WEBVTT
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36C3 preroll music
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Herald: OK, so the next talk for this
evening is on how to get to Mars and all
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in very interesting ways. Some of them
might be really, really slow. Our next
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speaker has studied physics and has a PhD
in maths and is currently working as a
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mission planner at the German Space
Operations Center. Please give a big round
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of applause to Sven.
Sven: Thank you.
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Hello and welcome to
"Thrust is not an option: How to get a
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Mars really slow". My name is Sven. I'm a
mission planner at the German Space
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Operations Center, which is a part of the
DLR, the Deutsches Zentrum für Luft- und
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Raumfahrt. And first of all, I have to
apologize because I kind of cheated a
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little bit in the title. The accurate
title would have been "Reducing thrust: How
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to get to Mars or maybe Mercury really
slow". The reason for this is that I will
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actually use Mercury as an example quite
a few times. And also we will not be able
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to actually get rid of all the maneuvers
that we want to do. So the goal of this
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talk is to give you an introduction to
orbital mechanics to see what we can do.
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What are the techniques that you can use
to actually get to another planet, to
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bring a spacecraft to another planet and
also go a few more, go a bit further into
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some more advanced techniques. So we will
start with gravity and the two body
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problem. So this is the basics, the
underlying physics that we need. Then we
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will talk about the two main techniques
maybe to get to Mars, for example, the
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Hohmann-transfer as well as gravity
assists. The third point will be an
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extension of that that's called a planar
circular restricted three body problem.
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Sounds pretty complicated, but we will see
in pictures what it is about. And then we
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will finally get a taste of certain ways
to actually be even better, be even more
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efficient by looking at what's called
ballistic capture and the weak stability
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boundary. All right, so let's start. First
of all, we have gravity and we need to
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talk about a two body problem. So I'm
standing here on the stage and I'm
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actually being well accelerated downwards,
right? The earth actually attracts me. And
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this is the same thing that happens for
any two bodies that have mass. OK. So they
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attract each other by gravitational force
and this force will actually accelerate
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the objects towards each other. Notice
that the force actually depends on the
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distance. OK. So we don't need
to know any details. But in principle, the
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force gets stronger the closer the objects
are. OK, good. Now, we can't really
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analyze this whole thing in every
detail. So we will make a few assumptions.
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One of them will be that all our bodies,
in particular, the Sun, Earth will
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actually be points, OK? So we will just
consider points because anything else is
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too complicated for me. Also, all our
satellites will actually be just points.
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One of the reasons is that, in principle,
you have to deal with the attitude of the
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satellites. For example, a solar panel
needs to actually point towards the sun,
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but of course that's complicated. So we
will skip this for this talk. Third point
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is that none of our planets will have an
atmosphere, so there won't be any
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friction anywhere in the space. And the
fourth point is that we will mostly
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restrict to movement within the plane. So
we only have like two dimensions during
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this talk. And also, I will kind of forget
about certain planets and other masses
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from time to time. Okay. I'm mentioning
this because I do not want you to go home
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this evening, start planning your own
interplanetary mission, then maybe
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building your spacecraft tomorrow,
launching in three days and then a week
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later I get an e-mail: "Hey, this
didn't work. I mean, what did you tell me?"
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OK. So if you actually want to do this at
home, don't try this just now but please
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consult your local flight dynamics department,
they will actually supply with the necessary
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details. All right. So what's the two body
problem about? So in principle we have
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some body - the Sun - and the spacecraft
that is being attracted by the Sun. Now,
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the Sun is obviously much heavier than a
spacecraft, meaning that we will actually
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neglect the force that the spacecraft
exerts on the Sun. So instead, the Sun
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will be at some place. It
might move in some way, or a
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planet. But we only care about a
spacecraft, in general. Furthermore,
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notice that if you specify the position
and the velocity of a spacecraft at some
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point, then the gravitational force will
actually determine the whole path of the
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spacecraft for all time. OK. So this path
is called the orbit and this is what we
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are talking about. So we want to determine
orbits. We want to actually find ways how
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to efficiently change orbits in order to
actually reach Mars, for example. There is
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one other thing that you may know from
your day to day life. If you actually take
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an object and you put it high up and you
let it fall down, then it will accelerate.
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OK. So one way to actually describe this
is by looking at the energy. There is a
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kinetic energy that's related to movement,
to velocity, and there is a potential
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energy which is related to this
gravitational field. And the sum of those
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energies is actually conserved. This means
that when the spacecraft moves, for
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example, closer to the Sun, then its
potential energy will decrease and thus
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the kinetic energy will increase. So it
will actually get faster. So you can see
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this, for example, here. We have
two bodies that rotate around their
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center of mass. And if you're careful, if
you're looking careful when they actually
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approach each other, then they are quite a
bit faster. OK. So it is important to keep
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in mind. All right, so how do spacecrafts
actually move? So we will now actually
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assume that we don't use any kind of
engine, no thruster. We just cruise along
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the gravitational field. And then there
are essentially three types of orbits that
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we can have. One of them are hyperbolas.
So this case happens if the velocity is
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very high, because those are not periodic
solutions. They're not closed. So instead,
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our spacecraft kind of approaches the Sun
or the planet in the middle and the center
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from infinity. It will kind of turn,
it will change its direction and then it
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will leave again to infinity. Another
orbit that may happen as a parabola, this
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is kind of similar. Actually, we won't
encounter parabolas during this talk. So I
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will skip this. And the probably most
common orbit that we all know are
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ellipses. In particular circles because,
well, we know that the Earth is actually
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moving around the sun approximately in a
circle. OK. So those are periodic
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solutions. They are closed. And in
particular, they are such that if a
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spacecraft is on one of those orbits and
it's not doing anything, then it will
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forever stay on that orbit, OK, in the two
body problem. So now the problem is we
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actually want to change this. So we need
to do something. OK. So we want to change
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from one circle around the Sun, which
corresponds to Earth orbit, for example, to
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another circle around the Sun, which
corresponds to Mars orbit. And in order to
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change this, we need to do some kind of
maneuver. OK. So this is an actual picture
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of a spacecraft. And what the spacecraft
is doing, it's emitting some kind of
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particles in some direction. They have a
mass m. Those particles might be gases or
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ions, for example. And because these gases
or these emissions, they carry some mass,
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they actually have some momentum due to
conservation of momentum. This means that
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the spacecraft actually has to accelerate
in the opposite direction. OK. So whenever
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we do this, we will actually accelerate
the spacecraft and change the velocity and
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this change of velocity as denoted by a
delta v. And delta v is sort of the basic
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quantity that we actually want to look at
all the time. OK. Because this describes
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how much thrust we need to actually fly
in order to change our orbit. Now,
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unfortunately, it's pretty expensive to,
well, to apply a lot of delta v. This is
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due to the costly rocket equation. So the
fuel that you need in order to reach or to
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change your velocity to some delta v this
depends essentially exponentially on the
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target delta v. So this means we really
need to take care that we use as few
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delta v as possible in order to reduce the
needed fuel. There's one reason for
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that is... we want to maybe reduce
costs because then we need to carry
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less fuel. However, we can also actually
think the other way round if we actually
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use less fuel than we can
bring more stuff for payloads, for
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missions, for science experiments. Okay.
So that's why in spacecraft mission
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design we actually have to take care of
reducing the amount of delta v that is
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spent during maneuvers. So let's see, what
can we actually do? So one example of a
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very basic maneuver is actually to, well,
sort of raise the orbit. So imagine you
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have a spacecraft on a circular orbit
around, for example, Sun here. Then you
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might want to raise the orbit
in the sense that you make it more
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elliptic and reach higher altitudes. For
this you just accelerate in the direction
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that you're flying. So you apply some
delta v and this will actually change the
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form of the ellipse. OK. So it's a very
common scenario. Another one is if you
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approach a planet from very far away, then
you might have a very high relative
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velocity such that with respect to the
planet, you're on a hyperbolic orbit. OK.
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So you would actually leave the planet.
However, if this is actually your
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target planet that you want to reach, then
of course you have to enter orbit. You
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have to somehow slow down. So the idea
here is that when you approach
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the closest point to the planet,
for example, then you actually slow down.
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So you apply delta v in sort of in the
opposite direction and change the orbit to
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something that you prefer, for example an
ellipse. Because now you will actually
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stay close to the planet forever. Well, if
relative it would a two body problem. OK,
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so. Let's continue. Now, we actually want
to apply this knowledge to well, getting,
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for example, to Mars. Let's start with
Hohmann transfers. Mars and Earth both
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revolve around the Sun in pretty much
circular orbits. And our spacecraft starts
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at the Earth. So now we want to reach
Mars. How do we do this? Well, we can fly
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what we just said. So we accelerate
when we are at the Earth orbit,
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such that our orbit touches the Mars orbit
on the other side. OK. So this gives us
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some amount of delta v we have to apply.
We need to calculate this. I'm not going
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to do this. Then we actually fly around
this orbit for half an ellipse. And once
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we have reached the Mars orbit, then we
can actually accelerate again in order to
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raise other side of the Ellipse until that
one reaches the Mars orbit. So with two
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maneuvers, two accelerations, we can
actually change from one circular orbit to
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another one. OK. This is the basic idea of
how you actually fly to Mars. So let's
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look at an animation. So this is the orbit
of the InSight mission. That's another Mars
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mission which launched and landed last
year. The blue circle is the Earth and the
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green one is Mars. And the pink is
actually the satellite or the probe.
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You can see that, well, it's flying in
this sort of half ellipse. However, there
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are two... well, there's just one problem,
namely when it actually reaches Mars, Mars
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needs to be there. I mean, that sounds
trivial. Yeah. But I mean, imagine you fly
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there and then well, Mars is somewhere
else, that's not good. I mean this happens
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pretty regularly when you begin playing a
Kerbal Space Program, for example.
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So we don't want to like play around
with this the whole time, we actually want
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to hit Mars. So we need to take care of
that Mars is at the right position when we
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actually launch. Because it will traverse
the whole green line during our transfer.
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This means that we can only launch such a
Hohmann transfer at very particular times.
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And sort of this time when you can do
this transfer is called the transfer
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window. And for Earth-Mars, for example.
This is possible every 26 months. So if
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you miss something, like, software's not
ready, whatever, then you have to wait for
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another twenty six months. So, the flight
itself takes about six months. All right.
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There is another thing that we kind of
neglected so far, namely when we start,
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when we depart from Earth, then well
there's Earth mainly. And so that's the
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main source of gravitational force. For
example, right now I'm standing here on
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the stage and I experience the Earth. I
also experience Sun and Mars. But I mean,
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that's very weak. I can ignore this. So at
the beginning of our mission to Mars, we
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actually have to take care that we
are close to Earth. Then during the
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flight, the Sun actually dominates the
gravitational force. So we will only
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consider this. But then when we approach
Mars, we actually have to take care about
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Mars. Okay. So we kind of forgot this
during the Hohmann transfer. So what you
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actually do is you patch together
solutions of these transfers. Yeah. So in
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this case, there are there are essentially
three sources of gravitational force so
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Earth, Sun, Mars. So we will have three two
body problems that we need to consider.
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Yeah. One for departing, one for the
actual Hohmann transfer. And then the third
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one when we actually approach Mars. So
this makes this whole thing a bit more
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complicated. But it's also nice because
actually we need less delta v than we
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would for the basic hohmann transfer. One
reason for this is that when we look at
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Mars. So the green line is now the Mars
orbit and the red one is again the
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spacecraft, it approaches Mars now we can
actually look at what happens at Mars by
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kind of zooming into the system of Mars.
OK. So Mars is now standing still. And
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then we see that the velocity of the
spacecraft is actually very high relative
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to Mars. So it will be on the hyperbolic
orbit and will actually leave Mars again.
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You can see this on the left side. Right.
Because it's leaving Mars again. So what
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you need to do is, in fact, you need to
slow down and change your orbit into an
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ellipse. Okay. And this delta v, is that
you that you need here for this maneuver
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it's actually less than the delta v you
would need to to circularize the orbit to
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just fly in the same orbit as Mars. So we
need to slow down. A similar argument
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actually at Earth shows that, well, if you
actually launch into space, then you do
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need quite some speed already to not fall
down back onto Earth. So that's something
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like seven kilometers per second or so.
This means that you already have some
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speed. OK. And if you align your orbit or
your launch correctly, then you already
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have some of the delta v that you need for
the Hohmann transfer. So in principle, you
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need quite a bit less delta v than than
you might naively think. All right. So
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that much about Hohmann transfer. Let's look
at Gravity assist. That's another major
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technique for interplanetary missions. The
idea is that we can actually use planets
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to sort of getting pulled along. So this
is an animation, on the lower animation
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you see kind of the picture when you look
at the planet. So the planets standing
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still and we assume that the spacecraft's
sort of blue object is on a hyperbolic
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orbit and it's kind of making a 90 degree
turn. OK. And the upper image, you
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actually see the picture when
you look from the Sun, so the planet is
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actually moving. And if you look very
carefully at the blue object then you can
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see that it is faster. So once it has
passed, the planet is actually faster.
00:18:45.030 --> 00:18:52.900
Well, we can actually look at this problem.
So this is, again, the picture. When
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Mars is centered, we have some entry
velocity. Then we are in this hyperbolic
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orbit. We have an exit velocity. Notice
that the lengths are actually equal. So
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it's the same speed. But just a turn
direction of this example. But then we can
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look at the whole problem with a moving
Mars. OK, so now Mars has some velocity
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v_mars. So the actual velocity that we see
is the sum of the entry and the Mars
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velocity before and afterwards exit, plus
Mars velocity. And if you look at those
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two arrows, then you see immediately that,
well, the lengths are different. Okay. So
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this is just the whole phenomenon here. So
we see that by actually passing close to
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such a planet or massive body, we
can sort of gain free delta v. Okay, so of
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course, it's not. I mean, the energy is
still conserved. Okay. But yeah, let's not
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worry about these details here. Now, the
nice thing is we can use this technique to
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actually alter course. We can speed up. So
this is the example that I'm shown here.
00:19:58.970 --> 00:20:02.790
But also, we can use this to slow down.
Okay. So that's a pretty common
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application as well. We can use this to
slow down by just changing the arrows,
00:20:08.160 --> 00:20:15.860
essentially. So just approaching Mars from
a different direction, essentially. So
00:20:15.860 --> 00:20:21.960
let's look at the example. And this is
Bepicolombo. That's actually the reason
00:20:21.960 --> 00:20:26.240
why I kind of changed the title, because
Bepicolombo is actually a mission to
00:20:26.240 --> 00:20:32.661
Mercury. So it was launched last year.
It's a combined ESA/JAXA mission and it
00:20:32.661 --> 00:20:38.390
consists of two probes and one thruster
centrally. So it's a through three stages
00:20:38.390 --> 00:20:43.780
that you can see in the picture. Yeah.
That's a pretty awesome mission, actually.
00:20:43.780 --> 00:20:49.930
It's really nice. But it has in
particular, a very cool orbit. So that's
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
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
00:21:00.180 --> 00:21:07.130
intermediate turquoise one - that's
Venus. And well the curve is
00:21:07.130 --> 00:21:10.790
Bepicolombo's orbit, so it looks pretty
complicated. Yeah, it's definitely not the
00:21:10.790 --> 00:21:16.020
Hohmann transfer. And in fact, this
mission uses nine Gravity assists to reach
00:21:16.020 --> 00:21:21.950
Mercury. And if you look at the
path so, for example, right now
00:21:21.950 --> 00:21:28.690
it actually is very close to Mercury
because the last five or six Gravity
00:21:28.690 --> 00:21:34.500
assists are just around Mercury or just
slow down. OK. And this saves a lot of
00:21:34.500 --> 00:21:41.760
delta v compared to the standard
Hohmann transfer. All right. But we
00:21:41.760 --> 00:21:45.810
want to do even better. OK. So let's now
actually make the whole problem more
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
00:21:53.830 --> 00:21:58.550
we will talk about a planar circular
restricted three body problem. All right.
00:21:58.550 --> 00:22:02.590
So in general, the three body problem just
means, hey, well, instead of two bodies,
00:22:02.590 --> 00:22:07.400
we have three. OK. They pairwise attract
each other and then we can solve this
00:22:07.400 --> 00:22:12.080
whole equation of motion. We can ask a
computer. And this is one animation of
00:22:12.080 --> 00:22:17.490
what it could look like. So the three
masses and their orbits are traced and we
00:22:17.490 --> 00:22:24.080
see immediately that we don't see anything
that's super complicated. There is no
00:22:24.080 --> 00:22:29.670
way we can really... I don't know,
formulate a whole solution theory for a
00:22:29.670 --> 00:22:33.650
general three body problem. That's
complicated. Those are definitely not
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
00:22:40.312 --> 00:22:44.520
we will now talk about a plane or circular
restricted three body problem. There are
00:22:44.520 --> 00:22:49.440
three words. So the first one is
restricted. Restricted means that in our
00:22:49.440 --> 00:22:54.350
application case, one of the bodies is
actually a spacecraft. Spacecrafts are
00:22:54.350 --> 00:22:58.440
much lighter than, for example, Sun and
Mars, meaning that we can actually ignore
00:22:58.440 --> 00:23:05.570
the force that the spacecraft exerts on
Sun and Mars. Okay. So we will actually
00:23:05.570 --> 00:23:11.740
consider Sun and Mars to be independent of
the spacecraft. OK. So in total, we only
00:23:11.740 --> 00:23:18.120
have like two gravitational forces now
acting on a spacecraft. So we neglect sort
00:23:18.120 --> 00:23:25.610
of this other force. Also, we will assume
that the whole problem is what's called
00:23:25.610 --> 00:23:30.800
circular. So we assume that Sun and Mars
actually rotate in circles around their
00:23:30.800 --> 00:23:37.081
center of mass. This assumption is
actually pretty okay. We will see a
00:23:37.081 --> 00:23:42.960
picture right now. So in this graph, for
example, in this image, you can see that
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
00:23:48.680 --> 00:23:54.520
later on, Sun and Mars actually have moved
to the red positions and the spacecraft is
00:23:54.520 --> 00:24:00.840
at some other place. And now, of course,
feels some other forces. OK. And also we
00:24:00.840 --> 00:24:04.330
will assume that this problem is plane,
meaning again that everything takes place
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
00:24:12.380 --> 00:24:19.610
frame rate, something like two images per
day. Maybe it's actually Pluto and Charon.
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
00:24:27.250 --> 00:24:34.360
2015 and it shows that they actually
rotate around the center of mass. Yeah. So
00:24:34.360 --> 00:24:40.270
both actually rotate. This also happens,
for example, for Sun and Earth or Sun and
00:24:40.270 --> 00:24:45.250
Mars or sun and Jupiter or also Earth and
Moon. However, in those other cases, the
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
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
00:24:57.910 --> 00:25:04.410
don't really see this extensive rotation.
OK. Now, this problem is still difficult.
00:25:04.410 --> 00:25:10.140
OK. So if you're putting out a mass in
there, then you don't really
00:25:10.140 --> 00:25:15.499
know what happens. However, there's a nice
trick to simplify this problem. And
00:25:15.499 --> 00:25:19.730
unfortunately, I can't do this here. But
maybe all the viewers at home, they can
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
00:25:25.080 --> 00:25:34.020
your laptop at the same speed as this
image actually rotates. OK. Well, then
00:25:34.020 --> 00:25:39.340
what happens? The two masses will actually
stand still from your point of view. OK.
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
00:25:45.080 --> 00:25:50.590
rotating point of view. OK, then the two
masses stand still. We still have the two
00:25:50.590 --> 00:25:56.020
gravitational forces towards Sun and Mars.
But because we kind of look at it from a
00:25:56.020 --> 00:26:00.670
rotated or from a moving point of view, we
get two new forces, those forces, you
00:26:00.670 --> 00:26:04.890
know, the centrifugal forces, of
course, the one that, for example, you
00:26:04.890 --> 00:26:11.510
have when you play with some
children or so, they want to be pulled in
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
00:26:17.440 --> 00:26:21.730
actually have this force acting on the
spacecraft. Okay. And also there is the
00:26:21.730 --> 00:26:26.790
Coriolis force, which is a bit less known.
This depends on the velocity of the
00:26:26.790 --> 00:26:31.660
spacecraft. OK. So if there is no velocity
in particular, then there will not be any
00:26:31.660 --> 00:26:38.270
Coriolis force. So our new problem
actually has four forces. OK, but the
00:26:38.270 --> 00:26:43.580
advantage is that Sun and Mars actually
are standing still. So we don't need to
00:26:43.580 --> 00:26:51.040
worry about their movement. OK. So now how
does this look like? Well, this might be
00:26:51.040 --> 00:26:55.990
an example for an orbit. Well, that looks
still pretty complicated. I mean, this is
00:26:55.990 --> 00:27:01.500
something that you can't have in a two
body problem. It has these weird wiggles.
00:27:01.500 --> 00:27:06.320
I mean, they're not really corners. And it
actually kind of switches from Sun to
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
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
00:27:15.650 --> 00:27:23.490
book "The Three-Body Problem". So there,
for example, the two masses might be a
00:27:23.490 --> 00:27:28.760
binary star system. OK. And then you have
a planet that's actually moving along such
00:27:28.760 --> 00:27:35.710
an orbit. OK, that looks pretty bad. So in
particular, the seasons might be somewhat
00:27:35.710 --> 00:27:41.960
messed up. Yeah. So this problem is, in
fact, in a strong mathematical sense,
00:27:41.960 --> 00:27:47.200
chaotic. OK. So chaotic means something
like if you start with very close initial
00:27:47.200 --> 00:27:51.610
conditions and you just let the system
evolve, then the solutions will look very,
00:27:51.610 --> 00:27:58.560
very different. OK. And this really
happens here, which is good. All right. So
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
00:28:03.950 --> 00:28:08.100
the system without any velocity, is it
possible that all the forces actually
00:28:08.100 --> 00:28:12.450
cancel out. And it turns out yes, it is
possible. And those points are called
00:28:12.450 --> 00:28:17.950
Lagrangian points. So if we have zero
velocity, there is no Coriolis force. So
00:28:17.950 --> 00:28:23.460
we have only these three forces. And as
you can see in this little schematics
00:28:23.460 --> 00:28:32.116
here, it's possible that all these forces
actually cancel out. Now imagine. Yeah. I
00:28:32.116 --> 00:28:36.940
give you a homework. Please calculate all
these possible points. Then you can do
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
00:28:42.280 --> 00:28:47.880
those are the five Lagrangian points in
this problem. OK, so we have L4 and L5
00:28:47.880 --> 00:28:52.150
which are at equilateral triangles with
Sun and Mars. Well, Sun - Mars in this
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
00:28:59.780 --> 00:29:05.250
the spacecraft precisely at L1 without any
velocity, then in relation to Sun and Mars
00:29:05.250 --> 00:29:10.150
it will actually stay at the same position.
Okay, that's pretty cool. However,
00:29:10.150 --> 00:29:15.770
mathematicians and physicists will
immediately ask well, OK, but what happens
00:29:15.770 --> 00:29:21.920
if I actually put a spacecraft close to a
Lagrangian point? OK, so this is
00:29:21.920 --> 00:29:28.200
related to what's called stability. And
you can calculate that around L4 and L5.
00:29:28.200 --> 00:29:33.330
The spacecraft will actually stay in the
vicinity. So it will essentially rotate
00:29:33.330 --> 00:29:38.980
around the Lagrangian points. So those are
called stable, whereas L1, L2 and L3 are
00:29:38.980 --> 00:29:43.990
actually unstable. This means that if you
put a spacecraft there, then it will
00:29:43.990 --> 00:29:50.600
eventually escape. However, this takes a
different amount of time depending on the
00:29:50.600 --> 00:29:55.330
Lagrangian points. For example, if you're
close to L2, this might take a few months,
00:29:55.330 --> 00:29:58.730
but if you're close to L3, this will
actually take something like a hundred
00:29:58.730 --> 00:30:08.140
years or so. Okay, so those points are
still different. All right. Okay. So
00:30:08.140 --> 00:30:10.950
is there actually any evidence that they
exist? I mean, maybe I'm just making this
00:30:10.950 --> 00:30:14.690
up and, you know, I mean, haven't shown
you any equations. I could just throw
00:30:14.690 --> 00:30:19.950
anything. However, we can actually look at
the solar system. So this is the inner
00:30:19.950 --> 00:30:23.570
solar system here. In the middle you see,
well, the center you see the Sun, of
00:30:23.570 --> 00:30:28.970
course. And to the lower left, there's
Jupiter. Now, if you imagine an
00:30:28.970 --> 00:30:35.250
equilateral triangle of Sun and Jupiter,
well, there are two of them. And then you
00:30:35.250 --> 00:30:40.920
see all these green dots there. And those
are asteroids. Those are the Trojans and
00:30:40.920 --> 00:30:47.770
the Greeks. And they accumulate there
because L4 and L5 are stable. Okay. So we
00:30:47.770 --> 00:30:55.140
can really see this dynamics gone on in
the solar system. However, there's also
00:30:55.140 --> 00:30:59.490
various other applications of Lagrangian
points. So in particular, you might want
00:30:59.490 --> 00:31:05.710
to put a space telescope somewhere in
space, of course, in such a way that the
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
00:31:11.520 --> 00:31:18.980
spacecraft behind Earth, then we might be
in the shadow. And this is the Lagrangian
00:31:18.980 --> 00:31:24.860
point L2, which is the reason why this is
actually being used for space telescopes
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
00:31:30.470 --> 00:31:35.091
really want to put the spacecraft just
there. But instead, we put it in an orbit
00:31:35.091 --> 00:31:40.730
close... in a close orbit, close to L2.
And this particular example is called the
00:31:40.730 --> 00:31:44.560
Halo orbit, and it's actually not
contained in the planes. I'm cheating a
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
00:31:48.030 --> 00:31:54.110
the the orbit from the side. So it
actually leaves the plane the blue dot is
00:31:54.110 --> 00:32:00.620
Earth and the left hand side you see
the actual orbit from the top. So
00:32:00.620 --> 00:32:06.230
it's rotating around this place. OK. So
that's the James Webb Space Telescope, by
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
00:32:11.360 --> 00:32:19.530
work out, unfortunately, but stay tuned.
Another example. That's how it has become
00:32:19.530 --> 00:32:26.200
pretty famous recently as the Chinese
Queqiao relay satellite. This one sits at
00:32:26.200 --> 00:32:31.090
the Earth - Moon L2 Lagrange point. And
the reason for this is that the Chinese
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.
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
00:32:41.560 --> 00:32:47.640
relay satellite behind the Moon, which
they could still see from Earth. And they
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
00:33:00.100 --> 00:33:07.510
technique: ballistic capture. Right. Okay.
So this whole business sort of started
00:33:07.510 --> 00:33:14.410
with a mission from the beginning of the
1990s, and that's the Hiten mission. So
00:33:14.410 --> 00:33:19.890
that was a Japanese well, Moon probe
consisted of a probe which had a small
00:33:19.890 --> 00:33:26.290
orbiter site which was separated, and then
it was supposed to actually enter orbit
00:33:26.290 --> 00:33:31.610
around Moon. Unfortunately, it missed its
maneuver. So it didn't apply enough delta v
00:33:31.610 --> 00:33:37.570
so it actually flew off. And the
mission was sort of lost at that point
00:33:37.570 --> 00:33:42.430
because Hiten itself, so the main probe
did not have enough fuel to reach the
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
00:33:47.701 --> 00:33:53.460
take. And they were probably pretty
devastated. However, there were two people
00:33:53.460 --> 00:34:00.780
from JPL, NASA, who actually heard about
this, Belbruno and Miller, and they were
00:34:00.780 --> 00:34:08.260
working on strange orbits, those ballistic
capture orbits. And they actually found
00:34:08.260 --> 00:34:14.609
one for the Hiten probe. They sent this to
the Japanese and they actually use that
00:34:14.609 --> 00:34:23.220
orbit to get the Hiten probe to the moon.
And it actually arrived in October 1991.
00:34:23.220 --> 00:34:26.450
And then it could do some
science, you know, maybe some
00:34:26.450 --> 00:34:31.389
different experiments, but it actually
arrived there. However, the transfer took
00:34:31.389 --> 00:34:37.070
quite a bit longer. So a normal Moon
transfer takes like three days or so. But
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
00:34:42.320 --> 00:34:48.600
looks pretty weird. So this is a
picture of the orbiter - schematic picture.
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
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
00:35:01.820 --> 00:35:07.430
pretty far out. And you can see that the
orbit sort of consists of two parts.
00:35:07.430 --> 00:35:13.100
First, it leaves Earth and it flies far
beyond the Moon. So somewhere completely
00:35:13.100 --> 00:35:18.910
different towards some other Lagrangian
point. That's really far away. Then it
00:35:18.910 --> 00:35:24.280
does some weird things. And in the upper
part of picture there it actually does a
00:35:24.280 --> 00:35:30.240
maneuver. So we apply some thrusts there
to be to change on a different orbit. And
00:35:30.240 --> 00:35:36.830
this orbit led the probe directly to the
moon where it was essentially captured for
00:35:36.830 --> 00:35:42.320
free. OK. So it just entered orbit around
the Moon. And this is, of course, not
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
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
00:35:56.530 --> 00:36:02.320
a bit more abstractly. The idea is...
we have Sun and Mars and we
00:36:02.320 --> 00:36:08.100
have a spacecraft that flies in this three
body problem. So the red orbit is actually
00:36:08.100 --> 00:36:14.960
the orbit of the spacecraft. Now, at any
point in time, we may decide to just
00:36:14.960 --> 00:36:20.970
forget about the Sun. OK. So instead we
consider the two body problem of Mars and
00:36:20.970 --> 00:36:26.760
a spacecraft. OK. Because at this point
in time, the spacecraft has a certain
00:36:26.760 --> 00:36:31.240
position relative to Mars and a certain
velocity. So this determines its orbit in
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
00:36:36.440 --> 00:36:43.269
orbit, which is denoted by the dashed line
here. OK. Or a dashed curve. So usually
00:36:43.269 --> 00:36:47.240
you happen to be in a hyperbolic orbit.
But of course, that orbit is only an
00:36:47.240 --> 00:36:50.280
approximation because in the three body
problem, the movement is actually
00:36:50.280 --> 00:36:57.490
different. But later on, it might happen
that we continue on the orbit. We can do
00:36:57.490 --> 00:37:01.530
the same kind of construction, but just
looking... but just ignoring the Sun
00:37:01.530 --> 00:37:09.710
essentially, and then we could find that
the spacecraft suddenly is in a elliptical
00:37:09.710 --> 00:37:14.190
orbit. This would mean that if you
forgot about the Sun, then the spacecraft
00:37:14.190 --> 00:37:19.670
would be stable and would be captured by
Mars. It would be there. That would be
00:37:19.670 --> 00:37:27.210
pretty nice. So this phenomenon, when this
happens, we call this a temporary capture.
00:37:27.210 --> 00:37:33.810
OK. Temporary because it might actually
leave that situation again later on. Now,
00:37:33.810 --> 00:37:37.320
because the actual movement depends on the
three body problem, which is super
00:37:37.320 --> 00:37:42.150
complicated. So it's possible that it
actually leaves again. But for that moment
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
00:37:46.300 --> 00:37:54.000
algorithm perhaps how we can find
this situation essentially. OK, and in a
00:37:54.000 --> 00:38:01.451
reasonable way, and the notion for this is
what's called, well, n-stability, the idea
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
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
00:38:13.250 --> 00:38:20.090
the Mars and we pick a perpendicular
speed, a perpendicular velocity to the
00:38:20.090 --> 00:38:25.810
line such that this corresponds to some
kind of elliptic orbit in the two body
00:38:25.810 --> 00:38:30.290
problem. Okay. So that's the dashed line.
But then we actually look at the problem
00:38:30.290 --> 00:38:37.820
in the three body problem and we just
evolve the spacecraft. And it's following
00:38:37.820 --> 00:38:46.080
the red orbit. It might follow the red
orbit. And it can happen that after going
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
00:38:54.200 --> 00:38:59.540
construction with forgetting the Sun again
and just look at the two body problem. And
00:38:59.540 --> 00:39:04.990
it's possible that this point actually
still corresponds to an elliptic orbit.
00:39:04.990 --> 00:39:10.870
That's somewhat interesting, right?
Because now this means that if we actually
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
00:39:17.120 --> 00:39:24.030
Mars once and are still sort of captured
in the corresponding two body problem.
00:39:24.030 --> 00:39:29.230
Okay. If we actually are able to wrap
around Mars twice, then we would call this
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
00:39:35.980 --> 00:39:39.170
because such an orbit corresponds to
something that's usable because we will
00:39:39.170 --> 00:39:45.370
wrap around Mars n times. However, it's
also possible that you have an unstable
00:39:45.370 --> 00:39:49.020
point, meaning that we again start in
something that corresponds to an ellipse
00:39:49.020 --> 00:39:54.170
around Mars. But if we actually follow the
orbit in a three body problem, it will,
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
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
00:40:03.470 --> 00:40:10.480
called unstable. And then there's another
thing we can do. That's actually a pretty
00:40:10.480 --> 00:40:18.280
common trick in finding orbits, etc. We
can instead of solving the problem in
00:40:18.280 --> 00:40:23.970
forward time we actually go back, okay. So
essentially in your program you just
00:40:23.970 --> 00:40:28.810
replace time by minus time, for example,
and then you just solve the thing and you
00:40:28.810 --> 00:40:37.070
go back in the past and it's possible
that a point that corresponds to such
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
00:40:41.641 --> 00:40:47.010
actually goes to the Sun, for example, we
call this unstable in the past. Okay. So
00:40:47.010 --> 00:40:56.680
that's just some random definition.
And we can use this. The reason for
00:40:56.680 --> 00:41:05.080
this is we can actually kind of take these
concepts together and build an orbit from
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
00:41:13.020 --> 00:41:19.710
might wrap around Mars six times, some
number that we like. This is the blue part
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
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
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
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
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
00:41:57.990 --> 00:42:03.650
transfer to get from Earth to that point
y. Okay? So our orbit actually consists of
00:42:03.650 --> 00:42:08.520
three parts now. Okay. So we have the
Hohmann transfer, but it's not actually
00:42:08.520 --> 00:42:14.390
aiming for Mars. It's actually aiming for
the point y. There we do a maneuver
00:42:14.390 --> 00:42:21.470
because we want to switch onto this red
orbit. Okay. And then this one will bring
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
00:42:28.780 --> 00:42:36.050
spacecraft will continue to rotate around
Mars for example six times. Okay. So in
00:42:36.050 --> 00:42:43.619
particular at x there is no maneuver
taking place. Okay. So that's a
00:42:43.619 --> 00:42:49.360
possible mission scenario. And the way
this is done then usually is you kind of..
00:42:49.360 --> 00:42:54.280
you calculate these points x that
are suitable for doing this. Okay. So they
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.
00:42:58.460 --> 00:43:02.500
And there's a lot of numerical
computations involved in that. But once we
00:43:02.500 --> 00:43:07.480
have this, you can actually build these
orbits. OK. So let's look at an actual
00:43:07.480 --> 00:43:16.021
example. So this is for Earth - Mars. On
the left, you see, well, that the two
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
00:43:23.380 --> 00:43:28.370
point of view centered around Mars. Okay.
And the colors correspond to each other.
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
00:43:32.500 --> 00:43:35.770
black line starting at Earth and then
hitting the point, which is called x_c
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
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
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
00:43:56.730 --> 00:44:01.310
will actually start rotating round Mars.
And the point x is actually at the top of
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
00:44:08.510 --> 00:44:13.550
pretty strangely. And also the red
orbit is when we kind of the capture orbit
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
00:44:19.180 --> 00:44:26.710
we actually rotate around Mars six
times. Okay. Now, during those six
00:44:26.710 --> 00:44:32.170
rotations around Mars, we could do
experiments. So maybe that is enough for
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
00:44:37.000 --> 00:44:44.797
actually execute another maneuver. OK. So
to actually stay around Mars. And I mean,
00:44:44.797 --> 00:44:48.060
the principle looks nice but of course,
you have to do some calculations. We have
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
00:44:55.640 --> 00:45:02.410
are few parameters that you can choose,
in particular the target point x, this has
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
00:45:07.310 --> 00:45:14.090
procedure here, for example, is only very
good if this altitude, this distance r is
00:45:14.090 --> 00:45:17.950
actually high enough. If it is high enough
then you can save - in principle - up to
00:45:17.950 --> 00:45:23.720
twenty three percent of the delta v, which
is enormous. OK. So that would
00:45:23.720 --> 00:45:29.050
be really good. However, in reality it's
not as good usually. Yeah. And for a
00:45:29.050 --> 00:45:34.530
certain lower distances, for example, you
cannot save anything, so there are
00:45:34.530 --> 00:45:40.810
certain tradeoffs to make. However, there
is another advantage. Remember this point y?
00:45:40.810 --> 00:45:45.990
We chose this along this capture orbit
along the red orbit. And the thing is, we
00:45:45.990 --> 00:45:51.380
can actually choose this freely. This
means that our Hohmann transfer doesn't
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
00:45:55.000 --> 00:46:02.970
particular point. It can actually aim for
any point on that capture orbit. This
00:46:02.970 --> 00:46:06.310
means that we have many more Hohmann
transfers available that we can actually
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
00:46:12.500 --> 00:46:17.730
cannot just start every 26 months. But now
we, with this technique, we could actually
00:46:17.730 --> 00:46:24.460
launch. Well, quite often. However,
there's a little problem. If you looked at
00:46:24.460 --> 00:46:33.950
the graph carefully, then you may have
seen that the red orbit actually took like
00:46:33.950 --> 00:46:39.350
three quarters of the rotation of Mars.
This corresponds to roughly something like
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
00:46:43.750 --> 00:46:49.060
with humans on board because they have to
actually wait for a long time. But in
00:46:49.060 --> 00:46:52.890
principle, there are ways to make this
shorter. So you can try to actually
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
00:46:58.450 --> 00:47:04.610
example for this. Again, that's
Bepicolombo. The green dot is now Mercury.
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
00:47:09.870 --> 00:47:20.640
orbit. And yeah, it looks strange. So the
first few movements around Mercury,
00:47:20.640 --> 00:47:28.300
they are actually the last gravity assists
for slowing down. And then it actually
00:47:28.300 --> 00:47:36.780
starts on the capture orbit. So now it
actually approaches Mercury. So this is
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.
00:47:41.010 --> 00:47:45.609
And once the animation actually ends,
this is when it actually reaches the point
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
00:47:52.240 --> 00:47:58.090
kilometers. So it is pretty high up above
Mercury, but it's enough for the mission.
00:47:58.090 --> 00:48:03.270
OK. And of course, then they do some
other maneuver to actually stay around
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
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
00:48:18.997 --> 00:48:23.600
can try to actually make this more general
to improve on this, this concept is then
00:48:23.600 --> 00:48:29.040
called the interplanetary transport
network. And it looks a bit similar to
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
00:48:36.290 --> 00:48:42.950
larger well, a set of orbits that have
these kinds of properties that wrap around
00:48:42.950 --> 00:48:48.520
Mars and then kind of leave Mars. And
they are very closely related to the
00:48:48.520 --> 00:48:53.280
dynamics of particular Lagrangian points,
in this case L1. So that was the one
00:48:53.280 --> 00:49:00.330
between the two masses. And if you
investigate this Lagrangian point a bit
00:49:00.330 --> 00:49:05.530
closer, you can see, well, you can see
different orbits of all kinds of
00:49:05.530 --> 00:49:10.650
behaviors. And if you understand this,
then you can actually try to do the same
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
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
00:49:21.440 --> 00:49:24.920
similar orbits that are wrapping around
the Sun and then going towards this
00:49:24.920 --> 00:49:31.859
Lagrangian point in a similar way. Well,
then we already have some orbits that are
00:49:31.859 --> 00:49:39.270
well, kind of meeting at L1. So we might
be able to actually connect them somehow,
00:49:39.270 --> 00:49:45.070
for example by maneuver. And then we only
need to reach the orbit around Earth or
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
00:49:50.130 --> 00:49:55.270
Hohmann transfer. And this way you reduce
your delta v even further. The problem is
00:49:55.270 --> 00:50:00.690
that this is hard to find because these
orbits they are pretty rare. And of
00:50:00.690 --> 00:50:07.320
course, you have to connect those orbits.
So they you approach the Lagrangian point
00:50:07.320 --> 00:50:14.330
from L1 from two sides, but you don't
really want to wait forever until they...
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
00:50:19.630 --> 00:50:24.490
not wait that long. So here's a picture
of how this might look like. Again
00:50:24.490 --> 00:50:28.960
very schematic. So we have Sun, we
have Mars and in between there is the
00:50:28.960 --> 00:50:35.240
Lagrangian point L1. The red orbit is sort
of an extension of one of those capture
00:50:35.240 --> 00:50:38.230
orbits that we have seen. OK, so that
wraps around Mars a certain number of
00:50:38.230 --> 00:50:45.350
times. And while in the past, for example,
it actually goes to Lagrangian point. I
00:50:45.350 --> 00:50:50.780
didn't explain this, but in fact, there
are many more orbits around L1, closed
00:50:50.780 --> 00:50:55.460
orbits, but they're all unstable. And
these orbits that are used in this
00:50:55.460 --> 00:51:05.030
interplanetary transport network they
actually approach those orbits around L1
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,
00:51:10.570 --> 00:51:15.590
we take these orbits, we connect
them. And when we are in the black orbit
00:51:15.590 --> 00:51:19.070
around L1, we actually apply some
maneuver, we apply some delta v to
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
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
00:51:28.500 --> 00:51:35.119
to this particular blue orbit around the
Sun. OK. So that's the general procedure.
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
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
00:51:44.840 --> 00:51:50.900
details. Please don't launch your
own mission tomorrow. OK. So with
00:51:50.900 --> 00:51:54.960
this, I want to thank you.
And I'm open to questions.
00:51:54.960 --> 00:52:05.640
Applause
00:52:05.640 --> 00:52:08.210
Herald: So thank you Sven for an
interesting talk. We have a few minutes
00:52:08.210 --> 00:52:11.160
for questions, if you have any questions
lined up next to the microphones, we'll
00:52:11.160 --> 00:52:18.400
start with microphone number one.
Mic1: Hello. So what are the problems
00:52:18.400 --> 00:52:22.680
associated? So you showed in the end is
going around to Lagrange Point L1?
00:52:22.680 --> 00:52:26.710
Although this is also possible for
00:52:26.710 --> 00:52:30.140
other Lagrange points. Could you do this
with L2?
00:52:30.140 --> 00:52:38.180
Sven: Yes, you can. Yeah. So in principle,
I didn't show the whole picture, but
00:52:38.180 --> 00:52:43.107
these kind of orbits, they exist at L1,
but they also exist at L2. And in
00:52:43.107 --> 00:52:49.080
principle you can this way sort of leave
this two body problem by finding similar
00:52:49.080 --> 00:52:53.650
orbits. But of course the the details are
different. So you cannot really take your
00:52:53.650 --> 00:52:58.640
knowledge or your calculations from L1
and just taking over to L2, you actually
00:52:58.640 --> 00:53:03.189
have to do the same thing again. You have
to calculate everything in detail.
00:53:03.189 --> 00:53:06.650
Herald: To a question from the Internet.
Signal Angel: Is it possible to use these
00:53:06.650 --> 00:53:11.350
kinds of transfers in Kerbal Space
Program?
00:53:11.350 --> 00:53:23.500
Sven: So Hohmann transfers, of course,
the gravity assists as well, but not the
00:53:23.500 --> 00:53:28.900
restricted three body problem because the
way Kerbal Space Program at least the
00:53:28.900 --> 00:53:33.450
default installation so without any mods
works is that it actually switches the
00:53:33.450 --> 00:53:40.270
gravitational force. So the thing that I
described as a patch solution where we
00:53:40.270 --> 00:53:46.220
kind of switch our picture, which
gravitational force we consider for our
00:53:46.220 --> 00:53:50.619
two body problem. This is actually the way
the physics is implemented in Kerbal space
00:53:50.619 --> 00:53:55.400
program. So we can't really do the
interplanetary transport network there.
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
00:54:00.090 --> 00:54:04.350
for this, I don't know.
Herald: If you're leaving please do so
00:54:04.350 --> 00:54:07.440
quietly. Small question and question from
microphone number four.
00:54:07.440 --> 00:54:12.619
Mic4: Hello. I have actually two
questions. I hope that's okay. First
00:54:12.619 --> 00:54:18.289
question is, I wonder how you do that in
like your practical calculations. Like you
00:54:18.289 --> 00:54:22.950
said, there's a two body problem and
there are solutions that you can
00:54:22.950 --> 00:54:27.440
calculate with a two body problem. And
then there's a three body problem. And I
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
00:54:32.050 --> 00:54:37.890
when you do that? And the second
question is: you said that reducing delta v
00:54:37.890 --> 00:54:47.770
about 15% is enormous. And I wonder what
effect does this have on the payload?
00:54:47.770 --> 00:54:57.550
Sven: Okay. So regarding the first
question. So in principle, I mean, you
00:54:57.550 --> 00:55:05.210
make a plan for a mission. So you have to
you calculate those things in these
00:55:05.210 --> 00:55:08.910
simplified models. Okay. You kind of you
patch together an idea of what you want to
00:55:08.910 --> 00:55:14.910
do. But of course, in the end, you're
right, there are actually many massive
00:55:14.910 --> 00:55:19.250
bodies in the solar system. And because we
want to be precise, we actually have to
00:55:19.250 --> 00:55:25.280
incorporate all of them. So in the end,
you have to do an actual numerical search
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
00:55:32.010 --> 00:55:37.800
and you have to incorporate other effects.
For example, the solar radiation might
00:55:37.800 --> 00:55:43.230
actually have a little influence on your
orbit. Okay. And there are many effects of
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
00:55:48.040 --> 00:55:53.260
your very good physics simulator for the
n-body problem, which actually has all
00:55:53.260 --> 00:55:57.050
these other effects as well. And then you
need to do a numerical search over this.
00:55:57.050 --> 00:56:01.410
Kind of, you know, where to start with
these ideas, where to look for solutions.
00:56:01.410 --> 00:56:06.680
But then you actually have to just try it
and figure out some algorithm to actually
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.
00:56:12.010 --> 00:56:16.500
Right. And the second question, can you
remind me again? Sorry.
00:56:16.500 --> 00:56:23.550
Mic4: Well, the second question was in
reducing delta v about 15%. What is the
00:56:23.550 --> 00:56:28.890
effect on the payload?
Sven: Right. So, I mean, if you need
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
00:56:35.750 --> 00:56:40.450
payload. Right. So you could put maybe
another instrument on there. Another thing
00:56:40.450 --> 00:56:46.119
you could do is actually keep the fuel but
actually use it for station keeping. So,
00:56:46.119 --> 00:56:52.550
for example, in the James Webb telescope
example, the James Webb telescope flies
00:56:52.550 --> 00:56:58.840
around this Halo orbit around L2, but the
orbit itself is unstable. So the James
00:56:58.840 --> 00:57:03.980
Webb Space Telescope will actually escape
from that orbit. So they have to do a few
00:57:03.980 --> 00:57:08.140
maneuvers every year to actually stay
there. And they have only a finite amount
00:57:08.140 --> 00:57:13.431
of fuels at some point. This won't be
possible anymore. So reducing delta v
00:57:13.431 --> 00:57:20.609
requirements might actually have increased
the mission lifetime by quite a bit.
00:57:20.609 --> 00:57:25.160
Herald: Number three.
Mic3: Hey. When you do such a
00:57:25.160 --> 00:57:29.869
mission, I guess you have to adjust the
trajectory of your satellite quite often
00:57:29.869 --> 00:57:34.190
because nothing goes exactly as you
calculated it. Right. And the question is,
00:57:34.190 --> 00:57:38.930
how precise can you measure the orbit?
Sorry, the position and the speed of a
00:57:38.930 --> 00:57:43.590
spacecraft at, let's say, Mars. What's the
resolution?
00:57:43.590 --> 00:57:48.300
Sven: Right. So from Mars, I'm not
completely sure how precise it is. But for
00:57:48.300 --> 00:57:52.300
example, if you have an Earth observation
mission, so something that's flying around
00:57:52.300 --> 00:57:58.500
Earth, then you can have a rather precise
orbit that's good enough for taking
00:57:58.500 --> 00:58:04.220
pictures on Earth, for example, for
something like two weeks or so. So
00:58:04.220 --> 00:58:12.080
you can measure the orbit well enough and
calculate the future something like two
00:58:12.080 --> 00:58:21.140
weeks in the future. OK. So that's good
enough. However. Yeah. The... I can't
00:58:21.140 --> 00:58:25.970
really give you good numbers on what the
accuracy is, but depending on the
00:58:25.970 --> 00:58:30.619
situation, you know, it can get pretty
well for Mars I guess that's pretty
00:58:30.619 --> 00:58:35.440
far, I guess that will be a bit less.
Herald: A very short question for
00:58:35.440 --> 00:58:38.780
microphone number one, please.
Mic1: Thank you. Thank you for the talk.
00:58:38.780 --> 00:58:44.540
I have a small question. As you said, you
roughly plan the trip using the three
00:58:44.540 --> 00:58:50.540
body and two body problems. And are there
any stable points like Lagrangian points
00:58:50.540 --> 00:58:54.060
in there, for example, four body problem?
And can you use them to... during the
00:58:54.060 --> 00:58:59.530
roughly planning stage of...
Sven: Oh, yeah. I actually wondered
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
00:59:03.830 --> 00:59:07.180
the three body problem is already
complicated enough from a mathematical
00:59:07.180 --> 00:59:12.270
point of view. So I have never actually
really looked at a four body problem.
00:59:12.270 --> 00:59:18.100
However with those many bodies, there
are at least very symmetrical solutions.
00:59:18.100 --> 00:59:22.210
So you can find some, but it's a different
thing than Lagrangian points, right.
00:59:22.210 --> 00:59:26.440
Herald: So unfortunately we're almost out
of time for this talk. If you have more
00:59:26.440 --> 00:59:29.910
questions, I'm sure Sven will be happy to
take them afterwards to talk. So please
00:59:29.910 --> 00:59:33.186
approach him after. And again, a big
round of applause for the topic.
00:59:33.186 --> 00:59:33.970
Sven: Thank you.
00:59:33.970 --> 00:59:39.658
Applause
00:59:39.658 --> 00:59:48.850
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