Using dimensional analysis

To estimate how deep thermal energy could have conducted through rock since the Solar System formed, we apply a simple dimensional analysis approach. Suppose heat conduction is the dominant process of energy loss from a planet’s interior. We aim to estimate the maximum depth, $L_{\text{max}}$, from which significant energy could have diffused away.

We assume that $L_{\text{max}}$ depends on a few key physical parameters:

• $K_T$ – the thermal conductivity of the rock
• $C_p$ – the specific heat capacity
• $\rho$ – the density of the material
• $\tau$ – the age of the Solar System (roughly $4.5 \times 10^9$ years)

Let’s assume:

$$ L_{\text{max}} \sim f(C_p,\, \rho,\, K_T,\, \tau) $$

We now use dimensional analysis to combine these variables into a length. The units are:

• $[K_T] = \mathrm{W\,m^{-1}\,K^{-1}} = \mathrm{J\,s^{-1}\,m^{-1}\,K^{-1}}$
• $[C_p] = \mathrm{J\,kg^{-1}\,K^{-1}}$
• $[\rho] = \mathrm{kg\,m^{-3}}$
• $[\tau] = \mathrm{s}$

We look for a combination of these that yields units of length ($\mathrm{m}$). The expression:

$$ \frac{K_T \tau}{\rho C_p} $$

has units of $\mathrm{m^2}$. Taking the square root gives a quantity with units of length:

$$ L_{\text{max}} \sim \sqrt{ \frac{K_T \tau}{\rho C_p} } $$

This gives a simple estimate of how far heat can travel by conduction in time $\tau$. Using typical rock values:

• $K_T \sim 3\, \mathrm{W\,m^{-1}\,K^{-1}}$
• $C_p \sim 1000\, \mathrm{J\,kg^{-1}\,K^{-1}}$
• $\rho \sim 3000\, \mathrm{kg\,m^{-3}}$
• $\tau \sim 4.5 \times 10^9\, \mathrm{years} \approx 1.4 \times 10^{17}\, \mathrm{s}$

We find:

$$ L_{\text{max}} \sim \sqrt{ \frac{3 \times 1.4 \times 10^{17}}{3000 \times 1000} } \approx \sqrt{1.4 \times 10^{11}} \approx 3.7 \times 10^5\, \mathrm{m} = 370\, \mathrm{km} $$

So, the maximum thermal diffusion depth is about 300–400 km, suggesting that small bodies like asteroids may have cooled completely by conduction over the Solar System’s lifetime, but larger planets could retain significant internal heat.