There is a ring outside of Saturn. In order to distinguish if the ring is actually a part of Saturn or is instead part of the satellites of Saturn, we need to know the relation between the velocity $v$ of each layer in the ring and the distance $R$ of the layer to the center of Saturn. Which of the following statements is correct?

$\textbf{(A) }$ If $v \propto R$, then the layer is part of Saturn.
$\textbf{(B) }$ If $v^2 \propto R$, then the layer is part of the satellites of Saturn.
$\textbf{(C) }$ If $v \propto 1/R$, then the layer is part of Saturn.
$\textbf{(D) }$ If $v^2 \propto 1/R$, then the layer is part of Saturn.
$\textbf{(E) }$ If $v \propto R^2$, then the layer is part of the satellites of Saturn.
To determine the correct relationship between the velocity \( v \) of each layer in the ring and the distance \( R \) of the layer to the center of Saturn, we need to consider the gravitational forces and the dynamics of circular motion.

1. **Gravitational Force and Circular Motion:**
   The gravitational force \( F_g \) acting on a mass \( m \) at a distance \( R \) from the center of Saturn is given by:
   \[
   F_g = \frac{G M m}{R^2}
   \]
   where \( G \) is the gravitational constant and \( M \) is the mass of Saturn.

2. **Centripetal Force:**
   For an object in circular motion, the centripetal force required to keep it moving in a circle of radius \( R \) with velocity \( v \) is:
   \[
   F_c = \frac{m v^2}{R}
   \]

3. **Equating Gravitational Force and Centripetal Force:**
   For an object in orbit around Saturn, the gravitational force provides the necessary centripetal force:
   \[
   \frac{G M m}{R^2} = \frac{m v^2}{R}
   \]

4. **Solving for \( v \):**
   Simplifying the above equation, we get:
   \[
   v^2 = \frac{G M}{R}
   \]
   Taking the square root of both sides, we obtain:
   \[
   v = \sqrt{\frac{G M}{R}}
   \]

5. **Analyzing the Options:**
   - **Option (A):** \( v \propto R \)
     \[
     v \propto R \quad \text{(Incorrect, as \( v \) is proportional to \( \frac{1}{\sqrt{R}} \))}
     \]
   - **Option (B):** \( v^2 \propto R \)
     \[
     v^2 \propto R \quad \text{(Incorrect, as \( v^2 \propto \frac{1}{R} \))}
     \]
   - **Option (C):** \( v \propto 1/R \)
     \[
     v \propto \frac{1}{R} \quad \text{(Incorrect, as \( v \) is proportional to \( \frac{1}{\sqrt{R}} \))}
     \]
   - **Option (D):** \( v^2 \propto 1/R \)
     \[
     v^2 \propto \frac{1}{R} \quad \text{(Correct, as derived above)}
     \]
   - **Option (E):** \( v \propto R^2 \)
     \[
     v \propto R^2 \quad \text{(Incorrect, as \( v \) is proportional to \( \frac{1}{\sqrt{R}} \))}
     \]

Conclusion:
The correct statement is:
\[
\boxed{\textbf{(D) } v^2 \propto 1/R}
\]