A spherical planet has an initial escape velocity v₀ . Consider two hypothetical scenarios for this planet. Scenario A: The planet shrinks such that its radius becomes half of its original value, but its total mass remains unchanged. The new escape velocity is v_A . Scenario B: The outer layers of the planet are stripped away until its radius becomes half of its original value, but its uniform density remains unchang

Step-by-step solution

The initial escape velocity is v₀ = 2GM R , which can also be written in terms of density as v₀ = R 8 G 3 . In Scenario A, the mass M is constant and the radius becomes R 2 . The new escape velocity v_A is: v_A = 2GM ( R 2 ) = 2 2GM R = 2 v₀ In Scenario B, the density is constant and the radius becomes R 2 . The new escape velocity v_B is: v_B = ( R 2 ) 8 G 3 = 1 2 ( R 8 G 3 ) = 1 2 v₀ The ratio of the two new escape velocities is: v_A v_B = 2 v₀ 1 2 v₀ = 2 2 Answer: 2 2