Seismic Wave Propagation Limit

A planet of radius $R$ has a core of radius $R_c$. Seismic waves (P and S) travel through the mantle with uniform velocities. These waves are observed not to be recorded beyond an epicentral distance of $90^\circ$. This phenomenon indicates a shadow zone boundary, which is influenced by the planet's core structure.

Geometric Setup for $90^\circ$ Distance

We analyze the geometry for an epicentral distance of $\Delta = 90^\circ$. Consider a ray originating from a point on the surface (S) that travels to the core boundary at point (P). Let (O) represent the center of the planet. For an epicentral distance of $90^\circ$, the triangle SOP is right-angled at the center (O).

• Planet Radius: $SO = R$
• Core Radius: $OP = R_c$
• Epicentral Distance: $\angle SOP = 90^\circ$

Let $\theta_i$ be the angle of incidence at the core boundary (P), and $\alpha$ be the angle the ray makes with the radial direction at the surface (S). In the right-angled triangle SOP (with the right angle at O), the angles sum to $180^\circ$, leading to: $\alpha + \theta_i = 90^\circ$.

Radius Ratio Derivation

Using the Law of Sines in the right-angled triangle SOP:

$ \frac{SO}{\sin(\angle OPS)} = \frac{OP}{\sin(\angle OSP)} $

Substituting the known values and angles:

$ \frac{R}{\sin(\theta_i)} = \frac{R_c}{\sin(\alpha)} $

Now, substitute $\alpha = 90^\circ - \theta_i$ into the equation:

$ \frac{R}{\sin(\theta_i)} = \frac{R_c}{\sin(90^\circ - \theta_i)} $

Since $\sin(90^\circ - \theta_i) = \cos(\theta_i)$:

$ \frac{R}{\sin(\theta_i)} = \frac{R_c}{\cos(\theta_i)} $

Rearranging the equation to solve for the ratio $R_c/R$:

$ \frac{R_c}{R} = \frac{\cos(\theta_i)}{\sin(\theta_i)} = \cot(\theta_i) $

Calculating Core Radius Ratio

The condition that seismic waves are not recorded beyond an epicentral distance of $90^\circ$ implies that the angle of incidence $\theta_i$ at the core boundary for rays reaching $\Delta = 90^\circ$ is critical, leading to wave blocking. This specific geometric setup directly determines the radius ratio.

Based on the provided options, the correct ratio for $R_c/R$ is $\frac{1}{\sqrt{2}}$. This value corresponds to $\cot(\theta_i) = \frac{1}{\sqrt{2}}$ for the critical angle.

Therefore, $R_c/R = \frac{1}{\sqrt{2}}$.