# Energy balance for Earth with regards to greenhouse effect

Following image (source: Norwegian website) portrays the energy which the Earth stores to maintain 14 degrees Celsius for the surface and the oceans : I tried to make sense of these numbers and quickly came to the conclusion that the numbers, if correct, would imply that the average temperature is not 14 degrees Celsius but almost 30 degrees Celsius! Here are the calculations I performed assuming Earth is a black body object (using the Stefan-Boltzmann Law) :

\begin{align} E &= k\cdot T^4\\ &\Downarrow\\ T &= \sqrt[\leftroot{1}\uproot{2}\frac{1}{4}]{\frac{E}{k}} = \sqrt[\leftroot{1}\uproot{2}\frac{1}{4}]{\frac{168 + 324}{5.67\times 10^{-8}}} = 305.2 K = 32.05 C \end{align}

Using the Stefan-Boltzmann Law to determine the energy for 14 degrees Celsius (i.e 287.15 Kelvin), the energy calculated becomes 385.49 W/$m^2$.

Are the numbers off on the image, or are my calculations wrong ? Am I wrong to make the assumption by approximating Earth as a blackbody ?

Let me know if anything is unclear (if a translation for the image is required).

Edit: According to following image (provided by NASA) The estimation for trapped heat is even higher (503.6 W/$m^2$).

Edit: From the following lecture and wiki-article, the amount of energy reflected (due to the albedo effect) is about 31 percent. On the NASA image, the amount reflected is at 40 percent. I am not sure how to interpret that. I would have expected the albedo number to be lower (over time) due to the ice melting near the poles. According to this source provided by NASA, there is no global trend on the albedo effect. The albedo is constantly shifting, there is simply a lot of variability.

• I can't explain the discrepancy, it wouldn't matter much if the Earth was not a blackbody because most of the incident energy is coming from a similar T as what the Earth is sending out. But maybe there's enough of a difference in the albedo for the sunlight portion to explain why the Earth's T is lower-- it would need to reflect more of that sunlight than it reflects the infrared. – Ken G Oct 6 '16 at 19:16
• @KenG If you look at the NASA image, they have accounted for the albedo effect. The amount of energy absorbed by the surface, plus back radiation (greenhouse effect) comes to a total of 503.6 W/$m^2$. Which is based on 10 years of data apparently, and is even higher than the Norwegian source. Again, using the Stefan-Boltzmann law, this amounts to 33.84 degrees Celsius in average temperature on Earth's surface! – imranal Oct 6 '16 at 19:26
• The NASA site raises even more questions. What appears to be happening is that not all of the 500 or so watts per square meter that the surface is taking in is getting radiated-- about 100 of it is lost in other ways, some through conduction directly into the air, and inexplicably, most of it goes into latent heat of phase changes. How could that be-- the phase cannot continuously keep changing! Are they really saying that a significant fraction of the Sun's heat is going into melting the icecaps? – Ken G Oct 7 '16 at 0:31
• @KenG It is evaporating water. – tfb Oct 16 '16 at 14:25
• It can't be doing that, for that must be a cycle. – Ken G Oct 17 '16 at 2:22

If you look at the NASA figure: The infrared radiation coming from the Earth's surface is only $398 W/m^2$. The reflected radiation does not figure into this calculation; only the absorbed and emitted radiation.

$398 W/m^2$ works out to about $16.5^\circ C$. That is indeed higher than $14^\circ$. The surface emission is the sum of temperature-driven black-body-like emission and an infrared contribution from the radioactivity inside the Earth's core and mantle.

• I checked the current estimation for the average temperature on Earth, and it was registered at around 16 degrees Celsius. – imranal Oct 6 '16 at 20:49
• Why isn't the average temperature determined from the other factors (ref. NASA image) ? – imranal Oct 9 '16 at 12:21