# Venus’ climate V: How scientists know Venus’ surface temperature is a result of greenhouse heating (corrected)

On Monday, I wrote that there were only two possibilities for why Venus’ surface temperature is so hot – either something internal to the planet’s crust and core was keeping Venus hot, or something about the atmosphere was. Tuesday I showed that it wasn’t internal heating. Wednesday I disproved the “Venus formed recently” hypothesis. And yesterday I ruled out a celestial collision that might have melted Venus’ crust, effectively absolving Venus’ core of any responsiblity for Venus’ surface temperature. Given the planet itself can’t be the source of the heat,the atmosphere has to be keeping the surface hot somehow.

There are two ways that Venus’ atmosphere could be responsible for keeping the surface hot, either individually or in combination. First, Venus’ atmosphere is very dense, and there is a physical relationship known as the ideal gas law that indicates that gases under pressure tend to be hotter. Second, robotic probes have measured Venus’ atmosphere to be about 97% CO2, and we can see from the image above (click for a larger version) that the absorption spectrum for CO2 (at Earth temperature and pressure – Venusian temperature and pressure increases the width of the absorption bands, making CO2 a stronger absorber in Venus’ atmosphere than in Earth’s) strongly overlaps the peak emission spectrum of Venus’ surface. The overlap in the spectra suggests that the greenhouse effect of so much CO2 is the cause Let’s investigate the ideal gas law first.

The ideal gas law states

$PV=nRT$ (Eqn 6.1)

where P is the pressure of the gas (in N·m2), V is the volume of the gas (in m3), n is the amount of the gas (in mols), R is the gas constant, and T is the temperature (in K).

This equation is a useful tool in many applications, but doesn’t apply universally. In fact, it only applies when the following conditions are met:

• the particles of gas are perfectly elastic (they bounce when they hit each other and don’t lose energy in the collision);
• the particles of gas have no size;
• if the gas is contained, the walls of the container are also perfectly elastic AND perfectly insulate the interior volume from the exterior.

Every atom or molecule of a gas has some volume, so we know right there that the ideal gas law is only a convenient approximation. Furthermore, gases like the Earth’s atmosphere are not composed of single molecules or single atoms, two conditions that tend to make a real gas closer to ideal. Instead, an atmosphere tends to be a mix of multiple different gases (nitrogen, CO2, oxygen, etc.), each of which behaves differently. Venus’ atmosphere is about 97% CO2, so it’s closer to ideal in this sense than the Earth’s atmosphere. But as we’ll see, that’s not enough.

The “perfectly insulating container” restriction is one of major problems with applying the ideal gas law to Venus’ atmosphere. An ideal gas is assumed to not lose energy that is added to the system. Increase the temperature of an ideal gas and the pressure and/or volume increases (assuming a constant amount of gas). But in a real gas, that added energy will leak out somewhere. As an example, imagine a balloon that floats into a sunbeam in a home. When the balloon is in the sun, the additional solar energy causes the volume to increase until the pressure inside the balloon equals the air pressure on the outside of the balloon. But if you take the balloon out of the sun, the volume doesn’t stay the same – the extra energy bleeds away as the air temperature inside the balloon drops to room temperature again (and the volume of the balloon drops as a result). Thermodynamics breaks the ideal-ness of the gas in the balloon.

The same thing happens with Venus’ atmosphere. The sun heats up Venus’ surface and then that surface heat flows into the atmosphere. But that heat doesn’t stay in one place. Some of the hot gas rises in the atmosphere until the heat can be radiated into space, some of the heat is re-radiated back to the ground where it’s reabsorbed, and some of it is radiated into the atmosphere where it is reabsorbed and re-radiated. All of this thermal interaction makes the atmosphere of any planet non-ideal.

Furthermore, convection creates other effects like the temperature lapse rate (the average change in temperature of the atmosphere as you change altitude in the troposphere) and state changes (such as the condensation of gases in to liquid clouds) are further reasons why we can’t view a planetary atmosphere as an ideal gas.

Finally, a planet’s atmosphere isn’t like a balloon or Druidia’s air shield, where some barrier prevents the atmosphere from being further out from the surface than that barrier. Instead, a planetary atmosphere is extends out to the point where the inherent energy of the gas molecules is more than the escape velocity of the planet’s gravity well, or until the solar wind strips the atoms away from the planet’s gravity. In either case, however, the volume of the atmosphere is highly variable and, in most respects, undefined.

Energy flow diagram assuming Venus’ surface temperature is due to internal heating instead of CO2. Note that conservation of energy is not maintained. (image corrected 2/4/2013)

Given that the ideal gas law can’t be used to define the temperature of Venus’ surface, we need to identify what does define the surface temperature. On Monday I identified two possibilities – something inherent in the planet itself, and something inherent in the planet’s atmosphere. The figure at right shows what the energy fluxes would be for a surface temperature generated by Venus’ core.

Note that this graphic shows that, in this case, conservation of energy would not be maintained – Venus’ core itself would be generating most of the energy in the system. But the problem is that I’ve proven that energy flow up from Venus’ core can’t be the source of the surface temperature, so we know that the dashed red line up to the surface is wrong. Furthermore, scientists don’t observe anything even close to 735 K as Venus’ effective temperature, so we know that the dashed red line showing nearly 19 kW·m2 exiting the atmosphere is also wrong. Without the massive flow of energy from Venus’ core, we need a system where conservation of energy is largely maintained, which means that this diagram does not correctly represent energy in Venus’ climate.

Energy flow diagram if Venus’ surface temperature is due to CO2. Note that conservation of energy is maintained. (image corrected 2/4/2013)

What we do see is closer to this diagram. Conservation of energy is still broken due to the small amount of energy flow from Venus’ core to the surface (0.17 W·m2), but because the effect is several orders of magnitude less than the other energies flowing in the diagram, we can essentially ignore it. When we do that, however, we are forced to apply conservation of energy to any interface – top of the atmosphere (TOA), surface, clouds, even the planet as a whole.

This diagram shows that, once the hot surface temperature was established who knows how long ago, it reached an thermal equilibrium state with a large amount of energy transfer between the surface and the lower levels of the atmosphere, and the planet as a whole reached an equilibrium state with the Sun. At this point, imagine Venus’ atmosphere as a space blanket that lets just enough heat through the atmosphere to keep the insides extra toasty.

Sherlock Holmes once said “when you’ve eliminated the impossible, whatever remains, however improbable, must be the truth.” We’ve eliminated all the other possibilities, reasonable and otherwise – the only one remaining hypothesis is that Venus’ surface temperature is due to the greenhouse effect as produced by an atmosphere that is 97% CO2.

It’s possible that there are other possibilities that I haven’t considered. I’m not worried about those hypotheses, because there is so much conservatism built into the calculations (especially those on Tuesday) that something could be wrong by a factor of 10 and my results wouldn’t be seriously affected in most cases. In addition, there is so much independent corroborating evidence that CO2 behaves exactly as climate scientists think it does that Venus’ climate is ultimately an interesting sideshow. There is a lot of technology today that relies directly or indirectly on CO2‘s infrared properties, and none of them would work correctly (or at all, in some cases) if CO2‘s infrared properties weren’t well known. Some examples include CO2 lasers used for metal cutting and welding, heat seeking missiles, IR imagers, and orbital CO2 imagers.

I’ll leave you with a demonstration of the thermal absorption properties of CO2 that makes the case for 2 being a strong IR absorber (and thus a greenhouse gas) stronger than all the physics I did this past week.

Special thanks to Arthur Smith, Michael Tobis, John Cook, Ray Weymann, Robert Rhode, Kevin Trenberth, and the many other scientists and experts who helped me refine this series through their unending patience with my odd and/or ignorant questions.

Common Constants and Variables:

• Speed of light in a vacuum: c = 299792458 m·s-1 (exact)
• Plank’s Constant: h = 6.62606896 × 10−34 J·s
• Boltzman’s Constant: k = 1.3806504 × 10−23 J·K-1
• Stefan-Boltzman constant: σ = 6.669 x 10-8 5.670 x 10-8 W·m-2·K-4)
• Universal Gravitation constant: G = 6.674 x 10-11 N·m2·kg-2
• Avagadro’s number: NA = 6.023 x 1023 items·mol-1
• Gas constant: R = 8.314 J·mol-1·K-1
• Mean radius of the Sun: rSun = 6.955 x 108 m
• Mean radius of Venus: rVenus = 6.052 x 106 m
• Mean orbital distance from Venus to the Sun: rVorbit = 1.0821 x 1011 m
• Mass of Venus: mVenus = 4.87 x 1024 kg
• Mass of the Sun: mSun = 1.99 x 1030 kg
• Approximate thickness of Venus’ crust: lcrust = 50 km
• Average surface temperature of the Sun: TSun = 5778 K
• Average surface temperature of Venus: TVenus = 735 K
• Estimated temperature of Venus’ core: TVcore = 7000 K
• Specific heat capacity of silica (SiO2): κsilica = 703 J·kg-1·K-1
• Thermal conductivity of silica: ksilica = 1.38 W·m-1·K-1
• Approximate density of silica: Dsilica = 2203 kg·m-3
• Molar mass of silica: Msilica = 2.81 x 10-2 kg·mol-1
• Heat of fusion of silicon: Hfuse-Si = 5.021 x 104 J·mol-1

### 6 Comments on “Venus’ climate V: How scientists know Venus’ surface temperature is a result of greenhouse heating (corrected)”

1. lgl May 8, 2011 at 7:38 am #

1. How much solar radiation (W/m2) reaches the surface of Venus directly?

2. Why didn’t you show the CO2 absorption spectrum under Venus conditions instead of Earths?

• Brian Angliss May 8, 2011 at 4:06 pm #

We’d need to add the Bond albedo into Equation 2.3 (making this a gray body instead of a black body in the process) and so about 10% of the radiation that reaches Venus is actually absorbed by the planet.

As for CO2 absorption under Venus conditions, I was able to get the spectrum from a colleague for the Earth, but unable to acquire the spectrum for Venus in anything resembling a timely fashion. However, you’re welcome to go through the HITEMP database at the HITRAN website to extract the Venus’ CO2 absorption spectrum. The main difference is that the Venus CO2 bands will be wider, and thus absorb more energy, than the Earath bands.

2. lgl May 9, 2011 at 4:36 am #

Thanks Brian, but that does not exactly answer my first question. I have found a presumably good paper however, with a lot of interesting numbers and figures. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.75.6539&rep=rep1&type=pdf fig. 2.17 showing only 17 W/m2 solar reaches the surface. The text also says “Approximately 2.5% of the sunlight incident on the planet makes it to the ground, with most of what is not reflected to space being absorbed in the clouds” This makes the ghg explanation very odd as it means the ghgs must amplify the solar input 1000 times. Viewing the system as a positive feedback loop with open loop amplification A=1 (no amplification without ghg) it means the feedback factor B [in output=A/(1-B) ] must be 0.999 i.e 0.999 of the LW radiation from the surface must be sent back to the surface. This seems impossible looking at all the windows in absorption spectrum.

Another oddity, because most of the solar input is absorbed by the upper cloud layer, its temperature would have been around 200K even without much gh-effect. Adding the lapse rate surface temp would be around 500 K even if the upper clouds were moved to 30 km (and I don’t think they would have moved that far down). Assuming the lapse-rate would not change much even with a modest gh-effect like that of the Earth, or is that wrong?