A planet has two moons of equal mass. Moon one orbits at radius r, and moon two orbits at radius 2r. How does the gravitational force on moon two compare to that on moon one?

For moon one, the masses are $M_p$ and $m$, and the distance is $r$.

For moon two, the masses are $M_p$ and $m$, and the distance is $2r$.

We can rewrite $F_2$ by factoring out $\frac{1}{4}$:

Since $F_1 = \frac{G M_p m}{r^2}$, we can substitute $F_1$ into the equation for $F_2$:

This means the gravitational force on moon two is one fourth as large as the gravitational force on moon one.

The result matches option D.

The final answer is $\boxed{\text{D) one fourth as large as that on moon one}}$.