USAAAO — 2025 First Round

Consider a giant star with radius R. The core has temperature Tc and the surface has temperature Ts ≪ Tc. Given R, Tc, and Ts, Alice and Bob estimate the core’s radius in two different ways, both assuming that the star is in thermal equilibrium: Alice: The core radiates energy. Bob: The inside of the star conducts heat with constant uniform thermal conductivity κ such that κ ≫ RσTc⁴/Ts. Let RA and RB be the core radii estimated by Alice and Bob. Which of the following is the ratio RA/RB proportional to?