USAAAO — 2026 First Round

Consider a cloud of hot gas with radius R and total (electromagnetic) power emission rate P. Keeping the average temperature of gas particles and average emissivity per particle constant, the cloud now expands to a radius 2R. How does the net total power P radiated away, observed from far away, change in the cases where the cloud is optically thin (optical depth τ ≪ 1) and optically thick (optical depth τ ≫ 1), respectively? Assume the total number of particles remains constant.