Consider an ideal family of rocky planets that all have the *same uniform average density* ρ but different radii. Treat each planet as a perfect sphere and neglect any rotation or atmosphere.
(a) Show that, for a planet of this family, the escape speed vesc from its surface can be expressed purely in terms of G, ρ, and the planet’s radius R. Provide the explicit algebraic expression.
(b) Explain how your result in part (a) indicates the way vesc scales with planet radius when density is held constant, and contrast this with how vesc scales when mass is held constant instead.
(c) Earth’s average density is 5.5gcm−3 and its radius is 6.37×106m. Using your expression from part (a), determine the escape speed for a hypothetical planet with the same density as Earth but radius 1.50 times larger. State your answer to an appropriate number of significant figures.