#scifi If we want a planet on which human beings can live, we’d better have a suitable temperature—not only one at which human beings are comfortable, but one at which water can exist in all three phases, allowing for temperature variations from equator to pole and through the height of the atmosphere. So what determines the temperature of a planet?
The starting point is what is called the equilibrium temperature of a planet. This is based on the idea that the planet is receiving energy from the sun at the same rate it is radiating it out to space.
Any object that is not at absolute zero temperature radiates energy, and at what we consider comfortable temperatures, most of that energy is in what we call the thermal infrared portion of the spectrum. We can’t see it, but we can feel it. On a very cold day, we may say that the ground and sky are radiating cold, but that is only because our skin is at a higher temperature and is radiating energy away faster than it is receiving it back.
In fact, the amount of energy an object radiates is directly proportional to the fourth power of the absolute temperature. For a planet, then, the total energy radiated is proportional to the surface area of the planet times the absolute temperature multiplied by itself four times.
But the energy received from the sun is proportional to the luminosity of the planet’s sun divided by the square of the planet’s distance from the sun times the cross sectional area of the planet. For spherical planets, the cross sectional area is one fourth of the surface area. A fraction of that radiation will be reflected out to space and can be ignored. This fraction is called the albedo, or reflectivity of the planet.
When all of these things are put together, we can get an equation for what is called the equilibrium temperature of the planet. The table shows this temperature (in degrees Farenheit) for three planets of the solar system: Venus, Earth, and Mars. It also shows their real surface temperatures.
What happened? These planets have atmospheres! The equilibrium temperature is the average temperature of the radiating layer, which is generally somewhere above the surface, in the colder part of the atmosphere. How high depends on the wavelength. In visible wavelengths, the outgoing radiation is from the visible surface—partly cloud tops (usually colder than the ground) and partly ground or ocean. Far more important is the fact that some of the gasses in our atmosphere—notably carbon dioxide, water vapor, and methane–are opaque in large parts of the thermal infrared. Most of the energy at temperatures we are comfortable at is radiated in the thermal infrared.
This is what is meant by the greenhouse effect, which actually has very little to do with why a greenhouse stays warm, especially during the day. The difference between the Earth’s equilibrium temperature and the actual temperature is a measure of the greenhouse effect with the current amounts of carbon dioxide, water vapor, methane and a few other radiatively active gasses in the atmosphere.
We are adding carbon dioxide to the atmosphere at an unprecedented rate. Further, the warming of the oceans is adding water vapor, a second important greenhouse gas, and agriculture, permafrost thawing and possibly the thawing of methane clathrates is adding methane as well. All of this means that the radiation the Earth is returning to space is coming from higher and higher in the atmosphere, and since the rate at which the air cools with elevation is unlikely to undergo major change, the surface gets hotter.
What does all this have to do with planet building? It suggests that the solar constant, proportional the luminosity of the star around which the fictional planet is circling divided by the square of the distance between the planet and its sun, had better be about the same as ours if the atmosphere and albedo are similar. If we make this assumption, we can say some things about the way the sun will look and the length of the year.