USAAAO 2026, I кръг — Задача 17
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
A quickly-moving black hole passes through the center of a large star cluster. While the black hole does not experience any direct collisions with any stars in the cluster or significantly disrupts any stellar atmospheres, the black hole imparts a fraction of its momentum to several stars in the cluster close to the black hole’s trajectory. (This is known as dynamic gravitational friction and can be explained purely with Newtonian mechanics.) Some time passes, and due to complex multi-body interactions, the additional energy and momentum get distributed over the entire star cluster.
After a sufficiently long time, which of the following statements are likely TRUE regarding the system (compared to before the black hole event), assuming no further perturbations occur?
(I) The average peak blackbody emission wavelength among stars in the cluster decreases significantly.
(II) Stars are on average further apart from each other than they were before.
(III) Stars have lower relative velocities with respect to each other than they had before.
(a) I only
(b) II only
(c) III only
(d) II and III
(e) I and II
Отговори
- D
Решение
Solution: The black hole initially increases the (relative) velocities of several stars in the cluster. This has the effect of increasing the total average energy of stars in the cluster. However, after a long time, this energy gets distributed over the entire cluster. By the virial theorem, this causes the average potential energy to increase (making stars further apart, so (II) is true) as well as causing the average kinetic energy to decrease, making (III) counterintuitively correct. (I) is trivially false because the additional energy manifests as a change in bulk kinetic energy of entire stars as opposed to their internal thermal energies, meaning that the spectra are essentially unchanged.
Answer: D
Оригинал в Архива: usaaao_first_exam_2026.pdf · официални решения: usaaao_first_exam_solutions_2026.pdf