====== Explain why Earth remains hot ======
Earth formed over 4.5 billion years ago from the gravitational collapse of planetesimals. That process released a large amount of energy, raising Earth’s initial temperature significantly. But surprisingly, even after billions of years, Earth’s interior is still hot and geologically active. Why?

This can be explained using three key physical ideas:

==== 1. Initial Heating from Accretion ====

The temperature of Earth immediately after its formation is estimated by:

$$
T = \frac{4\pi G R_p^2 \rho}{5 C_p}
$$

Where:
  * $G$ = gravitational constant
  * $R_p$ = Earth's radius
  * $\rho$ = Earth's average density
  * $C_p$ = specific heat of rock

Using typical values, this gives $T \approx 41000\,\text{K}$ — hot enough to melt rock. So, Earth started as a molten body.

==== 2. Limited Heat Loss via Conduction ====

Heat escapes from Earth's interior primarily by conduction through the lithosphere. The depth to which heat can escape through conduction and radiation over time $\tau$ is given by:

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

Where:
  * $K_T$ = thermal conductivity
  * $\tau$ = time (age of Solar System)
  * $\rho$ = density
  * $C_p$ = specific heat

This evaluates to about 300 km. Since Earth’s radius is over 6000 km, most of its interior lies **beneath** the conductive zone and remains well insulated. This means **cooling is very slow**.

==== 3. Ongoing Heating from Radioactive Decay ====

Earth's interior contains radioactive isotopes such as:
  * Uranium-238 ($^{238}$U) — half-life ≈ 4.5 billion years
  * Thorium-232 ($^{232}$Th) — half-life ≈ 14 billion years
  * Potassium-40 ($^{40}$K) — half-life ≈ 1.3 billion years

These decay slowly, releasing energy continuously. Because their half-lives are so long, these isotopes are **still active today**, providing a **steady internal heat source**.

==== How This Heat Keeps Earth Alive ====

The combination of initial heat, slow cooling, and radioactive heating powers:
  * **Mantle convection**
  * **Plate tectonics**
  * **Volcanic activity**
  * **The magnetic field** (via the core geodynamo)

Without these heat sources, Earth would be geologically inactive — like the Moon or Mercury.

==== Summary Table ====

^ Source of Heat ^ Role in Earth’s Interior ^
| Gravitational Accretion | Heated the early Earth to molten temperatures |
| Slow Thermal Conduction | Prevents deep interior from cooling quickly |
| Radioactive Decay | Continually replenishes internal heat |

**→ Together, these processes explain why Earth still has a hot interior and remains geologically active.**