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

(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)

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)

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: N_A = 6.023 x 10²³ items·mol^-1
• Gas constant: R = 8.314 J·mol^-1·K^-1
• Mean radius of the Sun: r_Sun = 6.955 x 10⁸ m
• Mean radius of Venus: r_Venus = 6.052 x 10⁶ m
• Mean orbital distance from Venus to the Sun: r_Vorbit = 1.0821 x 10¹¹ m
• Mass of Venus: m_Venus = 4.87 x 10²⁴ kg
• Mass of the Sun: m_Sun = 1.99 x 10³⁰ kg
• Approximate thickness of Venus’ crust: l_crust = 50 km
• Average surface temperature of the Sun: T_Sun = 5778 K
• Average surface temperature of Venus: T_Venus = 735 K
• Estimated temperature of Venus’ core: T_Vcore = 7000 K
• Specific heat capacity of silica (SiO₂): κ_silica = 703 J·kg^-1·K^-1
• Thermal conductivity of silica: k_silica = 1.38 W·m^-1·K^-1
• Approximate density of silica: D_silica = 2203 kg·m^-3
• Molar mass of silica: M_silica = 2.81 x 10^-2 kg·mol^-1
• Heat of fusion of silicon: H_fuse-Si = 5.021 x 10⁴ J·mol^-1