[worldbuilding] Greenhouse effect calculation?

[EDG]

Does anyone know of an accurate equation that would allow me to calculate the greenhouse effect for a planet, given at least the atmospheric composition and surface pressure? I'm looking for something to calculate the factor to multiply the temperature of the planet (the blackbody temperature modified by the albedo) by.

It's odd because greenhouse warming is pretty topical and yet I can't seem to track down any way to calculate how much it should increase the temperature of a planet by.

More specifically, I have a cold mars-like planet with a surface pressure of 1.18 atms (or 0.83 atms), with a 92% CO2/8% N2 atmosphere, albedo 0.22, and that orbits its star in roughly the equivalent orbit to Mars around our own sun today. I'm guessing a greenhouse factor of x1.2 would work here because the atmosphere is much thicker than Mars (that gives an increase of about 40K) but I have no idea if that's remotely accurate or not.

[parejkoj]

Unfortunately, what you are asking for isn't a straightforward calculation. But take a look at these posts for some direction on how to estimate it:

http://www.realclimate.org/index.php...ple-answer-no/
http://www.realclimate.org/index.php...nhouse-effect/
http://www.realclimate.org/index.php...-6-easy-steps/

[Murphy]

There is this nice little calculator programme that I stumbled upon, http://www.astro.indiana.edu/~gsimon...perature1.html.

It's relatively simple and I don't know how accurate it is, but I've found it useful.

[timb]

The degree to which a given amount of CO₂ warms the Earth is still controversial. See, for example, Warming the early Earth - CO2 reconsidered. Note that the response of the climate to CO₂ is far from linear, and beyond a certain amount adding more doesn't make the planet any warmer.

[Ronald Brak]

Generally if the greenhouse due to carbon dioxide equals X, then doubling the CO2 concentration will increase its greenhouse effect to 2X and doubling it again will increase it to 3X. So each percentage point increase in CO2 will have less of a total warming effect then the percentage point before, however you never reach a point where increasing CO2 concentration stops warming the planet.

[timb]

Generally if the greenhouse due to carbon dioxide equals X, then doubling the CO2 concentration will increase its greenhouse effect to 2X and doubling it again will increase it to 3X. So each percentage point increase in CO2 will have less of a total warming effect then the percentage point before, however you never reach a point where increasing CO2 concentration stops warming the planet.

For small changes in CO₂ the greenhouse response is approximately logarithmic. I don't think this could be true for large changes; otherwise you could make the surface temperature as large as you please -- say 3000K -- by adding enough CO₂. That doesn't sound likely.

[Murphy]

That certainly seems to be what's happening on Venus, doesn't it?

[EDG]

It's not just the CO2 (or CH4 or whatever) concentration though - it's partly down to the thickness of the atmosphere too isn't it? And I suspect that orbital distance is a factor too - if Venus was where Mars is today, would its greenhouse effect be the same?

[Ronald Brak]

The climate senstivity of CO2 is between 2-4.5 degrees with a best estimate around 3 degrees. Assuming for the sake of the argument that is is 3 degrees and that it will stay at 3 degrees as we increase CO2 concentrations. If we start with a CO2 concentration of 380 parts per million then it will take less than 12 doublings of concentration for the earth's entire atmosphere to be carbon dioxide. This would only increase the temperture by under 36 degrees, all else being equal, which seems surprising low and much less than 3,000K.

[timb]

The climate senstivity of CO2 is between 2-4.5 degrees with a best estimate around 3 degrees. Assuming for the sake of the argument that is is 3 degrees and that it will stay at 3 degrees as we increase CO2 concentrations. If we start with a CO2 concentration of 380 parts per million then it will take less than 12 doublings of concentration for the earth's entire atmosphere to be carbon dioxide. This would only increase the temperture by under 36 degrees, all else being equal, which seems surprising low and much less than 3,000K.

No, you can, arithmetically at least, double the amount of CO₂ in the atmosphere as long as you like. I didn't say anything about reducing the quantum of other gases. It's the amount of CO₂ that matters, not the proportion.

[Ronald Brak]

Well, beyond a certain point CO2 would become surface rather than atmosphere and if you kept adding more eventually it would fuse.

[parejkoj]

Ummm... read the links I posted. It's rather more complicated than either of you are considering, even for relatively small changes in the amount of CO₂.

[EDG]

Unfortunately, what you are asking for isn't a straightforward calculation. But take a look at these posts for some direction on how to estimate it:

http://www.realclimate.org/index.php...ple-answer-no/
http://www.realclimate.org/index.php...nhouse-effect/
http://www.realclimate.org/index.php...-6-easy-steps/

Thanks - interesting links, though somewhat depressing .

So it looks like I can't quantitatively figure this out (not without a handy cluster of parallel processors anyway) - fair enough.

What about qualitatively? I've got a really lousy grasp on this... first we have Venus with an atmosphere of 90+% CO2, at 90 atm surface pressure, and it has an enormous greenhouse effect (GFX) of +(several hundred kelvin).

Then we have Earth with a reasonably dense atmosphere, CO2 concentrations measured in a few hundred ppm, and it has a moderate GFX.

Then we have Mars with a very thin atmosphere of mostly CO2, and it has a small GFX.

So it seems to me that the GFX is strongly proportional to the thickness of the atmosphere (probably clouds reflecting radiation back to the surface, more air molecules to absorb it etc), and also to composition (if Venus' atmosphere was mostly Nitrogen instead of CO2, would it have as a high a GFX? I don't think it would, would it? Or would the thick atmosphere swamp any compositional contribution?).

Can we make any qualitative predictions here for other planets? If Mars had an atmosphere that had the same pressure as Earth, but the composition of modern day Mars, would it have a much higher temperature?

Or am I really just going to have to pull random numbers out of the air for this?

[timb]

Well, beyond a certain point CO2 would become surface rather than atmosphere and if you kept adding more eventually it would fuse.

So there is an upper limit to the amount of warming CO₂ can provide, which is what I said earlier and you denied. I'm pretty sure you would reach the limit well before CO₂ turned solid.

Ummm... read the links I posted. It's rather more complicated than either of you are considering, even for relatively small changes in the amount of CO2.

Where did I claim it was simple? We are discussing a fantasy planet. For that you want to be in the ballpark of plausibility. The OP's planet has roughly 11 doublings worth of CO₂ more than Earth (he was a little unclear about the atmospheric pressure, I used 1 bar), so adding around 11*3=33K to the irradiance adjusted terrestrial temperature probably wont see too many readers throwing the book down in disgust. Maybe you should discount the greenhouse term by the fourth root of the irradiance. Not a huge difference.

[Ronald Brak]

I certainly agree with you, parejkoj, that the situation is much more complex than the simple estimate I did. My rough estimate may definately be wildly off, but I hoped it might be helpful for the kind of world described in the OP.

[parejkoj]

Point taken, timb. Hmmm.... A ballpark guess is probably fine in this case. I guess i missed the forest for the trees.

The problem is, both the fractional content of CO₂ and the total amount matter: what is really important is how many IR absorption features are available to help trap heat. But N₂ doesn't have much overlap with CO₂ in terms of heat-trapping potential (in fact, it doesn't have much in the way of IR absorption features at all), so it can be mostly ignored in this context. So, timb's suggestion is probably the closest: each doubling of CO₂ should produce somewhere around 2.5-3ºC of warming (see that first realclimate link for references on this). And it looks like the CO₂ absorption on this imaginary planet is far from saturated (see here and raypierre's reply here), so no worries there.

That in mind, it sounds like it would be a chilly planet.

What's this planet for?

[EDG]

That in mind, it sounds like it would be a chilly planet.

That's what I'm figuring... it's going to be around 10°C at the equator during the summer, and around -20°C there in the winter.

What's this planet for?

Hopefully, eventual release in an SF RPG supplement . (I'll be adding a credit to BAUTForum too).

[Spaceman Spiff]

The degree to which a given amount of CO₂ warms the Earth is still controversial. See, for example, Warming the early Earth - CO2 reconsidered. Note that the response of the climate to CO₂ is far from linear, and beyond a certain amount adding more doesn't make the planet any warmer.

hmmm....I wonder when this occurs? (hint: Venus has ~90 atmospheres of CO2; Earth currently sports 386 ppm CO2 of a single atmosphere of pressure.) This is a fallacy. Here is all you might want to know about the atmosphere of Venus (you can skip down to "Peeking at the surface" with regards to the question at hand), and this and especially this explain one of the reasons why the above statement is a fallacy.

We see that for the pre-industrial CO2 concentration, it is only the wavelength range between about 13.5 and 17 microns (millionths of a meter) that can be considered to be saturated. Within this range, it is indeed true that adding more CO2 would not significantly increase the amount of absorption. All the red M&M's are already eaten. But waiting in the wings, outside this wavelength region, there's more goodies to be had. In fact, noting that the graph is on a logarithmic axis, the atmosphere still wouldn't be saturated even if we increased the CO2 to ten thousand times the present level.

As a matter of fact carbon dioxide never stops absorbing, albeit the "greenhouse" effects will certainly be non-linear (logarithmic) with respect to its concentration except at low concentrations.

The other reason is mentioned in the first article. It's because what matters is the temperature of the effective emitting layer (i.e., the photosphere), which lies somewhere high in the atmosphere. Piling on more CO2 not only increases the atmosphere's overall opacity, it increases the altitude of the layer that is the planet's effective photosphere, where T is lower and thus the thermal emission is lower. So the whole danged atmosphere (and the surface) must increase their temperatures to put the planet back into thermal equilibrium.

It's just an argument of the conservation of energy.

[timb]

hmmm....I wonder when this occurs? (hint: Venus has ~90 atmospheres of CO2; Earth currently sports 386 ppm CO2 of a single atmosphere of pressure.) This is a fallacy. Here is all you might want to know about the atmosphere of Venus (you can skip down to "Peeking at the surface" with regards to the question at hand), and this and especially this explain one of the reasons why the above statement is a fallacy.

As a matter of fact carbon dioxide never stops absorbing, albeit the "greenhouse" effects will certainly be non-linear (logarithmic) with respect to its concentration except at low concentrations.

The other reason is mentioned in the first article. It's because what matters is the temperature of the effective emitting layer (i.e., the photosphere), which lies somewhere high in the atmosphere. Piling on more CO2 not only increases the atmosphere's overall opacity, it increases the altitude of the layer that is the planet's effective photosphere, where T is lower and thus the thermal emission is lower. So the whole danged atmosphere (and the surface) must increase their temperatures to put the planet back into thermal equilibrium.

It's just an argument of the conservation of energy.

At some point the CO2 becomes the surface (CO2-III) and adding more just increases the gravity, which makes the photosphere lower.

[timb]

That's what I'm figuring... it's going to be around 10°C at the equator during the summer, and around -20°C there in the winter.

Your equator has a distinct summer? it must be a high obliquity planet, or in an eccentric orbit. If the average temperature is less than 0°C your planet will likely suffer a global snowball (if watery) and the CO₂ could even freeze out. I've read contradictory reports as to whether CO₂ can keep a Mars-like planet warm indefinitely.

[Murphy]

Going back to the original post, using the simple Planet Temp calculator I mentioned earlier (http://www.astro.indiana.edu/~gsimon...perature1.html), and plugging in the figures EDG provided (the Star's output wasn't mentioned, so I've just used the Sun)...

Star = 1 x Sol
Distance = 1.524 AU
Bond Albedo = 22
Greenhouse Effect = 1.2 x Earth's

I get an average surface temperature of -31 degrees C.

In order to get the +10 degrees C temp you mentioned, the planet would have to have a greenhouse effect 4.2 times stronger than Earth's.

Now I don't know how accurate that is, or how it converts into atmospheric composition and pressure, but I think it give a fair estimate of what you'd need.

[EDG]

Your equator has a distinct summer? it must be a high obliquity planet, or in an eccentric orbit.

It's an eccentric orbit.

[EDG]

Now I don't know how accurate that is, or how it converts into atmospheric composition and pressure, but I think it give a fair estimate of what you'd need.

It doesn't seem to be that accurate to me. Using its semimajor axis distance, the planet should have a blackbody temperature of 225K, and when you factor in the albedo of 0.22 that should drop to 212K. The GFX factor I'm using is 1.2, so the final temperature is 254K.

[Murphy]

Well I don't know, I've tried testing it out with some know planets in the solar system, and the programme seems to get it right most of the time, to within a few degrees of the actual values. It seems to get it right for Mars for instance. So, I don't know, try it out your self.

[EDG]

I tried using the values you used in my equations (even for luminosity and distance) and got an average surface temperature of -18.44°C (254.56 K). And my formulae are based directly on the blackbody temperature calculation.

[Spaceman Spiff]

At some point the CO2 becomes the surface (CO2-III) and adding more just increases the gravity, which makes the photosphere lower.

To the first point: you betcha; inquire with Venus. To the second point: if the CO2 mass becomes that significant, then the temperature must rise at all levels to generate sufficient pressure (gradient) to maintain hydrostatic equilibrium as well as thermal equilibrium. There is no free lunch.

But I suppose this discussion is a bit off target from the OP.

[ngc3314]

And at the risk of going farther OT, but relevant to the whole worldbuilding thing and definitely worth knowing - in the case of the Earth, by far the most important greenhouse gas is _________?

(Let's not just have the same people with their hands up, please!)

[parejkoj]

Oh... Oh... Teacher, Teacher! I know! I know! Call on me!

;-)

[Spaceman Spiff]

"Oooh! Oooh! Mr. Kotter! Mr. Kotter! Call on me! -- Is this a 'trick' question?"

Either way, an informed discussion of this issue, at least as it concerns our Earth, can be found here.

[EDG]

Radiative Forcing has something to do with this, right?

http://en.wikipedia.org/wiki/Radiative_forcing