A planet in orbit around a star absorbs infrared radiation from it. The average power of the infrared radiation on the planet’s surface is 450 watts per square meter. And the average surface temperature of the planet is a constant 33 degrees Celsius. The planet’s total surface area is five times 10 to the 12th square meters. How many watts of infrared radiation does the planet emit?
Okay, let’s say that this is our star and then this is our planet orbiting around the star. The star is giving off both light as well as heat. And this heat is infrared radiation. As the planet moves around the star, it absorbs some of that infrared radiation. And the average power per unit area of that absorption is 450 watts per square meter. This means that if we were to consider the area of the planet as a whole, every square meter of that area, on average, would receive 450 watts of infrared radiation. And again, we’re averaging that number over the whole planet, even though only one side receives infrared radiation at any one time, the side that faces the star.
Now, in general, when an object absorbs infrared radiation, it also emits it or gives off some of that radiation. Our planet behaves this way, which means we have two factors affecting the temperature of the planet. One factor is the infrared radiation it absorbs from the star, and the other is the infrared radiation it emits or gives off. We could say that when the planet absorbs infrared radiation, energy goes into it. And when it emits infrared radiation, energy is given off.
Now, we’re told something interesting about this planet in the problem statement. We’re told that the average surface temperature of the planet is a constant 33 degrees Celsius. Now, the particular temperature is not as important as the fact that this temperature is constant. This tells us that the rate at which the planet absorbs energy is equal to the rate at which it gives it off.
Note that this planet can only absorb or emit energy through receiving or giving off infrared radiation. If these two factors, the rate of energy coming in and the rate of energy going out, weren’t equal, then the temperature of the planet would either go up or go down. But it doesn’t; it stays constant. This means that, on average, 450 watts per square meter of infrared radiation is absorbed by the planet and also that 450 watts per square meter is given off, again on average.
And now, we’re ready to consider the question posed in this exercise. How many watts of infrared radiation does the planet emit? We’ve seen that the planet being at a constant temperature means that for every average square meter that absorbs 450 watts of power, every square meter on average also emits that same amount, 450 watts. So if every single one of the square meters of area on the planet gave off 450 watts. Then if we want to know simply how many watts of power are emitted, we can multiply that average power per square meter emitted by the total area of the planet.
We can see that if that area was expressed itself in square meters, then the units of square meters will cancel in this product. And we’ll be left simply with watts of power. And indeed, we’re told the total surface area of this planet, five times 10 to the 12th square meters. And as we mentioned, when we multiply these two numbers together, the units of square meters cancel out. And we’re left only with units of watts. The number we calculate is 2.25 times 10 to the 15 watts. That’s how many watts of infrared radiation the planet emits.