In the first post about CO2 I included a separate maths section which showed the energy budget for the earth and also derived how much energy we receive from the sun. A comment today reminded me that I should do a separate article about this topic. I’ve seen lots of comments on other blogs where people trip up over the basic numbers. It’s easy to get confused.
Don’t worry, there won’t be a lot of maths. This is to get you comfortable with some basics.
Energy from the Sun
It’s quite easy to derive how much energy we expect from the sun, but the good news is that since 1978 there have been satellites measuring it.
The solar “constant” is often written as S, so we’ll keep that convention. I put “constant” in quotes because it’s not really a constant, but that’s how it’s referred to. (And anyway, the changes year to year and decade to decade are very small – a subject for another post, another day).
The first important number, S = 1367 W/m2
Note the units – the amount of energy per second (the Watts) per unit area (the meters squared). By the way, sorry America, the science world moved on. We won’t convert it to ft2..
Just for illustration here’s the satellite measurements over 20 years:
For anyone a little confused, note that different satellites get different absolute measurements, it is the relative measurements that are more accurate.
Comparing Apples and Oranges? Surface Area vs Area of a Disc
The sun is really long way away from the earth – about 150M km (93M miles). We measure the incoming solar radiation at the top of the atmosphere in W/m2.
So how much total energy can be absorbed into the earth’s climate system from this solar radiation?
Hopefully the answer will become more obvious by looking at the image above. The solar radiation from a long way away strikes the effective 2d area that the earth cuts out.
A 2d area – or a flat disc – has area, A = πr2
Therefore, the total energy received by the earth = Sπr2
[Radius of the earth = 6.37 x 106 m (6,370 km) so Energy per second from the sun = 174,226,942,644,300,000 W also written as 1.74 x 1017 W]
It’s a really big number, so to make everything easier to visualize, climate scientists generally stay with W/m2, rather than numbers like 1.74 x 1017 W.
Now the real surface area of the earth is actually, Ae= 4πr2 (not πr2)
(Area of earth, Ae= 510M km2, or 5.1×1014m2)
Why isn’t the energy received by the sun = S x 4πr2?
Look back at the graphic – is the sun shining equally on every part of the earth every second, for all 24 hours of the day? It’s not. It’s shining onto one side of the earth. It’s night time for half the world at any given moment.
So think of it like this – the absolute maximum area receiving the sun’s energy on average can only be half of the surface area of the earth – 2πr2 (=4πr2/2)
But that’s not the end of the story. Picture someone where the sun is right down near the horizon. It’s still daytime but obviously that part of the earth is not receiving 1367W/m2 – they are receiving a lot less. In fact, the only spot on earth where someone receives 1367W/m2 is where the sun is directly overhead. So the effective area receiving the solar constant of 1367 W/m2 can’t even be as high as 2πr2.
So if the idea that solar radiation only strikes an effective area of πr2 is still causing you problems, this is the concept that might help you.
Linking Incoming Solar Radiation to the Earth’s Outgoing Radiation
The earth radiates out energy in a way that is linked to the surface temperature. In fact it is proportional to the fourth power of absolute temperature.
As we think about the earth radiating out energy, it might be clearer why we labored the point earlier about the area that the sun’s energy was received over.
Take a look at that graphic again. The energy from the sun hits an effective 2d disc with area = πr2.
The earth radiates out energy from its whole surface area = 4πr2.
So to be able to compare “apples and oranges”, when climate scientists talk about energy balance and the climate system they usually convert radiation from the sun into the effective radiation averaged across the complete surface of the earth.
This is simply 1367/4 = 342.
The second important number, incoming solar radiation at the top of atmosphere = 342 W/m2 (averaged across the whole surface of the earth).
Some energy is reflected but before we consider that note that this doesn’t mean that each square meter of the earth receives 342 W/m2 – it’s just the average. The equator receives more, the poles receive less.
Not all of this 342 W/m2 is absorbed. The clouds, aerosols, snow and ice reflect a lot of radiation. Even water reflects a few percent. On average, about 30% of the solar radiation is reflected back out. A lot of slightly different numbers are used because it’s difficult to measure average albedo.
The third important number, solar radiation absorbed into the climate system = 239 W/m2
This is simply 342 * (100% – 30%). You see slightly different numbers like 236, 240 – all related to the challenges of accurate measurement of albedo.
Some of the radiation is absorbed in the atmosphere, and the rest into the land and oceans.
Energy radiated out from the climate system must balance the energy received from the sun. This is energy balance. If it’s not true then the earth will be heating up or cooling down. Even with current concerns over global warming the imbalance is quite small. And so, as a starting point, we say that energy radiated out = energy absorbed from the sun.
Energy radiated from the earth, Ee = S (1- A) / 4 in W/m2
where A = albedo (as a number between 0 and 1, currently 0.3)
The solar constant, S = 1367 W/m2
The solar radiation at the top of atmosphere averaged over the whole surface of the earth = 342 W/m2
The solar radiation absorbed by the earth’s climate system = 239 W/m2 (about 28% into the atmosphere and 72% into the earth’s surface of land, oceans, ice, etc)
Therefore, the approximate radiation from the earth’s climate system at the top of atmosphere also equals 239 W/m2.
These numbers are useful to remember.
Update – new post The Earth’s Energy Budget – Part Two
Update – new post The Earth’s Energy Budget – Part Three