Earth Blackbody Temp: How Albedo Affects Result

[cepheid]

I saw a claim today that without the greenhouse effect, Earth's surface would be on average at a chilling -18°C (note: this is not a climate change thread).

I set about trying to reproduce this result, so at first I assumed that Earth was a perfect blackbody, and that in order to be in equilibrium, it would have to radiate away as much power as it received. So I took the solar irradiance of ~1400 W/m² and divided it by 2 (since I figured that the surface area over which it could be radiated away again would be twice the surface area over which it was received). Then I took this irradiance (or 'flux' in astronomy parlance) and divided it by the Stefan-Boltzmann constant in order to get the fourth power of the surface temperature that the Earth would have to have in order to have this surface flux (as a blackbody). The resulting surface temperature was T = 60°C.

Then I decided it was silly to assume that all of the incident solar radiation was absorbed, so I looked up the Albedo (reflectivity) of Earth on Wikipedia. Two numbers were stated: geometric Albedo of 0.367 and Bond Albedo of 0.306. Being too lazy to read more about them, I just tried them both. Assuming them to be the fraction of radiation reflected, I scaled my required output flux by (1-Albedo) and got results of 30°C and 22°C for the lower and higher albedos respectively. Neither of these is -18°C. What am I missing (aside from the obvious that Earth is not a blackbody). Shouldn't this method give something reasonably close? I assumed that an equally crude estimate was applied to arrive at the -18°C in the first place.

 

[phyzguy]

The area of sunlight that the Earth intercepts is pi*r^2, while the surface area that radiates energy away is 4*pi*r^2, so the ratio of the absorbing area to radiating area is 4, not 2. Figuring this, and the albedo of 0.37, gives a temperature of:
$$T=[\frac{1400*(1-.37)}{4*5.67*10^{-8}}]^{1/4} = 250K = -23C$$

 

[cepheid]

Okay so I looked here:

http://en.wikipedia.org/wiki/Climate_model#Zero-dimensional_models

and I see that I was

1. failing to take into account the effect of emissivity < 1
2. getting my geometric factor wrong (apparently the irradiated surface should be considered to be circular rather than hemispherical?)

I reproduced their result of about 15°C, but I don't know where -18 comes from still. They say that their albedo and emissivity are chosen to account for clouds and greenhouse effect already, so maybe that's it.

 

[cepheid]

The area of sunlight that the Earth intercepts is pi*r^2, while the surface area that radiates energy away is 4*pi*r^2, so the ratio of the absorbing area to radiating area is 4, not 2. Figuring this, and the albedo of 0.37, gives a temperature of:
$$T=[\frac{1400*(1-.37)}{4*5.67*10^{-8}}]^{1/4} = 250K = -23C$$

I agree. It looks like if you keep the emissivity factor out of it (i.e. assume it is 1), more will be radiated away, and the temperature will be colder (-23°C), whereas if I take their emissivity value, less is radiated away, and the temperature reaches 8°C (the discrespancy between this and their value of 15°C is because they used a lower albedo of 0.3). So it all makes sense. Thank you.

EDIT: In fact if I use their albedo value of 0.3, but an emissivity of 1, then I get -17°C, which is close enough for me!

 

[Andy Resnick]

I got 6°C, using the fact that the reflectivity + emissivity = 1 (I'm not sure how they justify albedo + emissivity <1)

Of course, the Earth also *generates* heat by radioactive decay:

http://physicsworld.com/cws/article/news/46592

but it's too early in the day for me to work through the (0th order) correction.

 

[phyzguy]

I think the interior heat is negligible in terms of the heat balance of the Earth. This article says that the heat flowing from the interior is ~ 4x10^13 W. The heat absorbed from the sun is ~ 1.4x10^3 W/m2 * pi * (6.4x10^6m)^2 * (0.7) ~ 10^17 W.

 

[klimatos]

I saw a claim today that without the greenhouse effect, Earth's surface would be on average at a chilling -18°C (note: this is not a climate change thread).

The consensus of informed opinion is that the blanket of greenhouse gases increases the Earth's surface temperature by an average of 39°C. That surface temperature is usually given as 288K (15°C). My calculations give an unblanketed surface temperature of -14°C, not -18C. You didn't specify your source.