USAAAO 2026, I кръг — Задача 14
USAAAO 2026 - First Round · 9 февруари 2026 г.
Problem writers
Problem writers: Srihari (Hari) Balaji, Vincent Bian, Ferdinand, Hagan Hensley, Elizabeth Lee, Joe McCarty, Lucas Pinheiro, Feodor Yevtushenko
Условие
Consider a cloud of hot gas with radius and total (electromagnetic) power emission rate . Keeping the average temperature of gas particles and average emissivity per particle constant, the cloud now expands to a radius . How does the net total power radiated away, observed from far away, change in the cases where the cloud is optically thin (optical depth ) and optically thick (optical depth ), respectively? Assume the total number of particles remains constant.
(a) Remains approximately constant, remains approximately constant
(b) Remains approximately constant, increases significantly
(c) Increases significantly, decreases significantly
(d) Increases significantly, remains approximately constant
(e) Increases significantly, increases significantly
Отговори
- B
Решение
Solution: If the gas is optically thin, then light emitted from one particle has a negligible probability of being absorbed and/or scattered by another particle, meaning that the output power will be essentially equal to the total power of all the particles both before and after, meaning that it remains approximately constant. (Note that because of Kirchoff’s laws, this curve may likely deviate significantly from that of a perfect blackbody.)
If the gas is optically thick, then by Kirchoff’s laws, we essentially have a blackbody emission curve. By the Stefan-Boltzmann Law, the power emitted is just . Since doubles throughout this process, the power increases by a factor of 4.
Answer: B
Оригинал в Архива: usaaao_first_exam_2026.pdf · официални решения: usaaao_first_exam_solutions_2026.pdf