36C3 - Thrust is not an Option: How to get to Mars really slow

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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