1 00:00:00,000 --> 00:00:20,679 *36C3 preroll music* 2 00:00:20,679 --> 00:00:25,929 Herald: OK, so the next talk for this evening is on how to get to Mars and all 3 00:00:25,929 --> 00:00:31,890 in very interesting ways. Some of them might be really, really slow. Our next 4 00:00:31,890 --> 00:00:36,640 speaker has studied physics and has a PhD in maths and is currently working as a 5 00:00:36,640 --> 00:00:41,180 mission planner at the German Space Operations Center. Please give a big round 6 00:00:41,180 --> 00:00:50,142 of applause to Sven. Sven: Thank you. 7 00:00:50,142 --> 00:00:52,551 Hello and welcome to "Thrust is not an option: How to get a 8 00:00:52,551 --> 00:00:56,910 Mars really slow". My name is Sven. I'm a mission planner at the German Space 9 00:00:56,910 --> 00:01:01,380 Operations Center, which is a part of the DLR, the Deutsches Zentrum für Luft- und 10 00:01:01,380 --> 00:01:05,190 Raumfahrt. And first of all, I have to apologize because I kind of cheated a 11 00:01:05,190 --> 00:01:11,461 little bit in the title. The accurate title would have been "Reducing thrust: How 12 00:01:11,461 --> 00:01:16,990 to get to Mars or maybe Mercury really slow". The reason for this is that I will 13 00:01:16,990 --> 00:01:22,750 actually use Mercury as an example quite a few times. And also we will not be able 14 00:01:22,750 --> 00:01:29,000 to actually get rid of all the maneuvers that we want to do. So the goal of this 15 00:01:29,000 --> 00:01:34,550 talk is to give you an introduction to orbital mechanics to see what we can do. 16 00:01:34,550 --> 00:01:37,860 What are the techniques that you can use to actually get to another planet, to 17 00:01:37,860 --> 00:01:44,000 bring a spacecraft to another planet and also go a few more, go a bit further into 18 00:01:44,000 --> 00:01:49,950 some more advanced techniques. So we will start with gravity and the two body 19 00:01:49,950 --> 00:01:54,900 problem. So this is the basics, the underlying physics that we need. Then we 20 00:01:54,900 --> 00:01:59,110 will talk about the two main techniques maybe to get to Mars, for example, the 21 00:01:59,110 --> 00:02:04,530 Hohmann-transfer as well as gravity assists. The third point will be an 22 00:02:04,530 --> 00:02:08,520 extension of that that's called a planar circular restricted three body problem. 23 00:02:08,520 --> 00:02:14,710 Sounds pretty complicated, but we will see in pictures what it is about. And then we 24 00:02:14,710 --> 00:02:21,920 will finally get a taste of certain ways to actually be even better, be even more 25 00:02:21,920 --> 00:02:26,190 efficient by looking at what's called ballistic capture and the weak stability 26 00:02:26,190 --> 00:02:32,770 boundary. All right, so let's start. First of all, we have gravity and we need to 27 00:02:32,770 --> 00:02:36,300 talk about a two body problem. So I'm standing here on the stage and I'm 28 00:02:36,300 --> 00:02:41,660 actually being well accelerated downwards, right? The earth actually attracts me. And 29 00:02:41,660 --> 00:02:47,537 this is the same thing that happens for any two bodies that have mass. OK. So they 30 00:02:47,537 --> 00:02:51,730 attract each other by gravitational force and this force will actually accelerate 31 00:02:51,730 --> 00:02:56,620 the objects towards each other. Notice that the force actually depends on the 32 00:02:56,620 --> 00:03:03,720 distance. OK. So we don't need to know any details. But in principle, the 33 00:03:03,720 --> 00:03:11,310 force gets stronger the closer the objects are. OK, good. Now, we can't really 34 00:03:11,310 --> 00:03:17,080 analyze this whole thing in every detail. So we will make a few assumptions. 35 00:03:17,080 --> 00:03:23,090 One of them will be that all our bodies, in particular, the Sun, Earth will 36 00:03:23,090 --> 00:03:27,470 actually be points, OK? So we will just consider points because anything else is 37 00:03:27,470 --> 00:03:32,680 too complicated for me. Also, all our satellites will actually be just points. 38 00:03:32,680 --> 00:03:38,569 One of the reasons is that, in principle, you have to deal with the attitude of the 39 00:03:38,569 --> 00:03:42,620 satellites. For example, a solar panel needs to actually point towards the sun, 40 00:03:42,620 --> 00:03:47,300 but of course that's complicated. So we will skip this for this talk. Third point 41 00:03:47,300 --> 00:03:51,790 is that none of our planets will have an atmosphere, so there won't be any 42 00:03:51,790 --> 00:03:58,760 friction anywhere in the space. And the fourth point is that we will mostly 43 00:03:58,760 --> 00:04:03,730 restrict to movement within the plane. So we only have like two dimensions during 44 00:04:03,730 --> 00:04:11,350 this talk. And also, I will kind of forget about certain planets and other masses 45 00:04:11,350 --> 00:04:16,070 from time to time. Okay. I'm mentioning this because I do not want you to go home 46 00:04:16,070 --> 00:04:20,590 this evening, start planning your own interplanetary mission, then maybe 47 00:04:20,590 --> 00:04:24,720 building your spacecraft tomorrow, launching in three days and then a week 48 00:04:24,720 --> 00:04:31,030 later I get an e-mail: "Hey, this didn't work. I mean, what did you tell me?" 49 00:04:31,030 --> 00:04:35,680 OK. So if you actually want to do this at home, don't try this just now but please 50 00:04:35,680 --> 00:04:40,539 consult your local flight dynamics department, they will actually supply with the necessary 51 00:04:40,539 --> 00:04:46,410 details. All right. So what's the two body problem about? So in principle we have 52 00:04:46,410 --> 00:04:51,229 some body - the Sun - and the spacecraft that is being attracted by the Sun. Now, 53 00:04:51,229 --> 00:04:55,520 the Sun is obviously much heavier than a spacecraft, meaning that we will actually 54 00:04:55,520 --> 00:05:01,699 neglect the force that the spacecraft exerts on the Sun. So instead, the Sun 55 00:05:01,699 --> 00:05:06,319 will be at some place. It might move in some way, or a 56 00:05:06,319 --> 00:05:12,309 planet. But we only care about a spacecraft, in general. Furthermore, 57 00:05:12,309 --> 00:05:16,469 notice that if you specify the position and the velocity of a spacecraft at some 58 00:05:16,469 --> 00:05:23,487 point, then the gravitational force will actually determine the whole path of the 59 00:05:23,487 --> 00:05:31,370 spacecraft for all time. OK. So this path is called the orbit and this is what we 60 00:05:31,370 --> 00:05:34,930 are talking about. So we want to determine orbits. We want to actually find ways how 61 00:05:34,930 --> 00:05:44,129 to efficiently change orbits in order to actually reach Mars, for example. There is 62 00:05:44,129 --> 00:05:51,380 one other thing that you may know from your day to day life. If you actually take 63 00:05:51,380 --> 00:05:55,680 an object and you put it high up and you let it fall down, then it will accelerate. 64 00:05:55,680 --> 00:06:01,879 OK. So one way to actually describe this is by looking at the energy. There is a 65 00:06:01,879 --> 00:06:05,620 kinetic energy that's related to movement, to velocity, and there is a potential 66 00:06:05,620 --> 00:06:10,680 energy which is related to this gravitational field. And the sum of those 67 00:06:10,680 --> 00:06:17,901 energies is actually conserved. This means that when the spacecraft moves, for 68 00:06:17,901 --> 00:06:23,370 example, closer to the Sun, then its potential energy will decrease and thus 69 00:06:23,370 --> 00:06:28,909 the kinetic energy will increase. So it will actually get faster. So you can see 70 00:06:28,909 --> 00:06:32,550 this, for example, here. We have two bodies that rotate around their 71 00:06:32,550 --> 00:06:37,550 center of mass. And if you're careful, if you're looking careful when they actually 72 00:06:37,550 --> 00:06:43,229 approach each other, then they are quite a bit faster. OK. So it is important to keep 73 00:06:43,229 --> 00:06:48,249 in mind. All right, so how do spacecrafts actually move? So we will now actually 74 00:06:48,249 --> 00:06:55,210 assume that we don't use any kind of engine, no thruster. We just cruise along 75 00:06:55,210 --> 00:07:00,180 the gravitational field. And then there are essentially three types of orbits that 76 00:07:00,180 --> 00:07:04,360 we can have. One of them are hyperbolas. So this case happens if the velocity is 77 00:07:04,360 --> 00:07:10,759 very high, because those are not periodic solutions. They're not closed. So instead, 78 00:07:10,759 --> 00:07:15,819 our spacecraft kind of approaches the Sun or the planet in the middle and the center 79 00:07:15,819 --> 00:07:21,210 from infinity. It will kind of turn, it will change its direction and then it 80 00:07:21,210 --> 00:07:27,990 will leave again to infinity. Another orbit that may happen as a parabola, this 81 00:07:27,990 --> 00:07:33,180 is kind of similar. Actually, we won't encounter parabolas during this talk. So I 82 00:07:33,180 --> 00:07:38,029 will skip this. And the probably most common orbit that we all know are 83 00:07:38,029 --> 00:07:44,509 ellipses. In particular circles because, well, we know that the Earth is actually 84 00:07:44,509 --> 00:07:49,449 moving around the sun approximately in a circle. OK. So those are periodic 85 00:07:49,449 --> 00:07:56,869 solutions. They are closed. And in particular, they are such that if a 86 00:07:56,869 --> 00:08:00,789 spacecraft is on one of those orbits and it's not doing anything, then it will 87 00:08:00,789 --> 00:08:09,120 forever stay on that orbit, OK, in the two body problem. So now the problem is we 88 00:08:09,120 --> 00:08:13,069 actually want to change this. So we need to do something. OK. So we want to change 89 00:08:13,069 --> 00:08:17,589 from one circle around the Sun, which corresponds to Earth orbit, for example, to 90 00:08:17,589 --> 00:08:21,509 another circle around the Sun, which corresponds to Mars orbit. And in order to 91 00:08:21,509 --> 00:08:27,319 change this, we need to do some kind of maneuver. OK. So this is an actual picture 92 00:08:27,319 --> 00:08:33,360 of a spacecraft. And what the spacecraft is doing, it's emitting some kind of 93 00:08:33,360 --> 00:08:40,500 particles in some direction. They have a mass m. Those particles might be gases or 94 00:08:40,500 --> 00:08:48,100 ions, for example. And because these gases or these emissions, they carry some mass, 95 00:08:48,100 --> 00:08:53,160 they actually have some momentum due to conservation of momentum. This means that 96 00:08:53,160 --> 00:08:58,050 the spacecraft actually has to accelerate in the opposite direction. OK. So whenever 97 00:08:58,050 --> 00:09:03,980 we do this, we will actually accelerate the spacecraft and change the velocity and 98 00:09:03,980 --> 00:09:12,660 this change of velocity as denoted by a delta v. And delta v is sort of the basic 99 00:09:12,660 --> 00:09:16,980 quantity that we actually want to look at all the time. OK. Because this describes 100 00:09:16,980 --> 00:09:26,009 how much thrust we need to actually fly in order to change our orbit. Now, 101 00:09:26,009 --> 00:09:32,440 unfortunately, it's pretty expensive to, well, to apply a lot of delta v. This is 102 00:09:32,440 --> 00:09:37,339 due to the costly rocket equation. So the fuel that you need in order to reach or to 103 00:09:37,339 --> 00:09:45,850 change your velocity to some delta v this depends essentially exponentially on the 104 00:09:45,850 --> 00:09:52,740 target delta v. So this means we really need to take care that we use as few 105 00:09:52,740 --> 00:10:00,490 delta v as possible in order to reduce the needed fuel. There's one reason for 106 00:10:00,490 --> 00:10:04,990 that is... we want to maybe reduce costs because then we need to carry 107 00:10:04,990 --> 00:10:10,009 less fuel. However, we can also actually think the other way round if we actually 108 00:10:10,009 --> 00:10:16,769 use less fuel than we can bring more stuff for payloads, for 109 00:10:16,769 --> 00:10:24,170 missions, for science experiments. Okay. So that's why in spacecraft mission 110 00:10:24,170 --> 00:10:28,399 design we actually have to take care of reducing the amount of delta v that is 111 00:10:28,399 --> 00:10:34,269 spent during maneuvers. So let's see, what can we actually do? So one example of a 112 00:10:34,269 --> 00:10:41,500 very basic maneuver is actually to, well, sort of raise the orbit. So imagine you 113 00:10:41,500 --> 00:10:48,100 have a spacecraft on a circular orbit around, for example, Sun here. Then you 114 00:10:48,100 --> 00:10:52,269 might want to raise the orbit in the sense that you make it more 115 00:10:52,269 --> 00:10:57,410 elliptic and reach higher altitudes. For this you just accelerate in the direction 116 00:10:57,410 --> 00:11:00,680 that you're flying. So you apply some delta v and this will actually change the 117 00:11:00,680 --> 00:11:08,029 form of the ellipse. OK. So it's a very common scenario. Another one is if you 118 00:11:08,029 --> 00:11:12,370 approach a planet from very far away, then you might have a very high relative 119 00:11:12,370 --> 00:11:18,570 velocity such that with respect to the planet, you're on a hyperbolic orbit. OK. 120 00:11:18,570 --> 00:11:22,540 So you would actually leave the planet. However, if this is actually your 121 00:11:22,540 --> 00:11:26,840 target planet that you want to reach, then of course you have to enter orbit. You 122 00:11:26,840 --> 00:11:31,290 have to somehow slow down. So the idea here is that when you approach 123 00:11:31,290 --> 00:11:37,449 the closest point to the planet, for example, then you actually slow down. 124 00:11:37,449 --> 00:11:41,830 So you apply delta v in sort of in the opposite direction and change the orbit to 125 00:11:41,830 --> 00:11:45,709 something that you prefer, for example an ellipse. Because now you will actually 126 00:11:45,709 --> 00:11:54,760 stay close to the planet forever. Well, if relative it would a two body problem. OK, 127 00:11:54,760 --> 00:12:02,230 so. Let's continue. Now, we actually want to apply this knowledge to well, getting, 128 00:12:02,230 --> 00:12:08,829 for example, to Mars. Let's start with Hohmann transfers. Mars and Earth both 129 00:12:08,829 --> 00:12:16,589 revolve around the Sun in pretty much circular orbits. And our spacecraft starts 130 00:12:16,589 --> 00:12:21,220 at the Earth. So now we want to reach Mars. How do we do this? Well, we can fly 131 00:12:21,220 --> 00:12:27,270 what we just said. So we accelerate when we are at the Earth orbit, 132 00:12:27,270 --> 00:12:36,810 such that our orbit touches the Mars orbit on the other side. OK. So this gives us 133 00:12:36,810 --> 00:12:40,990 some amount of delta v we have to apply. We need to calculate this. I'm not going 134 00:12:40,990 --> 00:12:47,939 to do this. Then we actually fly around this orbit for half an ellipse. And once 135 00:12:47,939 --> 00:12:53,139 we have reached the Mars orbit, then we can actually accelerate again in order to 136 00:12:53,139 --> 00:12:59,680 raise other side of the Ellipse until that one reaches the Mars orbit. So with two 137 00:12:59,680 --> 00:13:04,839 maneuvers, two accelerations, we can actually change from one circular orbit to 138 00:13:04,839 --> 00:13:09,960 another one. OK. This is the basic idea of how you actually fly to Mars. So let's 139 00:13:09,960 --> 00:13:16,339 look at an animation. So this is the orbit of the InSight mission. That's another Mars 140 00:13:16,339 --> 00:13:25,199 mission which launched and landed last year. The blue circle is the Earth and the 141 00:13:25,199 --> 00:13:33,130 green one is Mars. And the pink is actually the satellite or the probe. 142 00:13:33,130 --> 00:13:40,381 You can see that, well, it's flying in this sort of half ellipse. However, there 143 00:13:40,381 --> 00:13:47,339 are two... well, there's just one problem, namely when it actually reaches Mars, Mars 144 00:13:47,339 --> 00:13:51,779 needs to be there. I mean, that sounds trivial. Yeah. But I mean, imagine you fly 145 00:13:51,779 --> 00:13:57,449 there and then well, Mars is somewhere else, that's not good. I mean this happens 146 00:13:57,449 --> 00:14:05,439 pretty regularly when you begin playing a Kerbal Space Program, for example. 147 00:14:05,439 --> 00:14:11,050 So we don't want to like play around with this the whole time, we actually want 148 00:14:11,050 --> 00:14:16,760 to hit Mars. So we need to take care of that Mars is at the right position when we 149 00:14:16,760 --> 00:14:21,779 actually launch. Because it will traverse the whole green line during our transfer. 150 00:14:21,779 --> 00:14:27,980 This means that we can only launch such a Hohmann transfer at very particular times. 151 00:14:27,980 --> 00:14:31,579 And sort of this time when you can do this transfer is called the transfer 152 00:14:31,579 --> 00:14:39,599 window. And for Earth-Mars, for example. This is possible every 26 months. So if 153 00:14:39,599 --> 00:14:44,639 you miss something, like, software's not ready, whatever, then you have to wait for 154 00:14:44,639 --> 00:14:53,000 another twenty six months. So, the flight itself takes about six months. All right. 155 00:14:53,000 --> 00:14:59,399 There is another thing that we kind of neglected so far, namely when we start, 156 00:14:59,399 --> 00:15:04,450 when we depart from Earth, then well there's Earth mainly. And so that's the 157 00:15:04,450 --> 00:15:11,009 main source of gravitational force. For example, right now I'm standing here on 158 00:15:11,009 --> 00:15:19,800 the stage and I experience the Earth. I also experience Sun and Mars. But I mean, 159 00:15:19,800 --> 00:15:24,899 that's very weak. I can ignore this. So at the beginning of our mission to Mars, we 160 00:15:24,899 --> 00:15:29,410 actually have to take care that we are close to Earth. Then during the 161 00:15:29,410 --> 00:15:34,379 flight, the Sun actually dominates the gravitational force. So we will only 162 00:15:34,379 --> 00:15:38,029 consider this. But then when we approach Mars, we actually have to take care about 163 00:15:38,029 --> 00:15:44,430 Mars. Okay. So we kind of forgot this during the Hohmann transfer. So what you 164 00:15:44,430 --> 00:15:49,970 actually do is you patch together solutions of these transfers. Yeah. So in 165 00:15:49,970 --> 00:15:55,240 this case, there are there are essentially three sources of gravitational force so 166 00:15:55,240 --> 00:15:59,389 Earth, Sun, Mars. So we will have three two body problems that we need to consider. 167 00:15:59,389 --> 00:16:04,639 Yeah. One for departing, one for the actual Hohmann transfer. And then the third 168 00:16:04,639 --> 00:16:09,449 one when we actually approach Mars. So this makes this whole thing a bit more 169 00:16:09,449 --> 00:16:14,649 complicated. But it's also nice because actually we need less delta v than we 170 00:16:14,649 --> 00:16:19,589 would for the basic hohmann transfer. One reason for this is that when we look at 171 00:16:19,589 --> 00:16:25,930 Mars. So the green line is now the Mars orbit and the red one is again the 172 00:16:25,930 --> 00:16:31,509 spacecraft, it approaches Mars now we can actually look at what happens at Mars by 173 00:16:31,509 --> 00:16:40,480 kind of zooming into the system of Mars. OK. So Mars is now standing still. And 174 00:16:40,480 --> 00:16:46,050 then we see that the velocity of the spacecraft is actually very high relative 175 00:16:46,050 --> 00:16:50,399 to Mars. So it will be on the hyperbolic orbit and will actually leave Mars again. 176 00:16:50,399 --> 00:16:55,270 You can see this on the left side. Right. Because it's leaving Mars again. So what 177 00:16:55,270 --> 00:17:00,459 you need to do is, in fact, you need to slow down and change your orbit into an 178 00:17:00,459 --> 00:17:04,770 ellipse. Okay. And this delta v, is that you that you need here for this maneuver 179 00:17:04,770 --> 00:17:12,220 it's actually less than the delta v you would need to to circularize the orbit to 180 00:17:12,220 --> 00:17:18,400 just fly in the same orbit as Mars. So we need to slow down. A similar argument 181 00:17:18,400 --> 00:17:24,640 actually at Earth shows that, well, if you actually launch into space, then you do 182 00:17:24,640 --> 00:17:29,780 need quite some speed already to not fall down back onto Earth. So that's something 183 00:17:29,780 --> 00:17:33,700 like seven kilometers per second or so. This means that you already have some 184 00:17:33,700 --> 00:17:38,810 speed. OK. And if you align your orbit or your launch correctly, then you already 185 00:17:38,810 --> 00:17:43,350 have some of the delta v that you need for the Hohmann transfer. So in principle, you 186 00:17:43,350 --> 00:17:52,080 need quite a bit less delta v than than you might naively think. All right. So 187 00:17:52,080 --> 00:17:57,280 that much about Hohmann transfer. Let's look at Gravity assist. That's another major 188 00:17:57,280 --> 00:18:03,530 technique for interplanetary missions. The idea is that we can actually use planets 189 00:18:03,530 --> 00:18:10,570 to sort of getting pulled along. So this is an animation, on the lower animation 190 00:18:10,570 --> 00:18:16,300 you see kind of the picture when you look at the planet. So the planets standing 191 00:18:16,300 --> 00:18:21,320 still and we assume that the spacecraft's sort of blue object is on a hyperbolic 192 00:18:21,320 --> 00:18:27,120 orbit and it's kind of making a 90 degree turn. OK. And the upper image, you 193 00:18:27,120 --> 00:18:32,320 actually see the picture when you look from the Sun, so the planet is 194 00:18:32,320 --> 00:18:38,820 actually moving. And if you look very carefully at the blue object then you can 195 00:18:38,820 --> 00:18:45,030 see that it is faster. So once it has passed, the planet is actually faster. 196 00:18:45,030 --> 00:18:52,900 Well, we can actually look at this problem. So this is, again, the picture. When 197 00:18:52,900 --> 00:18:56,260 Mars is centered, we have some entry velocity. Then we are in this hyperbolic 198 00:18:56,260 --> 00:19:03,160 orbit. We have an exit velocity. Notice that the lengths are actually equal. So 199 00:19:03,160 --> 00:19:08,580 it's the same speed. But just a turn direction of this example. But then we can 200 00:19:08,580 --> 00:19:13,410 look at the whole problem with a moving Mars. OK, so now Mars has some velocity 201 00:19:13,410 --> 00:19:19,610 v_mars. So the actual velocity that we see is the sum of the entry and the Mars 202 00:19:19,610 --> 00:19:25,870 velocity before and afterwards exit, plus Mars velocity. And if you look at those 203 00:19:25,870 --> 00:19:31,910 two arrows, then you see immediately that, well, the lengths are different. Okay. So 204 00:19:31,910 --> 00:19:37,650 this is just the whole phenomenon here. So we see that by actually passing close to 205 00:19:37,650 --> 00:19:43,250 such a planet or massive body, we can sort of gain free delta v. Okay, so of 206 00:19:43,250 --> 00:19:49,080 course, it's not. I mean, the energy is still conserved. Okay. But yeah, let's not 207 00:19:49,080 --> 00:19:53,550 worry about these details here. Now, the nice thing is we can use this technique to 208 00:19:53,550 --> 00:19:58,970 actually alter course. We can speed up. So this is the example that I'm shown here. 209 00:19:58,970 --> 00:20:02,790 But also, we can use this to slow down. Okay. So that's a pretty common 210 00:20:02,790 --> 00:20:08,160 application as well. We can use this to slow down by just changing the arrows, 211 00:20:08,160 --> 00:20:15,860 essentially. So just approaching Mars from a different direction, essentially. So 212 00:20:15,860 --> 00:20:21,960 let's look at the example. And this is Bepicolombo. That's actually the reason 213 00:20:21,960 --> 00:20:26,240 why I kind of changed the title, because Bepicolombo is actually a mission to 214 00:20:26,240 --> 00:20:32,661 Mercury. So it was launched last year. It's a combined ESA/JAXA mission and it 215 00:20:32,661 --> 00:20:38,390 consists of two probes and one thruster centrally. So it's a through three stages 216 00:20:38,390 --> 00:20:43,780 that you can see in the picture. Yeah. That's a pretty awesome mission, actually. 217 00:20:43,780 --> 00:20:49,930 It's really nice. But it has in particular, a very cool orbit. So that's 218 00:20:49,930 --> 00:20:56,627 it. What can we see here? So first of all, the blue line, that's actually Earth. The 219 00:20:56,627 --> 00:21:00,180 green one, that's Mercury. So that's where we want to go. And we have this 220 00:21:00,180 --> 00:21:07,130 intermediate turquoise one - that's Venus. And well the curve is 221 00:21:07,130 --> 00:21:10,790 Bepicolombo's orbit, so it looks pretty complicated. Yeah, it's definitely not the 222 00:21:10,790 --> 00:21:16,020 Hohmann transfer. And in fact, this mission uses nine Gravity assists to reach 223 00:21:16,020 --> 00:21:21,950 Mercury. And if you look at the path so, for example, right now 224 00:21:21,950 --> 00:21:28,690 it actually is very close to Mercury because the last five or six Gravity 225 00:21:28,690 --> 00:21:34,500 assists are just around Mercury or just slow down. OK. And this saves a lot of 226 00:21:34,500 --> 00:21:41,760 delta v compared to the standard Hohmann transfer. All right. But we 227 00:21:41,760 --> 00:21:45,810 want to do even better. OK. So let's now actually make the whole problem more 228 00:21:45,810 --> 00:21:53,830 complicated in order to hope for some kind of nice tricks that we can do. OK, so now 229 00:21:53,830 --> 00:21:58,550 we will talk about a planar circular restricted three body problem. All right. 230 00:21:58,550 --> 00:22:02,590 So in general, the three body problem just means, hey, well, instead of two bodies, 231 00:22:02,590 --> 00:22:07,400 we have three. OK. They pairwise attract each other and then we can solve this 232 00:22:07,400 --> 00:22:12,080 whole equation of motion. We can ask a computer. And this is one animation of 233 00:22:12,080 --> 00:22:17,490 what it could look like. So the three masses and their orbits are traced and we 234 00:22:17,490 --> 00:22:24,080 see immediately that we don't see anything that's super complicated. There is no 235 00:22:24,080 --> 00:22:29,670 way we can really... I don't know, formulate a whole solution theory for a 236 00:22:29,670 --> 00:22:33,650 general three body problem. That's complicated. Those are definitely not 237 00:22:33,650 --> 00:22:40,312 ellipses. So let's maybe go a step back and make the problem a bit easier. OK. So 238 00:22:40,312 --> 00:22:44,520 we will now talk about a plane or circular restricted three body problem. There are 239 00:22:44,520 --> 00:22:49,440 three words. So the first one is restricted. Restricted means that in our 240 00:22:49,440 --> 00:22:54,350 application case, one of the bodies is actually a spacecraft. Spacecrafts are 241 00:22:54,350 --> 00:22:58,440 much lighter than, for example, Sun and Mars, meaning that we can actually ignore 242 00:22:58,440 --> 00:23:05,570 the force that the spacecraft exerts on Sun and Mars. Okay. So we will actually 243 00:23:05,570 --> 00:23:11,740 consider Sun and Mars to be independent of the spacecraft. OK. So in total, we only 244 00:23:11,740 --> 00:23:18,120 have like two gravitational forces now acting on a spacecraft. So we neglect sort 245 00:23:18,120 --> 00:23:25,610 of this other force. Also, we will assume that the whole problem is what's called 246 00:23:25,610 --> 00:23:30,800 circular. So we assume that Sun and Mars actually rotate in circles around their 247 00:23:30,800 --> 00:23:37,081 center of mass. This assumption is actually pretty okay. We will see a 248 00:23:37,081 --> 00:23:42,960 picture right now. So in this graph, for example, in this image, you can see that 249 00:23:42,960 --> 00:23:48,680 the black situation. So this might be at some time, at some point in time. And then 250 00:23:48,680 --> 00:23:54,520 later on, Sun and Mars actually have moved to the red positions and the spacecraft is 251 00:23:54,520 --> 00:24:00,840 at some other place. And now, of course, feels some other forces. OK. And also we 252 00:24:00,840 --> 00:24:04,330 will assume that this problem is plane, meaning again that everything takes place 253 00:24:04,330 --> 00:24:12,380 in the plane. OK. So let's look at the video. That's a video with a very low 254 00:24:12,380 --> 00:24:19,610 frame rate, something like two images per day. Maybe it's actually Pluto and Charon. 255 00:24:19,610 --> 00:24:27,250 So the larger one, this is the ex-planet Pluto. It was taken by New Horizons in 256 00:24:27,250 --> 00:24:34,360 2015 and it shows that they actually rotate around the center of mass. Yeah. So 257 00:24:34,360 --> 00:24:40,270 both actually rotate. This also happens, for example, for Sun and Earth or Sun and 258 00:24:40,270 --> 00:24:45,250 Mars or sun and Jupiter or also Earth and Moon. However, in those other cases, the 259 00:24:45,250 --> 00:24:50,650 center of mass is usually contained in the larger body. And so this means that in the 260 00:24:50,650 --> 00:24:57,910 case of Sun-Earth, for example, the Sun will just wiggle a little bit. OK. So you 261 00:24:57,910 --> 00:25:04,410 don't really see this extensive rotation. OK. Now, this problem is still difficult. 262 00:25:04,410 --> 00:25:10,140 OK. So if you're putting out a mass in there, then you don't really 263 00:25:10,140 --> 00:25:15,499 know what happens. However, there's a nice trick to simplify this problem. And 264 00:25:15,499 --> 00:25:19,730 unfortunately, I can't do this here. But maybe all the viewers at home, they can 265 00:25:19,730 --> 00:25:25,080 try to do this. You can take your laptop. Please don't do this. And you can rotate 266 00:25:25,080 --> 00:25:34,020 your laptop at the same speed as this image actually rotates. OK. Well, then 267 00:25:34,020 --> 00:25:39,340 what happens? The two masses will actually stand still from your point of view. OK. 268 00:25:39,340 --> 00:25:45,080 If you do it carefully and don't break anything. So we switch to this sort of 269 00:25:45,080 --> 00:25:50,590 rotating point of view. OK, then the two masses stand still. We still have the two 270 00:25:50,590 --> 00:25:56,020 gravitational forces towards Sun and Mars. But because we kind of look at it from a 271 00:25:56,020 --> 00:26:00,670 rotated or from a moving point of view, we get two new forces, those forces, you 272 00:26:00,670 --> 00:26:04,890 know, the centrifugal forces, of course, the one that, for example, you 273 00:26:04,890 --> 00:26:11,510 have when you play with some children or so, they want to be pulled in 274 00:26:11,510 --> 00:26:17,440 a circle very quickly and then they start flying and that's pretty cool. But here we 275 00:26:17,440 --> 00:26:21,730 actually have this force acting on the spacecraft. Okay. And also there is the 276 00:26:21,730 --> 00:26:26,790 Coriolis force, which is a bit less known. This depends on the velocity of the 277 00:26:26,790 --> 00:26:31,660 spacecraft. OK. So if there is no velocity in particular, then there will not be any 278 00:26:31,660 --> 00:26:38,270 Coriolis force. So our new problem actually has four forces. OK, but the 279 00:26:38,270 --> 00:26:43,580 advantage is that Sun and Mars actually are standing still. So we don't need to 280 00:26:43,580 --> 00:26:51,040 worry about their movement. OK. So now how does this look like? Well, this might be 281 00:26:51,040 --> 00:26:55,990 an example for an orbit. Well, that looks still pretty complicated. I mean, this is 282 00:26:55,990 --> 00:27:01,500 something that you can't have in a two body problem. It has these weird wiggles. 283 00:27:01,500 --> 00:27:06,320 I mean, they're not really corners. And it actually kind of switches from Sun to 284 00:27:06,320 --> 00:27:10,650 Mars. Yes. So it stays close to Sun for some time and it moves somewhere else as 285 00:27:10,650 --> 00:27:15,650 it, it's still pretty complicated. I don't know. Maybe some of you have have read the 286 00:27:15,650 --> 00:27:23,490 book "The Three-Body Problem". So there, for example, the two masses might be a 287 00:27:23,490 --> 00:27:28,760 binary star system. OK. And then you have a planet that's actually moving along such 288 00:27:28,760 --> 00:27:35,710 an orbit. OK, that looks pretty bad. So in particular, the seasons might be somewhat 289 00:27:35,710 --> 00:27:41,960 messed up. Yeah. So this problem is, in fact, in a strong mathematical sense, 290 00:27:41,960 --> 00:27:47,200 chaotic. OK. So chaotic means something like if you start with very close initial 291 00:27:47,200 --> 00:27:51,610 conditions and you just let the system evolve, then the solutions will look very, 292 00:27:51,610 --> 00:27:58,560 very different. OK. And this really happens here, which is good. All right. So 293 00:27:58,560 --> 00:28:03,950 one thing we can ask is, well, is it possible that if we put a spacecraft into 294 00:28:03,950 --> 00:28:08,100 the system without any velocity, is it possible that all the forces actually 295 00:28:08,100 --> 00:28:12,450 cancel out. And it turns out yes, it is possible. And those points are called 296 00:28:12,450 --> 00:28:17,950 Lagrangian points. So if we have zero velocity, there is no Coriolis force. So 297 00:28:17,950 --> 00:28:23,460 we have only these three forces. And as you can see in this little schematics 298 00:28:23,460 --> 00:28:32,116 here, it's possible that all these forces actually cancel out. Now imagine. Yeah. I 299 00:28:32,116 --> 00:28:36,940 give you a homework. Please calculate all these possible points. Then you can do 300 00:28:36,940 --> 00:28:42,280 this. But we won't do this right here. Instead, we just look at the result. So 301 00:28:42,280 --> 00:28:47,880 those are the five Lagrangian points in this problem. OK, so we have L4 and L5 302 00:28:47,880 --> 00:28:52,150 which are at equilateral triangles with Sun and Mars. Well, Sun - Mars in this 303 00:28:52,150 --> 00:28:59,780 case. And we have L1, L2 and L3 on the line through Sun and Mars. So if you put 304 00:28:59,780 --> 00:29:05,250 the spacecraft precisely at L1 without any velocity, then in relation to Sun and Mars 305 00:29:05,250 --> 00:29:10,150 it will actually stay at the same position. Okay, that's pretty cool. However, 306 00:29:10,150 --> 00:29:15,770 mathematicians and physicists will immediately ask well, OK, but what happens 307 00:29:15,770 --> 00:29:21,920 if I actually put a spacecraft close to a Lagrangian point? OK, so this is 308 00:29:21,920 --> 00:29:28,200 related to what's called stability. And you can calculate that around L4 and L5. 309 00:29:28,200 --> 00:29:33,330 The spacecraft will actually stay in the vicinity. So it will essentially rotate 310 00:29:33,330 --> 00:29:38,980 around the Lagrangian points. So those are called stable, whereas L1, L2 and L3 are 311 00:29:38,980 --> 00:29:43,990 actually unstable. This means that if you put a spacecraft there, then it will 312 00:29:43,990 --> 00:29:50,600 eventually escape. However, this takes a different amount of time depending on the 313 00:29:50,600 --> 00:29:55,330 Lagrangian points. For example, if you're close to L2, this might take a few months, 314 00:29:55,330 --> 00:29:58,730 but if you're close to L3, this will actually take something like a hundred 315 00:29:58,730 --> 00:30:08,140 years or so. Okay, so those points are still different. All right. Okay. So 316 00:30:08,140 --> 00:30:10,950 is there actually any evidence that they exist? I mean, maybe I'm just making this 317 00:30:10,950 --> 00:30:14,690 up and, you know, I mean, haven't shown you any equations. I could just throw 318 00:30:14,690 --> 00:30:19,950 anything. However, we can actually look at the solar system. So this is the inner 319 00:30:19,950 --> 00:30:23,570 solar system here. In the middle you see, well, the center you see the Sun, of 320 00:30:23,570 --> 00:30:28,970 course. And to the lower left, there's Jupiter. Now, if you imagine an 321 00:30:28,970 --> 00:30:35,250 equilateral triangle of Sun and Jupiter, well, there are two of them. And then you 322 00:30:35,250 --> 00:30:40,920 see all these green dots there. And those are asteroids. Those are the Trojans and 323 00:30:40,920 --> 00:30:47,770 the Greeks. And they accumulate there because L4 and L5 are stable. Okay. So we 324 00:30:47,770 --> 00:30:55,140 can really see this dynamics gone on in the solar system. However, there's also 325 00:30:55,140 --> 00:30:59,490 various other applications of Lagrangian points. So in particular, you might want 326 00:30:59,490 --> 00:31:05,710 to put a space telescope somewhere in space, of course, in such a way that the 327 00:31:05,710 --> 00:31:11,520 Sun is not blinding you. Well, there is Earth, of course. So if we can put the 328 00:31:11,520 --> 00:31:18,980 spacecraft behind Earth, then we might be in the shadow. And this is the Lagrangian 329 00:31:18,980 --> 00:31:24,860 point L2, which is the reason why this is actually being used for space telescopes 330 00:31:24,860 --> 00:31:30,470 such as, for example, this one. However, it turns out L2 is unstable. So we don't 331 00:31:30,470 --> 00:31:35,091 really want to put the spacecraft just there. But instead, we put it in an orbit 332 00:31:35,091 --> 00:31:40,730 close... in a close orbit, close to L2. And this particular example is called the 333 00:31:40,730 --> 00:31:44,560 Halo orbit, and it's actually not contained in the planes. I'm cheating a 334 00:31:44,560 --> 00:31:48,030 little bit. It's on the right hand side to you. And in the animation you actually see 335 00:31:48,030 --> 00:31:54,110 the the orbit from the side. So it actually leaves the plane the blue dot is 336 00:31:54,110 --> 00:32:00,620 Earth and the left hand side you see the actual orbit from the top. So 337 00:32:00,620 --> 00:32:06,230 it's rotating around this place. OK. So that's the James Webb Space Telescope, by 338 00:32:06,230 --> 00:32:11,360 the way. You can see in the animation it's supposed to launch in 2018. That didn't 339 00:32:11,360 --> 00:32:19,530 work out, unfortunately, but stay tuned. Another example. That's how it has become 340 00:32:19,530 --> 00:32:26,200 pretty famous recently as the Chinese Queqiao relay satellite. This one sits at 341 00:32:26,200 --> 00:32:31,090 the Earth - Moon L2 Lagrange point. And the reason for this is that the Chinese 342 00:32:31,090 --> 00:32:37,650 wanted to or actually did land the Chang'e 4 Moon lander on the backside of the Moon. 343 00:32:37,650 --> 00:32:41,560 And in order to stay in contact, radio contact with the lander, they had to put a 344 00:32:41,560 --> 00:32:47,640 relay satellite behind the Moon, which they could still see from Earth. And they 345 00:32:47,640 --> 00:33:00,100 chose some similar orbit around L2. OK. So let's now go to some other more advanced 346 00:33:00,100 --> 00:33:07,510 technique: ballistic capture. Right. Okay. So this whole business sort of started 347 00:33:07,510 --> 00:33:14,410 with a mission from the beginning of the 1990s, and that's the Hiten mission. So 348 00:33:14,410 --> 00:33:19,890 that was a Japanese well, Moon probe consisted of a probe which had a small 349 00:33:19,890 --> 00:33:26,290 orbiter site which was separated, and then it was supposed to actually enter orbit 350 00:33:26,290 --> 00:33:31,610 around Moon. Unfortunately, it missed its maneuver. So it didn't apply enough delta v 351 00:33:31,610 --> 00:33:37,570 so it actually flew off. And the mission was sort of lost at that point 352 00:33:37,570 --> 00:33:42,430 because Hiten itself, so the main probe did not have enough fuel to reach the 353 00:33:42,430 --> 00:33:47,701 Moon. All right. That's, of course, a problem. I mean, that's a risk you have to 354 00:33:47,701 --> 00:33:53,460 take. And they were probably pretty devastated. However, there were two people 355 00:33:53,460 --> 00:34:00,780 from JPL, NASA, who actually heard about this, Belbruno and Miller, and they were 356 00:34:00,780 --> 00:34:08,260 working on strange orbits, those ballistic capture orbits. And they actually found 357 00:34:08,260 --> 00:34:14,609 one for the Hiten probe. They sent this to the Japanese and they actually use that 358 00:34:14,609 --> 00:34:23,220 orbit to get the Hiten probe to the moon. And it actually arrived in October 1991. 359 00:34:23,220 --> 00:34:26,450 And then it could do some science, you know, maybe some 360 00:34:26,450 --> 00:34:31,389 different experiments, but it actually arrived there. However, the transfer took 361 00:34:31,389 --> 00:34:37,070 quite a bit longer. So a normal Moon transfer takes like three days or so. But 362 00:34:37,070 --> 00:34:42,320 this one actually took a few months. All right. And the reason for this is that it 363 00:34:42,320 --> 00:34:48,600 looks pretty weird. So this is a picture of the orbiter - schematic picture. 364 00:34:48,600 --> 00:34:54,260 And you can see the Earth. Well, there in the middle sort of. And the Moon a bit to 365 00:34:54,260 --> 00:35:01,820 the left at the L2 is the Lagrangian point of the Sun - Earth system. OK. So it's 366 00:35:01,820 --> 00:35:07,430 pretty far out. And you can see that the orbit sort of consists of two parts. 367 00:35:07,430 --> 00:35:13,100 First, it leaves Earth and it flies far beyond the Moon. So somewhere completely 368 00:35:13,100 --> 00:35:18,910 different towards some other Lagrangian point. That's really far away. Then it 369 00:35:18,910 --> 00:35:24,280 does some weird things. And in the upper part of picture there it actually does a 370 00:35:24,280 --> 00:35:30,240 maneuver. So we apply some thrusts there to be to change on a different orbit. And 371 00:35:30,240 --> 00:35:36,830 this orbit led the probe directly to the moon where it was essentially captured for 372 00:35:36,830 --> 00:35:42,320 free. OK. So it just entered orbit around the Moon. And this is, of course, not 373 00:35:42,320 --> 00:35:46,470 possible in the two body problem, but we may find a way for doing this in the three 374 00:35:46,470 --> 00:35:56,530 body problem. OK, so what do we mean by capture? Now we have to sort of think 375 00:35:56,530 --> 00:36:02,320 a bit more abstractly. The idea is... we have Sun and Mars and we 376 00:36:02,320 --> 00:36:08,100 have a spacecraft that flies in this three body problem. So the red orbit is actually 377 00:36:08,100 --> 00:36:14,960 the orbit of the spacecraft. Now, at any point in time, we may decide to just 378 00:36:14,960 --> 00:36:20,970 forget about the Sun. OK. So instead we consider the two body problem of Mars and 379 00:36:20,970 --> 00:36:26,760 a spacecraft. OK. Because at this point in time, the spacecraft has a certain 380 00:36:26,760 --> 00:36:31,240 position relative to Mars and a certain velocity. So this determines its orbit in 381 00:36:31,240 --> 00:36:36,440 the two body problem. Usually it would be very fast. So it would be on a hyperbolic 382 00:36:36,440 --> 00:36:43,269 orbit, which is denoted by the dashed line here. OK. Or a dashed curve. So usually 383 00:36:43,269 --> 00:36:47,240 you happen to be in a hyperbolic orbit. But of course, that orbit is only an 384 00:36:47,240 --> 00:36:50,280 approximation because in the three body problem, the movement is actually 385 00:36:50,280 --> 00:36:57,490 different. But later on, it might happen that we continue on the orbit. We can do 386 00:36:57,490 --> 00:37:01,530 the same kind of construction, but just looking... but just ignoring the Sun 387 00:37:01,530 --> 00:37:09,710 essentially, and then we could find that the spacecraft suddenly is in a elliptical 388 00:37:09,710 --> 00:37:14,190 orbit. This would mean that if you forgot about the Sun, then the spacecraft 389 00:37:14,190 --> 00:37:19,670 would be stable and would be captured by Mars. It would be there. That would be 390 00:37:19,670 --> 00:37:27,210 pretty nice. So this phenomenon, when this happens, we call this a temporary capture. 391 00:37:27,210 --> 00:37:33,810 OK. Temporary because it might actually leave that situation again later on. Now, 392 00:37:33,810 --> 00:37:37,320 because the actual movement depends on the three body problem, which is super 393 00:37:37,320 --> 00:37:42,150 complicated. So it's possible that it actually leaves again. But for that moment 394 00:37:42,150 --> 00:37:46,300 at least, it's captured and we want to find a way or describe some kind of 395 00:37:46,300 --> 00:37:54,000 algorithm perhaps how we can find this situation essentially. OK, and in a 396 00:37:54,000 --> 00:38:01,451 reasonable way, and the notion for this is what's called, well, n-stability, the idea 397 00:38:01,451 --> 00:38:08,020 is the following: we look at the three body probleme, we want to go to Mars. So we 398 00:38:08,020 --> 00:38:13,250 pick a line there. And on the line we take a point x, which has some distance r to 399 00:38:13,250 --> 00:38:20,090 the Mars and we pick a perpendicular speed, a perpendicular velocity to the 400 00:38:20,090 --> 00:38:25,810 line such that this corresponds to some kind of elliptic orbit in the two body 401 00:38:25,810 --> 00:38:30,290 problem. Okay. So that's the dashed line. But then we actually look at the problem 402 00:38:30,290 --> 00:38:37,820 in the three body problem and we just evolve the spacecraft. And it's following 403 00:38:37,820 --> 00:38:46,080 the red orbit. It might follow the red orbit. And it can happen that after going 404 00:38:46,080 --> 00:38:54,200 around Mars for one time, it hits again the line. Okay, then we can do the same 405 00:38:54,200 --> 00:38:59,540 construction with forgetting the Sun again and just look at the two body problem. And 406 00:38:59,540 --> 00:39:04,990 it's possible that this point actually still corresponds to an elliptic orbit. 407 00:39:04,990 --> 00:39:10,870 That's somewhat interesting, right? Because now this means that if we actually 408 00:39:10,870 --> 00:39:17,120 hit the point x, then we can follow the orbit and we know that we wrap around 409 00:39:17,120 --> 00:39:24,030 Mars once and are still sort of captured in the corresponding two body problem. 410 00:39:24,030 --> 00:39:29,230 Okay. If we actually are able to wrap around Mars twice, then we would call this 411 00:39:29,230 --> 00:39:35,980 2-stable and, well, for more rotations that it is n-stable. Okay, so that's good 412 00:39:35,980 --> 00:39:39,170 because such an orbit corresponds to something that's usable because we will 413 00:39:39,170 --> 00:39:45,370 wrap around Mars n times. However, it's also possible that you have an unstable 414 00:39:45,370 --> 00:39:49,020 point, meaning that we again start in something that corresponds to an ellipse 415 00:39:49,020 --> 00:39:54,170 around Mars. But if we actually follow the orbit in a three body problem, it will, 416 00:39:54,170 --> 00:39:58,110 for example, not come back. It will not wrap around Mars, it will go to the Sun or 417 00:39:58,110 --> 00:40:03,470 somewhere else. OK. So that's that's of course, not a nice point. This one's 418 00:40:03,470 --> 00:40:10,480 called unstable. And then there's another thing we can do. That's actually a pretty 419 00:40:10,480 --> 00:40:18,280 common trick in finding orbits, etc. We can instead of solving the problem in 420 00:40:18,280 --> 00:40:23,970 forward time we actually go back, okay. So essentially in your program you just 421 00:40:23,970 --> 00:40:28,810 replace time by minus time, for example, and then you just solve the thing and you 422 00:40:28,810 --> 00:40:37,070 go back in the past and it's possible that a point that corresponds to such 423 00:40:37,070 --> 00:40:41,641 an ellipse when you go back into the past and it does not wrap around, but it 424 00:40:41,641 --> 00:40:47,010 actually goes to the Sun, for example, we call this unstable in the past. Okay. So 425 00:40:47,010 --> 00:40:56,680 that's just some random definition. And we can use this. The reason for 426 00:40:56,680 --> 00:41:05,080 this is we can actually kind of take these concepts together and build an orbit from 427 00:41:05,080 --> 00:41:13,020 that. The idea being we pick a point x that is n-stable. So, for example, it 428 00:41:13,020 --> 00:41:19,710 might wrap around Mars six times, some number that we like. This is the blue part 429 00:41:19,710 --> 00:41:24,130 here in the picture. So it wraps around Mars six times. But the way we go back in 430 00:41:24,130 --> 00:41:30,560 time, it actually leaves Mars or at least it doesn't come back in such a way that 431 00:41:30,560 --> 00:41:42,070 it's again on an ecliptic curve. So this is the red part. Okay. So we can 432 00:41:42,070 --> 00:41:48,390 just follow this and then we pick a point y on that curve. Okay. So this one will be 433 00:41:48,390 --> 00:41:57,990 pretty far away from Mars or we can choose it. And then we sort of use a Hohmann 434 00:41:57,990 --> 00:42:03,650 transfer to get from Earth to that point y. Okay? So our orbit actually consists of 435 00:42:03,650 --> 00:42:08,520 three parts now. Okay. So we have the Hohmann transfer, but it's not actually 436 00:42:08,520 --> 00:42:14,390 aiming for Mars. It's actually aiming for the point y. There we do a maneuver 437 00:42:14,390 --> 00:42:21,470 because we want to switch onto this red orbit. Okay. And then this one will bring 438 00:42:21,470 --> 00:42:28,780 us to the point x where we know because it was constructed in this way that the 439 00:42:28,780 --> 00:42:36,050 spacecraft will continue to rotate around Mars for example six times. Okay. So in 440 00:42:36,050 --> 00:42:43,619 particular at x there is no maneuver taking place. Okay. So that's a 441 00:42:43,619 --> 00:42:49,360 possible mission scenario. And the way this is done then usually is you kind of.. 442 00:42:49,360 --> 00:42:54,280 you calculate these points x that are suitable for doing this. Okay. So they 443 00:42:54,280 --> 00:42:58,460 have to be stable and unstable in the past at the same time. So we have to find them. 444 00:42:58,460 --> 00:43:02,500 And there's a lot of numerical computations involved in that. But once we 445 00:43:02,500 --> 00:43:07,480 have this, you can actually build these orbits. OK. So let's look at an actual 446 00:43:07,480 --> 00:43:16,021 example. So this is for Earth - Mars. On the left, you see, well, that the two 447 00:43:16,021 --> 00:43:23,380 circular orbits of Earth, Mars, and on the right you see the same orbit, but from a 448 00:43:23,380 --> 00:43:28,370 point of view centered around Mars. Okay. And the colors correspond to each other. 449 00:43:28,370 --> 00:43:32,500 So the mission starts on the left side by doing a Hohmann transfer. So that's the 450 00:43:32,500 --> 00:43:35,770 black line starting at Earth and then hitting the point, which is called x_c 451 00:43:35,770 --> 00:43:42,290 here. So that's the y that I had on the other slide. So this point y 452 00:43:42,290 --> 00:43:47,930 or x_c here is pretty far away still from Mars. There we do a maneuver and we switch 453 00:43:47,930 --> 00:43:56,730 under the red orbit. Which brings us to the point x closer to Mars, after which we 454 00:43:56,730 --> 00:44:01,310 will actually start rotating round Mars. And the point x is actually at the top of 455 00:44:01,310 --> 00:44:08,510 this picture. Okay. And then on the right you can see the orbit and it's looking 456 00:44:08,510 --> 00:44:13,550 pretty strangely. And also the red orbit is when we kind of the capture orbit 457 00:44:13,550 --> 00:44:19,180 our way to actually get to Mars. And then if you look very carefully, you can count 458 00:44:19,180 --> 00:44:26,710 we actually rotate around Mars six times. Okay. Now, during those six 459 00:44:26,710 --> 00:44:32,170 rotations around Mars, we could do experiments. So maybe that is enough for 460 00:44:32,170 --> 00:44:37,000 whatever we are trying to do. OK. However, if we want to stay there, we need to 461 00:44:37,000 --> 00:44:44,797 actually execute another maneuver. OK. So to actually stay around Mars. And I mean, 462 00:44:44,797 --> 00:44:48,060 the principle looks nice but of course, you have to do some calculations. We have 463 00:44:48,060 --> 00:44:55,640 to find some ways to actually quantify how good this is. And it turns out that there 464 00:44:55,640 --> 00:45:02,410 are few parameters that you can choose, in particular the target point x, this has 465 00:45:02,410 --> 00:45:07,310 a certain distance that you're aiming for at around Mars. And it turns out that this 466 00:45:07,310 --> 00:45:14,090 procedure here, for example, is only very good if this altitude, this distance r is 467 00:45:14,090 --> 00:45:17,950 actually high enough. If it is high enough then you can save - in principle - up to 468 00:45:17,950 --> 00:45:23,720 twenty three percent of the delta v, which is enormous. OK. So that would 469 00:45:23,720 --> 00:45:29,050 be really good. However, in reality it's not as good usually. Yeah. And for a 470 00:45:29,050 --> 00:45:34,530 certain lower distances, for example, you cannot save anything, so there are 471 00:45:34,530 --> 00:45:40,810 certain tradeoffs to make. However, there is another advantage. Remember this point y? 472 00:45:40,810 --> 00:45:45,990 We chose this along this capture orbit along the red orbit. And the thing is, we 473 00:45:45,990 --> 00:45:51,380 can actually choose this freely. This means that our Hohmann transfer doesn't 474 00:45:51,380 --> 00:45:55,000 need to hit Mars directly when it's there. So it doesn't need to aim for that 475 00:45:55,000 --> 00:46:02,970 particular point. It can actually aim for any point on that capture orbit. This 476 00:46:02,970 --> 00:46:06,310 means that we have many more Hohmann transfers available that we can actually 477 00:46:06,310 --> 00:46:12,500 use to get there. This means that we have a far larger transfer window. OK. So we 478 00:46:12,500 --> 00:46:17,730 cannot just start every 26 months. But now we, with this technique, we could actually 479 00:46:17,730 --> 00:46:24,460 launch. Well, quite often. However, there's a little problem. If you looked at 480 00:46:24,460 --> 00:46:33,950 the graph carefully, then you may have seen that the red orbit actually took like 481 00:46:33,950 --> 00:46:39,350 three quarters of the rotation of Mars. This corresponds to roughly something like 482 00:46:39,350 --> 00:46:43,750 400 days. OK. So this takes a long time. So you probably don't want to use this 483 00:46:43,750 --> 00:46:49,060 with humans on board because they have to actually wait for a long time. But in 484 00:46:49,060 --> 00:46:52,890 principle, there are ways to make this shorter. So you can try to actually 485 00:46:52,890 --> 00:46:58,450 improve on this, but in general, it takes a long time. So let's look at a real 486 00:46:58,450 --> 00:47:04,610 example for this. Again, that's Bepicolombo. The green dot is now Mercury. 487 00:47:04,610 --> 00:47:09,870 So this is kind of a zoom of the other animation and the purple line is the 488 00:47:09,870 --> 00:47:20,640 orbit. And yeah, it looks strange. So the first few movements around Mercury, 489 00:47:20,640 --> 00:47:28,300 they are actually the last gravity assists for slowing down. And then it actually 490 00:47:28,300 --> 00:47:36,780 starts on the capture orbit. So now it actually approaches Mercury. So this is 491 00:47:36,780 --> 00:47:41,010 the part that's sort of difficult to find, but which you can do with this stability. 492 00:47:41,010 --> 00:47:45,609 And once the animation actually ends, this is when it actually reaches the point 493 00:47:45,609 --> 00:47:52,240 when it's temporarily captured. So in this case, this is at an altitude of 180,000 494 00:47:52,240 --> 00:47:58,090 kilometers. So it is pretty high up above Mercury, but it's enough for the mission. 495 00:47:58,090 --> 00:48:03,270 OK. And of course, then they do some other maneuver to actually stay around 496 00:48:03,270 --> 00:48:12,230 Mercury. Okay, so in the last few minutes, let's have a look. Let's have a brief look 497 00:48:12,230 --> 00:48:18,997 at how we can actually extend this. So I will be very brief here, because while we 498 00:48:18,997 --> 00:48:23,600 can try to actually make this more general to improve on this, this concept is then 499 00:48:23,600 --> 00:48:29,040 called the interplanetary transport network. And it looks a bit similar to 500 00:48:29,040 --> 00:48:36,290 what we just saw. The idea is that, in fact, this capture orbit is part of a 501 00:48:36,290 --> 00:48:42,950 larger well, a set of orbits that have these kinds of properties that wrap around 502 00:48:42,950 --> 00:48:48,520 Mars and then kind of leave Mars. And they are very closely related to the 503 00:48:48,520 --> 00:48:53,280 dynamics of particular Lagrangian points, in this case L1. So that was the one 504 00:48:53,280 --> 00:49:00,330 between the two masses. And if you investigate this Lagrangian point a bit 505 00:49:00,330 --> 00:49:05,530 closer, you can see, well, you can see different orbits of all kinds of 506 00:49:05,530 --> 00:49:10,650 behaviors. And if you understand this, then you can actually try to do the same 507 00:49:10,650 --> 00:49:16,880 thing on the other side of the Lagrangian point. OK. So we just kind of switch from 508 00:49:16,880 --> 00:49:21,440 Mars to the Sun and we do a similar thing there. Now we expect to actually find 509 00:49:21,440 --> 00:49:24,920 similar orbits that are wrapping around the Sun and then going towards this 510 00:49:24,920 --> 00:49:31,859 Lagrangian point in a similar way. Well, then we already have some orbits that are 511 00:49:31,859 --> 00:49:39,270 well, kind of meeting at L1. So we might be able to actually connect them somehow, 512 00:49:39,270 --> 00:49:45,070 for example by maneuver. And then we only need to reach the orbit around Earth or 513 00:49:45,070 --> 00:49:50,130 around Sun from Earth. OK. If you find a way to do this, you can get rid of the 514 00:49:50,130 --> 00:49:55,270 Hohmann transfer. And this way you reduce your delta v even further. The problem is 515 00:49:55,270 --> 00:50:00,690 that this is hard to find because these orbits they are pretty rare. And of 516 00:50:00,690 --> 00:50:07,320 course, you have to connect those orbits. So they you approach the Lagrangian point 517 00:50:07,320 --> 00:50:14,330 from L1 from two sides, but you don't really want to wait forever until they... 518 00:50:14,330 --> 00:50:19,630 it's very easy to switch or so, so instead you apply some delta v, OK, in order to 519 00:50:19,630 --> 00:50:24,490 not wait that long. So here's a picture of how this might look like. Again 520 00:50:24,490 --> 00:50:28,960 very schematic. So we have Sun, we have Mars and in between there is the 521 00:50:28,960 --> 00:50:35,240 Lagrangian point L1. The red orbit is sort of an extension of one of those capture 522 00:50:35,240 --> 00:50:38,230 orbits that we have seen. OK, so that wraps around Mars a certain number of 523 00:50:38,230 --> 00:50:45,350 times. And while in the past, for example, it actually goes to Lagrangian point. I 524 00:50:45,350 --> 00:50:50,780 didn't explain this, but in fact, there are many more orbits around L1, closed 525 00:50:50,780 --> 00:50:55,460 orbits, but they're all unstable. And these orbits that are used in this 526 00:50:55,460 --> 00:51:05,030 interplanetary transport network they actually approach those orbits around L1 527 00:51:05,030 --> 00:51:10,570 and we do the same thing on the other side of the Sun now and then the idea is, OK, 528 00:51:10,570 --> 00:51:15,590 we take these orbits, we connect them. And when we are in the black orbit 529 00:51:15,590 --> 00:51:19,070 around L1, we actually apply some maneuver, we apply some delta v to 530 00:51:19,070 --> 00:51:22,450 actually switch from one to the other. And then we have sort of a connection of how 531 00:51:22,450 --> 00:51:28,500 to get from Sun to Mars. So we just need to do a similar thing again from for Earth 532 00:51:28,500 --> 00:51:35,119 to this particular blue orbit around the Sun. OK. So that's the general procedure. 533 00:51:35,119 --> 00:51:38,050 But of course, it's difficult. And in the end, you have to do a lot of numerics 534 00:51:38,050 --> 00:51:44,840 because as I said at the beginning, this is just a brief overview. It's not all the 535 00:51:44,840 --> 00:51:50,900 details. Please don't launch your own mission tomorrow. OK. So with 536 00:51:50,900 --> 00:51:54,960 this, I want to thank you. And I'm open to questions. 537 00:51:54,960 --> 00:52:05,640 *Applause* 538 00:52:05,640 --> 00:52:08,210 Herald: So thank you Sven for an interesting talk. We have a few minutes 539 00:52:08,210 --> 00:52:11,160 for questions, if you have any questions lined up next to the microphones, we'll 540 00:52:11,160 --> 00:52:18,400 start with microphone number one. Mic1: Hello. So what are the problems 541 00:52:18,400 --> 00:52:22,680 associated? So you showed in the end is going around to Lagrange Point L1? 542 00:52:22,680 --> 00:52:26,710 Although this is also possible for 543 00:52:26,710 --> 00:52:30,140 other Lagrange points. Could you do this with L2? 544 00:52:30,140 --> 00:52:38,180 Sven: Yes, you can. Yeah. So in principle, I didn't show the whole picture, but 545 00:52:38,180 --> 00:52:43,107 these kind of orbits, they exist at L1, but they also exist at L2. And in 546 00:52:43,107 --> 00:52:49,080 principle you can this way sort of leave this two body problem by finding similar 547 00:52:49,080 --> 00:52:53,650 orbits. But of course the the details are different. So you cannot really take your 548 00:52:53,650 --> 00:52:58,640 knowledge or your calculations from L1 and just taking over to L2, you actually 549 00:52:58,640 --> 00:53:03,189 have to do the same thing again. You have to calculate everything in detail. 550 00:53:03,189 --> 00:53:06,650 Herald: To a question from the Internet. Signal Angel: Is it possible to use these 551 00:53:06,650 --> 00:53:11,350 kinds of transfers in Kerbal Space Program? 552 00:53:11,350 --> 00:53:23,500 Sven: So Hohmann transfers, of course, the gravity assists as well, but not the 553 00:53:23,500 --> 00:53:28,900 restricted three body problem because the way Kerbal Space Program at least the 554 00:53:28,900 --> 00:53:33,450 default installation so without any mods works is that it actually switches the 555 00:53:33,450 --> 00:53:40,270 gravitational force. So the thing that I described as a patch solution where we 556 00:53:40,270 --> 00:53:46,220 kind of switch our picture, which gravitational force we consider for our 557 00:53:46,220 --> 00:53:50,619 two body problem. This is actually the way the physics is implemented in Kerbal space 558 00:53:50,619 --> 00:53:55,400 program. So we can't really do the interplanetary transport network there. 559 00:53:55,400 --> 00:54:00,090 However, I think there's a mod that allows this, but your computer might be too slow 560 00:54:00,090 --> 00:54:04,350 for this, I don't know. Herald: If you're leaving please do so 561 00:54:04,350 --> 00:54:07,440 quietly. Small question and question from microphone number four. 562 00:54:07,440 --> 00:54:12,619 Mic4: Hello. I have actually two questions. I hope that's okay. First 563 00:54:12,619 --> 00:54:18,289 question is, I wonder how you do that in like your practical calculations. Like you 564 00:54:18,289 --> 00:54:22,950 said, there's a two body problem and there are solutions that you can 565 00:54:22,950 --> 00:54:27,440 calculate with a two body problem. And then there's a three body problem. And I 566 00:54:27,440 --> 00:54:32,050 imagine there's an n-body problem all the time you do things. So how does it look 567 00:54:32,050 --> 00:54:37,890 when you do that? And the second question is: you said that reducing delta v 568 00:54:37,890 --> 00:54:47,770 about 15% is enormous. And I wonder what effect does this have on the payload? 569 00:54:47,770 --> 00:54:57,550 Sven: Okay. So regarding the first question. So in principle, I mean, you 570 00:54:57,550 --> 00:55:05,210 make a plan for a mission. So you have to you calculate those things in these 571 00:55:05,210 --> 00:55:08,910 simplified models. Okay. You kind of you patch together an idea of what you want to 572 00:55:08,910 --> 00:55:14,910 do. But of course, in the end, you're right, there are actually many massive 573 00:55:14,910 --> 00:55:19,250 bodies in the solar system. And because we want to be precise, we actually have to 574 00:55:19,250 --> 00:55:25,280 incorporate all of them. So in the end, you have to do an actual numerical search 575 00:55:25,280 --> 00:55:32,010 in a much more complicated n-body problem. So with, I don't know, 100 bodies or so 576 00:55:32,010 --> 00:55:37,800 and you have to incorporate other effects. For example, the solar radiation might 577 00:55:37,800 --> 00:55:43,230 actually have a little influence on your orbit. Okay. And there are many effects of 578 00:55:43,230 --> 00:55:48,040 this kind. So once you have a rough idea of what you want to do, you need to take 579 00:55:48,040 --> 00:55:53,260 your very good physics simulator for the n-body problem, which actually has all 580 00:55:53,260 --> 00:55:57,050 these other effects as well. And then you need to do a numerical search over this. 581 00:55:57,050 --> 00:56:01,410 Kind of, you know, where to start with these ideas, where to look for solutions. 582 00:56:01,410 --> 00:56:06,680 But then you actually have to just try it and figure out some algorithm to actually 583 00:56:06,680 --> 00:56:12,010 approach a solution that has to behaviors that you want. But it's a lot of numerics. 584 00:56:12,010 --> 00:56:16,500 Right. And the second question, can you remind me again? Sorry. 585 00:56:16,500 --> 00:56:23,550 Mic4: Well, the second question was in reducing delta v about 15%. What is the 586 00:56:23,550 --> 00:56:28,890 effect on the payload? Sven: Right. So, I mean, if you need 587 00:56:28,890 --> 00:56:35,750 15% less fuel, then of course you can use 15% more weight for more mass for the 588 00:56:35,750 --> 00:56:40,450 payload. Right. So you could put maybe another instrument on there. Another thing 589 00:56:40,450 --> 00:56:46,119 you could do is actually keep the fuel but actually use it for station keeping. So, 590 00:56:46,119 --> 00:56:52,550 for example, in the James Webb telescope example, the James Webb telescope flies 591 00:56:52,550 --> 00:56:58,840 around this Halo orbit around L2, but the orbit itself is unstable. So the James 592 00:56:58,840 --> 00:57:03,980 Webb Space Telescope will actually escape from that orbit. So they have to do a few 593 00:57:03,980 --> 00:57:08,140 maneuvers every year to actually stay there. And they have only a finite amount 594 00:57:08,140 --> 00:57:13,431 of fuels at some point. This won't be possible anymore. So reducing delta v 595 00:57:13,431 --> 00:57:20,609 requirements might actually have increased the mission lifetime by quite a bit. 596 00:57:20,609 --> 00:57:25,160 Herald: Number three. Mic3: Hey. When you do such a 597 00:57:25,160 --> 00:57:29,869 mission, I guess you have to adjust the trajectory of your satellite quite often 598 00:57:29,869 --> 00:57:34,190 because nothing goes exactly as you calculated it. Right. And the question is, 599 00:57:34,190 --> 00:57:38,930 how precise can you measure the orbit? Sorry, the position and the speed of a 600 00:57:38,930 --> 00:57:43,590 spacecraft at, let's say, Mars. What's the resolution? 601 00:57:43,590 --> 00:57:48,300 Sven: Right. So from Mars, I'm not completely sure how precise it is. But for 602 00:57:48,300 --> 00:57:52,300 example, if you have an Earth observation mission, so something that's flying around 603 00:57:52,300 --> 00:57:58,500 Earth, then you can have a rather precise orbit that's good enough for taking 604 00:57:58,500 --> 00:58:04,220 pictures on Earth, for example, for something like two weeks or so. So 605 00:58:04,220 --> 00:58:12,080 you can measure the orbit well enough and calculate the future something like two 606 00:58:12,080 --> 00:58:21,140 weeks in the future. OK. So that's good enough. However. Yeah. The... I can't 607 00:58:21,140 --> 00:58:25,970 really give you good numbers on what the accuracy is, but depending on the 608 00:58:25,970 --> 00:58:30,619 situation, you know, it can get pretty well for Mars I guess that's pretty 609 00:58:30,619 --> 00:58:35,440 far, I guess that will be a bit less. Herald: A very short question for 610 00:58:35,440 --> 00:58:38,780 microphone number one, please. Mic1: Thank you. Thank you for the talk. 611 00:58:38,780 --> 00:58:44,540 I have a small question. As you said, you roughly plan the trip using the three 612 00:58:44,540 --> 00:58:50,540 body and two body problems. And are there any stable points like Lagrangian points 613 00:58:50,540 --> 00:58:54,060 in there, for example, four body problem? And can you use them to... during the 614 00:58:54,060 --> 00:58:59,530 roughly planning stage of... Sven: Oh, yeah. I actually wondered 615 00:58:59,530 --> 00:59:03,830 about this very recently as well. And I don't know the answer. I'm not sure. So 616 00:59:03,830 --> 00:59:07,180 the three body problem is already complicated enough from a mathematical 617 00:59:07,180 --> 00:59:12,270 point of view. So I have never actually really looked at a four body problem. 618 00:59:12,270 --> 00:59:18,100 However with those many bodies, there are at least very symmetrical solutions. 619 00:59:18,100 --> 00:59:22,210 So you can find some, but it's a different thing than Lagrangian points, right. 620 00:59:22,210 --> 00:59:26,440 Herald: So unfortunately we're almost out of time for this talk. If you have more 621 00:59:26,440 --> 00:59:29,910 questions, I'm sure Sven will be happy to take them afterwards to talk. So please 622 00:59:29,910 --> 00:59:33,186 approach him after. And again, a big round of applause for the topic. 623 00:59:33,186 --> 00:59:33,970 Sven: Thank you. 624 00:59:33,970 --> 00:59:39,658 *Applause* 625 00:59:39,658 --> 00:59:48,850 *36C3 postroll music* 626 00:59:48,850 --> 01:00:06,000 Subtitles created by c3subtitles.de in the year 2020. Join, and help us!