USAAAO 2024, I кръг — Задача 3
USAAAO 2024 - First Round · 10 февруари 2024 г.
Problem writers
Wesley Andrade, Abhay Bestrapalli, Austin Chen, Justin Chen, Ved Gund, Hagan Hensley, Sandesh Kalantre, Evan Kim, David Lee, Joe McCarty, Arjun Patel, Sahil Pontula, Tara Rezaei, Raul Teixeira, Leo Yao, David Zhang
Условие
Aliens in a nearby star system (located in a random direction from Earth) are looking for nearby planets using the transit method. What is the probability that they can see the Earth transit across the Sun?
(Assume they observe our Sun over multiple years and their instruments are sensitive enough to detect any transit that occurs.)
(a) 100%
(b) 5.8%
(c) 0.93%
(d) 0.47%
(e) 0.15%
Отговори
- D
Решение
Solution:
Using the small-angle approximation, a planet orbiting a distance around a star of radius can be seen to transit if the inclination of the orbital plane is less than . We can think of this as tracing out a strip around the celestial sphere of width (since the inclination can be in either direction), and the aliens will only see the transit if their star lies in this strip. The solid-angle area of this strip is , and the total solid angle of the celestial sphere is , so the chance that a randomly selected star is within the strip is
For Earth, this is about 0.47%.
Answer: D
Оригинал в Архива: usaaao_first_exam_2024-3.pdf · официални решения: usaaao_first_exam_2024_solutions-1.pdf