.

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So here's where it goes.

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T minus zero point one one times.

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Then this time factor and then this heavy side function will be the point where the engine is turned

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off again.

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And I don't know, I just take zero point twelve.

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So basically, we have the engine on four one hundredths of a year, which is, of course not how it

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would be done.

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In reality, they're they would really just boost for a few, I don't know, few seconds.

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Few minutes.

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I don't know.

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I'm not an astrophysicist, but definitely not for one hundredth of a year, which is almost four days.

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But anyway, this is what I just found to work.

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So we boost here for four days with a force of twenty seven Meghan Newton divided by the mass of the

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object, of course.

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So this is here an acceleration then, and this is our heavy side function.

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So this means the engine is turned on just in this range from zero one two zero point twelve years.

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OK, and if I did not make a mistake, then I think it should work now.

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Let's see.

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I run this, run this and run this all three cells.

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And I think, no, it will take a bit of time.

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This one worked, but now it's running here.

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Maybe we have to think for how long we want to run.

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Actually, I want to run only for 0.3 years as before.

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So before we look, let me rerun this zero point three and this is what we get.

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So first of all, let's look again at the trajectory without the engine.

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This was this oval shape.

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And then once we encountered the moon, we leave it, which we also saw here at the distance plot.

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And now with the engine turned on at this short amount of time, we get such a trajectory.

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First, the Oval and then you see it moves somewhere close to the trajectory of the Moon.

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And you see this is what happens when you plot the distance spaceship moon.

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It goes first on this funky oval shape and then here and 0.11.

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Maybe here to here we turn on the engine and we break the spaceship a whole lot.

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And then you see, this is what we end up with.

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So that's pretty cool, I think.

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It's not the last thing that I want to show you is a plot similar to this one where we have plotted

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the trajectory of the spaceship with respect to the Earth, where the Earth was fixed here.

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So as well as with respect to Earth.

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So I copied this code here and adapted it.

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And instead, now we are in the reference frame of the Moon.

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So we have X and Y with respect to the Moon and the X and Y limits are a bit zoomed in, of course,

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because the radius of the trajectory is, of course, pretty, pretty small here.

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As you can see, it's oscillating pretty closely around the Moon compared to the previous rotation around

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the Earth.

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And here I changed the indices.

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So previously we had here three and four for the Earth, and now it's six and seven for the Moon and

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you see already the results.

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This is what we get.

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We have a point where we enter the vicinity of the Moon, which is positioned here at zero zero and

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then we oscillate on the rather strange looking trajectory.

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So it looks pretty funny, actually very, very oval.

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And also, since these points here are quite far still from the position of the Moon, it will still

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very strongly feel the influence of the Earth.

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So this is why it will not move on the same oval, the whole time of why it will be shifted around the

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whole time.

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So we could fix this if we want to make our orbits here are circular.

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And if we would make it pretty small, this we could achieve, for example, by breaking another time

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when we are here, so we could now add here in our plots, for example, when we are here at this point

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or also maybe at this point we could break another time so we could change here on a break function

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and add another two heavy side functions and another brake maneuver.

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But I want to leave it as it is, even though it looks a bit funny, it is a trajectory and an orbit

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around the Moon, even though it's not very, very stable.

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I think if you would simulate this for several years, you would eventually leave the Moon once again.

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But you could easily avoid this in reality by just breaking another time, as I just told you.

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And we could go even one step further by breaking at this point, even stronger.

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And then we would hit the Moon, we would crash into the Moon.

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So now it's really just a matter of optimizing the parameters or adding more break maneuvers and making

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everything a bit more complicated.

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And then you can really achieve everything we want.

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So if you want to do it, that's nice exercise to play a bit more with the code and the parameters.

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But eventually, everything comes down to using the integrator to solve IBP, and it comes also down

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to programming this differential equation here.

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And for this one, we have a four body problem.

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We had 12 plus 12 equations, 12 equations for the coordinates of the four objects and 12 equations

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for the velocities.

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So that's the end of this lecture about the three body problem, which we have actually turned into

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a full body problem.

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And we have learned quite a lot about in America and also about the physics, especially.

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I think this is a very nice lecture because we have achieved this moon orbit, we have achieved the

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surface K by a moon and counter.

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We have achieved the regular purpose escape and we have achieved a circular and an overall trajectory

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around the Earth.