36C3 preroll music
Herald: OK, so the next talk for this evening is on how to get to Mars and all
in very interesting ways. Some of them might be really, really slow. Our next
speaker has studied physics and has a PhD in maths and is currently working as a
mission planner at the German Space Operations Center. Please give a big round
of applause to Sven. Sven: Thank you.
Hello and welcome to "Thrust is not an option: How to get a
Mars really slow". My name is Sven. I'm a mission planner at the German Space
Operations Center, which is a part of the DLR, the Deutsches Zentrum für Luft- und
Raumfahrt. And first of all, I have to apologize because I kind of cheated a
little bit in the title. The accurate title would have been "Reducing thrust: How
to get to Mars or maybe Mercury really slow". The reason for this is that I will
actually use Mercury as an example quite a few times. And also we will not be able
to actually get rid of all the maneuvers that we want to do. So the goal of this
talk is to give you an introduction to orbital mechanics to see what we can do.
What are the techniques that you can use to actually get to another planet, to
bring a spacecraft to another planet and also go a few more, go a bit further into
some more advanced techniques. So we will start with gravity and the two body
problem. So this is the basics, the underlying physics that we need. Then we
will talk about the two main techniques maybe to get to Mars, for example, the
Hohmann-transfer as well as gravity assists. The third point will be an
extension of that that's called a planar circular restricted three body problem.
Sounds pretty complicated, but we will see in pictures what it is about. And then we
will finally get a taste of certain ways to actually be even better, be even more
efficient by looking at what's called ballistic capture and the weak stability
boundary. All right, so let's start. First of all, we have gravity and we need to
talk about a two body problem. So I'm standing here on the stage and I'm
actually being well accelerated downwards, right? The earth actually attracts me. And
this is the same thing that happens for any two bodies that have mass. OK. So they
attract each other by gravitational force and this force will actually accelerate
the objects towards each other. Notice that the force actually depends on the
distance. OK. So we don't need to know any details. But in principle, the
force gets stronger the closer the objects are. OK, good. Now, we can't really
analyze this whole thing in every detail. So we will make a few assumptions.
One of them will be that all our bodies, in particular, the Sun, Earth will
actually be points, OK? So we will just consider points because anything else is
too complicated for me. Also, all our satellites will actually be just points.
One of the reasons is that, in principle, you have to deal with the attitude of the
satellites. For example, a solar panel needs to actually point towards the sun,
but of course that's complicated. So we will skip this for this talk. Third point
is that none of our planets will have an atmosphere, so there won't be any
friction anywhere in the space. And the fourth point is that we will mostly
restrict to movement within the plane. So we only have like two dimensions during
this talk. And also, I will kind of forget about certain planets and other masses
from time to time. Okay. I'm mentioning this because I do not want you to go home
this evening, start planning your own interplanetary mission, then maybe
building your spacecraft tomorrow, launching in three days and then a week
later I get an e-mail: "Hey, this didn't work. I mean, what did you tell me?"
OK. So if you actually want to do this at home, don't try this just now but please
consult your local flight dynamics department, they will actually supply with the necessary
details. All right. So what's the two body problem about? So in principle we have
some body - the Sun - and the spacecraft that is being attracted by the Sun. Now,
the Sun is obviously much heavier than a spacecraft, meaning that we will actually
neglect the force that the spacecraft exerts on the Sun. So instead, the Sun
will be at some place. It might move in some way, or a
planet. But we only care about a spacecraft, in general. Furthermore,
notice that if you specify the position and the velocity of a spacecraft at some
point, then the gravitational force will actually determine the whole path of the
spacecraft for all time. OK. So this path is called the orbit and this is what we
are talking about. So we want to determine orbits. We want to actually find ways how
to efficiently change orbits in order to actually reach Mars, for example. There is
one other thing that you may know from your day to day life. If you actually take
an object and you put it high up and you let it fall down, then it will accelerate.
OK. So one way to actually describe this is by looking at the energy. There is a
kinetic energy that's related to movement, to velocity, and there is a potential
energy which is related to this gravitational field. And the sum of those
energies is actually conserved. This means that when the spacecraft moves, for
example, closer to the Sun, then its potential energy will decrease and thus
the kinetic energy will increase. So it will actually get faster. So you can see
this, for example, here. We have two bodies that rotate around their
center of mass. And if you're careful, if you're looking careful when they actually
approach each other, then they are quite a bit faster. OK. So it is important to keep
in mind. All right, so how do spacecrafts actually move? So we will now actually
assume that we don't use any kind of engine, no thruster. We just cruise along
the gravitational field. And then there are essentially three types of orbits that
we can have. One of them are hyperbolas. So this case happens if the velocity is
very high, because those are not periodic solutions. They're not closed. So instead,
our spacecraft kind of approaches the Sun or the planet in the middle and the center
from infinity. It will kind of turn, it will change its direction and then it
will leave again to infinity. Another orbit that may happen as a parabola, this
is kind of similar. Actually, we won't encounter parabolas during this talk. So I
will skip this. And the probably most common orbit that we all know are
ellipses. In particular circles because, well, we know that the Earth is actually
moving around the sun approximately in a circle. OK. So those are periodic
solutions. They are closed. And in particular, they are such that if a
spacecraft is on one of those orbits and it's not doing anything, then it will
forever stay on that orbit, OK, in the two body problem. So now the problem is we
actually want to change this. So we need to do something. OK. So we want to change
from one circle around the Sun, which corresponds to Earth orbit, for example, to
another circle around the Sun, which corresponds to Mars orbit. And in order to
change this, we need to do some kind of maneuver. OK. So this is an actual picture
of a spacecraft. And what the spacecraft is doing, it's emitting some kind of
particles in some direction. They have a mass m. Those particles might be gases or
ions, for example. And because these gases or these emissions, they carry some mass,
they actually have some momentum due to conservation of momentum. This means that
the spacecraft actually has to accelerate in the opposite direction. OK. So whenever
we do this, we will actually accelerate the spacecraft and change the velocity and
this change of velocity as denoted by a delta v. And delta v is sort of the basic
quantity that we actually want to look at all the time. OK. Because this describes
how much thrust we need to actually fly in order to change our orbit. Now,
unfortunately, it's pretty expensive to, well, to apply a lot of delta v. This is
due to the costly rocket equation. So the fuel that you need in order to reach or to
change your velocity to some delta v this depends essentially exponentially on the
target delta v. So this means we really need to take care that we use as few
delta v as possible in order to reduce the needed fuel. There's one reason for
that is... we want to maybe reduce costs because then we need to carry
less fuel. However, we can also actually think the other way round if we actually
use less fuel than we can bring more stuff for payloads, for
missions, for science experiments. Okay. So that's why in spacecraft mission
design we actually have to take care of reducing the amount of delta v that is
spent during maneuvers. So let's see, what can we actually do? So one example of a
very basic maneuver is actually to, well, sort of raise the orbit. So imagine you
have a spacecraft on a circular orbit around, for example, Sun here. Then you
might want to raise the orbit in the sense that you make it more
elliptic and reach higher altitudes. For this you just accelerate in the direction
that you're flying. So you apply some delta v and this will actually change the
form of the ellipse. OK. So it's a very common scenario. Another one is if you
approach a planet from very far away, then you might have a very high relative
velocity such that with respect to the planet, you're on a hyperbolic orbit. OK.
So you would actually leave the planet. However, if this is actually your
target planet that you want to reach, then of course you have to enter orbit. You
have to somehow slow down. So the idea here is that when you approach
the closest point to the planet, for example, then you actually slow down.
So you apply delta v in sort of in the opposite direction and change the orbit to
something that you prefer, for example an ellipse. Because now you will actually
stay close to the planet forever. Well, if relative it would a two body problem. OK,
so. Let's continue. Now, we actually want to apply this knowledge to well, getting,
for example, to Mars. Let's start with Hohmann transfers. Mars and Earth both
revolve around the Sun in pretty much circular orbits. And our spacecraft starts
at the Earth. So now we want to reach Mars. How do we do this? Well, we can fly
what we just said. So we accelerate when we are at the Earth orbit,
such that our orbit touches the Mars orbit on the other side. OK. So this gives us
some amount of delta v we have to apply. We need to calculate this. I'm not going
to do this. Then we actually fly around this orbit for half an ellipse. And once
we have reached the Mars orbit, then we can actually accelerate again in order to
raise other side of the Ellipse until that one reaches the Mars orbit. So with two
maneuvers, two accelerations, we can actually change from one circular orbit to
another one. OK. This is the basic idea of how you actually fly to Mars. So let's
look at an animation. So this is the orbit of the InSight mission. That's another Mars
mission which launched and landed last year. The blue circle is the Earth and the
green one is Mars. And the pink is actually the satellite or the probe.
You can see that, well, it's flying in this sort of half ellipse. However, there
are two... well, there's just one problem, namely when it actually reaches Mars, Mars
needs to be there. I mean, that sounds trivial. Yeah. But I mean, imagine you fly
there and then well, Mars is somewhere else, that's not good. I mean this happens
pretty regularly when you begin playing a Kerbal Space Program, for example.
So we don't want to like play around with this the whole time, we actually want
to hit Mars. So we need to take care of that Mars is at the right position when we
actually launch. Because it will traverse the whole green line during our transfer.
This means that we can only launch such a Hohmann transfer at very particular times.
And sort of this time when you can do this transfer is called the transfer
window. And for Earth-Mars, for example. This is possible every 26 months. So if
you miss something, like, software's not ready, whatever, then you have to wait for
another twenty six months. So, the flight itself takes about six months. All right.
There is another thing that we kind of neglected so far, namely when we start,
when we depart from Earth, then well there's Earth mainly. And so that's the
main source of gravitational force. For example, right now I'm standing here on
the stage and I experience the Earth. I also experience Sun and Mars. But I mean,
that's very weak. I can ignore this. So at the beginning of our mission to Mars, we
actually have to take care that we are close to Earth. Then during the
flight, the Sun actually dominates the gravitational force. So we will only
consider this. But then when we approach Mars, we actually have to take care about
Mars. Okay. So we kind of forgot this during the Hohmann transfer. So what you
actually do is you patch together solutions of these transfers. Yeah. So in
this case, there are there are essentially three sources of gravitational force so
Earth, Sun, Mars. So we will have three two body problems that we need to consider.
Yeah. One for departing, one for the actual Hohmann transfer. And then the third
one when we actually approach Mars. So this makes this whole thing a bit more
complicated. But it's also nice because actually we need less delta v than we
would for the basic hohmann transfer. One reason for this is that when we look at
Mars. So the green line is now the Mars orbit and the red one is again the
spacecraft, it approaches Mars now we can actually look at what happens at Mars by
kind of zooming into the system of Mars. OK. So Mars is now standing still. And
then we see that the velocity of the spacecraft is actually very high relative
to Mars. So it will be on the hyperbolic orbit and will actually leave Mars again.
You can see this on the left side. Right. Because it's leaving Mars again. So what
you need to do is, in fact, you need to slow down and change your orbit into an
ellipse. Okay. And this delta v, is that you that you need here for this maneuver
it's actually less than the delta v you would need to to circularize the orbit to
just fly in the same orbit as Mars. So we need to slow down. A similar argument
actually at Earth shows that, well, if you actually launch into space, then you do
need quite some speed already to not fall down back onto Earth. So that's something
like seven kilometers per second or so. This means that you already have some
speed. OK. And if you align your orbit or your launch correctly, then you already
have some of the delta v that you need for the Hohmann transfer. So in principle, you
need quite a bit less delta v than than you might naively think. All right. So
that much about Hohmann transfer. Let's look at Gravity assist. That's another major
technique for interplanetary missions. The idea is that we can actually use planets
to sort of getting pulled along. So this is an animation, on the lower animation
you see kind of the picture when you look at the planet. So the planets standing
still and we assume that the spacecraft's sort of blue object is on a hyperbolic
orbit and it's kind of making a 90 degree turn. OK. And the upper image, you
actually see the picture when you look from the Sun, so the planet is
actually moving. And if you look very carefully at the blue object then you can
see that it is faster. So once it has passed, the planet is actually faster.
Well, we can actually look at this problem. So this is, again, the picture. When
Mars is centered, we have some entry velocity. Then we are in this hyperbolic
orbit. We have an exit velocity. Notice that the lengths are actually equal. So
it's the same speed. But just a turn direction of this example. But then we can
look at the whole problem with a moving Mars. OK, so now Mars has some velocity
v_mars. So the actual velocity that we see is the sum of the entry and the Mars
velocity before and afterwards exit, plus Mars velocity. And if you look at those
two arrows, then you see immediately that, well, the lengths are different. Okay. So
this is just the whole phenomenon here. So we see that by actually passing close to
such a planet or massive body, we can sort of gain free delta v. Okay, so of
course, it's not. I mean, the energy is still conserved. Okay. But yeah, let's not
worry about these details here. Now, the nice thing is we can use this technique to
actually alter course. We can speed up. So this is the example that I'm shown here.
But also, we can use this to slow down. Okay. So that's a pretty common
application as well. We can use this to slow down by just changing the arrows,
essentially. So just approaching Mars from a different direction, essentially. So
let's look at the example. And this is Bepicolombo. That's actually the reason
why I kind of changed the title, because Bepicolombo is actually a mission to
Mercury. So it was launched last year. It's a combined ESA/JAXA mission and it
consists of two probes and one thruster centrally. So it's a through three stages
that you can see in the picture. Yeah. That's a pretty awesome mission, actually.
It's really nice. But it has in particular, a very cool orbit. So that's
it. What can we see here? So first of all, the blue line, that's actually Earth. The
green one, that's Mercury. So that's where we want to go. And we have this
intermediate turquoise one - that's Venus. And well the curve is
Bepicolombo's orbit, so it looks pretty complicated. Yeah, it's definitely not the
Hohmann transfer. And in fact, this mission uses nine Gravity assists to reach
Mercury. And if you look at the path so, for example, right now
it actually is very close to Mercury because the last five or six Gravity
assists are just around Mercury or just slow down. OK. And this saves a lot of
delta v compared to the standard Hohmann transfer. All right. But we
want to do even better. OK. So let's now actually make the whole problem more
complicated in order to hope for some kind of nice tricks that we can do. OK, so now
we will talk about a planar circular restricted three body problem. All right.
So in general, the three body problem just means, hey, well, instead of two bodies,
we have three. OK. They pairwise attract each other and then we can solve this
whole equation of motion. We can ask a computer. And this is one animation of
what it could look like. So the three masses and their orbits are traced and we
see immediately that we don't see anything that's super complicated. There is no
way we can really... I don't know, formulate a whole solution theory for a
general three body problem. That's complicated. Those are definitely not
ellipses. So let's maybe go a step back and make the problem a bit easier. OK. So
we will now talk about a plane or circular restricted three body problem. There are
three words. So the first one is restricted. Restricted means that in our
application case, one of the bodies is actually a spacecraft. Spacecrafts are
much lighter than, for example, Sun and Mars, meaning that we can actually ignore
the force that the spacecraft exerts on Sun and Mars. Okay. So we will actually
consider Sun and Mars to be independent of the spacecraft. OK. So in total, we only
have like two gravitational forces now acting on a spacecraft. So we neglect sort
of this other force. Also, we will assume that the whole problem is what's called
circular. So we assume that Sun and Mars actually rotate in circles around their
center of mass. This assumption is actually pretty okay. We will see a
picture right now. So in this graph, for example, in this image, you can see that
the black situation. So this might be at some time, at some point in time. And then
later on, Sun and Mars actually have moved to the red positions and the spacecraft is
at some other place. And now, of course, feels some other forces. OK. And also we
will assume that this problem is plane, meaning again that everything takes place
in the plane. OK. So let's look at the video. That's a video with a very low
frame rate, something like two images per day. Maybe it's actually Pluto and Charon.
So the larger one, this is the ex-planet Pluto. It was taken by New Horizons in
2015 and it shows that they actually rotate around the center of mass. Yeah. So
both actually rotate. This also happens, for example, for Sun and Earth or Sun and
Mars or sun and Jupiter or also Earth and Moon. However, in those other cases, the
center of mass is usually contained in the larger body. And so this means that in the
case of Sun-Earth, for example, the Sun will just wiggle a little bit. OK. So you
don't really see this extensive rotation. OK. Now, this problem is still difficult.
OK. So if you're putting out a mass in there, then you don't really
know what happens. However, there's a nice trick to simplify this problem. And
unfortunately, I can't do this here. But maybe all the viewers at home, they can
try to do this. You can take your laptop. Please don't do this. And you can rotate
your laptop at the same speed as this image actually rotates. OK. Well, then
what happens? The two masses will actually stand still from your point of view. OK.
If you do it carefully and don't break anything. So we switch to this sort of
rotating point of view. OK, then the two masses stand still. We still have the two
gravitational forces towards Sun and Mars. But because we kind of look at it from a
rotated or from a moving point of view, we get two new forces, those forces, you
know, the centrifugal forces, of course, the one that, for example, you
have when you play with some children or so, they want to be pulled in
a circle very quickly and then they start flying and that's pretty cool. But here we
actually have this force acting on the spacecraft. Okay. And also there is the
Coriolis force, which is a bit less known. This depends on the velocity of the
spacecraft. OK. So if there is no velocity in particular, then there will not be any
Coriolis force. So our new problem actually has four forces. OK, but the
advantage is that Sun and Mars actually are standing still. So we don't need to
worry about their movement. OK. So now how does this look like? Well, this might be
an example for an orbit. Well, that looks still pretty complicated. I mean, this is
something that you can't have in a two body problem. It has these weird wiggles.
I mean, they're not really corners. And it actually kind of switches from Sun to
Mars. Yes. So it stays close to Sun for some time and it moves somewhere else as
it, it's still pretty complicated. I don't know. Maybe some of you have have read the
book "The Three-Body Problem". So there, for example, the two masses might be a
binary star system. OK. And then you have a planet that's actually moving along such
an orbit. OK, that looks pretty bad. So in particular, the seasons might be somewhat
messed up. Yeah. So this problem is, in fact, in a strong mathematical sense,
chaotic. OK. So chaotic means something like if you start with very close initial
conditions and you just let the system evolve, then the solutions will look very,
very different. OK. And this really happens here, which is good. All right. So
one thing we can ask is, well, is it possible that if we put a spacecraft into
the system without any velocity, is it possible that all the forces actually
cancel out. And it turns out yes, it is possible. And those points are called
Lagrangian points. So if we have zero velocity, there is no Coriolis force. So
we have only these three forces. And as you can see in this little schematics
here, it's possible that all these forces actually cancel out. Now imagine. Yeah. I
give you a homework. Please calculate all these possible points. Then you can do
this. But we won't do this right here. Instead, we just look at the result. So
those are the five Lagrangian points in this problem. OK, so we have L4 and L5
which are at equilateral triangles with Sun and Mars. Well, Sun - Mars in this
case. And we have L1, L2 and L3 on the line through Sun and Mars. So if you put
the spacecraft precisely at L1 without any velocity, then in relation to Sun and Mars
it will actually stay at the same position. Okay, that's pretty cool. However,
mathematicians and physicists will immediately ask well, OK, but what happens
if I actually put a spacecraft close to a Lagrangian point? OK, so this is
related to what's called stability. And you can calculate that around L4 and L5.
The spacecraft will actually stay in the vicinity. So it will essentially rotate
around the Lagrangian points. So those are called stable, whereas L1, L2 and L3 are
actually unstable. This means that if you put a spacecraft there, then it will
eventually escape. However, this takes a different amount of time depending on the
Lagrangian points. For example, if you're close to L2, this might take a few months,
but if you're close to L3, this will actually take something like a hundred
years or so. Okay, so those points are still different. All right. Okay. So
is there actually any evidence that they exist? I mean, maybe I'm just making this
up and, you know, I mean, haven't shown you any equations. I could just throw
anything. However, we can actually look at the solar system. So this is the inner
solar system here. In the middle you see, well, the center you see the Sun, of
course. And to the lower left, there's Jupiter. Now, if you imagine an
equilateral triangle of Sun and Jupiter, well, there are two of them. And then you
see all these green dots there. And those are asteroids. Those are the Trojans and
the Greeks. And they accumulate there because L4 and L5 are stable. Okay. So we
can really see this dynamics gone on in the solar system. However, there's also
various other applications of Lagrangian points. So in particular, you might want
to put a space telescope somewhere in space, of course, in such a way that the
Sun is not blinding you. Well, there is Earth, of course. So if we can put the
spacecraft behind Earth, then we might be in the shadow. And this is the Lagrangian
point L2, which is the reason why this is actually being used for space telescopes
such as, for example, this one. However, it turns out L2 is unstable. So we don't
really want to put the spacecraft just there. But instead, we put it in an orbit
close... in a close orbit, close to L2. And this particular example is called the
Halo orbit, and it's actually not contained in the planes. I'm cheating a
little bit. It's on the right hand side to you. And in the animation you actually see
the the orbit from the side. So it actually leaves the plane the blue dot is
Earth and the left hand side you see the actual orbit from the top. So
it's rotating around this place. OK. So that's the James Webb Space Telescope, by
the way. You can see in the animation it's supposed to launch in 2018. That didn't
work out, unfortunately, but stay tuned. Another example. That's how it has become
pretty famous recently as the Chinese Queqiao relay satellite. This one sits at
the Earth - Moon L2 Lagrange point. And the reason for this is that the Chinese
wanted to or actually did land the Chang'e 4 Moon lander on the backside of the Moon.
And in order to stay in contact, radio contact with the lander, they had to put a
relay satellite behind the Moon, which they could still see from Earth. And they
chose some similar orbit around L2. OK. So let's now go to some other more advanced
technique: ballistic capture. Right. Okay. So this whole business sort of started
with a mission from the beginning of the 1990s, and that's the Hiten mission. So
that was a Japanese well, Moon probe consisted of a probe which had a small
orbiter site which was separated, and then it was supposed to actually enter orbit
around Moon. Unfortunately, it missed its maneuver. So it didn't apply enough delta v
so it actually flew off. And the mission was sort of lost at that point
because Hiten itself, so the main probe did not have enough fuel to reach the
Moon. All right. That's, of course, a problem. I mean, that's a risk you have to
take. And they were probably pretty devastated. However, there were two people
from JPL, NASA, who actually heard about this, Belbruno and Miller, and they were
working on strange orbits, those ballistic capture orbits. And they actually found
one for the Hiten probe. They sent this to the Japanese and they actually use that
orbit to get the Hiten probe to the moon. And it actually arrived in October 1991.
And then it could do some science, you know, maybe some
different experiments, but it actually arrived there. However, the transfer took
quite a bit longer. So a normal Moon transfer takes like three days or so. But
this one actually took a few months. All right. And the reason for this is that it
looks pretty weird. So this is a picture of the orbiter - schematic picture.
And you can see the Earth. Well, there in the middle sort of. And the Moon a bit to
the left at the L2 is the Lagrangian point of the Sun - Earth system. OK. So it's
pretty far out. And you can see that the orbit sort of consists of two parts.
First, it leaves Earth and it flies far beyond the Moon. So somewhere completely
different towards some other Lagrangian point. That's really far away. Then it
does some weird things. And in the upper part of picture there it actually does a
maneuver. So we apply some thrusts there to be to change on a different orbit. And
this orbit led the probe directly to the moon where it was essentially captured for
free. OK. So it just entered orbit around the Moon. And this is, of course, not
possible in the two body problem, but we may find a way for doing this in the three
body problem. OK, so what do we mean by capture? Now we have to sort of think
a bit more abstractly. The idea is... we have Sun and Mars and we
have a spacecraft that flies in this three body problem. So the red orbit is actually
the orbit of the spacecraft. Now, at any point in time, we may decide to just
forget about the Sun. OK. So instead we consider the two body problem of Mars and
a spacecraft. OK. Because at this point in time, the spacecraft has a certain
position relative to Mars and a certain velocity. So this determines its orbit in
the two body problem. Usually it would be very fast. So it would be on a hyperbolic
orbit, which is denoted by the dashed line here. OK. Or a dashed curve. So usually
you happen to be in a hyperbolic orbit. But of course, that orbit is only an
approximation because in the three body problem, the movement is actually
different. But later on, it might happen that we continue on the orbit. We can do
the same kind of construction, but just looking... but just ignoring the Sun
essentially, and then we could find that the spacecraft suddenly is in a elliptical
orbit. This would mean that if you forgot about the Sun, then the spacecraft
would be stable and would be captured by Mars. It would be there. That would be
pretty nice. So this phenomenon, when this happens, we call this a temporary capture.
OK. Temporary because it might actually leave that situation again later on. Now,
because the actual movement depends on the three body problem, which is super
complicated. So it's possible that it actually leaves again. But for that moment
at least, it's captured and we want to find a way or describe some kind of
algorithm perhaps how we can find this situation essentially. OK, and in a
reasonable way, and the notion for this is what's called, well, n-stability, the idea
is the following: we look at the three body probleme, we want to go to Mars. So we
pick a line there. And on the line we take a point x, which has some distance r to
the Mars and we pick a perpendicular speed, a perpendicular velocity to the
line such that this corresponds to some kind of elliptic orbit in the two body
problem. Okay. So that's the dashed line. But then we actually look at the problem
in the three body problem and we just evolve the spacecraft. And it's following
the red orbit. It might follow the red orbit. And it can happen that after going
around Mars for one time, it hits again the line. Okay, then we can do the same
construction with forgetting the Sun again and just look at the two body problem. And
it's possible that this point actually still corresponds to an elliptic orbit.
That's somewhat interesting, right? Because now this means that if we actually
hit the point x, then we can follow the orbit and we know that we wrap around
Mars once and are still sort of captured in the corresponding two body problem.
Okay. If we actually are able to wrap around Mars twice, then we would call this
2-stable and, well, for more rotations that it is n-stable. Okay, so that's good
because such an orbit corresponds to something that's usable because we will
wrap around Mars n times. However, it's also possible that you have an unstable
point, meaning that we again start in something that corresponds to an ellipse
around Mars. But if we actually follow the orbit in a three body problem, it will,
for example, not come back. It will not wrap around Mars, it will go to the Sun or
somewhere else. OK. So that's that's of course, not a nice point. This one's
called unstable. And then there's another thing we can do. That's actually a pretty
common trick in finding orbits, etc. We can instead of solving the problem in
forward time we actually go back, okay. So essentially in your program you just
replace time by minus time, for example, and then you just solve the thing and you
go back in the past and it's possible that a point that corresponds to such
an ellipse when you go back into the past and it does not wrap around, but it
actually goes to the Sun, for example, we call this unstable in the past. Okay. So
that's just some random definition. And we can use this. The reason for
this is we can actually kind of take these concepts together and build an orbit from
that. The idea being we pick a point x that is n-stable. So, for example, it
might wrap around Mars six times, some number that we like. This is the blue part
here in the picture. So it wraps around Mars six times. But the way we go back in
time, it actually leaves Mars or at least it doesn't come back in such a way that
it's again on an ecliptic curve. So this is the red part. Okay. So we can
just follow this and then we pick a point y on that curve. Okay. So this one will be
pretty far away from Mars or we can choose it. And then we sort of use a Hohmann
transfer to get from Earth to that point y. Okay? So our orbit actually consists of
three parts now. Okay. So we have the Hohmann transfer, but it's not actually
aiming for Mars. It's actually aiming for the point y. There we do a maneuver
because we want to switch onto this red orbit. Okay. And then this one will bring
us to the point x where we know because it was constructed in this way that the
spacecraft will continue to rotate around Mars for example six times. Okay. So in
particular at x there is no maneuver taking place. Okay. So that's a
possible mission scenario. And the way this is done then usually is you kind of..
you calculate these points x that are suitable for doing this. Okay. So they
have to be stable and unstable in the past at the same time. So we have to find them.
And there's a lot of numerical computations involved in that. But once we
have this, you can actually build these orbits. OK. So let's look at an actual
example. So this is for Earth - Mars. On the left, you see, well, that the two
circular orbits of Earth, Mars, and on the right you see the same orbit, but from a
point of view centered around Mars. Okay. And the colors correspond to each other.
So the mission starts on the left side by doing a Hohmann transfer. So that's the
black line starting at Earth and then hitting the point, which is called x_c
here. So that's the y that I had on the other slide. So this point y
or x_c here is pretty far away still from Mars. There we do a maneuver and we switch
under the red orbit. Which brings us to the point x closer to Mars, after which we
will actually start rotating round Mars. And the point x is actually at the top of
this picture. Okay. And then on the right you can see the orbit and it's looking
pretty strangely. And also the red orbit is when we kind of the capture orbit
our way to actually get to Mars. And then if you look very carefully, you can count
we actually rotate around Mars six times. Okay. Now, during those six
rotations around Mars, we could do experiments. So maybe that is enough for
whatever we are trying to do. OK. However, if we want to stay there, we need to
actually execute another maneuver. OK. So to actually stay around Mars. And I mean,
the principle looks nice but of course, you have to do some calculations. We have
to find some ways to actually quantify how good this is. And it turns out that there
are few parameters that you can choose, in particular the target point x, this has
a certain distance that you're aiming for at around Mars. And it turns out that this
procedure here, for example, is only very good if this altitude, this distance r is
actually high enough. If it is high enough then you can save - in principle - up to
twenty three percent of the delta v, which is enormous. OK. So that would
be really good. However, in reality it's not as good usually. Yeah. And for a
certain lower distances, for example, you cannot save anything, so there are
certain tradeoffs to make. However, there is another advantage. Remember this point y?
We chose this along this capture orbit along the red orbit. And the thing is, we
can actually choose this freely. This means that our Hohmann transfer doesn't
need to hit Mars directly when it's there. So it doesn't need to aim for that
particular point. It can actually aim for any point on that capture orbit. This
means that we have many more Hohmann transfers available that we can actually
use to get there. This means that we have a far larger transfer window. OK. So we
cannot just start every 26 months. But now we, with this technique, we could actually
launch. Well, quite often. However, there's a little problem. If you looked at
the graph carefully, then you may have seen that the red orbit actually took like
three quarters of the rotation of Mars. This corresponds to roughly something like
400 days. OK. So this takes a long time. So you probably don't want to use this
with humans on board because they have to actually wait for a long time. But in
principle, there are ways to make this shorter. So you can try to actually
improve on this, but in general, it takes a long time. So let's look at a real
example for this. Again, that's Bepicolombo. The green dot is now Mercury.
So this is kind of a zoom of the other animation and the purple line is the
orbit. And yeah, it looks strange. So the first few movements around Mercury,
they are actually the last gravity assists for slowing down. And then it actually
starts on the capture orbit. So now it actually approaches Mercury. So this is
the part that's sort of difficult to find, but which you can do with this stability.
And once the animation actually ends, this is when it actually reaches the point
when it's temporarily captured. So in this case, this is at an altitude of 180,000
kilometers. So it is pretty high up above Mercury, but it's enough for the mission.
OK. And of course, then they do some other maneuver to actually stay around
Mercury. Okay, so in the last few minutes, let's have a look. Let's have a brief look
at how we can actually extend this. So I will be very brief here, because while we
can try to actually make this more general to improve on this, this concept is then
called the interplanetary transport network. And it looks a bit similar to
what we just saw. The idea is that, in fact, this capture orbit is part of a
larger well, a set of orbits that have these kinds of properties that wrap around
Mars and then kind of leave Mars. And they are very closely related to the
dynamics of particular Lagrangian points, in this case L1. So that was the one
between the two masses. And if you investigate this Lagrangian point a bit
closer, you can see, well, you can see different orbits of all kinds of
behaviors. And if you understand this, then you can actually try to do the same
thing on the other side of the Lagrangian point. OK. So we just kind of switch from
Mars to the Sun and we do a similar thing there. Now we expect to actually find
similar orbits that are wrapping around the Sun and then going towards this
Lagrangian point in a similar way. Well, then we already have some orbits that are
well, kind of meeting at L1. So we might be able to actually connect them somehow,
for example by maneuver. And then we only need to reach the orbit around Earth or
around Sun from Earth. OK. If you find a way to do this, you can get rid of the
Hohmann transfer. And this way you reduce your delta v even further. The problem is
that this is hard to find because these orbits they are pretty rare. And of
course, you have to connect those orbits. So they you approach the Lagrangian point
from L1 from two sides, but you don't really want to wait forever until they...
it's very easy to switch or so, so instead you apply some delta v, OK, in order to
not wait that long. So here's a picture of how this might look like. Again
very schematic. So we have Sun, we have Mars and in between there is the
Lagrangian point L1. The red orbit is sort of an extension of one of those capture
orbits that we have seen. OK, so that wraps around Mars a certain number of
times. And while in the past, for example, it actually goes to Lagrangian point. I
didn't explain this, but in fact, there are many more orbits around L1, closed
orbits, but they're all unstable. And these orbits that are used in this
interplanetary transport network they actually approach those orbits around L1
and we do the same thing on the other side of the Sun now and then the idea is, OK,
we take these orbits, we connect them. And when we are in the black orbit
around L1, we actually apply some maneuver, we apply some delta v to
actually switch from one to the other. And then we have sort of a connection of how
to get from Sun to Mars. So we just need to do a similar thing again from for Earth
to this particular blue orbit around the Sun. OK. So that's the general procedure.
But of course, it's difficult. And in the end, you have to do a lot of numerics
because as I said at the beginning, this is just a brief overview. It's not all the
details. Please don't launch your own mission tomorrow. OK. So with
this, I want to thank you. And I'm open to questions.
Applause
Herald: So thank you Sven for an interesting talk. We have a few minutes
for questions, if you have any questions lined up next to the microphones, we'll
start with microphone number one. Mic1: Hello. So what are the problems
associated? So you showed in the end is going around to Lagrange Point L1?
Although this is also possible for
other Lagrange points. Could you do this with L2?
Sven: Yes, you can. Yeah. So in principle, I didn't show the whole picture, but
these kind of orbits, they exist at L1, but they also exist at L2. And in
principle you can this way sort of leave this two body problem by finding similar
orbits. But of course the the details are different. So you cannot really take your
knowledge or your calculations from L1 and just taking over to L2, you actually
have to do the same thing again. You have to calculate everything in detail.
Herald: To a question from the Internet. Signal Angel: Is it possible to use these
kinds of transfers in Kerbal Space Program?
Sven: So Hohmann transfers, of course, the gravity assists as well, but not the
restricted three body problem because the way Kerbal Space Program at least the
default installation so without any mods works is that it actually switches the
gravitational force. So the thing that I described as a patch solution where we
kind of switch our picture, which gravitational force we consider for our
two body problem. This is actually the way the physics is implemented in Kerbal space
program. So we can't really do the interplanetary transport network there.
However, I think there's a mod that allows this, but your computer might be too slow
for this, I don't know. Herald: If you're leaving please do so
quietly. Small question and question from microphone number four.
Mic4: Hello. I have actually two questions. I hope that's okay. First
question is, I wonder how you do that in like your practical calculations. Like you
said, there's a two body problem and there are solutions that you can
calculate with a two body problem. And then there's a three body problem. And I
imagine there's an n-body problem all the time you do things. So how does it look
when you do that? And the second question is: you said that reducing delta v
about 15% is enormous. And I wonder what effect does this have on the payload?
Sven: Okay. So regarding the first question. So in principle, I mean, you
make a plan for a mission. So you have to you calculate those things in these
simplified models. Okay. You kind of you patch together an idea of what you want to
do. But of course, in the end, you're right, there are actually many massive
bodies in the solar system. And because we want to be precise, we actually have to
incorporate all of them. So in the end, you have to do an actual numerical search
in a much more complicated n-body problem. So with, I don't know, 100 bodies or so
and you have to incorporate other effects. For example, the solar radiation might
actually have a little influence on your orbit. Okay. And there are many effects of
this kind. So once you have a rough idea of what you want to do, you need to take
your very good physics simulator for the n-body problem, which actually has all
these other effects as well. And then you need to do a numerical search over this.
Kind of, you know, where to start with these ideas, where to look for solutions.
But then you actually have to just try it and figure out some algorithm to actually
approach a solution that has to behaviors that you want. But it's a lot of numerics.
Right. And the second question, can you remind me again? Sorry.
Mic4: Well, the second question was in reducing delta v about 15%. What is the
effect on the payload? Sven: Right. So, I mean, if you need
15% less fuel, then of course you can use 15% more weight for more mass for the
payload. Right. So you could put maybe another instrument on there. Another thing
you could do is actually keep the fuel but actually use it for station keeping. So,
for example, in the James Webb telescope example, the James Webb telescope flies
around this Halo orbit around L2, but the orbit itself is unstable. So the James
Webb Space Telescope will actually escape from that orbit. So they have to do a few
maneuvers every year to actually stay there. And they have only a finite amount
of fuels at some point. This won't be possible anymore. So reducing delta v
requirements might actually have increased the mission lifetime by quite a bit.
Herald: Number three. Mic3: Hey. When you do such a
mission, I guess you have to adjust the trajectory of your satellite quite often
because nothing goes exactly as you calculated it. Right. And the question is,
how precise can you measure the orbit? Sorry, the position and the speed of a
spacecraft at, let's say, Mars. What's the resolution?
Sven: Right. So from Mars, I'm not completely sure how precise it is. But for
example, if you have an Earth observation mission, so something that's flying around
Earth, then you can have a rather precise orbit that's good enough for taking
pictures on Earth, for example, for something like two weeks or so. So
you can measure the orbit well enough and calculate the future something like two
weeks in the future. OK. So that's good enough. However. Yeah. The... I can't
really give you good numbers on what the accuracy is, but depending on the
situation, you know, it can get pretty well for Mars I guess that's pretty
far, I guess that will be a bit less. Herald: A very short question for
microphone number one, please. Mic1: Thank you. Thank you for the talk.
I have a small question. As you said, you roughly plan the trip using the three
body and two body problems. And are there any stable points like Lagrangian points
in there, for example, four body problem? And can you use them to... during the
roughly planning stage of... Sven: Oh, yeah. I actually wondered
about this very recently as well. And I don't know the answer. I'm not sure. So
the three body problem is already complicated enough from a mathematical
point of view. So I have never actually really looked at a four body problem.
However with those many bodies, there are at least very symmetrical solutions.
So you can find some, but it's a different thing than Lagrangian points, right.
Herald: So unfortunately we're almost out of time for this talk. If you have more
questions, I'm sure Sven will be happy to take them afterwards to talk. So please
approach him after. And again, a big round of applause for the topic.
Sven: Thank you.
Applause
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