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!