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.

Now let's apply these ideas to see what determines the temperature of planets.

And we're going to particular move towards the studying

three particularly interesting cases.

What sets the temperature of Mars, Earth, and Venus?

As we'll see, this triplet sort of a Goldielocks set.

Venus is too hot, Mars too cold, and Earth just right for life.

So now, let's try to apply energy conservation to estimate the temperature

of a planet and see what determines its basic properties.

The idea for energy conservation applying to planets is to say

that the energy that flows into a planet balances the energy that goes out.

Or I can express this as the energy in per unit time or

the luminosity in balances the luminosity out.

This is what we call thermal equilibrium.

This is a pretty good way of describing the overall evolution of a planet's

temperature versus time, is to assume that it's close to constant.

Now in doing this we're making some approximations.

You might have a planet that stores up some energy over time by heating up its

oceans for a little bit, but eventually that energy's going to flow out.

So this is a pretty good way of estimating the basic properties of a planet.

To equate the energy that comes in,

hits it from the star, to the energy that radiates outwards.

So first, we need to compute how much energy from a star strikes a planet.

What's the flux coming from a star?

So here's our star sending energy out in all directions.

The energy that flows out of the star per unit time is its luminosity.

We've already made some estimates of that.

That energy flows out and at a given moment,

that energy flows into a sphere around the star.

If we look at a sphere, Of radius D,

the flux that we measure at this distance is

the luminosity divided by the surface area of the sphere,

that surface area is 4 pi D squared.

So the flux that would reach a planet right here is

the star's luminosity divided by 4 pi D squared.

Recall that we showed In the previous section that the star's luminosity

goes as its temperature times its surface area 4 pi R squared.

So we can plug the star's luminosity into this equation and

find that the flux hitting a planet at distance D

goes to the temperature of the star to the fourth

power times the radius of the star squared,

divided by the distance from the planet to the star.

So this is our star.

So these basic astronomical quantities, the star's temperature,

the radius of a star, the distance from the star to the planet

will determine how much energy hits the planets surface.

So now let's look at the energy that comes into the Earth.

The energy in is going to equal the amount of light that hits

the Earth's surface minus the amount of light that bounces back.

So we have a flux coming into the surface F and some fraction

that we call the albedo times f, bounces back.

So the energy in is one minus the flux, and a times flux,

this quantity hits every part of the Earth's surface.

And the fraction of the surface that is exposed to the sunlight

will go as the area of the Earth seen in projection towards the sun.

The area of the earth is going to go as pi R planet squared.

So we now have all the ingredients we need to compute

how much energy flows into the Earth or any other planet per unit time.

The energy that flows in

is going to depend upon the reflectivity of the surface.

If the planet was completely reflective, all the light hit the planet and

bounce back immediately, nothing would be converted to heating the planet.

On the other hand, if the planet was black so a black planet would have a of zero.

Nothing would bounce back.

All the energy would go in.

So this term here describes the reflectivity of the planet,

how much energy is put in.

The amount of energy that's put in is going to depend upon the temperature

of the star.

A hotter star will give out more luminosity, so it'll heat the planet more.

It will depend upon the size of the star.

A bigger star puts out more energy.

It depends on the distance, the further you are away from the star,

the less energy you get.

This term here means that Pluto receives much less energy from the sun than we do.

Turning it around, Mercury, which is very close to the sun,

gets much more energy per unit area than the Earth does.

And then the total amount of energy absorbed by the planet

will depend on its size.

A bigger planet will present a larger surface to the sun,

it will absorb more light.

So we can see each of these terms

comes in to determining how much energy flows into the planet.

How much energy does the planet radiate?

Well not we can apply the same equation that we applied to radiation from the sun.

The energy this planet radiates depends on the planet's temperature and

the surface area of the planet.

A bigger planet will radiate more, a hotter planet will radiate more.

Now you'll notice in doing this estimate I'm making some simplifying assumptions.

I'm assuming the planet has the same temperature everywhere,

of course this is just an approximation.

I don't know what time you're watching this video, but

regardless of when you're watching it, the temperature

in Singapore I'm willing to bet is higher than the temperature at the South Pole.

But the approximation we're going to make here

is that the Earth has a single uniform temperature.

I would have to do a much more detailed model, though the answer wouldn't change

that much to account for variations in temperature across the Earth's surface.

So here's the total luminosity the Earth sends out.

We want to balance that against the luminosity in, and

that's what we'll do right now.

We're going to work through some equations that will tell us

what temperature planets will have.

It's going to be a balance.

The visible white comes in from the sun, a little bit bounces right back out.

It's only the difference between these two that gets absorbed in the planet surface.

The planet then emits radiation.

We are going to balance these arrows.

We're going to equate the power flowing in from the star to the power flowing out.

Well here's the power flowing in.

There's our reflectivity term, the star's temperature, its size, its distance, and

the radius of the planet.

Here's the energy flowing out.

We're going to equate this with this, it's what we do right here.

Now we can start to simplify this equation.

This term appears on both sides, we'll cancel it.

The radius of the planet appears on both sides.

That pi appears on both sides.

Good, you can now write down a simpler equation here.

We're laying the planet's temperature to the star's temperature.

Let's bring the factor 4 onto this side, take the fourth root of this equation,

and what this gives us is a relatively simple equation that is very general.

This is the beautiful thing about physics is we know we can

compute the temperature of the planet if we know the temperature of the star,

how reflective the planet surface is,

the size of the star and the distance from the star to the planet.

This can be applied to Earth, to Venus, or to any of

the thousand planets that astronomers have discovered around other stars.

And let's look carefully at the dependences here.

A hotter star, everything else being equal means a hotter planet.

It just scales linearly.

You double the temperature of the star, you double the temperature of the planet.

If you have a planet that's more reflective,

more of the light bounces back.

A planet, say, you paint the surface white, that planet will be cooler.

That increases the albido by painting it white.

You could also see that the planet's temperature is going to depend upon

the radius of the star and its distance.

This ratio by the way is the angular size, how big the star seems in the sky.

So if the star appeared to someone as twice as big

as the sun does to the earth, that would mean this ratio here is twice as large.

You can use this by the way when you're watching science fiction movies.

And you'll realize that sometimes stars are simply much too big or much

too small to be consistent with the planet temperature in your science fiction movie.

So I'm going to now use this equation a lot

because it's going to let us study and invent planets and look at how

its properties change as we change the properties of the system as a whole.

Now let's apply this in some questions.

You should now exit this and answer the following two questions, and

then return and we'll discuss more about planets, thanks.

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