USAAAO 2026, I кръг — Задача 2
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 planet of mass orbits a star of mass in an elliptical orbit with semi-major axis and eccentricity where . Which of the following statements is true?
(a) The orbital speed is maximized at aphelion due to conservation of energy
(b) The planet moves in a perfect circle around the star regardless of eccentricity
(c) The orbital period depends on eccentricity through Kepler’s Third Law
(d) The Roche limit increases if the planet’s density decreases
(e) All five Lagrange points correspond to stable equilibria
Отговори
- D
Решение
Solution:
(a) Orbital speed is maximized at aphelion. FALSE. Speed is maximized at perihelion due to conservation of angular momentum.
(b) The planet moves in a perfect circle around the star regardless of eccentricity. FALSE. The problem states the orbit is elliptical, so the planet does not move in a perfect circle.
(c) Orbital period depends on eccentricity. FALSE. Kepler’s Third Law gives , independent of eccentricity.
(d) Roche limit increases if planet density decreases. TRUE. Roche limit scales as
Lower planet density increases .
(e) All Lagrange points stable. FALSE. are unstable; only may be stable.
Answer: D
Оригинал в Архива: usaaao_first_exam_2026.pdf · официални решения: usaaao_first_exam_solutions_2026.pdf