Are we alone? This course introduces core concepts in astronomy, biology, and planetary science that enable the student to speculate scientifically about this profound question and invent their own solar systems.

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From the course by Princeton University

Imagining Other Earths

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Are we alone? This course introduces core concepts in astronomy, biology, and planetary science that enable the student to speculate scientifically about this profound question and invent their own solar systems.

From the lesson

Quantum Mechanics and Fingerprinting Planets

This lecture introduces the Pauli exclusion principle, which requires that only one electron can be in any state. We use this principle to understand the properties of materials and the atomic lines seen in planetary and stellar structure.

- David SpergelCharles Young Professor of Astronomy on the Class of 1897 Foundation and Chair

Department of Astrophysics

Welcome back.

Now we're going to talk about Quantum

Mechanics, introduce some of its basic ideas, and

then apply it to brown dwarfs, the

height of mountains, and how we fingerprint planets.

So, first we'll talk about quantum mechanics and the heights of mountains.

Then, quantum mechanics and how it determines some of the structure

of atoms, and then we'll apply it to Jupiters and brown dwarfs.

So, quantum mechanics.

Let's begin with two important ideas in quantum mechanics.

The first is what's called the Heisenberg Uncertainty Principal.

That you can't simultaneously know the position and momentum of a particle.

That there's an intrinsic uncertainty in figuring out exactly

where a particle is, and exactly what it's momentum is.

And no matter what you do, you can't determine

the position and momentum better than to a constant.

Actually this constant h we've seen before, that's

the Planck's constant involved in properties of radiation.

Then Planck's constant divided by two pi

As the joke goes, when Heisenberg was ticketed for speeding, the cop asked

him do you know how fast you were going, and he said no, but I know where I am.

As Heisenberg knew you couldn't simultaneously

know your position and your momentum.

The second important principle we want to use is called Pauli's Exclusion Principle.

The idea here is that you can only put a single electron in any given state.

And by a state, we're going to mean a

position, a given value of position and momentum.

So what does that imply?

It has an interesting effect when we think about properties of things like metals.

In a metal, you should think about electrons.

Some electrons tightly bound around each nucleus.

And then, a set of electrons that are are free, they're

floating around in what we call a sea of electrons in the middle.

These electrons are whizzing around, back and forth, through the metal.

That's why metals are such good conductors, because electrons, there

are some electrons that could move freely through a metal.

Now, these electrons, as they move around, they have a position and a momentum.

And we can think of each electron

occupying a different value of position and momentum.

If we were to make a plot showing where

the different states we can put the electrons in.

You can imagine making a graph where you write

down the momentum of each electron and its position.

And each electron

occupies a different position in momentum position space.

And we can't put them closer together

than that, because of the Heisenberg Uncertainty Principle.

And we can't put two electrons in

each block because of the Pauli Exclusion Principle.

So if we add more and more electrons to the material, they take up,

they have larger and larger momentum, and with more momentum, we have more energy.

We could quantify this, and that's going to determine, the

properties of metals and as we'll see white dwarf stars.

Let's say I have a certain number of electrons,

a 1,000 electrons I want to fit in a box.

Given the number of electrons I have in my metal.

That's going to determine the average distance between them.

The more electrons, the closer together they'll have to be.

So as the electron density goes up,

the average space between the particles goes down.

They get more and more tightly bunched together.

As they get more and more tightly bunched together.

Their momentum goes up.

So we can solve for the momentum in terms of the spacing and positions.

The momentum goes here h bar is also the same as h over two pi.

It's a notation we use a lot.

We can solve for the momentum.

In terms of plancks constant and the number density of electrons.

The more we pack the electrons in, the higher the density.

The faster they have to move because of the Uncertainty Principle,

so the momentum goes up and hence their velocity goes up.

So the more we pack them in, the more momentum they have, the faster they move.

Eventually, as they move faster and faster, their pressure goes up.

The pressure in an electron is going to depend on the

number of electrons, times their mass, times the velocity squared.

So that implies the pressure in a metal, depends

on the electron number density to the five thirds power.

This is the important result that we're going to need from Quantum Mechanics.

This is different from the Ideal Gas Law.

Remember, the Ideal Gas Law told us, in a

gas, the pressure depends on the density and the temperature.

The hotter it is, the higher the pressure.

Cold gas has low pressure.

That's not true of our solids.

The pressure in the solid depends only on the electron

number density You'll notice, it doesn't depend on the temperature.

That's because the electron velocity is

determined by the Heisenberg Uncertainty Principle.

This is also called the generosity pressure.

So the pressure goes as density to five thirds, doesn't care about temperature.

That has a big impact on the behavior of materials and the structure of planets

like Jupiter and low temperature stars called brown dwarfs.

It's also the source of support for things like white dwarfs.

So let's see how this affects the properties of materials.

Let's think about our metal.

As the, the binding energy of that metal is

associated with the pressure you put the metal on.

If you put more and more pressure on the metal, make it denser and denser.

As you make the squeezed material more and

more tightly, eventually you squeeze it so much

it's under so much pressure that the electrons

move around so quickly, that they're no longer bound.

So if you take a metal and squeeze it enough, it will heat up so much

the electrons will gain so much energy, that,

that pressure will overcome the atomic binding energy.

Typically about electron volt for atoms and it will cause the metal to melt.

The fact that there's a maximum pressure that

material can handle and this is kind of

a rough explanation of it, means

that there's a maximum height, for the tallest mountain.

Let's work out what the tallest mountain on Earth could be.

In order to work out the height

of the mountain, we want to balance the characteristic

velocity of the material against the strength

of gravity and the height of the mountain.

So we can equate the height of the mountain to the

melting temperature divided by the electron

mass, times the strength of gravity,

or equivalently, the sound speed of the material, which is about

600 meters per second, divided by the strength of gravity on earth.

The strength of gravity is on the surface, is about ten meters per second squared.

Sticking in these numbers, that implies that the height of the

tallest mountain you can make for the base of the mantel,

the crust where it melts to the top, is around 30

kilometers and if you look at the Earth's crust, that's about right.

You stack things higher than that, it's going to melt.

So, you can see that basic physics determines the height of mountains, and

the height of mountains are set by the balance between the strength of materials.

How much you can compress something before it melts, and the strength of gravity.

Now, this is interesting implications when we

look elsewhere in our solar system or in space.

The strength of materials are going to be the same everywhere.

What's going to vary as we go from the Earth to Mars, to an

asteroid, to an extrasolar planet, is the strength of gravity.

And you can see if gravity's stronger, the tallest mountains are shorter.

If gravity's weaker, mountains could be quite high.

So, given this relationship, I'd like you to now go off and figure out,

what's going to be the tallest mountain that you expect on Mars or on Pluto.

Work out the strength of gravity on Mars's surface Right?

Remember, that's going to go as GMmars divided by radius of Mars squared.

That's the strength of gravity on Mars.

I want you to compare that, to the earth and figure out, do you think

mountains will be higher on Mars or on the Earth and then go check that.

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