USAAAO 2018, II кръг — Задача 1
2018 National Astronomy Olympiad (NAO) · 5 т.
Instructions (Please Read Carefully)
(Start the exam after reading the instructions carefully and affirming that the contestant understood the instructions given.)
The top 5 eligible scorers on the NAO will be invited to represent USA at the next IOAA. In order to qualify for the national team, you must be a high school student with US citizenship or permanent residency.
This exam consists of 2 parts: Short Questions (5 points each) and Long Questions (15 points each). The maximum score is 100 points.
The test must be completed within 2.5 hours (150 minutes). The proctor should mark the start and end time of the exam on the front page.
Please solve each problem on a blank piece of paper and mark the number of the problem at the top of the page. The contestants full name in capital letters should appear at the top of each solution page. If the contestant uses scratch papers, those should be labeled with the contestants name as well and marked as ”scratch paper” at the top of the page. Scratch paper will not be graded. Partial credit will be available given that correct and legible work was displayed in the solution. This exam document, solution, and all the scratch paper used should be turned in to the proctor at the end of the exam.
This is a written exam. Contestants can only use a scientific calculator for this exam. A table of physical constants will be provided. Discussing the problems with other people is strictly prohibited in any way until the end of the examination period on April 6th. Receiving any external help during the exam is strictly prohibited. This means that the only allowed items are: a scientific calculator, the provided table of constants, a pencil (or pen), an eraser, blank sheets of papers, and the exam. No books, notes, laptops, mobile phones, or any other devices are allowed while taking the exam.
After reading the instructions, please make sure to sign at the bottom of this page, affirming that:
- All work on this exam is mine.
- I took this exam under a proctors supervision.
- I did not receive any external aids besides the materials provided.
- I am not allowed to discuss the test with others throughout the period of this examination.
- Failure to follow these rules will lead to disqualification from the exam.
Условие
In order to detect an Earth-twin, we need significant advances in the precision of spectrographs to detect the periodic Doppler shift of nearby stars. Estimate the radial velocity semi-amplitude, in m/s, that a planet with the mass, radius, and semi-major axis of Earth would cause in the motion of a star with the mass of the Sun. Assume that the Earth-twin has zero eccentricity. Note that the mass of Earth is kg, and the distance from the Earth to the Sun is m.
Отговори
0.09 m/s
The official solution gives vs = 0.090 m/s, i.e. about 0.1 m/s. The problems document prints the Earth–Sun distance as 1.5 × 10⁸ m; the solutions document restates it as 1.5 · 10⁸ km (the correct unit).
Решение
Since , and from Kepler’s law . Calculating it ends up being
where is the mass of the Earth and is the mass of the Sun. Plugging in the numbers we get . Thus the radial velocity semi-amplitude of .
Оригинал в Архива: usaaao-nao-2018.pdf · официални решения: usaaao_second_round_2018.pdf