Hemispheric view of Venus produced by Magellan.

One of the hypotheses proposed by climate disruption deniers for Venus’ hot surface temperature is that Venus has an unusually hot core. The logic goes like this – if the core is hot enough, then the surface temperature would be from heat bleeding through the crust instead of from the greenhouse effect of a 97% carbon dioxide (CO_{2}) atmosphere. This hypothesis can be quickly disproved by running three simple calculations.

Scientists estimate that Venus’ solid crust is about 50 km thick, and data from robotic probes indicates that it’s of similar composition to the Earth’s crust. The Earth’s crust is largely silicates, and so I’ve simplified the following calculations by assuming that the entire surface of Venus is composed of silica (silicon dioxide, SiO_{2}, aka quartz), which makes up about 60% of the Earth’s crust. The physical property that we care about at the moment is thermal conductivity, or how easily a material conducts heat through it. Silica’s thermal conductivity is 1.38 W·m^{-1}·K^{-1} (source).

The thermal resistivity (how much a material resists heat moving through it) is equal to the reciprocal of the thermal conductivity, therefore the thermal resistivity of silica *ρ _{silica}* is 0.72 m·K·W

^{-1}.

Yesterday I calculated how much power the surface of a black body Venus has to radiate in order for the surface temperature to be at the observed 735 K: 1.946 W·m^{-2} x 10^{4}. Therefore, this much power has to pass through every square meter of the crust. Furthermore, that much power must also pass through the entire 50 km thick block of the crust, and that 50 km thick block of crust resists the flow of heat according to its thermal resistance. Thermal resistance is calculated as follows

(Eqn 3.1)

where *l* is the length of the column of crust (50 km) and *A* is 1 square meter. The thermal resistance of a single block of crust is therefore 3.62 x 10^{4} K·W^{-1}.

We know the power through this block of crust, so we know that the temperature of Venus’ core (really the mantle, but it won’t really matter) is

(Eqn 3.2)

Blue supergiant Rigel illuminating the Witchhead Nebulae

(Clark Planetarium)

This gives us a core temperature of Venus of greater than 700 *million* K. For comparison, the core of the Sun is estimated to be only about 14 million K. And given that Venus is not a measurable source of X-rays or gamma radiation (the types of electromagnetic radiation generated by extremely high temperatures), we can safely rule out the possibility that Venus houses a star burning 50 times hotter than the sun (a blue supergiant maybe?) within the planet’s crust.

Maybe the data on Venus’ composition and estimates of the crust’s thickness are wrong. In order to be thorough, we should rule out both of those possibilities as well, starting with the thickness of Venus’ crust.

Replacing R_{crust} in Equation 3.2 with Equation 3.1, we get

(Eqn 3.3)

Solving Equation 3.3 for *l*, we get

(Eqn 3.4)

where *l _{crust}* is the thickness of Venus’ crust and T

_{Vcore}is estimated to be roughly equal to that of the Earth’s core, or between 4000 and 7000 K (Source). For simplicity’s sake I’ll assume the higher value and ignore the fact that the Earth’s mantle is quite a bit cooler than the core. From Eqn 3.4, we get that the crust would be a grand total of 0.49 meters thick. Yes, you read that right – approximately a half-meter (18.5 inches) thick. Using the lower temperature of 4000 degrees produces a crust thickness that’s still less than a meter.

Radar mapping of Venus’ topology doesn’t support this conclusion. Radar altimetry has detected volcanic mountains, impact craters, and other features that simply would not exist if the crust was only 0.5 m thick. In addition, a crust this thin would support massive amounts of active volcanism (in fact, one could argue that the entire surface would essentially qualify as a lava lake) and show tidal deformation due to the effects of the Sun’s gravity. The presence of volcanoes has been verified on Venus’ surface, but actual eruptions have not been observed, never mind confirmed. Similarly, Venus is too round to have the kinds of tidal variations that would occur with such a thin crust.

Computer generated surface view of Eistla Regio, with exaggerated vertical relief (from the southwest). The Gula Mons

volcano at left is approximately 3 km high.

So Venus isn’t harboring a star in it’s core and it’s crust is thicker than 0.5 meters. But maybe the crust isn’t mostly silica. We can solve Equation 3.3 for ρ instead and calculate what the thermal resistance (conductivity) of the crust would have to be for a 50 km thick crust and a reasonable core temperature.

(Eqn 3.5)

From Equation 3.5, we calculate that the thermal conductivity of Venus’ crust would have to be about 139,000 W/m·K. Only one thing comes close to having a thermal conductivity this high – heated diamond. It would be difficult for a probe analyzing the composition of the crust to mistake silica rock for diamond.

So for Venus’ surface to be hot because of internal heat bleeding out through the crust, one of three things has to be possible: Venus’ crust surrounds a blue supergiant star, Venus’ crust is less than 1 meter thick, or the crust is composed entirely of diamond. Since each of these options is obviously impossible, Venus’ surface heat is **not** from internal heating.

Tomorrow, how scientists know Venus isn’t geologically young.

**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}W·m^{-2}·K^{-4}) - Universal Gravitation constant:
*G*= 6.674 x 10^{-11}N·m^{2}·kg^{-2} - Avagadro’s number:
*N*= 6.023 x 10_{A}^{23}items·mol^{-1} - Gas constant:
*R*= 8.314 J·mol^{-1}·K^{-1} - Mean radius of the Sun:
*r*= 6.955 x 10_{Sun}^{8}m - Mean radius of Venus:
*r*= 6.052 x 10_{Venus}^{6}m - Mean orbital distance from Venus to the Sun:
*r*= 1.0821 x 10_{Vorbit}^{11}m - Mass of Venus:
*m*= 4.87 x 10_{Venus}^{24}kg - Mass of the Sun:
*m*= 1.99 x 10_{Sun}^{30}kg - Approximate thickness of Venus’ crust:
*l*= 50 km_{crust} - Average surface temperature of the Sun:
*T*= 5778 K_{Sun} - Average surface temperature of Venus:
*T*= 735 K_{Venus} - Estimated temperature of Venus’ core:
*T*= 7000 K_{Vcore} - Specific heat capacity of silica (SiO
_{2}):*κ*= 703 J·kg_{silica}^{-1}·K^{-1} - Thermal conductivity of silica:
*k*= 1.38 W·m_{silica}^{-1}·K^{-1} - Approximate density of silica:
*D*= 2203 kg·m_{silica}^{-3} - Molar mass of silica:
*M*= 2.81 x 10_{silica}^{-2}kg·mol^{-1} - Heat of fusion of silicon:
*H*= 5.021 x 10_{fuse-Si}^{4}J·mol^{-1}

*Environment & Nature, Science*

Anonymous

May 3, 2011

Brian–brave attempt, but even Stefan Bolzman blew his brains out because he couldnt explain thermodynamics to his fellow scientists. Good luck explaining it to the rest of us.

klem

May 3, 2011

“One of the hypotheses proposed by climate disruption deniers for Venus’ hot surface temperature is that Venus has an unusually hot core.”

What the hell? I know of no climate disruption deniers who say this. They know that the atmosphere of Venus is 97% CO2 and it is hot there. They also know that Venus is 40 million kms closer to the sun than earth. The location of Venus is the reason it is so hot, if we dropped the earth 40 million kms closer to the sun we’d fry too. Mars is 40 million km further away from the sun than earth. It too has an atmosphere 95% Co2 but it is freezing cold there. It is cold due to it’s location. If we dropped the earth into Mars orbit we’d freeze. All of this talk about Venus being hot due to it’s atmospheric CO2 content is a red herring. It’s all about location location location. Go back to Russia.

Brian Angliss

May 3, 2011

I refer you to yesterday’s post, where I show that Venus’ surface temperature, minus its atmosphere, would be a balmy 327.5 K, or approximately 56 °C, not the 735 K it actually is. In addition, as is the case with this post, all the requisite equations are available for you to verify the calculation yourself.

If Venus’ surface temperature were all “location location location,” then Venus’ surface would be much colder than it actually is.

BTW, you can run the calculations yourself to determine how close to the Sun Venus would have to be in order for Venus’ surface to the temperature it is. You’ll find that Venus would have to orbit closer than the orbit of Mercury.

Ubertramp

May 4, 2011

If I recall correctly, the diameter of Venus is roughly the same as Earth, maybe a little smaller. But it’s gravity is lower. So the composition would have to be different (presumably less dense, on average). How would that affect your calculations?

Brian Angliss

May 4, 2011

Not much. Thermal conductivity of solids doesn’t change a lot due to changes in density simply because mineral forms tend to be density-dependent. Looking at the atomic scale, however, it would be logical that lower densities would tend to lower thermal conductivity (fewer atoms with which to conduct heat) and thus increase thermal resistance. So lower density minerals would tend to result in core temperatures that had to be greater than 700 million degrees, rather than less (which is the direction we’d need to move this calculation).

We don’t need to go to this level of detail, though, because we can see that we could boost the thermal conductivity of the crust by a massive amount – 10x, say – and we’d still have a core temp of something like 70 million K, 5x the estimated core temperature of the Sun. Still totally unrealistic. And if the general “less dense means higher thermal resistance” claim I made holds, then the calculated temperature would be more than 700 million K, not less.

Ross McLeod

September 4, 2011

As CO2 doesn’t generate its own energy are you saying that the temperature simply increased year after year because the energy was trapped ?

So then is the hypothesis that this process could continue indefinitely and Venus could keep heating up ?

At the temperature of Venus the radiative flux must be ~ 16,500 W/ sq m – it is difficult to see where this comes from and why Venus doesn’t radiate more.

Brian Angliss

September 6, 2011

No, the process could not continue indefinitely because radiation is still being emitted, and the emission increases as T^4. Venus’ surface temperature will increase until the overall energy balance of the system is restored, as I discuss in greater detail in part 5.