It's important to distinguish between the surface temperature of a planet and the planet's interior temperature. Before, when we were talking about small planets cooling off faster, we were talking about their insides cooling off. The insides are kept hot by decay of radioactive minerals, but the inside of a planet like Earth doesn't do much to keep the outside hot.

The heat that we get from the inside of the Earth is much less than the heat that we get, living here on the surface, from the Sun's light.

We're going to be talking about the mean surface temperature of a planet--mean meaning average, because the side in day is hotter, and it's hotter near the equator. But what controls the average temperature of a planet?

There are mainly 3 things that go into determining the mean surface temperature of a planet:

1. How far that planet is from the Sun

   Closer to the Sun: a planet gets hotter, further from the Sun: a planet gets colder.

   This is roughly what we observe, although there are exceptions. Venus is hotter than Mercury, even though it is further from the Sun.

   Mathematically, the temperature of a planet depends on 1/sqrt(distance from Sun). This first approximation assumes that all of the Sun's light reaching a planet gets absorbed. The size of the planet doesn't matter in determining how hot it is. A bigger planet gets more heat from the Sun, but has a larger area to heat up.

   The mathematical relationship between distance of a planet from the Sun and temperature--you can see it proved on the lab site or in your book, and I did it in class--the physical reason for this is as follows.

   The amount of heat from the Sun reaching the Earth must equal the amount of heat coming out of the Earth. The heat from the Sun is L/(4 pi R_orb²) times pi R_Earth².

   What's this? Well, let's call L the luminosity of the Sun, how much energy comes out in a second. The energy hitting every square meter at the distance of the Earth's orbit is L/(4 pi R_orb²). We're just dividing that energy by the surface area of a sphere surrounding the Sun reaching to Earth's orbit to find the energy per unit area.

   Then we multiply that energy per second per square meter by this area of the Earth (as viewed from the Sun, that's just pi R_Earth² because the Sun sees the Earth as a circle, as we see the Sun). That tells us the total energy reaching the Earth.

   Now here's the physical principle behind the temperature of the Earth (using this first approximation). The total amount of heat reaching the Earth must equal the total amount of heat energy leaving the Earth. Otherwise, the Earth would either keep heating up over time, or keep cooling down.

   But in a steady state that doesn't happen. The Earth heats up, and the hotter it becomes, the more heat it gives off, until the heat coming in and going out are equal. So if the Earth's temperature is T,

   Heat In = Heat Out

   L / (4 pi R_orbit²) (pi R_Earth² = 4 pi R_Earth² sigma T⁴

   What does all this mean? Well, the right hand side of the equation is how much heat energy the Earth is giving off. Remember the Stefan-Boltzmann law? It says that something with surface area A and temperature T, if it is thick enough to radiate like a blackbody, will give off a total amount of light of A sigma T⁴. sigma is a constant, in MKS units it is 5.7x10^-8, but don't worry about that! Remember this is the law that's just saying that something that's hotter is giving off more energy (per unit area).

   The result fo setting these 2 formulas equal to each other (heat coming in to the Earth, and heat going off) is that the temperature of the Earth depends on 1/sqrt(R_Orbit) and does not depend on the size of our planet at all.

2. One big thing that this approximation ignores is the albedo of the planet. It assumes that all of the Sun's energy that reaches a planet ends up heating it up. In fact, much of the sunlight will just be reflected into space. For example, clouds are very good at reflecting light.

   The measure of how much of the Sun's light is reflected is called the albedo. The same root as the word "albino", or the "albumin" in an egg, or "Albus Dumbledore" from Harry Potter (who has white hair.) It means white or reflective. The albedo is how much light is coming out, divided by how much light went in. For complete reflection the albedo is 1, and for complete absorption the albedo is 0.

3. Another thing that affects the surface temperature of a planet is the Greenhouse Effect.

   We're used to thinking of this as only a very bad thing here on Earth. And yes, it could be very bad. But without any Greenhouse Effect at all, the Earth would be completely frozen over!

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What gases can a planet hold onto? That depends on the planet's escape velocity! The old idea comes back to haunt us. If you throw up something from a planet at its escape velocity or above, then it can break free of the gravity and never fall back down.

If there's some gas in the upper atmosphere that has the escape velocity, it can leave the planet. Now, the velocity of the atoms in a gas depends on the temperature. That's what we understand temperature to be: a measure of the average energy of random motion. When something is hot, all the atoms are whizzing around randomly bumping in to each other and whatever container they're in. That's why when you heat up a gas, its pressure goes up too--the atoms are bumping against the sides of the container with more velocity.

The relationship between temperature and average energy is given by E = kT. E is the average energy of an atom or molecule, and T is the temperature in Kelvins. k is a constant, Boltzmann's constant, equal to 1.38x10^-23 in MKS units. This just says the more heat, the more the average energy.

In the graph below, we see the distribution of velocities in 2 gases with different temperatures. The hotter gas has a higher average velocity.

So a hot gas is more likely to escape from a planet. The gas doesn't escape all the way directly from the surface, but gas in the upper atmosphere has some chance of having the escape velocity and being aimed out towards space--and sometimes it breaks free.

A light element is also more likely to escape from a planet. This is one reason why there isn't so much hydrogen and helium on the Earth, even though these elements make up 98% of most matter in the Universe. (Another reason is that ices couldn't condense at the Earth's distance from the Sun.)

Why is this? It's because in a gas, the atoms tend to all have the same energy. Energy of motion is called kinetic energy and is related to the mass and velocity of the object: E=1/2 m v². The faster or more massive the particles in motion, the more kinetic energy they have.

So hydrogen and oxygen gas atoms both have the same amount of energy in an atmosphere, but because hydrogen is lighter, in order to have the same energy it will have to move faster. That means that it is more likely to reach the escape velocity and escape!