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====== Plot Earth’s density profile ======
We can estimate how density varies inside the Earth by using seismic data, particularly the speeds of **P-waves** and **S-waves**. These waves travel through the Earth’s interior during earthquakes and carry information about its internal structure.

==== Seismic Wave Velocities ====

The velocities of seismic waves depend on the elastic properties and density of the material.

  * **P-wave (primary, compressional) speed:**
  $$
  v_p^2 = \frac{K + \frac{4}{3} \mu}{\rho}
  $$

  * **S-wave (secondary, shear) speed:**
  $$
  v_s^2 = \frac{\mu}{\rho}
  $$

Here:
  * $K$ = bulk modulus (resistance to compression)
  * $\mu$ = shear modulus (resistance to shear)
  * $\rho$ = density

By eliminating $\mu$, we can express the **bulk modulus per unit density** as:

$$
\frac{K}{\rho} = v_p^2 - \frac{4}{3} v_s^2
$$

==== Hydrostatic Equilibrium and Pressure Gradient ====

The pressure at a given depth in the Earth must balance the weight of overlying layers. The pressure difference between the top and bottom of a spherical shell is given by:

$$
\frac{\Delta P}{\Delta r} \simeq -\frac{G \rho M(<r)}{r^2}
$$

where:
  * $G$ = gravitational constant
  * $M(<r)$ = mass enclosed within radius $r$

The **bulk modulus** is also defined as:

$$
K = \frac{\Delta P}{\Delta \rho}
$$

Combining the two equations, we get the **density gradient** equation:

$$
\Delta \rho = - \frac{\rho^2 G M(<r)}{K r^2} \Delta r
$$

This tells us how density changes with depth, provided we know seismic wave speeds (to get $K/\rho$) and mass distribution.

==== Williams–Adams Algorithm: Step-by-Step ====

To compute the density profile from the surface to the center of the Earth:

  - **Step 1**: Use P- and S-wave travel time data to calculate $K/\rho$ at each depth.
  - **Step 2**: Assume outermost density (surface rock) $\rho \approx 3000$ kg/m³.
  - **Step 3**: Estimate total mass $M = 5.972 \times 10^{24}$ kg.
  - **Step 4**: Discretize the Earth into thin shells (e.g., 100 layers).
  - **Step 5**: Use the density gradient equation to compute $\Delta \rho$ for each shell.
  - **Step 6**: Recompute the mass enclosed within each radius.
  - **Step 7**: Repeat inward to the core.

==== Python ====
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