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USAAAO 2020, II кръг — Задача 6

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от USAAAO 2020

2020 National Astronomy Competition · 15 т.

Instructions (Please Read Carefully)

The top 5 eligible scorers on the NAC will be invited to represent USA at the next IOAA. The next 5 eligible scorers will be invited to join the USA guest team 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 3 parts: Short Questions, Medium Questions and Long Questions.

The test must be completed within 2.5 hours (150 minutes).

Please solve each problem on a blank piece of paper and mark the number of the problem at the top of the page. The contestant’s 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 contestant’s 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 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 5th. 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 or notes are allowed during the exam.

Условие

An astronomer used his f/5 telescope with a diameter of 130 mm to observe a binary system. He is using an eyepiece with a field of view of $45^{o}$ and a focal length of 25 mm. In this system, star A has a mass of 18.9 solar masses, and an apparent magnitude in the V filter of 9.14. Star B has a mass of 16.2 solar masses, and an apparent magnitude in the V filter of 9.60. The period of the system is 108 days, and the distance between the binary stars and the Solar System is 2.29 kpc. The binary system has an edge-on orbit relative to the Solar System.

(a) What is the field of view of the telescope?

(b) What is the limiting magnitude of the telescope?

(c) What is the angular resolution of the telescope?

(d) What is the angular separation between the stars?

(e) Is the astronomer able to observe both stars as distinct points in the telescope? Answer as YES or NO.

The limiting magnitude for the human eye is 6.0, and the diameter of the pupil is equal to 7.0 mm. Also consider that visible light has a wavelength of 550 nm.

Отговори

Решение

Solution:

(a) It is possible to divide the field of view of the eyepiece by the magnification of the telescope in order to obtain an approximation for the field of view of the telescope:

FOV_{telescope} = \frac{FOV_{eyepiece}}{m}

FOV_{telescope} = \frac{FOV_{eyepiece}}{f_{telescope}/f_{eyepiece}}

FOV_{telescope} = \frac{45^{\circ}}{(130 \times 5)/(25)}

FOV_{telescope} = 1.7^{\circ}

(b)

m_{eye} - m_{telescope} = 2.5\log\frac{D_{eye}^2}{D_{telescope}^2}

6.0 - m_{telescope} = 2.5\log\frac{7^2}{130^2}

m_{telescope} = 12.3

Since both of stars are brighter than the limiting magnitude, they both can be detected by the telescope.

\theta = \frac{1.22\lambda}{D_{telescope}}

\theta = \frac{1.22 \times 550 \times 10^{-9}}{130 \times 10^{-3}}

\theta = 5.16 \times 10^{-6}\mathrm{rad}

(d) Using Kepler’s Third Law, it is possible to determine the separation between star A and star B:

T^2/a^3 = 1/M

(108/365.25years)^2/a^3 = 1/((18.9 + 16.2))

a = 1.45\,AU

Dividing the separation between the stars by the distance to the binary system, it is possible to obtain the angular separation between the stars:

\alpha = a/d

\alpha = (1.45AU)/(2.29 \times 10^{3} \times 206265AU)

\alpha = 3.08 \times 10^{-9}\mathrm{rad}

(e) Since the angular separation between the stars is smaller than the angular resolution, the astronomer will observe both stars as a single point in his telescope. Therefore, the correct answer is NO.

Оригинал в Архива: usaaao_second_exam_2020__apr4.pdf · официални решения: usaaao_second_exam_2020__apr4sol.pdf