WEBVTT
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36c3 preroll music
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Herald: Ok, I have to say, I'm always
deeply impressed about how much we already
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learned about space, about the universe
and about our place in the universe,
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our solar system. But the next speakers
will explain us how we can use
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computational methods to simulate the
universe and actually grow planets. The
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speakers will be Anna Penzlin (miosta).
She is PHC student in computational
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astrophysics in Tübingen and Carolin
Kimmich (caro). She is a physics master's
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student at Heidelberg University. And the
talk is entitled "Grow Your Own Planets
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How Simulations Help us understand the
universe." Thank you!
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applause
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caro: So hi, everyone. It's a cool
animation right? And the really cool thing
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is that there's actually physics going on
there. So this object could really be out
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there in space but was created on a
computer. So this is how a star is
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forming, how our solar system could have
looked like in the beginning. Thank you
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for being here and that you're interested
in how we make such an animation. Anna and
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I are researchers in astrophysics. And
we're concentrating on how planets form
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and evolve. She's doing her PHD and in
Tübingen and I'm doing my masters in
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Heidelberg. And in this talk, we want to
show you a little bit of physics and how
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we can translate that in such a way that a
computer can calculate it. So, let's ask a
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question first. What is the universe or
what's in the universe? The most part of
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the universe is something we don't
understand, yet. It's dark matter and dark
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energy and we don't know what it is, yet.
And that's everything we cannot see in
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this picture here. What we can see are
stars and galaxies, and that's what we
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want to concentrate on in this talk. But
if we can see it, why would we want to
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watch a computer? Well, everything in
astronomy takes a long time. So each of
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these tiny specs you see here are galaxies
just like ours. This is how the Milkyway
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looks like. And we are living in this tiny
spot here. And as you all know, our earth
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takes one year to orbit around the sun.
Now, think about how long it takes for the
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sun to orbit around the center of the
galaxy. It's four hundred million years.
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And even the star formation is 10 million
years. We cannot wait 10 million years to
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watch how a star is forming, right? That's
why we need computational methods or
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simulations on a computer to understand
these processes. So, when we watch to the
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night sky, what do we see? Of course we
see stars and those beautiful nebulas.
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They are a gas and dust. And all of these
images are taken with Hubble Space
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Telescope. Oh, so there's one image that
does belong in there. But it looks very
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similar, right? This gives us the idea
that we can describe the gases in the
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universe as a fluid. It's really
complicated to describe the gas in every
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single particle. So, we cannot track every
single molecule in the gas that moves
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around. It's way easier to describe it as
a fluid. So remember that for later, we
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will need that. But first, let's have a
look how stars form. A star forms from a
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giant cloud of dust and gas. Everything
moves in that cloud. So, eventually more
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dense regions occur and they get even
denser. And these clams can eventually
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collapse to one star. So, this is how a
star forms. They collapse due to their own
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gravity. And in this process, a disc
forms. And in this disc, planets can form.
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So why a disc? As I said, everything moves
around in the cloud. So it's likely that
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the cloud has a little bit of an initial
rotation. As it collapses, this rotation
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gets larger and faster. And now you can
think of making a pizza. So when you make
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a pizza and spin your dough on your
finger, you get a flat disc like a star,
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like a disc around a star. That's the same
process, actually. In this disc, we have
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dust and gas. From this dust in the disc
the planet can form. But how do we get
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from tiny little dust particles to a big
planet? Well, it somehow has to grow and
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grow even further and compact until we
have rocks. And even grow further until we
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reach planets. How does it grow? Well,
that dust grows we know that. At least
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that's what I observed when I took those
images in my flat. Well, so dust can grow
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and grow even further and compact. But
when you take two rocks, we're now at this
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in this stage. When you take two rocks and
throw them together, you don't expect them
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to stick, right? You expect them to crash
and crack into a thousand pieces. So,
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we're standing on the proof that planets
exist. How does this happen? And it's not
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quite solved yet in research. So, this is
a process that is really hard to observe
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because planets are very, very tiny
compared to stars. And even stars are only
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small dots in the night sky. Also, as I
said, planets form in a disc. And it's
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hard to look inside the disc. So this is
why we need computation to understand a
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process that how planets form and other
astronomical processes. So let's have a
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look at how this simulated on a computer.
miosta: OK. So, somehow we have seen
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nature. It's beautiful and it's just like
a tank of water and a bubbly fluid we
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already have. So, now we have this bubbly
fluid and here in the middle demonstrated.
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But now we have to teach our computer to
deal with the bubbly fluid. And that's way
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too much single molecules to simulate
them, as we already said. So there are two
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ways to discretize it in a way that we
just look at smaller pieces. One is the
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Lagrangian description, just like taking
small bubbles or balls of material that
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have a fixed mass. They have a certain
velocity that varies between each particle
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and they have, of course, a momentum
because they have a velocity and a mass.
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And we've created a number of those
particles and then just see how they move
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around and how they collide with each
other. That would be one way. And that was
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described last year in a very good talk. I
can highly recommend to hear this talk if
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you're interested in this method. However,
there's a second way to also describe
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this. Not just going with the flow of the
particles, but we are a bit lazy, we just
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box it. So we create a grid. And as you
see down here in this grid, you have the
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certain filling level, a bit of a slope.
So, what's the trend there? And then we
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just look for each box, what flows in what
flows out through the surfaces of this
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box. And then we have a volume or a mass
filled within this box. And this is how we
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discretize what is going on in the disc.
And actually, since we are usually in the
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system of a disc, we do not do it in this
nice box way like this. But we use boxes
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like those because they are already almost
like a disc and we just keep exactly the
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same boxes all the time and you just
measure what goes through the surface in
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these boxes. So, this is how these two
methods look like if you compute with both
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of them. So, one was done by me. I'm
usually using this boxing method and the
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other was done by my colleague. You see
this like when you look at them, at the
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colors, at the structure here, you have
the slope inwards, you have the same slope
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inwards here. You have even this silly
structure here. The same here. But what
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you notice is you have this enlarge dots
that are really the mass particles we saw
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before, these bubbles. And here you have
this inner cutout. This is because when
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you create this grid, you have the very
region at the inner part of the disc where
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the boxes become tiny and tinier. And
well, we can't compute that. So, we have
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to cut out at some point in inner part So, here
when you go to low densities, these
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bubbles blow up and distribute their mass
over a larger area. So, it's not very
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accurate for these areas. And here we have
the problem we can't calculate the inner
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area. So both methods have their pros and
cons. And are valid. But now, for most we
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will focus on this one. Just so we have
this nice stream features. So, again,
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going back to the boxes, we have to
measure the flow between the boxes. This
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flow, in physics we call it flux, and we
have a density row one, density row too.
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And the flux is the description of what
mass moves through the surface here from
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one box to the next. So, if we write this
in math terms, it looks like this. This
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says the time derivative of the density,
meaning the change over time. So how much
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faster or slower, the velocity would be a
change in time. And then this weird
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triangle symbol it's called nabla is a
positional derivative. So, it's like a
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slope. So, how do we change our position,
actually. So, if we change, look at the
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density over time, it should correlate to
what inflow we have over position. That is
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what that says. So and then we have in
physics a few principles that we have
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always to obey because that is just almost
common sense. One of them is, well, if we
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have mass in a box. Well, like this, the
mass should not go anywhere unless someone
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takes it out. So, if we have a closed box
and mass in that box, nothing should
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disappear magically. It should all stay in
this box. So, even if these particles jump
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around in our box with a certain velocity,
it's the same number of particles in the
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end. That's again, the same equation just
told in math. So, a second very
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rudimentary principle is if we have energy
in it, in a completely closed box. So, for
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example, this nice chemicals here and we
have a certain temperature. So, in this
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case, our temperature is low, maybe like
outside of around zero degree Celsius. And
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then we have this nice chemicals down here
and at some point they react very heavily.
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We suddenly end up with much less chemical
energy and a lot more thermal energy. But
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overall, the complete energy summed up
here, like the thermal and the chemical
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energy, also the energy of the movement
and the energy of potential added up to
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this variable "U". That should not change
over time if you sum up everything.
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Because our energy is conserved within our
clothed box. And then the third thing is I
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think you all know this. If you have like
a small mass with a certain velocity, a
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very high velocity in this case and it
bumps into someone very large, what
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happens? Well, you get a very small
velocity in this large body and the
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smaller mass stops. And the principle here
is that momentum is conserved, meaning
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that the velocity times the mass of one
object is the same as then later for the
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other one. But since it's larger, this
product has to be the same. That doesn't
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change. And we have also in our
simulations to obey these rules and we
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have to code that in so that we have
physics in them. So you say, ok, this is
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really simple, these rules, right? But
actually, well, it's not quite as simple.
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So, this is the Navier-Stokes equation, a
very complicated equation is not
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completely solved. And we have here all
that is marked red are derivatives. Here
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we have our conservation law that was the
nice and simple part. But now we have to
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take other physical things into accounting
for pressure, accounting for viscosity,
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for compression. So squeezing. And like
how sticky is our fluid? And also gravity.
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So, we have a lot of additional factors,
additional physics we also have to get in
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somehow. And all of these also depend
somehow on the change of position or the
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change of time. And these derivatives
aren't really nice for our computers
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because they well, they don't understand
this triangle. So, we need to find a way
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to write an algorithm so that it can
somehow relate with these math formula in
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a way that the computer likes. And one of
the way to do this is, well, the simplest
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solution actually is just we say, OK, we
have now this nasty derivatives and we
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want to get rid of them. So, if we look
just at one box now and we say that in
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this box, the new value for the density in
this box would be the previous density,
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plus the flux in and out times the time
stepover which we measure this flux,
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right? So, and we have to somehow get to
this flux and we just say, OK, this flux
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now is if we start here and the slope of
this curve, the trends so to say, where
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this curve is going right now, it would
look like this. So, in our next step, time
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step, we would have a density down here.
And well, then we do this again. We again
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look at this point, where's the trend
going, where's the line going? And then we
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end up here. Same here. So, again, we just
try to find this flax and this is the
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trend at this position in time. So, this
goes up here. And then if we are here now,
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look at this point, it should go up here.
So this is what our next trend would be.
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And we do this over all the times. And
this is how our simulation then would
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calculate the density for one box over a
different time steps. So, that kind of
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works. So, the blue curve is the
analytical one, the red curve, well it
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kind of similar, it works. But can we do
better? It's not perfect, yet, right? So,
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what we can do is we refine this a bit,
taking a few more steps, making it a bit
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more computationally heavy, but trying to
get a better resolution. So, first we
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start with the same thing as before. We go
to this point, find the trend in this
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point. That point like the line would go
in this direction from this point. And
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then we go just half a step now. Sorry!
And now we look at this half a step to
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this point now. And again, the same
saying, OK, where's the trend going now?
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And then we take where this point would go
and added to this trend. So that would be
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that. The average of this trend, of this
exact point and this trend, this dark
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orange curve. And then we go back to the
beginning with this trend now and say this
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is a better trend than the one we had
before. We now use that and go again and
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search the point for half a time step. And
then again, we do the same thing. Now we
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again try to find actually the trend and
average it with the arrow before. So it's
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not exactly the trend. It's a bit below
the trend because we averaged it with the
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arrow before. And now we take this
averaging trend from the beginning to the
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top like this. Okay. This is already quite
good, but we can still do a little bit
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better if we averaged with our ending
point. So, we go here, look, where is the
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trend going that would go quite up like
this and we average this and this together
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and then we end up with a line like this.
This is so much better than what we had
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before. It's a bit more complicated, to be
fair. But actually it's almost on the
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line. So, this is what we wanted. So, if
you compare both of them, we have here our
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analytical curve. So, over time in one
box, this is how the densities should
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increase. And now with it both of the
numerical method, the difference looks
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like this. So, if we have smaller and
smaller time steps, even the Euler gets
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closer and closer to the curve. But
actually the Runge Kutta this four step process
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works much better and much faster.
However, it's a bit more computationally
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and difficult.
caro: When we simulate objects in
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astronomy, we always want to compare that
to objects that are really out there. So,
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this is a giant telescope, well consisting
of a lot of small telescopes. But they can
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be connected and used as a giant telescope
and it takes photos of dust in the sky.
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And this is used to take images of discs
around stars. And these discs look like
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this. So, these images were taken last
year and they are really cool. Before we
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had those images, we only had images with
less resolution. So, they were just
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blurred blobs. And we could say, yeah,
that might be a disc. But now we really
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see the discs and we see rings here, thin
rings and we see thicker rings over here.
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And even some spiraly structures here. And
also some features that are not really
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radial symmetric like this arc here. And
it's not completely solved how these
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structures formed. And to find that out a
colleague of mine took this little object
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with the asymmetry here. And so, this is
image we just saw. And this is his
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simulation. So, this is how the disc
looked like in the beginning, probably.
00:22:37.590 --> 00:22:43.980
And we put in three planets and let the
simulation run. And so, what we see here
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is that the star is cut out as Anna said.
So, the grid cells in the inner part are
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very, very small. And it would take a long
time to compute them all. So, that's why
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we're leaving out that spot in the middle.
And what we see here is three planets
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interacting with the material in the disc.
And we can see that these planets can make
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this thing here appear so that in the end
we have something looking very similar to
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what we want to have or what we really
observe. So, we can say three planets
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could explain how these structures formed
in this disc. It's a little bit
00:23:37.379 --> 00:23:42.409
elliptical, you see that. That's because
it's tilted from our side of line. It
00:23:42.409 --> 00:23:47.430
would be round if you watched at it face
on. But it's a little bit tilted. That's
00:23:47.430 --> 00:23:55.269
why it looks elliptical.
miosta: So, we already saw we can put
00:23:55.269 --> 00:24:02.080
planets in the gas and then we create
structures. One very exciting thing that
00:24:02.080 --> 00:24:08.740
we found in the last year - or two years
ago it started but then we found more - is
00:24:08.740 --> 00:24:15.690
this system PDS 70. In this system, for
the very first time, we found a planet
00:24:15.690 --> 00:24:24.249
that was still embedded completely within
the disc. So, the gas and dust. Usually,
00:24:24.249 --> 00:24:32.259
because the gas and dust is the main thing
that creates this signal of some radiation
00:24:32.259 --> 00:24:37.749
because of heat. We only observe that and
then we can't observe the planet embedded.
00:24:37.749 --> 00:24:41.629
But in this case, the planet was large
enough. And in the right position that we
00:24:41.629 --> 00:24:48.940
actually were able to observe some
signature of accretion on this planet that
00:24:48.940 --> 00:24:57.440
was brighter than the rest of the disc.
And then later, just this year, just a few
00:24:57.440 --> 00:25:03.739
months ago, we actually found out well,
this is not the only object here. This is
00:25:03.739 --> 00:25:10.850
very clearly a planet. But actually,
like this spot here is also something. So,
00:25:10.850 --> 00:25:17.299
we can see it in different grains. Every
picture here is a different set of grains
00:25:17.299 --> 00:25:26.950
observed. And we can see
this in five different kinds of
00:25:26.950 --> 00:25:32.799
observations. So, there is a planet here.
And then there is also something we don't
00:25:32.799 --> 00:25:37.710
know what it is yet, but its point like
and actually creates the feature that we
00:25:37.710 --> 00:25:43.240
reproduce in different kinds of
observational bands or different kinds of
00:25:43.240 --> 00:25:52.070
signals here. This is very interesting.
For the first time, we actually see a
00:25:52.070 --> 00:25:58.030
planet forming right now within the disc.
And so a colleague of mine also is very
00:25:58.030 --> 00:26:04.929
interested in the system and started to
simulate how do two planets in a disc
00:26:04.929 --> 00:26:13.149
change the dynamics of a disc? So here we
have, of course, this disc is again tilted
00:26:13.149 --> 00:26:20.230
because it's not phase on, it's like 45
degrees tilted, not like this, but like
00:26:20.230 --> 00:26:27.289
this. And so he had it face on. This is
what a simulation looks like. So, there
00:26:27.289 --> 00:26:33.880
are two planets: these blobs here, again,
as in this simulation. Here we have a
00:26:33.880 --> 00:26:39.289
close up. You can actually see this little
boxes are actually our simulation boxes in
00:26:39.289 --> 00:26:47.429
which we have our own densities. And then
he just looked at how the planets would
00:26:47.429 --> 00:26:52.620
change the structure and the gas and also
how the gas would interact with the
00:26:52.620 --> 00:26:59.249
planets, shifting them around. And it's
interesting. So, the planets tend to clear
00:26:59.249 --> 00:27:05.259
out an area, open a gap, and within the
disk, that block has a lot of gas around
00:27:05.259 --> 00:27:11.039
here. So, you have the brighter ring here
again and then clearing out more and more.
00:27:11.039 --> 00:27:23.390
And at some point in the simulation you
saw they get a bit jumpy. So it's very nice.
00:27:23.390 --> 00:27:29.570
You also see that planets induce in the
whole disc some kind of features like
00:27:29.570 --> 00:27:36.740
spiral features. And so a single planet
will change the symmetry and the
00:27:36.740 --> 00:27:40.989
appearance of a whole disc.
caro: So, the reason why the planet is
00:27:40.989 --> 00:27:46.489
staying at this point is because we're
rotating with the planet. So it's actually
00:27:46.489 --> 00:27:53.499
going around the disc, but the like camera
is rotating with the planet. So, it's
00:27:53.499 --> 00:28:00.360
staying at that fixed place we put it in.
miosta: Exactly. But there's more because
00:28:00.360 --> 00:28:04.600
as I already said, in the Navier-Stokes
equation, we have a lot of different kinds
00:28:04.600 --> 00:28:08.970
of physics that we all have to include in
our simulations. One of the things, of
00:28:08.970 --> 00:28:15.149
course, is we maybe don't have just a star
and a disc. We have planets in there and
00:28:15.149 --> 00:28:20.600
maybe two stars in there. And all of these
larger bodies have also an interaction
00:28:20.600 --> 00:28:27.360
between each other. So, if we have the
star, every planet will have an
00:28:27.360 --> 00:28:32.600
interaction with the star, of course. But
then also the planets between each other,
00:28:32.600 --> 00:28:40.381
they have also an interaction, right? So,
in the end, you have to take into account
00:28:40.381 --> 00:28:48.820
all of these interactions. And then also
we have accretion just looking like this.
00:28:48.820 --> 00:28:59.350
So, accretion means that the gas is bound
by some objects. It can be the disc, the
00:28:59.350 --> 00:29:07.009
planet or the star that takes up the mass,
the dust or the gas and bounce it to this
00:29:07.009 --> 00:29:14.840
object. And then it's lost to the disc or
the other structures because it's
00:29:14.840 --> 00:29:22.309
completely bound to that. So, the
principle of this would be the simulation
00:29:22.309 --> 00:29:29.279
I did last year and published, we have
here a binary star. So, these two dots are
00:29:29.279 --> 00:29:38.809
stars. I kind of kept them in the same
spot. But every picture will be one orbit
00:29:38.809 --> 00:29:42.759
of this binary, but since we have
interactions, you actually see them
00:29:42.759 --> 00:29:48.539
rotating because of the interactions, with
each other. And then also we have here a
00:29:48.539 --> 00:29:52.669
planet and here a planet. And the
interesting thing was that these two
00:29:52.669 --> 00:30:00.361
planets interact in such a way that they
end up on exactly the same orbit. So, one
00:30:00.361 --> 00:30:06.179
star's further out, the orange one, and then
very fast they go in. And they end up on
00:30:06.179 --> 00:30:28.419
exactly the same orbit. If it now play nicely.
So, another thing is with the accretion here,
00:30:28.419 --> 00:30:36.600
we actually see clouds from above dropping
down onto the new forming star here. So,
00:30:36.600 --> 00:30:44.409
all of this, what you see here would be
gas, hydrogen. And it's a very early phase
00:30:44.409 --> 00:30:49.499
so that disc is not completely flat. It
has a lot of material. And then we
00:30:49.499 --> 00:30:55.779
actually have this infall from above
towards the star and then the star keeps
00:30:55.779 --> 00:31:01.767
the mass. And we have to take this also
into account in our simulations. Another
00:31:01.767 --> 00:31:07.220
thing we have to take into account up till
now, we just cared about masses and
00:31:07.220 --> 00:31:12.929
densities. But of course what we actually
see is that stars are kind of warm,
00:31:12.929 --> 00:31:21.759
hopefully. Otherwise, temperatures here
would also not be nice. And different
00:31:21.759 --> 00:31:27.929
chemicals have different condensation
points. And this is also true in a system.
00:31:27.929 --> 00:31:35.019
So, we start with the start temperature at
the surface of the star. We have a
00:31:35.019 --> 00:31:41.479
temperature around 4.000 Kelvin. And then
we go a bit into the disc. And there is a
00:31:41.479 --> 00:31:47.889
point where we for the first time reach a
point where we have any material at all.
00:31:47.889 --> 00:31:52.169
Because it starts to condensate and we
actually have something solid like iron.
00:31:52.169 --> 00:31:58.179
For example, at a 1500 Kelvin. And then if
we go further in, we reach a point where
00:31:58.179 --> 00:32:07.690
we have solid water and this is at 200
Kelvin. This is what we then would need
00:32:07.690 --> 00:32:12.590
actually to have a planet that also has
water on it. Because if we don't get the
00:32:12.590 --> 00:32:18.889
water in the solid state, it will not fall
onto a terrestrial planet and be bound
00:32:18.889 --> 00:32:24.899
there, right? So, this is important for
our Earth, actually. And then if we go
00:32:24.899 --> 00:32:33.340
even further out, we have also other gases
condensating to solids like CO2 or methane
00:32:33.340 --> 00:32:40.591
or things like that. And since we only get
water on a planet when we have a
00:32:40.591 --> 00:32:48.009
temperature that is low enough so that the
water actually forms is solid and it's
00:32:48.009 --> 00:32:54.269
important for us to think about where that
is in our forming disc. Where do we start?
00:32:54.269 --> 00:32:59.769
We have a planet like Earth that could
have some water, right? But it's not just
00:32:59.769 --> 00:33:07.570
the simple picture, where we have all these
nice ring structures, where we have a clear
00:33:07.570 --> 00:33:13.619
line. Actually, it gets more complicated
because we have pressure and shocks. And
00:33:13.619 --> 00:33:19.539
thermodynamics is a lot like pogo dancing,
right? You crash into each other. And it's
00:33:19.539 --> 00:33:25.629
all about collisions. So, the gas
temperature is determined by the speed of
00:33:25.629 --> 00:33:31.299
your gas molecules. Like you bouncing and
crashing into each other, exchanging
00:33:31.299 --> 00:33:39.340
momentum. So, there's two ways to heat up
such dance. First thing is you get a large
00:33:39.340 --> 00:33:45.944
amount of velocity from the outside like a
huge kick, a shock into your system. A
00:33:45.944 --> 00:33:51.519
second way would be if we have a higher
pressure, like more molecules, then also
00:33:51.519 --> 00:33:55.909
you of course have more collisions and
then a higher temperature. So, if you
00:33:55.909 --> 00:34:02.529
change - because you have a planet now in
the system - the pressure at some point,
00:34:02.529 --> 00:34:08.700
you actually get a higher temperature. So,
that is not then we lose this nice line
00:34:08.700 --> 00:34:19.136
because suddenly we have different
pressures at different locations. And a
00:34:19.136 --> 00:34:24.700
colleague of mine also simulated this.
So, this is the initial condition we
00:34:24.700 --> 00:34:28.860
just assumed: OK, if we have no
disturbance whatsoever, we have our nice
00:34:28.860 --> 00:34:36.890
planet here at 1au. So, same distance as
earth to the sun. Here, too. But here we
00:34:36.890 --> 00:34:46.670
assume that less heat gets transferred
from the surface of the disc. And here we
00:34:46.670 --> 00:34:52.030
have the planet far out like Jupiter or
something. And now we actually let this
00:34:52.030 --> 00:34:59.590
planet change the structure of the disc.
And what happens is - we found these spirals
00:34:59.590 --> 00:35:05.800
and within these spirals, we change
pressure. And with this actually, if you
00:35:05.800 --> 00:35:11.590
see this orange, everywhere where it's
orange it's hotter than the iceline. So,
00:35:11.590 --> 00:35:17.020
we don't have water where it's orange. And
where it's blue we can have water. And the
00:35:17.020 --> 00:35:22.350
interesting thing is, even if we put a
planet out here like Jupiter, we still
00:35:22.350 --> 00:35:32.569
form these regions in the inner system
where we have less water.
00:35:32.569 --> 00:35:38.022
caro: One problem in astrophysical
simulations is that we don't always know
00:35:38.022 --> 00:35:47.940
how to shape our boxes or how small these
boxes have to be. So, we use a trick to
00:35:47.940 --> 00:35:54.670
reshape the boxes as we need them. It's
called adaptive mesh. And this is a
00:35:54.670 --> 00:35:58.890
simulation of the red fluid flowing in
this direction and the blue fluid in the
00:35:58.890 --> 00:36:06.581
other one. So, at the boundary, the two
fluid shear and they mix up somehow and we
00:36:06.581 --> 00:36:12.990
don't know how in advance. So, we start a
simulation and as the simulation starts,
00:36:12.990 --> 00:36:19.640
we reshape those boxes here. So, in the
middle we don't need much. We reshape
00:36:19.640 --> 00:36:25.400
because it's not that complicated here.
It's just the flow. But at the boundary we
00:36:25.400 --> 00:36:35.060
see those mixing up of the two fluids. And
so, we reshape the cells as we need them.
00:36:35.060 --> 00:36:44.760
This is done in a program, in an
astrophysical program called AREPO. We
00:36:44.760 --> 00:36:52.750
will later show you some more programs to
use for simulations. But another
00:36:52.750 --> 00:36:59.020
simulation I want to show you first is
also done with AREPO and it's a simulation
00:36:59.020 --> 00:37:04.710
of the universe. So, from here to here,
it's very big. It's 30 million light
00:37:04.710 --> 00:37:12.210
years. So each of these dots you see here
is the size of a galaxy or even more. And
00:37:12.210 --> 00:37:17.840
here you can actually see that at some
regions it's very empty. So, we're
00:37:17.840 --> 00:37:23.420
rotating around this universe, this
simulated universe here. And these regions
00:37:23.420 --> 00:37:28.990
here are empty. And we don't need a lot of
boxes there. The big boxes are enough
00:37:28.990 --> 00:37:35.010
here. But in this dense regions where we
have a lot of material, we need smaller
00:37:35.010 --> 00:37:42.380
boxes. And this method I showed you where
we reshape the boxes as we need them is
00:37:42.380 --> 00:37:53.420
used for this simulation.
miosta: So, actually, you see the
00:37:53.420 --> 00:37:56.340
beginning of the universe there.
caro: Yes!
00:37:56.340 --> 00:38:01.000
miosta: Basically, the initial mass
collapsing to the first galaxies and first
00:38:01.000 --> 00:38:07.030
supernovae starting. Very beautiful
simulation.
00:38:07.030 --> 00:38:19.820
caro: So, there are different programs, as
I already mentioned, in astrophysics.
00:38:19.820 --> 00:38:24.970
Three of them, those three are all open
source, so you can download them and use
00:38:24.970 --> 00:38:31.090
them on your own machine, if you like. But
there are more, a lot more. Some of them
00:38:31.090 --> 00:38:38.630
are open source, some of them are not.
Sometimes it's hard to get them. In the
00:38:38.630 --> 00:38:43.700
following, we will present the tool
FARGO3D and PLUTO in a detailed version or
00:38:43.700 --> 00:38:53.160
a more detailed vision than AREPO
because we usually use those two for our
00:38:53.160 --> 00:38:58.380
simulations. What I want to show you with
this slide is that depending on what you
00:38:58.380 --> 00:39:04.520
want to simulate, you need to choose a
different program. And one thing is that
00:39:04.520 --> 00:39:10.250
in astrophysics we sometimes call the
whole program code. So, if I use the word
00:39:10.250 --> 00:39:19.170
code. Sorry about that. I mean, the whole
program. So, let's have a look at FARGO3D.
00:39:19.170 --> 00:39:27.870
It's a hydro dynamics code and what you
see here is an input parameter file. There
00:39:27.870 --> 00:39:35.180
you define how the disc looks like. How
much mass does it have? How big is it? And
00:39:35.180 --> 00:39:43.140
what planet? So, here at Jupiter, do you
see that? Jupiter is put in. And we also
00:39:43.140 --> 00:39:51.280
define how big our boxes are. This
program is written in C, which is quite
00:39:51.280 --> 00:39:57.500
nice because a lot of astrophysical
programs are still written in Fortran. So,
00:39:57.500 --> 00:40:05.600
this is good for me because I don't know
any Fortran. We can run this and what's
00:40:05.600 --> 00:40:11.010
typical for FARGO3D. So that's a compilation
actually on my computer. So, I don't need
00:40:11.010 --> 00:40:18.840
a fancy computer. I just did it on my
small laptop and now we run it. Now,
00:40:18.840 --> 00:40:26.130
typical for FARGO3D, as you will see are lot
of dots. So, here it will print out a lot
00:40:26.130 --> 00:40:33.810
of dots and it will create at certain
times some outputs. And these outputs are
00:40:33.810 --> 00:40:38.300
huge files containing numbers. So, if you
look at them they are not really
00:40:38.300 --> 00:40:44.290
interesting. They just are a numbers in
something like a text file. So, a big part
00:40:44.290 --> 00:40:50.430
of astrophysics is also to visualize the
data. Not only to create it but also to
00:40:50.430 --> 00:40:57.080
make images so that we can make movies out
of them. For that, I prefer to use Python
00:40:57.080 --> 00:41:01.600
but there are a lot of tools Python
Matplotlib, but there are a lot of
00:41:01.600 --> 00:41:09.290
different tools to visualize the data. So,
this is actually that output. That first
00:41:09.290 --> 00:41:16.350
one we just saw. The Jupiter planet in the
disc that I defined in this parameter file
00:41:16.350 --> 00:41:23.280
and it's already started to do some
spirals. And if I would have let it
00:41:23.280 --> 00:41:33.680
run further than the spirals were more
prominent. And yeah, now we have a planet
00:41:33.680 --> 00:41:45.230
here on our computer.
miosta: OK, so we also have PLUTO. PLUTO
00:41:45.230 --> 00:41:53.590
somehow has a bit more setup files. So,
what I need is three files here. Looks a
00:41:53.590 --> 00:41:59.320
bit complicated to break it down. This
file defines my grid and initial values.
00:41:59.320 --> 00:42:04.770
And this simulation time here we input
actually what physics do we want to need?
00:42:04.770 --> 00:42:13.020
What is our coordinate system? So, do we
want to have a disc or just like spherical
00:42:13.020 --> 00:42:20.660
boxes or like squared boxes? And how is
the time defined? And here we then
00:42:20.660 --> 00:42:26.720
actually write a bit of code to say, OK,
now how do I want a gravitational
00:42:26.720 --> 00:42:34.580
potential? So, what's the source of
gravity or what will happen at the inner
00:42:34.580 --> 00:42:39.890
region where we have this dark spot? We
have somehow to define what happens if gas
00:42:39.890 --> 00:42:45.090
reaches this boundary. Is it just falling
in? Is it bouncing back or something? Or
00:42:45.090 --> 00:42:50.890
is it looping through the one end to the
next? This is also something we then just
00:42:50.890 --> 00:43:01.530
have to code in. And if we then make it
and let run, it looks like this. So,
00:43:01.530 --> 00:43:08.590
again, our nice thing we hopefully put in
or wanted to put in: the time steps, what
00:43:08.590 --> 00:43:14.890
our boundaries were, parameters of
physics. Hopefully, the right ones and
00:43:14.890 --> 00:43:21.396
then nicely we start with our time steps
and then we see this. It's hooray! It
00:43:21.396 --> 00:43:27.240
worked actually. Because it's actually not
quite simple usually to set up a running
00:43:27.240 --> 00:43:32.000
program. A running problem, because you
have to really think about what should be
00:43:32.000 --> 00:43:38.170
the physics. What's the scale of your
problem? What's the timescale of your
00:43:38.170 --> 00:43:44.990
problem? And specify this in a good way.
But in principle, this is how it works.
00:43:44.990 --> 00:43:49.320
There are few test problems if you
actually want to play around with this to
00:43:49.320 --> 00:43:56.390
make it easy for the beginning. And this
is how we do simulations. So, as I already
00:43:56.390 --> 00:44:02.320
set, we can just start them on our
laptops. So, here this is my laptop. I
00:44:02.320 --> 00:44:07.859
just type a dot slash FARGO3D and that
should run, right? And then I just wait
00:44:07.859 --> 00:44:16.450
for ten years to finish the simulations of
500 timesteps or outputs. Well, that's not the best
00:44:16.450 --> 00:44:27.660
idea. So, we need more power. And both of
us, for example, are using a cluster for
00:44:27.660 --> 00:44:36.880
Baden-Württemberg and that takes down our
computation time by a lot. Usually, like a
00:44:36.880 --> 00:44:45.050
factor of maybe 20, which is a lot. So, I
would need on my computer maybe a year and
00:44:45.050 --> 00:44:53.040
then I just need maybe 5 hours, a few days
or a week on this cluster, which is
00:44:53.040 --> 00:44:56.380
usually the simulation time about a week
for me.
00:44:56.380 --> 00:45:04.440
caro: So, what you see here is that we use
GPUs, yes. But we do not or mostly not use
00:45:04.440 --> 00:45:09.630
them for gaming. We use them for actually
actual science. Yeah, would be nice to
00:45:09.630 --> 00:45:20.614
play on that, right? That just said!
miosta: So, back to our Earth, actually.
00:45:20.614 --> 00:45:27.670
So, can we now? We wanted to grow our own
planet. We can do that, yes of course. Can
00:45:27.670 --> 00:45:31.600
we grow Earth? Well, Earth is a very
special planet. We have a very nice
00:45:31.600 --> 00:45:37.720
temperature here, right? And we have not a
crushing atmosphere like Jupiter, like a
00:45:37.720 --> 00:45:43.440
huge planet that we could not live under.
We have a magnetic field that shields us
00:45:43.440 --> 00:45:53.760
from the radiation from space and we have
water. But just enough water so that we
00:45:53.760 --> 00:46:00.170
still have land on this planet where we
can live on. So, even if we fine tune
00:46:00.170 --> 00:46:05.230
simulations, the probability that we
actually hit Earth and have all the
00:46:05.230 --> 00:46:12.800
parameters right is actually tiny. It's
not that easy to simulate an Earth. And
00:46:12.800 --> 00:46:17.320
there are a lot of open questions, too.
How did we actually manage to get just
00:46:17.320 --> 00:46:24.240
this sip of water on our surface? How did
we manage to collide enough mass or
00:46:24.240 --> 00:46:30.060
aggregate enough mass to form this
terrestrial planet without Jupiter is
00:46:30.060 --> 00:46:35.740
sweeping up all the mass in our system?
How could we be stable in this orbit when
00:46:35.740 --> 00:46:42.660
there are seven other planets swirling
around and interacting with us? All of
00:46:42.660 --> 00:46:48.660
this is open in our field of research
actually, and not completely understood.
00:46:48.660 --> 00:46:54.620
This is the reason why we still need to
do astrophysics and even in all our
00:46:54.620 --> 00:47:01.010
simulations there is no planet B. And the
earth is quite unique and perfect for
00:47:01.010 --> 00:47:06.570
human life. So, please take care of the
Earth and take care of yourself and of all
00:47:06.570 --> 00:47:12.270
the others people on the Congress. And
thank you for listening and thank you to
00:47:12.270 --> 00:47:20.380
everyone who helped us make this possible.
And to the people who actually coded our
00:47:20.380 --> 00:47:24.210
programs with which we simulate.
Thank you!
00:47:24.210 --> 00:47:37.370
applause
00:47:37.370 --> 00:47:42.320
Herald: Thank you for the beautiful talk
and for the message at the end, the paper
00:47:42.320 --> 00:47:47.970
is open for discussion, so if you guys
have any questions, please come to the
00:47:47.970 --> 00:47:57.160
microphones. I'm asking my Signal Angel?
No questions right now. But microphone two
00:47:57.160 --> 00:48:00.160
please!
Mic2: Oh, yeah. Thank you very much.
00:48:00.160 --> 00:48:05.690
Really beautiful talk. I can agree. I have
two questions. The first is short. You are
00:48:05.690 --> 00:48:10.980
using Navier-Stokes equation, but you have
on the one hand, you have the dust disc
00:48:10.980 --> 00:48:14.940
and on the other hand, you have solid
planets in it. And so are you using the
00:48:14.940 --> 00:48:18.620
same description for both
or is it a hybrid?
00:48:18.620 --> 00:48:23.550
miosta: It very much depends. This is one
of the things I showed you that for PLUTO,
00:48:23.550 --> 00:48:31.300
we write this C file that specifies some
things and about every physicist has
00:48:31.300 --> 00:48:39.090
somewhat his or her own version of things.
So, some usually the planets, if they are
00:48:39.090 --> 00:48:47.030
large, they will be put in as a gravity
source. And possibly one that can accrete
00:48:47.030 --> 00:48:54.090
and pebbles are usually then put in a
different way. However, also pebbles are
00:48:54.090 --> 00:48:57.540
at the moment a bit complicated. There are
special groups specializing in
00:48:57.540 --> 00:49:04.080
understanding pebbles because as we said
in the beginning, when they collide,
00:49:04.080 --> 00:49:10.450
usually they should be destroyed. If you
hit two rocks very together, they don't
00:49:10.450 --> 00:49:14.870
stick. If you hit them hard together, they
splatter around and we don't end up with an bigger object
00:49:14.870 --> 00:49:23.390
caro: Just to explain pebbles are small
rocks or like big sand stones or something
00:49:23.390 --> 00:49:28.710
like that. Yeah. So bigger rocks,
but not very big, yet.
00:49:28.710 --> 00:49:33.190
miosta: Yes!
caro: It depends on which code you use.
00:49:33.190 --> 00:49:38.370
Mic2: Thank you. Very short, maybe one.
Do you also need to include relativistic
00:49:38.370 --> 00:49:46.520
effects. Or is that completely out?
miosta: It's a good question. Mostly if
00:49:46.520 --> 00:49:54.680
you have a solar type system, you're in
the arrange where this is not necessary.
00:49:54.680 --> 00:50:00.010
For example, with the binaries, if they
got very close together, then at the inner
00:50:00.010 --> 00:50:05.200
part of the disc, that is something we
could consider. And actually, I know for
00:50:05.200 --> 00:50:10.500
PLUTO, it has modules to include
relativistic physics, too, yes!
00:50:10.500 --> 00:50:14.000
Mic2: Thank you!
Herald: OK, we have quite some questions,
00:50:14.000 --> 00:50:19.700
so keep them short. Number one, please!
Mic1: Thank you. Yeah. Thank you very
00:50:19.700 --> 00:50:24.490
much for your interesting talk. And I
think you had it on your very first slides
00:50:24.490 --> 00:50:31.780
that about 70 percent of the universe
consists of dark matter and energy. Is that
00:50:31.780 --> 00:50:37.000
somehow considered in your
simulations or how do you handle this?
00:50:37.000 --> 00:50:43.020
caro: Well in the simulations we make, we
are doing planets and discs around stars.
00:50:43.020 --> 00:50:47.440
It's not considered there. In the
simulation we showed you about the
00:50:47.440 --> 00:50:52.670
universe at the beginning, the blueish
things were all dark matter. So, that was
00:50:52.670 --> 00:50:56.260
included in there.
Mic1: OK, thank you.
00:50:56.260 --> 00:51:00.510
Herald: OK. Microphone 3.
Mic3: Hi, thanks. Sorry, I think you
00:51:00.510 --> 00:51:05.740
talked about three different programs. I
think PLUTO, FARGO3D and a third one. So,
00:51:05.740 --> 00:51:09.620
for a complete beginner: which program
would you suggest is like you more use
00:51:09.620 --> 00:51:12.570
like if you want to learn more?
Which one is user friendly or good?
00:51:12.570 --> 00:51:18.590
miosta: I would suggest FARGO3D first. It's
kind of user friendly, has a somewhat good
00:51:18.590 --> 00:51:26.240
support and they are always also very
thankful for actual comments and additions
00:51:26.240 --> 00:51:32.030
if people actually are engaged in trying
to improve on that. Because we are
00:51:32.030 --> 00:51:37.328
physicists. We're not perfect programmers
and we're also happy to learn more. So
00:51:37.328 --> 00:51:42.720
yeah, FARGO3D I would suggest, it has some
easy ways of testing some systems and
00:51:42.720 --> 00:51:45.440
getting something done.
caro: And it also has a very good
00:51:45.440 --> 00:51:53.567
documentation and also a manual "How to
make the first steps on the Internet". So,
00:51:53.567 --> 00:51:56.980
you can look that up.
Mic3: Awesome. Thank you.
00:51:56.980 --> 00:52:00.150
Herald: Let's get one question from
outside, from my Signal Angel.
00:52:00.150 --> 00:52:05.600
Signal Angel: Thank you for your talk.
There's one question from IRC: How do you
00:52:05.600 --> 00:52:09.510
know your model is good when you can only
observe snapshots?
00:52:09.510 --> 00:52:17.770
caro: Oh, that's a good question. As we
said, we're in theoretical astrophysics.
00:52:17.770 --> 00:52:25.170
So, there are theoretical models and these
models cannot include everything. So,
00:52:25.170 --> 00:52:32.610
every single process, it's not possible
because then we would calculate for years.
00:52:32.610 --> 00:52:37.480
Yeah, to know if a model is
good you have to…
00:52:37.480 --> 00:52:46.430
miosta: Usually, you have a hypothesis or
an observation that you somehow want to
00:52:46.430 --> 00:52:54.064
understand. With most of the necessary
physics at this stage to reproduce this
00:52:54.064 --> 00:53:01.660
image. So, also from the observation we
have to take into the account what our
00:53:01.660 --> 00:53:07.650
parameters kind of should be, how dense
this end of the simulation should be and
00:53:07.650 --> 00:53:13.150
things like this. So, by comparing two
observations, that's the best measure we
00:53:13.150 --> 00:53:21.790
can get. If we kind of agree. Of course,
if we do something completely wrong, then
00:53:21.790 --> 00:53:26.600
it will just blow up or we will get a
horribly high density. So, this is how we
00:53:26.600 --> 00:53:34.270
know. Physics will just go crazy if we do
too large mistakes. Otherwise, we would
00:53:34.270 --> 00:53:39.330
try to compare two observations that it
actually is sensible what we did.
00:53:39.330 --> 00:53:44.440
caro: Yeah, that's one of the most
complicated tasks to include just enough
00:53:44.440 --> 00:53:52.400
physics that the system is represented in
a good enough way. But not too much that
00:53:52.400 --> 00:53:57.400
our simulation would blow up in time.
Herald: Number two, please.
00:53:57.400 --> 00:54:03.210
Mic2: I've got a question about the
adaptive grids. How does the computer
00:54:03.210 --> 00:54:10.800
decide how to adapt the grid? Because the
data where's the high density comes after
00:54:10.800 --> 00:54:17.660
making the grid...
miosta: Yes, this is actually quite an
00:54:17.660 --> 00:54:25.470
interesting and also not quite easy to
answer question. Let me try to give a
00:54:25.470 --> 00:54:34.300
breakdown nutshell answer here.
The thing is, you measure and evaluate the
00:54:34.300 --> 00:54:39.380
velocities. Or in the flux, you also
evaluate the velocity. And if the velocity
00:54:39.380 --> 00:54:44.840
goes high, you know there's a lot
happening. So, we need a smaller grid then
00:54:44.840 --> 00:54:50.420
there. So, we try to create more grid
cells where we have a higher velocity. In
00:54:50.420 --> 00:54:55.050
a nutshell, this is of course in an
algorithm a bit harder to actually
00:54:55.050 --> 00:55:00.000
achieve. But this is the idea. We measured
the velocities at each point. And then if
00:55:00.000 --> 00:55:03.510
we measure a high velocity,
we change to a smaller grid.
00:55:03.510 --> 00:55:08.640
Mic2: So, you can predict where the mass
will go and whether densities are getting high.
00:55:08.640 --> 00:55:12.600
miosta: Exactly. Step by step so to say.
00:55:12.600 --> 00:55:15.890
Mic2: Thanks
Herald: We stay with Microphone 2.
00:55:15.890 --> 00:55:20.640
Mic2: Okay. I've got a bit of a classical
question. So, I guess a lot relies on your
00:55:20.640 --> 00:55:25.201
initial conditions and I have two
questions related to that. So first, I
00:55:25.201 --> 00:55:30.670
guess they are inspired by observations.
What are the uncertainties that you have?
00:55:30.670 --> 00:55:33.850
And B, then what is the impact if you
change your initial conditions like the
00:55:33.850 --> 00:55:41.170
density in the disc?
miosta: Yeah, right now my main research
00:55:41.170 --> 00:55:46.110
is actually figuring out a sensible
initial conditions or parameters for a
00:55:46.110 --> 00:55:53.220
disc. If you just let it have an initial
set of conditions and a sensible set of
00:55:53.220 --> 00:56:00.420
parameters and let it run very long, you
expect a system hopefully to convert to
00:56:00.420 --> 00:56:05.130
the state that it should be in. But your
parameters are of course very important.
00:56:05.130 --> 00:56:12.240
And here we go back to what we can
actually understand from observations. And
00:56:12.240 --> 00:56:17.880
what we need for example is the density,
for example. And that is something we try
00:56:17.880 --> 00:56:24.900
to estimate from the light we see in these
discs that you saw in this nice grid with
00:56:24.900 --> 00:56:31.110
all these discs we estimate OK, what's the
average light there? What should then be
00:56:31.110 --> 00:56:37.790
the average densities of dust
and gas in comparable disks.
00:56:37.790 --> 00:56:42.890
Mic2: Okay, thanks.
Herald: Okay, one more at number two.
00:56:42.890 --> 00:56:50.150
Mic2: Yes. Thank you for the talk. When
you increase the detail on the grid and
00:56:50.150 --> 00:56:59.550
you learn more. When you want to compute
the gravitational force in one cell, you
00:56:59.550 --> 00:57:05.090
have to somehow hold masses from the all
the other cells. So, the complexity of the
00:57:05.090 --> 00:57:07.090
calculus grows.
miosta: Yes
00:57:07.090 --> 00:57:13.820
Mic2: Quadratically, at the square of the...
how do you solve that? With more CPUs?
00:57:13.820 --> 00:57:20.930
caro: Well, that would be one way to do
that. But there are ways to simplify if
00:57:20.930 --> 00:57:26.290
you have a lot of particles in one
direction and they are far away from the
00:57:26.290 --> 00:57:34.400
object you're looking at. So, yeah. So, if
you have several balls here and one ball
00:57:34.400 --> 00:57:41.710
here, then you can include all these balls
or you can think of them as one ball. So,
00:57:41.710 --> 00:57:48.770
it depends on how you look at it. So, how
you define how many particles you can take
00:57:48.770 --> 00:58:02.230
together is when you look at the angle of
this... many particles we'll have from the
00:58:02.230 --> 00:58:08.040
seen from the object you're looking at.
And you can define a critical angle. And
00:58:08.040 --> 00:58:14.230
if an object gets smaller or if lot of
objects get smaller than this angle, you
00:58:14.230 --> 00:58:20.200
can just say, OK, that's one object. So,
that's a way to simplify this method. And
00:58:20.200 --> 00:58:23.390
there are some, yeah,
I think that's the main idea.
00:58:23.390 --> 00:58:30.920
Herald: Okay, we have another one.
Mic2: Do you have a strategy to check if
00:58:30.920 --> 00:58:35.890
the simulation will give a valuable
solution or does it happen a lot that you
00:58:35.890 --> 00:58:42.060
wait one week for the calculation and find
out OK it's total trash or it crashed in
00:58:42.060 --> 00:58:45.400
the time.
caro: So, that also depends on the program
00:58:45.400 --> 00:58:53.240
you're using. So, in FARGO3D, it gives
these outputs after a certain amount of
00:58:53.240 --> 00:58:58.980
calculation steps and you can already look
at those outputs before the simulation is
00:58:58.980 --> 00:59:05.210
finished. So, that would be a way to
control if it's really working. Yeah, but
00:59:05.210 --> 00:59:11.530
I think...
miosta: It's the same for PLUTO. So, there
00:59:11.530 --> 00:59:18.040
is a difference between timesteps and
actually output steps. So and you could
00:59:18.040 --> 00:59:23.490
define your output steps not and as the
whole simulation, but you can look at each
00:59:23.490 --> 00:59:31.150
output step as soon as it's produced. So,
I usually get like 500 outputs, but I
00:59:31.150 --> 00:59:36.630
already can look at the first and second after
maybe half an hour or something like that.
00:59:36.630 --> 00:59:39.950
caro: Yeah, but it also happens that you
00:59:39.950 --> 00:59:44.060
start a simulation and wait, and wait, and
wait and then see you put something wrong
00:59:44.060 --> 00:59:48.760
in there and well then you have to do it
again. So, this happens as well.
00:59:48.760 --> 00:59:53.070
Mic2: Thanks.
Herald: Okay. One final question.
00:59:53.070 --> 01:00:02.240
Mic2: Yeah, OK. Is there a program in
which you can calculate it backwards? So
01:00:02.240 --> 01:00:07.180
that you don't have the starting
conditions but the ending conditions
01:00:07.180 --> 01:00:15.220
and you can calculate how it had started?
miosta: Not for hydrodynamic. If you go to
01:00:15.220 --> 01:00:22.240
n-body, there is a way to go backwards in
time. But for hydrodynamics, the thing is
01:00:22.240 --> 01:00:31.580
that you have turbulent and almost like
chaotic conditions. So, you cannot really
01:00:31.580 --> 01:00:38.810
turn them back in time. With n-body you
can it because actually it's kind of... Well,
01:00:38.810 --> 01:00:44.890
it's not analytically solved, but it's
much closer than like turbulences,
01:00:44.890 --> 01:00:50.470
streams, spirals and all the
things you saw in the simulations.
01:00:50.470 --> 01:00:57.561
Herald: OK, I guess that brings us to the
end of the talk and of the session. Thank
01:00:57.561 --> 01:01:03.266
you for the discussion and of course,
thank you guys for the presentation.
01:01:03.266 --> 01:01:16.730
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