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 (CO2) 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, SiO2, 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 ρsilica 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 104. 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
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 104 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
Blue supergiant Rigel illuminating the Witchhead Nebulae
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 Rcrust in Equation 3.2 with Equation 3.1, we get
Solving Equation 3.3 for l, we get
where lcrust is the thickness of Venus’ crust and TVcore 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.
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.
- 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·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