Answer
The acceleration due to gravity at a planet's surface would depend linearly on the planet’s radius.
Work Step by Step
We see from equation 13.4 that the acceleration due to gravity is proportional to a planet’s mass, and inversely proportional to the square of its radius: $g=\frac{GM}{R^2}$.
If all planets had the same average density $\rho$, then the planet's mass is the density multiplied by the volume of the planet.
$$M=\rho \frac{4\pi}{3}R^3$$
In that case, relate the acceleration due to gravity to the planet’s size.
$$g=\frac{GM}{R^2}=\frac{G}{R^2}(\rho \frac{4\pi}{3}R^3)$$
$$g=\frac{4G\pi\rho}{3} R$$
The acceleration due to gravity at a planet's surface would depend linearly on the planet’s radius.
