Calculating Planetary Energy Balance & Temperature

Calculating Planetary Energy Balance & Temperature

Calculating Planetary Energy BalanceHow hot or cold is the surface of a planet? Using some fairly simple physics and math, you can calculate the expected temperature of a planet, including Earth. This page explains how!
 
What do we mean by the "expected temperature" of a planet? Mostly it means that we'll simplify things by leaving out the effects of an atmosphere or oceans on the average global temperature. It turns out that oceans and atmospheres can have a big influence on a planet's temperature... we'll have more to say about that later. For now, lets look at the simple case of a planet without air or water. We'll discover along the way that without certain chemicals in Earth's atmosphere, our home planet would not be the most comfortable abode for life.
 
Visible light from the Sun carries energy to the planets in our solar system. That sunlight is absorbed by the planet's surface, heating the ground. Any object with a temperature above absolute zero emits electromagnetic radiation. For planets, that outgoing EM radiation takes the form of infrared "light". The planet will continue to warm until the outgoing infrared energy exactly balances the incoming energy from sunlight. Scientists call this balance "thermal equilibrium". With a bit of basic physics we can calculate the temperature at which this state of thermal equilibrium is reached.
 
Satellites have directly measured the amount of energy arriving at Earth from the Sun as sunlight. Although this value varies slightly over time, it is usually very close to 1,361 watts of power per square meter. To visualize this, imagine lighting a small closet with about 13 or 14 one hundred watt light bulbs. Scientist refer to the amount of incoming energy from sunlight as "insolation". The specific value at Earth of 1,361 W/m2 is called the "solar constant".
 
In order to calculate the total amount of energy arriving at Earth, we need to know how much area is being lit. We then multiply the area by the insolation (in units of energy flow per unit area) to find out the total amount of incoming energy.
 
It turns out that we can simplify our calculation of area by noticing that the amount of light intercepted by our spherical planet is exactly the same as the amount that would be blocked by a flat disk with the same diameter as Earth, as shown in the diagram below.
 

 

The area of a circle is pi times the radius of the circle squared. In this case, the circle's radius is simply the radius of Earth, which is about 6,371 km (3,959 miles) on average. If we multiply this area by the amount of energy per unit area - the solar "insolation" mentioned above, we can determine the total amount of energy intercepted by Earth:

  • E = total energy intercepted (technically, energy flux = energy per unit time, in watts)
  • KS = solar insolation ("solar constant") = 1,361 watts per square meter
  • RE = radius of Earth = 6,371 km = 6,371,000 meters

Plugging in values and solving for E, we find that our planet intercepts about 174 petawatts of sunlight... quite a lot of energy!

Since Earth is not completely black, some of this energy is reflected away and not absorbed by our planet. Scientists use the term albedo to describe how much light a planet or surface reflects away. A planet completely covered with snow or ice would have an albedo close to 100%, while a completely dark planet would have an albedo close to zero. To determine how much energy Earth absorbs from sunlight, we must multiply the energy intercepted (that we calculated above) times one minus the albedo value; since albedo represents the light reflected away, one minus albedo equals the light energy absorbed. Our equation for total energy absorbed becomes:

Now that we have a value for the energy flowing into the Earth system, let's calculate the energy flowing out.

The sunlight Earth absorbs heats our planet. Any object with a temperature above absolute zero emits electromagnetic (EM) radiation. In the case of Earth, this EM radiation takes the form of longwave, infrared radiation (or IR "light").

Earth emits energy as longwave infrared radiation

In the 1800s, two scientists determined that the amount of radiation emitted by an object depends on the temperature of the object. The equation for this relationship is called the Stefan-Boltzmann law. It was determined experimentally by Joseph Stefan in 1879 and theoretically derived by Ludwig Boltzmann in 1844. Notice that the amount of energy emitted is proportional to the 4th power of the temperature. Energy emissions go up A LOT as temperature rises!

Stefan-Boltzmann law

  • j* = energy flux = energy per unit time per unit area (joules per second per square meter or watts per square meter)
  • σ = Stefan-Boltzmann constant = 5.670373 x 10-8 watts / m2 K4 (m = meters, K = kelvins)
  • T = temperature (in the Kelvin scale)

The Stefan-Boltzmann law tells us how much infrared energy Earth will emit per unit area. We need to multiply this by the total area of Earth's surface to calculate the total amount of energy emitted by Earth. Since Earth rotates, all of it's surface is heated by sunlight. Therefore, the whole surface of the spherical planet emits infrared radiation. We can't use the same shortcut we used for incoming sunlight by treating Earth as equivalent to a disk. Geometry tells us that the surface area of a sphere is four times pi times the radius of the sphere squared. Multiplying the energy emissions per unit area times the surface area of Earth, we derive an expression for Earth's total infrared energy emissions:

The law of conservation of energy tells us that the energy emitted must be equal to the energy absorbed. Setting these two values equal, we can substitute in the expressions for each. Noting that pi times Earth's radius squared appears on both sides of the equation, we can use a little algebra to simplify the result:

Conservation of Energy: Energy In = Energy Out

Since the values for the solar constant (KS), Earth's albedo, and the Stefan-Boltzmann constant (σ) are all known, it is possible to solve this equation for temperature (T). Using a little more algebra, we can write the expression above as:

Earth's overall, average albedo is about 0.31 (or 31%). The value of the Stefan-Boltzmann constant (σ) is 5.6704 x 10-8 watts / m2 K4. Plugging these numbers and the value for KS into the equation, we can calculate Earth's expected temperature:

Earth Temperature equation

Converting to the more familiar Celcius and Fahrenheit temperature scales, we get:

Based on this calculation, Earth's expected average global temperature is well below the freezing point of water!

Earth's actual average global temperature is around 14° C (57° F). Our planet is warmer than predicted by 34° C (60° F). That's a pretty big difference!

Greenhouse EffectWhy is Earth's temperature so much warmer than our calculations predicted? Certain gases in the atmosphere trap some extra heat, warming our planet like a blanket. This extra warming is called the greenhouse effect. Without it, our planet would be a frozen ball of ice. Thanks to the natural greenhouse effect, Earth is comfortable place for life as we know it. However, too much of a good thing can cause problems. In recent decades, a rise in the amount of greenhouse gases has begun to warm Earth a bit too much.

This calculation of the expected temperature can be done for other planets as well. To do so, you need to adjust the value of the solar insolation, KS. A planet closer to the Sun receives more energy, so KS is larger. Planets further from the Sun than Earth receive less sunlight, so KS has a smaller value. Knowing the planet's distance from the Sun, you can make a ratio with Earth's distance and determine the solar insolation at that planet's distance. Since the amount of sunlight decreases as the square of the distance from the Sun, a planet twice as far from the Sun as Earth would receive on one-fourth as much solar energy.

This energy balance calculation also helped scientists discover a bit of a puzzle from Earth's history. Based on observations of similar stars, astronomers think our Sun is brighter now than it was early in its lifetime. The early Sun was probably only about 70% as bright as it is in modern times. If you multiply KS by 0.7 and use the result in the equations above for the solar insolation of the early Sun, you'll find that Earth would have been much, much colder than it is today. However, there is a lot of geologic evidence that there was liquid - not frozen - water on Earth even very early in our planet's history. How could Earth have liquid water if it was so cold due to the dimmer Sun? This puzzle is called the Faint Young Sun Paradox. This paradox is an area of active scientific research. Some scientists think early Earth may have had much, much higher concentrations of greenhouse gases in its atmosphere; enough to warm the planet above freezing despite the dimmer Sun.

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