Gravitation Near Earth's Surface — Question 3

Question 3

Surface gravity and radius constrain mean density

A spherical nonrotating planet has measured radius $R$ and surface gravity $g_s$. No assumption about uniform interior density is made.

1. Infer its total mass and mean density.

2. A second planet has the same $g_s$ but twice the radius. Compare mass and mean density.

Question 3 - Solution

Surface gravity and radius constrain mean density

(a) Derivation and calculation

Spherical symmetry fixes the exterior field from total mass: $$g_s=\frac{GM}{R^2}\quad\Longrightarrow\quad \boxed{M=\frac{g_sR^2}{G}}.$$ Dividing by total volume, $$\boxed{\bar\rho=\frac{M}{4\pi R^3/3}=\frac{3g_s}{4\pi GR}}.$$ No constant-density interior assumption is needed.

(b) Result and interpretation

For $R_2=2R_1$ and equal $g_s$, $$\boxed{\frac{M_2}{M_1}=\left(\frac{R_2}{R_1}\right)^2=4},$$ $$\boxed{\frac{\bar\rho_2}{\bar\rho_1}=\frac{M_2/M_1}{(R_2/R_1)^3}=\frac48=\frac12}.$$ These measurements determine mean density, not its radial distribution.

Physical check

The inferred density has units $\mathrm{kg/m^3}$ and decreases with radius at fixed surface gravity.