 for this evening, is on how to get to Mars in all very interesting ways, some of them might be really, really slow. Our next speaker has studied physics and has a PhD in math and is currently working as a mission planner at the German Space Operations Center. Please give a big round of applause to Sven. Thank you. Hello and welcome to Thrust is not an option, how to get to Mars really slow. My name is Sven. I'm a mission planner at the German Space Operations Center, which is a part of the DLR, the Deutsches Zentrum für Luft und Raumfahrt. And first of all, I have to apologize because I kind of cheated a little bit in the title. The accurate title would have been reducing Thrust, how to get to Mars or maybe Mercury really slow. The reason for this is that I will actually use Mercury as an example quite a few times. And also we will not be able to actually get rid of all the maneuvers that we want to do. So the goal of this talk is to give you an introduction to orbital mechanics, to see what we can do, what are the techniques that you can use to actually get to another planet, to bring a spacecraft to another planet. And also go a few more, go a bit further into some more advanced techniques. So we will start with gravity and the two-body problem. So this is the basics, the underlying physics that we need. Then we will talk about the two main techniques, maybe to get to Mars, for example, the Hohmann transfer, as well as gravity assists. The third point will be an extension of that that's called a planar circular restricted three-body problem. Sounds pretty complicated, but we will see in pictures what it is about. And then we will finally get the taste of certain ways to actually be even better, be even more efficient by looking at what's called ballistic capture and the weak stability boundary. All right, so let's start. First off, we have gravity and we need to talk about the two-body problem. So I'm standing here on the stage and I'm actually being, well, accelerated downwards, right? The Earth actually attracts me and this is the same thing that happens for any two bodies that have mass, okay? So they attract each other by gravitational force and this force will actually accelerate the objects towards each other. Notice that the force actually depends on the distance, okay? So we don't need to know any details, but in principle, the force gets stronger the closer the objects are, okay? Good. Now, we can't really analyze this whole thing in every detail. So we will make a few assumptions. One of them will be that all our bodies, so in particular the sun, Earth, will actually be points, okay? So we will just consider points because anything else is too complicated for me. Also, all our satellites will actually be just points. One of the reasons is that in principle you have to deal with the attitude of the satellites. For example, a solar panel needs to actually point towards the sun. Of course, that's complicated, so we will skip this for this talk. Third point is that none of our planets will have an atmosphere, so there won't be any friction anywhere in the space. And the fourth point is that we will mostly restrict to movement within the plane, so we only have like two dimensions during this talk and also I will kind of forget about certain planets and other masses from time to time, okay? I'm mentioning this because I do not want you to go home this evening. Start planning your own interplanetary mission, then maybe building your spacecraft tomorrow, launching in three days, and then a week later I get an email, hey, you just didn't work. I mean, what did you tell me? Okay, so if you actually want to do this at home, don't try this just now, but please consult your local flight dynamics department. They will actually supply you with the necessary details. All right, so what's the two-body problem about? So in principle, we have some body, the sun and the spacecraft that is being attracted by the sun. Now, the sun is obviously much heavier than a spacecraft, meaning that we will actually neglect the force that the spacecraft exerts on the sun. So instead, the sun will just be like, it will be at some place, it might move in some way or a planet, but we only care about a spacecraft in general. Furthermore, notice that if you specify the position and the velocity of a spacecraft at some point, then the gravitational force will actually determine the whole path of the spacecraft for all time, okay? So this path is called the orbit, and this is what we are talking about. So we want to determine orbits, we want to actually find ways how to efficiently change orbits in order to actually reach Mars, for example. There's one other thing that you may know from your day-to-day life. If you actually take an object and you put it high up and you let it fall down, then it will accelerate, okay? So one way to actually describe this is by looking at the energy, yeah? There is kinetic energy that's related to movement, to velocity, and there is a potential energy which is related to this gravitational field. And the sum of those energies is actually conserved. This means that when the spacecraft moves, for example, closer to the sun, then its potential energy will decrease and thus the kinetic energy will increase, so it will actually get faster. So you can see this, for example, here we have two bodies that rotate around their center of mass, and if you're looking careful when they actually approach each other, then they are quite a bit faster, okay? So this is important to keep in mind. All right, so how do spacecrafts actually move? So we will now actually assume that we don't use any kind of engine, no thruster, we just cruise along the gravitational field. Then there are essentially three types of orbits that we can have. One of them are hyperbolas. So this case happens if the velocity is very high, because those are not periodic solutions, they're not closed, so instead our spacecraft kind of approaches the sun or the planet in the middle, the center. From infinity, it will kind of, well, turn, it will change its direction, and then it will leave again to infinity. Another orbit that may happen is a parabola. This is kind of similar. Actually, we won't encounter parabolas during this talk, so I will skip this. And probably most common orbit that we all know are ellipses, in particular, circles, because, well, we know that the Earth is actually moving around the sun approximately in a circle, okay? So those are periodic solutions, they are closed, and in particular, they are such that if a spacecraft is on one of those orbits, and it's not doing anything, then it will forever stay on that orbit, okay? In the two-body problem. So now the problem is we actually want to change this, so we need to do something, okay? So we want to change from one circle around the sun, which corresponds to the Earth orbit, for example, to another circle around the sun, which corresponds to the Mars orbit. And in order to change this, we need to do some kind of maneuver, okay? So this is an actual picture of a spacecraft. And what the spacecraft is doing, it's emitting some kind of particles in some direction. They have a mass, M. Those particles might be gases or ions, for example. And because these gases or these emissions, they carry some mass, they actually have some momentum due to conservation of momentum, this means that the spacecraft actually has to accelerate in the opposite direction, okay? So whenever we do this, we will actually accelerate the spacecraft and change the velocity. And this change of velocity is denoted by delta V. And delta V is sort of the basic quantity that we actually want to look at all the time, okay? Because this describes how much thrust we need to actually apply in order to, well, change our orbit. Now, unfortunately, it's pretty expensive to, well, to apply a lot of delta V. This is due to the Tarkovsky rocket equation. So the fuel that you need in order to reach or to change your velocity with some delta V, this depends essentially exponentially on the target delta V. So this means we really need to take care that we use as a few delta V as possible in order to reduce the needed fuel. There's, well, one reason for that is, well, we want to maybe reduce costs because then we need to carry less fuel. However, we can also actually think the other way around if we actually use less delta, less fuel, then we can bring more stuff for payloads, for missions, for science experiments, okay? So that's why in spacecraft mission design, we actually have to take care of reducing the amount of delta V that is spent during maneuvers. So let's see, what can we actually do? So one example of a very basic maneuver is actually to, well, sort of raise the orbit. So imagine you have a spacecraft on a circular orbit around, for example, sun here. Then you might want to, well, raise the orbit in the sense that you make it more elliptic and reach higher altitudes, okay? And for this, you just accelerate in the direction that you're flying. So you apply some delta V and this will actually change the form of the ellipse, okay? So that's a very common scenario. Another one is, if you approach a planet from a very far way, then you might have a very high relative velocity, such that with respect to the planet, you're on a hyperbolic orbit, okay? So you would actually leave the planet. However, if this is actually your target planet that you want to reach, then, of course, you have to enter orbit, you have to somehow slow down. So the idea here is that when you approach the closest point to the planet, for example, then you actually slow down. So you apply delta V sort of in the opposite direction and change the orbit to something that you prefer, for example, an ellipse, because now you will actually stay close to the planet forever, okay? Well, if reality would be a two-body problem. Okay, so let's continue. Now we actually want to apply this knowledge to, well, getting, for example, to Mars. Let's start with moment transfers. Mars and Earth both, well, evolve around the sun in pretty much circular orbits, and our spacecraft starts at the Earth. So now we want to reach Mars. How do we do this? Well, we can apply what we just said. So we accelerate when we are at the Earth orbit, such that our orbit touches the Mars orbit on the other side. Okay, so this gives us some amount of delta V that we have to apply. We need to calculate this. I'm not going to do this. Then we actually fly around this orbit for half an ellipse, and once we have reached the Mars orbit, then we can actually accelerate again in order to raise the other side of the ellipse until that one reaches the Mars orbit. So with two maneuvers, two accelerations, we can actually change from one circular orbit to another one. Okay, this is the basic idea of how you actually fly to Mars. So let's look at an animation. So this is the orbit of the inside mission. That's a NASA Mars mission, which launched and landed last year. The blue circle is the Earth, and the green one is Mars, and the pink is actually the satellite of the probe. You can see that, well, it's flying in this sort of half ellipse. However, there are two, well, there's just one problem. Namely, when it actually reaches Mars, Mars needs to be there. I mean, that sounds trivial, but I mean, imagine you fly there and then, well, Mars is somewhere else. That's not good. I mean, this happens pretty regularly when you begin playing a Kerber space program, for example. But yeah, so we don't want to play around with this the whole time. We actually want to hit Mars. So we need to take care that Mars is at the right position when we actually launch, because it will traverse the whole green line during our transfer. This means that we can only launch such a home and transfer at very particular times. This time when you can do this transfer is called the transfer window. For Earth Mars, for example, this is possible every 26 months. So if you miss something, software's not ready, whatever, then you have to wait for another 26 months. The flight itself takes about six months. All right. There's another thing that we kind of neglected so far. Namely, when we start, when we depart from Earth, then, well, there's Earth mainly. So that's the main source of gravitational force. For example, right now, I'm standing here on the stage and I experience the Earth. I also experience Sun and Mars, but I mean, that's very weak. I can't ignore this. So at the beginning of our mission to Mars, we actually have to take care that we're close to Earth. Then during the flight, the Sun actually dominates the gravitational force, so we will only consider this. But then when we approach Mars, we actually have to take care about Mars. So we kind of forgot this during the home and transfer. So what you actually do is you patch together solutions of these transfers. So in this case, there are essentially three sources of gravitational force or Earth, Sun and Mars. So we will have three two-body problems that we need to consider. One for departing, one for the actual home and transfer, and then the third one when we actually approach Mars. So this makes this whole thing a bit more complicated, but it's also nice because actually we need less delta V than we would for the basic home and transfer. One reason for this is that when we look at Mars, so the green line is now the Mars orbit and the red one is again the spacecraft. It approaches Mars. Then we can actually look at what happens at Mars by kind of zooming into the system of Mars. So Mars is now standing still and then we see that the velocity of the spacecraft is actually very high relative to Mars. So it will be on a hyperbolic orbit. It will actually leave Mars again. You can see this on the left side, because it's leaving Mars again. So what you need to do is, in fact, you need to slow down and change your orbit into an ellipse. And this delta V that you need here for this maneuver is actually less than the delta V you would need to circularize the orbit to just fly on the same orbit as Mars. So we need to slow down. A similar argument actually at Earth shows that, well, if you actually launch into space, then you need quite some speed already to not fall down back onto Earth. So that's something like seven kilometers per second or so. This means that you already have some speed. And if you align your orbit or your launch correctly, then you already have some of the delta V that you need for the Hohman transfer. So in principle, you need quite a bit less delta V than you might naively think. All right. So that much about Hohman transfer, let's look at gravity assists. That's another major technique for interplanetary missions. The idea is that we can actually use planets to sort of getting pulled along. So this is an animation. On the lower, in the lower animation, you see kind of the picture when you look at the planet. So the planet is standing still. And we assume that the spacecraft, so the blue object, is on a hyperbolic orbit. And it's kind of making a 90 degree turn. Okay. In the upper image, you actually see the picture when you look from the sun. So the planet is actually moving. And if you look very carefully at the blue object, then you can see that it is faster. So once it has passed, the planet is actually faster. Well, we can actually look at this problem. So this is again the picture. When Mars is centered, we have some entry velocity. Then we are in this hyperbolic orbit. We have an exit velocity. Notice that the lengths are actually equal. So it's the same, it's the same speed, but just a turn direction in this example. But then we can look at the whole problem with the moving Mars. Okay. So now Mars has some velocity V Mars. So the actual velocity that we see is the sum of the entry in the Mars velocity before and afterwards exit plus Mars velocity. And if you look at those two errors, then you see immediately that, well, yeah, the lengths are different. Okay. So this is just the whole phenomenon here. So we see that by actually passing close to such a, well, planet or massive body, we can sort of gain free delta V. Okay. So of course it's not, I mean, energy is still conserved. Okay. But yeah, let's not worry about these details here. Now, the nice thing is we can use this technique to actually alter course. We can speed up. So this is the example that I'm shown here. But also we can use this to slow down. Okay. So that's a pretty common application as well. And we can use this to slow down by just changing the arrows essentially. So just approaching Mars from, in this case, Mars from a different direction essentially. So let's look at an example. And this is a baby Colombo. That's actually the reason why I kind of changed the title because baby Colombo is actually a mission to Mercury. Okay. So it was launched last year. It's a combined ESA JAXA mission. And it consists of two probes and one thrust essentially. So it's the three stages that you can see in the picture. Yeah. And that's a pretty awesome mission actually. It's really nice. But it has in particular a very cool orbit. So that's it. What can we see here? So first of all, the blue line, that's actually earth. The green one, that's Mercury. So that's where we want to go. And we have this intermediate trachease one. That's Venus. And well, the pink curve is baby Colombo's orbit. So it looks pretty complicated. It's definitely not a Hohmann transfer. And in fact, this mission uses nine gravity assists to reach Mercury. And if you look at the path, so for example, right now, it actually is very close to Mercury because the last five or six gravity assists are just around Mercury in order to slow down. Okay. And this saves a lot of Delta V compared to the standard Hohmann transfer. All right. But we want to do, we want to do even better. Okay. So let's now actually make the whole problem more complicated in order to hope for some kind of nice tricks that we can do. Okay. So now we will talk about a planar circular restricted three-body problem. All right. So in general, the three-body problem just means, hey, well, instead of two bodies, we have three. Okay. And they pairwise attract each other. And then we can solve this equation of motion. We can ask a computer. And this is one animation of what this could look like. So there are three masses and their orbits are traced. And we see immediately that we don't see anything. That's super complicated. Yeah. There is no way we can really, well, I don't know, formulate a whole solution theory for a general three-body problem that's complicated. Those are definitely not ellipses. So let's maybe go a step back and make the problem a bit easier. Okay. So we will now talk about a planar circular restricted three-body problem. There are three words. So the first one is restricted. Restricted means that in our application case, one of the bodies is actually a spacecraft. Spacecrafts are much lighter than, for example, sun and Mars, meaning that we can actually ignore the force that the spacecraft exerts on sun and Mars. Okay. So we will actually consider sun and Mars to be independent of the spacecraft. Okay. So in total, we only have like two gravitational forces now acting on a spacecraft now. So we neglect sort of this other force. Also, we will assume that the whole problem is what's called circular. So we assume that sun and Mars actually rotate in circles around their center of mass. This assumption is actually pretty okay. We will see a picture right now. So in this graph, for example, in this image, you can see that the black situation, so this might be at some point in time. And then later on, the sun and Mars actually have moved to the red positions and the spacecraft is at some other place and now, of course, field some other forces. Okay. And also, we will assume that problem is planar, meaning again that everything takes place in the plane. Okay. So let's look at a video. That's a video with a very low frame rate, something like two images per day maybe. It's actually Pluto and Charon. So the larger one is the, well, X-planet Pluto. It was taken by New Horizons in 2015 and it shows that they actually rotate around their, well, center of mass. So both actually rotate. This also happens, for example, for sun and earth or sun and Mars or sun and Jupiter or also earth and moon. However, in those other cases, the center of mass is usually contained in the larger body. So this means that in the case of sun, earth, for example, the sun will just wiggle a little bit. Okay. So you don't really see this extensive rotation. Okay. Now, this problem is still difficult. Okay. So if I put now a mass in there, then, well, yeah, you don't really know what happens. However, there's a nice trick to simplify this problem. And unfortunately, I can't do this here, but maybe all the viewers at home, they can try to do this. You can take your laptop. Please don't do this. And you can rotate your laptop at the same speed as this image actually rotates. Okay. Well, then what happens? The two masses will actually stand still from your point of view. Okay. If you do it carefully and don't break anything. So we switch to this sort of rotating point of view. Okay. Then the two masses stand still. We still have the two gravitational forces towards sun and Mars. But because we kind of look at it from a rotated or from a moving point of view, we get two new forces. Those forces you know. So centrifugal forces, of course, the one that, for example, you have when you play with, I don't know, with some children or so, and they want to be pulled in a circle very quickly, and then they start flying. And that's pretty cool. But here we actually have this force acting on the spacecraft. Okay. And also there is the Coriolis force, which is a bit less known. This depends on the velocity of the spacecraft. Okay. So if there's no velocity in particular, then there will not be any Coriolis force. So our new problem actually has four forces. Okay. But the advantages of sun and Mars actually are standing still. So we don't we don't need to worry about their movement. Okay. So now how does this look like? Well, this might be an example for an orbit. Well, that looks still pretty complicated. Now, I mean, this is something that you can't have in a two-body problem. It has these weird wiggles. I mean, they're not really corners. It actually kind of switches from sun to Mars. Yeah. So it stays close to sun for some time and then it moves somewhere else. So that's still pretty complicated. I don't know. Maybe some of you have read the book, the three-body problem. Yeah. So there, for example, the two masses might be a binary star system. Okay. And then you have a planet that's actually moving along such an orbit. Okay. That looks pretty bad. So in particular, the seasons might be somewhat messed up. Yeah. So this problem is in fact in a strong mathematical sense, chaotic. Okay. So chaotic means something like if you start with very close initial conditions and you just let the system evolve, then the solutions will look very, very different. Okay. And this really happens here, which is good. All right. So one thing we can ask is, well, is it possible that if we put a spacecraft into the system without any velocity, is it possible that all the forces actually cancel out? And turns out, yes, this is possible. And those points are called Lagrangian points. So if we have zero velocity, there's no Coriolis force. So we have only these three forces. And as you can see in this little schematics here, it's possible that all these forces actually cancel out. Now, imagine, yeah, I gave you homework. Please calculate all these possible points. Then you can do this, but we won't do this right here. Instead, we just look at the result. So those are the five Lagrangian points in this problem. Okay. So we have L4 and L5, which are at equilateral triangles with Sun and Mars. Well, Sun and Mars in this case. And we have L1 and 2 and L3 on the line through Sun and Mars. So if you put a spacecraft precisely at L1 without any velocity, then in relation to Sun and Mars, it will actually stay in the same position. Okay. That's pretty cool. However, mathematicians and physicists will immediately ask, well, okay, but what happens if I actually put a spacecraft close to a Lagrangian point? Okay. So this is related to what's called stability. And you can calculate that around L4 and L5, the spacecraft will actually stay in the vicinity. So it will essentially rotate around the Lagrangian points. So those are called stable. Whereas L1, L2, and L3 are actually unstable. This means that if you put a spacecraft there, then it will eventually escape. However, it just takes a different amount of time depending on the Lagrangian points. So, for example, if you're close to L2, this might take a few months. But if you're close to L3, this will actually take something like 100 years or so. Okay. So those points are still different. All right. Okay. So is there actually any evidence that they exist? I mean, maybe I'm just making this up and, you know, I mean, I haven't shown you any equations. I could just show anything. However, we can actually look at the solar system. So this is the inner solar system here. In the middle you see, well, in the center, you see the Sun, of course. And to the lower left, there's Jupiter. Now, if you imagine an equilateral triangle of Sun and Jupiter, well, there are two of them. And then you see all these green dots there. And those are asteroids. Those are the Trojans and the Greeks. And they accumulate there because L4 and L5 are stable. Okay. So we can really see this dynamics going on in the solar system. However, there's also various other applications of Lagrangian points. So in particular, you might want to put a space telescope somewhere in space, of course, in such a way that the Sun is not blinding you. Well, there's Earth, of course. So if we can put the spacecraft behind Earth, then we might be in the shadow. And this is the Lagrangian point L2, which is the reason why this is actually being used for space telescopes, such as, for example, this one. However, it turns out that L2 is unstable. So we don't really want to put the spacecraft just there. But instead, we put it in a closed orbit close to L2. This particular example is called a halo orbit. And it's actually not contained in the planes. I'm cheating a little bit. So on the right-hand side, in the animation, you actually see the orbit from the side. So it actually leaves the plane. The blue dot is Earth. And on the left-hand side, you see the actual orbit from the top. So it's rotating around this place. Okay. So that's the James Webb Space Telescope. By the way, you can see in the animation, it's supposed to launch in 2018. That didn't work out, unfortunately. But stay tuned. Another example that has become pretty famous recently is the Chinese Keqiao Relay Satellite. This one sits at the Earth-Moon L2 Lagrangian point. And the reason for this is that the Chinese wanted to, or actually did, land the Changi-4 Moon land on the backside of the Moon. And in order to stay in contact, radio contact with the lander, they had to put a relay satellite behind the Moon, which they could still see from Earth. And they chose some similar orbit around L2. Okay. So let's now go to some other, more advanced technique. Ballistic capture. Right. Okay. So this whole business sort of started with a mission from the beginning of the 1990s. That's the Hiton mission. So that was a Japanese Moon probe, consisted of a probe which had a small orbiter inside, which was separated. And then it was supposed to actually enter orbit around Moon. Unfortunately, it missed its maneuver. So it didn't apply enough Delta-V. So it actually flew off. And the mission was sort of lost at that point because Hiton itself, so the main probe, did not have enough fuel to reach the Moon. All right. That's, of course, a problem. I mean, it's a risk you have to take. And they were probably pretty devastated. However, there were two people from JPL, NASA, who actually heard about this, the Bruno and Miller. And they were working on strange orbits, those ballistic capture orbits. And they actually found one for the Hiton probe. They sent this to the Japanese. And they actually used that orbit to get the Hiton probe to the Moon. And it actually arrived in October 1991. And then it could do some science, maybe some different experiments, but it actually arrived there. However, the transfer took quite a bit longer. So a normal Moon transfer takes like three days or so. But this one actually took a few months. All right. And the reason for this is that it looks pretty weird. So this is a picture of the orbit or a schematic picture. You can see the Earth, well, they're in the middle sort of, and the Moon a bit to the left of it. The L2 is the Lagrangian point of the Sun-Earth system. So it's pretty far out. And you can see that the orbit sort of consists of two parts. First, it leaves Earth, and it flies far beyond the Moon. So somewhere completely different towards some other Lagrangian point that's really far away. Then it does some weird things. And in the upper part of the picture, there it actually does a maneuver. So we apply some thrust, some delta B to change on a different orbit. And this orbit led the probe directly to the Moon, where it was essentially captured for free. So it just entered orbit around the Moon. And this is of course not possible in the two-body problem, but we may find a way for doing this in the three-body problem. Okay. So what do we mean by capture? Now we have to sort of think a bit more abstractly. The idea is, well, we have Sun and Mars, and we have a spacecraft that flies in this three-body problem. So the red orbit is actually the orbit of the spacecraft. Now, at any point in time, we may decide to just forget about the Sun. Okay. So instead, we consider the two-body problem of Mars and the spacecraft. Okay. Because at this point in time, the spacecraft has a certain position relative to Mars and a certain velocity. So this determines its orbit in the two-body problem. Usually, it would be very fast. So it would be on a hyperbolic orbit, which is denoted by the dashed line here, okay, or a dashed curve. So usually, you happen to be on a hyperbolic orbit. But of course, the orbit is only an approximation because in the three-body problem, the movement is actually different. But later on, it might happen that we continue on the orbit. We can do the same kind of construction by just ignoring the Sun, essentially. And then we could find that the spacecraft suddenly is in an elliptic orbit. This would mean that, well, if we forgot about the Sun, then the spacecraft would be stable. It would be captured by Mars. It would be there. It would be pretty nice. So this phenomenon, when this happens, we call this a temporary capture, okay? Temporary because it might actually leave that situation again later on, because the actual movement depends on the three-body problem, which is super complicated. So it's possible that it actually leaves again. But for that moment, at least, it's captured. And we want to find a way or describe some kind of algorithm, perhaps, how we can, well, find this situation, essentially, okay, in a reasonable way. And the notion for this is what's called, well, instability. The idea here is the following. We look at the three-body problem. We want to go to Mars. So we pick a line there. And on that line, we take a point x, which has some distance r to the Mars. And we pick a perpendicular speed, a perpendicular velocity to the line, such that this corresponds to some kind of elliptic orbit in the two-body problem, okay? So that's the dashed line. But then we actually look at the problem in the three-body problem. And we just evolve the spacecraft. And it's following the red orbit. It might follow the red orbit, okay? And it can happen that after going around Mars for one time, it hits again the line, okay? Then we can do the same construction with forgetting the sun again and just look at the two-body problem. And it's possible that this point actually still corresponds to an elliptic orbit. That's somewhat interesting, right? Because now this means that if we actually hit the point x, then we can follow the orbit. And we know that we wrap around Mars once and are still sort of captured in the corresponding two-body problem, okay? If we actually are able to wrap around Mars twice, then we would call this two-stable and, well, for more rotations than it's n-stable, okay? So that's good because such an orbit corresponds to something that's usable because we will wrap around Mars n times. However, it's also possible that you have an unstable point, meaning that we again start on something that corresponds to an ellipse around Mars. But if we actually follow the orbit in a three-body problem, it will, for example, not come back, it will not wrap around Mars, it will go to the sun or somewhere else, okay? So that's, of course, not a nice point. This one's called unstable. And then there's another thing we can do. That's actually a pretty common trick in finding orbits, et cetera. We can, instead of solving the problem in forward time, we actually go back, okay? So essentially in your programming, you just replace time by minus time, for example. And then you just solve the thing and you go back in the past. And it's possible that the point that corresponds to such an ellipse, when you go back into the past and it does not wrap around, but it actually goes to the sun, for example. We call this unstable in the past, okay? So that's just some random definition. And we can, well, we can use this. The reason for this is we can actually kind of take these concepts together and build an orbit from that. The idea being we pick a point x that is unstable. So for example, it might wrap around Mars six times. Well, some number that we like. This is the blue part here in the picture. So it wraps around Mars six times. But when we go back in time, it actually leaves Mars or at least it doesn't come back in such a way that it's again on an elliptic curve. So this is the red part, okay? Well, so we can just follow this. And then we pick a point y on that curve, okay? So this one will be pretty far away from Mars, or we can choose it. And then we sort of use a home and transfer to get from Earth to that point y, okay? So our orbit actually consists of three parts now, okay? So we have the home and transfer, but it's not actually aiming for Mars. It's actually aiming for the point y. Then there we do a maneuver because we want to switch onto this red orbit, okay? And then this one will bring us to the point x where we know, because it was constructed in this way, that the spacecraft will continue to rotate around Mars, for example, six times, okay? So in particular, at x, there's no maneuver taking place, okay? So that's a possible mission scenario. The way this is done then usually is you kind of, you calculate these points x that are suitable for doing this, okay? So they have to be stable and unstable in the past at the same time, so you have to find them. And there's a lot of numerical computations involved in that. But once we have this, you can actually build these orbits. Okay, so let's look at an actual example. So this is for Earth Mars. On the left, you see, well, the two circular orbits of Earth and Mars. And on the right, you see the same orbit, but from a point of view centered around Mars, okay? And the colors correspond to each other. So the mission starts on the left side by doing a Hohmann transfer. So that's the black line starting at Earth and then hitting the point which is called xc here. So that's the y that I had on the other slide. So this is this point y or xc here is pretty far away still from Mars. There we do a maneuver and we switch onto the red orbit, which brings us to the point x closer to Mars, after which we will actually start rotating around Mars. The point x is actually at the top of this picture, okay? And then on the right, you can see the orbit. And it's looking pretty strangely. So the red orbit is when we capture orbit, our way to actually get to Mars. And then if you look very carefully, you can count we actually rotate around Mars six times, okay? Now, during those six rotations around Mars, we could do experiments. So maybe that is enough for whatever we are trying to do, okay? However, if we want to stay there, we need to actually execute another maneuver, okay? So to actually stay around Mars. And I mean, the principle looks nice, but of course, we have to do some calculations. We have to find some ways to actually quantify how good this is. And it turns out that there are a few parameters that you can choose. So in particular, the target point x, this has a certain distance that you're aiming for around Mars. And it turns out that this procedure here, for example, is only very good if this altitude, this distance r, is actually high enough. If it's high enough, then you can save, in principle, up to 23% of the data we, which is enormous, okay? So that would be really good. However, in reality, it's not as good usually. And for certain lower distances, for example, you cannot save anything. So there are certain tradeoffs to make. However, there's another advantage. Remember this point y. We chose this along this capture orbit, along the red orbit. And the thing is, we can actually choose this freely. This means that our home untransfer doesn't need to hit Mars directly when it's there. So it doesn't need to aim for that particular point. It can actually aim for any point on that capture orbit. This means that we have many more home untransfer available that we can actually use to get there. This means that we have a far larger transfer window, okay? So we cannot just start every 26 months. But now, with this technique, we could actually launch. Well, quite often. However, there's a little problem. If you looked at the graph carefully, then you may have seen that the red orbit actually took like three quarters of the rotation of Mars. This corresponds to roughly something like 400 days, okay? So this takes a long time. So you probably don't want to use this with humans on board because they have to actually wait for a long time. But in principle, there are ways to make this shorter. So you can try to actually improve on this. But in general, it takes a long time. So let's look at a real example for this. Again, that's baby Colombo. The green dot is now Mercury. So this is kind of a zoom of the other animation. And the purple line is the orbit. And yeah, it looks strange. So the first few, well, movements around Mercury, they are actually the last gravity assets for slowing down, okay? And then it actually starts on the capture orbit. So now it actually approaches Mercury. So this is the part that's sort of difficult to find, but which you can do with this stability. And once the animation actually ends, this is when it actually reaches the point when it's temporarily captured. So in this case, this is at an altitude of 180,000 kilometers. So this is pretty high up above Mercury. But it's enough for the mission, okay? And of course, then they do some other maneuver to actually stay around Mercury. Okay. So in the last few minutes, let's have a look, let's have a brief look at how we can actually extend this. So I will be very brief here because well, we can try to actually make this more general to improve on this. This concept is then called the Interplanetary Transport Network. And it looks a bit similar to what we just saw. The idea is that in fact, this capture orbit is part of a larger, well, a set of orbits that have these kinds of properties that are wrapped around Mars and then kind of leave Mars. And they are very closely related to the dynamics of a particular Lagrangian point, in this case L1. So that was the one between the two masses. And if you investigate this Lagrangian point a bit closer, you can see, well, you can see different orbits of all kinds of behaviors. And if you understand this, then you can actually try to do the same thing on the other side of the Lagrangian point. Okay. So we just kind of switch from Mars to the Sun and we do a similar thing there. We expect to actually find similar orbits that are wrapping around the Sun and then going towards this Lagrangian point in a similar way. Well, then we already have some orbits that are, well, kind of meeting at L1. So we might be able to actually connect them somehow, for example, by maneuver. And then we only need to reach the orbit around Sun from Earth. Okay. If you find a way to do this, you can get rid of the Hohmann transfer. And this way, you reduce your Delta V even further. The problem is that this is hard to find, because these orbits, they are pretty rare. And of course, you have to connect those orbits. So you approach the Lagrangian point from L1, from two sides, but you don't really want to wait forever until it's very easy to switch. So instead, you apply some Delta V in order to, well, not wait that long. So here's a picture of how this map looked like. Well, again, very schematic. So we have Sun, we have Mars, and in between there's a Lagrangian point L1. The red orbit is sort of an extension of one of those capture orbits that we have seen. Okay. So it wraps around Mars a certain number of times. And well, in the past, for example, it actually goes to Lagrangian point. I didn't explain this, but in fact, there are many more orbits around L1, closed orbits, but they're all unstable. And these orbits that are used in this interplanetary transport network, they actually approach those orbits around L1. And we do the same thing on the other side, on the Sun. And then the idea is, okay, well, we take these orbits, we connect them. And when we are on the black orbit around L1, we actually apply some maneuver, we apply some Delta V to actually switch from one to the other. And then we have sort of a connection of how to get from Sun to Mars. So we just need to do a similar thing again from Earth to this particular blue orbit around the Sun. Okay. So that's the general procedure. But of course, it's difficult. And in the end, you have to do a lot of numerics because, yeah, as I said at the beginning, this is just a brief overview. It's not all the details. Please don't launch your own mission tomorrow. Okay. So with this, I want to thank you. And I'm open to questions. So thank you, Sven, for an interesting talk. We have a few minutes for questions. If you have any questions, line up next to the microphones, we'll start with microphone number one. Hello. So what are the problems associated? So you showed in the end this going around the Lagrange point L1. Is this also possible for other Lagrange points? So could you do this with L2? Yes, you can. Yeah. So in principle, I didn't show the whole picture, but these kind of orbits, they exist at L1, but they also exist at L2. And in principle, you can this way sort of leave this two-body problem by finding similar orbits. But of course, the details are different. So you cannot really take your knowledge or your calculations from L1 and just take them over to L2. You actually have to do the same thing again. You have to calculate everything in detail. I'll take a question from the internet. Is it possible to use these kinds of transfers in Kerbal space program? Okay. So, well, so home and transfers, of course, the gravity assist as well, but not the restricted three-body problem because the way Kerbal space program, at least the default installation, so without any mods, works is that it actually switches the gravitational force. So the thing that I described as a patched solution where we kind of switch our picture, which gravitational force we consider for our two-body problem, this is actually the way the physics is implemented in Kerbal space program. So we can't really do the interplanetary transport network there. However, I think there's a mod that allows this, but your computer might be too slow for this. I don't know. If you're leaving, please do so quietly. Small question. And a question from microphone number four. Hello. I have actually two questions. I hope that's okay. First question is, I wonder how you do that in like your practical calculations. Like you said, there's a two-body problem and there's solutions that you can calculate with a two-body problem and then there's a three-body problem. And I imagine there's an n-body problem all the time you do things. So how does it look when you do that? And a second question is you said that reducing delta V about 15% is enormous. And I wonder what effect does this have on the payload? Okay. So regarding the first question. So in principle, I mean, you make a plan for a mission. So you have to, well, you calculate those things in the simplified models. Okay. You kind of patch together an idea of what you want to do. But of course, in the end, you're right. There are actually many massive bodies in the solar system. And because we want to be precise, we actually have to incorporate all of them. So in the end, you have to do an actual numerical search in a much more complicated n-body problem with, I don't know, 100 bodies or so. And you have to incorporate other effects. For example, the solar radiation might actually have a little influence on your orbit. Okay. And there are many effects of this kind. So once you have a rough idea of what you want to do, you need to take your very good physics simulator for the n-body problem, which actually has all these other effects as well. And then you need to do a numerical search over this. You kind of, you know where to start with these ideas, where to look for solutions. But then you actually have to just try it and figure out some algorithm to actually, well, approach a solution that has the behavior that you want. But it's a lot of numerics. Right. And the second question, can you write me again? Sorry. Second question was reducing delta V by 15%. What is the effect on the payload? Right. So I mean, well, if you need 15% less fuel, then of course you can use 15% more weight for more mass for the payload, right? So you could put maybe another instrument on there. And another thing you could do is actually keep the fuel, but actually use it for station keeping. So for example, in the James Webb telescope example, the James Webb telescope flies around this halo orbit around L2, but that orbit itself is unstable. So the James Webb telescope will actually escape from that orbit. So they have to do a few maneuvers every year to actually stay there. And they have only a finite amount of fuels at some point. This won't be possible anymore. So reducing delta V requirements might actually have increased the mission lifetime by quite a bit. Number three. Hey, when you do such a mission, I guess you have to adjust the directory of your satellite quite often because nothing goes exactly as you calculated it, right? My question is how precise can you measure the orbit, sorry, the position and the speed of a spacecraft at, let's say, Mars? What's the resolution? Right. So for Mars, I'm not completely sure how precise it is, but for example, if you have an Earth observation mission or something that's flying around Earth, then you can have an orbit, a rather precise orbit that's good enough for taking pictures on Earth, for example, for something like two weeks or so. So you can measure the orbit well enough and calculate the future, something like two weeks in the future. So that's good enough. However, I can't really give you good numbers on what the accuracy is, but depending on the situation, it can get pretty well for Mars, I guess. Well, that's pretty far, I guess, that will be a bit less. A very short question for microphone number one, please. Thank you. Thank you for the talk. I have a small question. As you said, you roughly plan the trip using the three body and two body problems, and are there any stable points like gradient points in, for example, four body problem, and can you use them during the roughly planning stage? Yeah, I actually wondered about this very recently as well, and I don't know the answer. I'm not sure. So the three body problem is already complicated enough from a mathematical point of view. So I have never actually really looked at the four body problem. However, with those many bodies, there are at least very symmetrical solutions, so you can find. But it's a different thing than the gradient points. So unfortunately, we're almost out of time for this talk. If you have more questions, I'm sure Sven will be happy to take them after the talk, so please approach him after. And again, a big round of applause for an interesting talk. Thank you.