====== 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.