Halliday, Resnick & Walker — Gravitation: Problem Problem_96

Compactness. The compactness of an astronomical body is the ratio of its Schwarzschild radius \( R_s \) to its actual radius \( R \). What is the compactness of (a) Earth, (b) the Sun, (c) a neutron star with density \( \rho = 4.0 \times 10^{17} \text{ kg/m}^3 \) and radius \( R = 20.0 \text{ km} \), and (d) a black hole (take \( R \) to be the Schwarzschild radius, its only measurable radius)? A black hole is the most compact object in the universe.