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

Why is Mercury Hot?

This lecture uses energy balance to determine the effective temperature of planets. The lecture introduces the idea of temperature, black body spectrum and luminosity. We then calculate the location of the habitable zone, the range of distances where planets are likely to have liquid water.

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

Department of Astrophysics

Welcome back.

Today's lecture, we are going to talk

about what sets the temperature of planets.

Why is Mercury hot? Uranus cold?

Why is Venus so damn hot?

So, let's look at the three parts of our lecture.

We're going to begin in the first section with an overview of the

law of the basic physics talking about the temperature, what temperature means.

What's a black body?

What do we mean when we talk about luminosity?

Then we'll use energy conservation to compute the temperature of the

planets, and then apply it to see what makes planets habitable.

This will be a much more quantitative lecture than the previous lecture.

What, what we're going to try to do is introduce a lot of the

basic physics, show how we can use

that physics to understand the properties of planets.

For some of you who are familiar with this physics, this may seem simple.

For others of you who are not used

to quantitative reasoning, this may be more challenging.

Let me encourage you, if things don't make sense, to go back, listen to the

lecture again, work through the underlying physics

and algebra, and see what you can understand.

I think you'll find it

to be very rewarding, because you'll be able to see how we

can use basic physical principles to

make general statements about properties of planets.

So let's begin by reviewing temperature.

There are three different temperature scales that are often used.

Here in the United States, we use the Fahrenheit scale.

That gets a sad face because the Fahrenheit scale not

well adapted to doing much in the way scientifically.

It's a funny scale in that water freezes

at 32, water boils at 212 degrees Fahrenheit.

And absolute zero, the minimum temperature that things can

reach in the universe, is minus 459 degrees Fahrenheit.

In much of the ret, rest of the world, people use the Celsius scale.

The Celsius scale has the advantage of water

freezing at 0, boiling at a 100 degrees.

So we have a nice factor of 100, over which water exists as a liquid.

For those, as we think about habitability in

planets, this is an interesting range to think about.

All life that we know about on Earth requires water.

And while it's possible life could exist in another

medium, water seems to be so fundamental to life that

we often start by assuming that we need to have

conditions, to have liquid water, in order to have habitability.

This may be the wrong assumption, but since we only know

of life on our planet, we try to extrapolate to other systems.

One of the things we do is start by thinking about whether water could exist.

Most physicists don't like to use either the

Fahrenheit or Celsius scale, but the Kelvin scale.

The Kelvin scale is easily related to Celsius.

The temperature in Celsius is the temperature in Kelvin minus 273 degrees.

In, on the Kelvin scale, the coldest temperatures can get is absolute zero.

That's the nice thing about the

Kelvin scale, you never have negative temperatures.

Temperature starts at 0, water freezes at 273 degrees

Kelvin, and water boils at 373 degrees Kelvin.

You notice we just have to subtract 273 to go from Kelvin to Celsius.

273 minus 273 is

0, so water freezes at 273 degrees Kelvin.

We'll refer back to this diagram later, but it's a

scale that I think is important for you to understand.

because it's going to be much easier to talk about

properties of stars and of planets using the Kelvin scale.

And of course, we're all familiar with what's hot and what's

cold, in terms of fire being hot and ice cream being cold.

But I also want to remind you when you think

about hot and cold, you should think about the

atoms in a gas when it's hot moving very

quickly, or when it's cold, they move very slowly.

We go to absolute zero, atoms or molecules

don't move at all.

The next concept I want to talk about is luminosity.

Luminosity is the total energy that an object emits per unit time.

We can think about a light bulb emitting light.

Rather the energy it puts out in terms of number of watts.

A typical light bulb might emit about 100 watts of energy.

And since luminosity is energy

per unit time, that's 100 joules per second.

As I stand here talking to you, my body's

also emitting about 100 watts of heat, a comparable number.

So then you can think about watts as a

unit where things that you encounter in daily life.

Your computer, light, your own body are

emitting tens or hundreds of watts of energy.

We look at the Sun, the Sun emits 4 times 10 to the 26th watts.

Or equivalently, if I don't want to use scientific notation, 400,

followed by another 24 zeroes, watts.

So that's a lot of light bulbs. So the Sun is much more energetic.

But still these, what we want to express this all in the same units.

Watts or equivalently, joules per second.

The light that comes from the Sun is mostly visible on your infrared light.

I want to remind you about the

basic properties of the electromagnetic spectrum.

I think most of us are familiar with the basic

properties of visible light that if we look at a

rainbow or look at light through a prism, we can

divide light into red, orange, yellow, green, blue, or violet light.

The properties of light depend on its wavelength.

If light has a shorter wavelength, visible light might appear blue or violet.

The wavelengths here are expressed in nanometers, or billionths of a meter.

Blue light is emitted about 500 nanometers.

Yellow light, the wavelength is a bit longer, 600 nanometers.

While red light is red atom, is about 700 nanometers.

The visible spectrum is but a small portion of the overall spectrum of light.

As we go towards shorter wavelengths, we move to higher energies.

We can look, and we'll begin with

visible, then ultraviolet, then X-ray, then gamma rays.

These X-rays and gamma rays

carry an enormous amount of energy.

Most of them fortunately do not reach the ground.

These X-rays and gamma rays are absorbed by our atmosphere.

As we move towards longer wavelengths, we're looking at less energetic photons.

So we can move from the red to infrared light, to microwaves, to radio waves.

Most of the Sun's light

is emitted at visible and infrared light.

The Earth itself is radiating as is our bodies.

Our bodies are mostly radiated in the infrared.

An important idea when we look at objects is

they emit what's, usually what's called a black body spectrum.

So what's plotted here is how much energy is emitted as

a functional wavelength when you look at objects of different temperatures.

This yellow curve here shows the energy emitted by the Sun.

The Sun emits most of its energy here around

visible wavelengths, but you'll notice the Sun also emits ultraviolet wavelength.

These ultraviolet waves emitted by the Sun is what causes skin cancer.

And one of the reasons we put on suntan lotion

or try to stay out of the direct sun is

to protect our skin from this energetic light coming from

the Sun that are able to produce mutations in our cells.

While the Sun emits a lot of its light at visible wavelengths, you'll notice the

Sun, the yellow line in this plot, also emits a lot of radiation in the infrared.

The red curve shows radiation emitted by the Earth.

Recall that 0 degrees centigrade is 273

Kelvin, so this red curve at 300 Kelvin

are equivalently 27 degrees centigrade or Celsius, corresponds

to the radiation emitted by the Earth. You could see that the Earth emits

very little in this visible range. Most of the Earth's

radiation is coming out with, at wavelengths of around 10 microns.

So the Earth is emitting primarily in the infrared.

And you can see the characteristic wavelength is a function of energy.

As we go from a planet like Saturn to the Earth to

Mercury, the characteristic wavelength that emits that moves upward.

Finally, we get to stars like a red dwarf star,

the Sun, or a blue star, they say an A star.

As we move up in temperature, the

characteristic wavelength shifts this way toward shorter wavelengths.

These curves also show the total energy emitted.

And you'll

notice there's much more area under the yellow curve than the red curve.

The Sun, per unit area, emits a lot more energy than the Earth does.

We want to look at how much energy the Sun or the Earth emits.

We have, want to calculate the flux per unit area, and then

figure out how much area the Sun or the Earth has.

So before we go further, we do a little quick geometry review.

Let me remind you that if I draw a circle, the circumference of a circle

is 2 pi times its radius. The radius of a circle recalls

the distance from its center to its edge.

The diameter goes all the way across, the diameter of a circle is twice its radius.

So the circumference of a circle is 2 pi R.

If I draw a sphere like the Earth, the

surface area of the sphere is 4 pi R squared.

And since we want to compute the amount of energy emitted by the

Sun or the Earth, we want to emit the, compute the

energy per unit area, then multiply it by the surface area.

Okay, now let's turn to one of the technical parts of this discussion.

The Stefan-Boltzmann Law.

If we take this curve and ask how much energy is emitted per unit area by a

star or by the Earth, that flux goes

as temperature to the fourth power times a constant.

This constant written

here is called the Stefan-Boltzmann constant, and this is the flux submitted

by the Sun or by the Earth.

If I want to compute the luminosity, how bright the Sun

is, I take its surface area, I multiply by the flux.

Surface area, recall is 4 pi R squared. The flux is sigma temperature to the 4th.

This gives me one of the most important relationships we want

to write down to characterize what's going on with the star.

The luminosity, the energy a star emits per unit time, depends on how

big the star is and how hot the star is. If a star is bigger,

it's going to emit more radiation.

A red giant star is going to emit a

lot more energy than a small little red dwarf star.

A hotter star also emits more energy than a cold star.

And it's going to scale this radius squared, and temperature to the 4th power.

This equation will tell us the basic properties we'll observe of stars.

Just notice before we go further

to something about constant, you'll notice this

constants in units of watts per meter squared per Kelvin to the 4th power.

If I want to compute the luminosity of the Sun, I need

to stick in its radius in meters and its temperature in Kelvin.

If I want to use this equation with the constant

expressed this way to compute the luminosity of the Sun.

Good. Let's apply this.

And let's compute how much energy does the Sun radiate.

The sun's temperature is about 6,000 degrees

Kelvin, or to be precise, 5,758 degrees Kelvin.

The radius of the Sun is about 7 times 10 to the 8th meters.

I can plug this into this equation,

sticking in the radius squared, the constant, and the temperature of the Sun.

Plugging in these numbers, you'll find that the Sun's

luminosity is about 4 times 10 to the 26th watts.

Thus we can relate the Sun's temperature in

radius to the amount of energy that comes out.

For the problem, I'd like you to apply this to the Earth.

So you should now exit the lecture, look up the radius of the Earth,

take as its typical temperature, about 16

degrees centigrade or equivalently 289 degrees Kelvin.

Plug in the temperature, the radius, and compute the luminosity of the Earth.

And then compare that to the luminosity of the Sun.

See you in a few moments.

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