Simulation of planetary accretion (Ken Rice, UC-Riverside)

Yesterday we found that, for Venus to be hot due to internal heating, either the planet’s core would have to be a star, its crust would have to be less than a meter thick, or its crust would have be composed of diamond. None of these is remotely possible, so the internal heating hypothesis has been entirely disproved. Another hypothesis proposed by climate disruption deniers for why the surface temperature of Venus is so hot is that Venus might have formed recently (geologically speaking) instead of having formed about 4.5 billion years ago along with the Earth and the rest of the planets.

We can test this hypothesis a couple of different ways. The first is to again use the mathematics of a black body. In the case of an ideal black body the size of Venus, we can calculate the amount of time it would take to cool from one temperature to another as Venus radiates energy into space. This calculation will determine the absolute minimum age of Venus because the ideal black body assumption ignores all the effects of Venus’ atmosphere as well as the physical limits of heat transfer through the planet’s crust.

Equation 4.1 below defines how long it would take for a hypothetical black body to cool from a starting temperature to a final temperature, in seconds:

(Eqn 4.1)

where N_atoms is the number of atoms in Venus, k, σ, r_Venus, and T_Venus are defined here, and the initial temperature is to be determined.

In order to determine the number of atoms in Venus, we need to know some data about Venus and make a simplifying assumption. First, we need to know the mass of Venus, then we need to assume that the entire planet is made up of silica (silicon dioxide), and then we need to know the mass of silica. Once we know these things, calculating the number of atoms in Venus is relatively simple.

(Eqn 4.2)

where N_A is Avogadro’s Number and we multiply by three because there are three atoms in every molecule of silica. Substituting in the mass of Venus (in kg), the mass of silica (in kg/mol), and Avagadro’s Number (in molecules/mol) and we get 3.1 x 10⁵⁰ atoms.

Assuming a starting black body temperature of about 7000 K (which we used yesterday) for T_initial, and substituting values back into Equation 4.1, we get that the time to cool a black body Venus from 7000 K to 735 K is 2.1 x 10¹¹ seconds. Divide that by 31,536,000 to convert seconds into years and we get about 6500 years.

Because we made a whole bunch of simplifying assumptions that don’t hold for real objects like a planet, we know that this is far too young for the age of Venus. Even if we hadn’t made a lot of assumptions, however, we know that the geologic record of Earth doesn’t support Venus being this young either. If Venus had formed less than 10,000 years ago, we’d see a lot of young and large impact craters on the Earth due to debris being scattered out of Venus’ orbit and colliding with the Earth. However, there are few young impact craters and no young and major impact craters.

Cutaway view of Venus’ interior and atmosphere.
(Calvin J. Hamilton, 2000)

The next step is to analyze how long a non-ideal black body Venus would have taken to cool down. We’ll do this by including the thermal properties of silica as the limiting factors.

All materials conduct and store heat. Thermal conductivity is the property of a metal spoon that results in a nasty burn if you leave one end in boiling water too long – the heat in the water conducts up the handle. Heat capacity is how much energy it takes to heat up, or cool down, a material. That same spoon takes some time to cool down enough that it’s safe to touch after you take it out of the water. These two properties of materials limit how fast a hypothetical Venus could actually cool off.

Instead of calculating heat capacity and thermal conductivity of the entire crust, however, we’ll simplify the math and assume that we can use a simple column of the crust instead of the whole thing. Equation 4.3 shows how we calculate the heat capacity of a 50 km tall x 1 m x 1 m square column of silica:

(Eqn 4.3)

where D_silica is the density of silica in kg/m³, v is the volume of the column of crust we’re working with in m³, and κ_silica is the specific heat capacity of silica. Plugging in these values from this list, we get that the heat capacity of a 50 km tall column of crust is 7.74 x 10¹⁰ J·K^-1.

From Equation 3.1 yesterday we know the thermal resistance R_crust of the same column of crust is 3.62 x 10⁴ K·W^-1.

We know from experience that anything that stores heat and resists its movement from one place to another cannot instantly move that heat from one place to another. Using the same hot spoon example, if we put just the handle in an ice bath immediately after we remove the spoon from the boiling water, we can still burn ourselves on the spoon end for several seconds, until the heat stored in the spoon has a chance to move to the ice bath (warming up the bath as the spoon cools). We can calculate how long this process takes for a chunk of Venus’ crust using Equation 4.4:

(Eqn 4.4)

Solving for time t, we get Equation 4.5:

(Eqn. 4.5)

where T_start is 7000 K (the assumed formation temperature of Venus and its assumed present core temperature) and the other values have been previously calculated. From this calculation, we find that the time to cool the crust alone is about 201 million years. If we only cool from the melting temperature of silica (1923 K), then we calculate that the crustal cooling time is approximately 86 million years.

Manicouagan Impact Crater on Earth (STS-9 Crew, Dec 13,
2000)

These calculations indicate that Venus must have formed sometime between 86 and 201 million years ago in order for the planet’s surface to still be hot from the heat of formation. There are a number of major impact craters on the Earth’s sufrace that fall within this period, but there’s no obvious clusters of impacts that we might expect due to the formation of a new planet. The only mass extinction (an event that could reasonably coincide with a formation of a new planet within the Earth’s orbit) that’s even close to this range is the Triassic-Jurassic (end-Triassic) extinction, but no impact crater has yet been located that is close enough to the boundary to have been the cause. Instead, scientists currently believe this particular mass extinction was caused by increased volcanic activity leading to major climate changes.

While we could possibly imagine that the formation of Venus could stretch the Earth enough to increase volcanic activity, keep in mind that planet formation is thought to take tens to hundreds of millions of year. That entire period would have bombarded the Earth with major impacts (as it was when the Solar System was believed to have formed about 4-4.5 billion years ago), something that is not observed in the geologic record.

In summary, scientists don’t see the geologic evidence we would expect to see if Venus formed recently (in the last billion years or so). And this hypothesis also doesn’t solve the problem we identified yesterday, namely that the surface is too hot for all of that heat today to be coming from the core.

Tomorrow, how scientists know that Venus’ surface temperature isn’t from a recent astronomical collision.

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
• Avogadro’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