SAO 2021, Писмен тур — Задача 8. Distance Parameters

Singapore Astronomy Olympiad 2021 · 9 т.

Указания
  1. The entire Olympiad lasts for 3.5 hours and is worth a total of 200 marks.
  2. Fill in these details on the cover page (after the Contents page), the Summary Answer Sheet and each side of an SAO answer sheet:
    • Year of competition
    • Your participant code
    • The page number – which should be continuous from 1 to N
    • The part of the paper, and the question number
  3. Cross out all workings or answers you do not wish to be evaluated.
  4. You are allowed to leave the venue early upon submission of your answer script (see Instructions on Cover Page), but only after one hour after the start of the paper and not within 30 minutes to completion of the paper.
  5. Please raise your hand to seek assistance from the invigilators (e.g. to visit the restroom, request for more answer sheets, enquire about an ambiguity, etc.)
Competition Rules and Regulation
  1. Only the use of scientific calculators is permitted. No graphing or programmable calculators are allowed.
  2. Disruptive behaviour, cheating, collusion to cheat or any integrity-related offences are grounds for immediate disqualification.
  3. You may opt to retain the question paper and constants sheet for personal use. Return all unused answer sheets to the Organising Team.
References for Data Analysis Question
  1. MS Star Details are derived from: Zombeck, Martin V. (1990). Handbook of Space Astronomy and Astrophysics (2nd ed.). Cambridge University Press
  2. Additional MS Star Details are derived from: Vardavas, Ilias M.; et al. (2011), "Chapter 5. Incoming Solar Radiation" Radiation and Climate: Atmospheric Energy Budget from Satellite Remote Sensing.

Astronomy.SG would like to thank the organizing committee and the question setters. This document is typeset in LaTeX.

Cover Page
  1. Fill in all required details below, before time is up.
  2. Ensure that all materials (answer sheets, graphing paper, practical question paper) you intend to submit for grading have been attached. Amendments to any answer script after it has been collected will not be entertained.
  3. Details on each side of answer sheet:
    • Your participant code.
    • Question number.
    • Page number – continuous from 1 to N throughout your answer script.
    • DO NOT WRITE YOUR NAME ON YOUR ANSWER SCRIPT.
  4. Order of answer script:
    • Stack 1: i. This cover page.
      ii. Summary Answer Sheet iii. Answer sheets from Part 1 to 3, in continuous order from page 1 to M.
    • Stack 2: i. Answer sheets for Part 4, in continuous order from M to N.
      ii. Graph(s) from Part 4.
    • Stack 3: i. Practical cover page.
      ii. Practical Component from Part 5.
1 Section A [12]

Write all your responses in the summary answer sheet provided.

Section B(1): Short Questions [36]

This section contains a total of 5 short-length questions worth 9 marks each.
You are to choose 4 to do.

Section B(2): Medium Questions [18]

This section contains a total of 2 medium-length questions worth 18 marks each.
You are to choose 1 to do.

Practical Cover Page

INSTRUCTIONS TO PARTICIPANTS:

  1. Fill in all required details below, before time is up.
  2. You are to detach this page and all subsequent pages as part of your response.
  3. All detached pages are to be re-stapled and be part of your 3rd stack of your answer script.
  4. All responses to this section of the paper are to be written on the question paper itself, in the spaces provided.
  5. There will be a star chart provided as part of the question. Responses to the corresponding questions in that section are to be answered on the star chart itself.
  6. All final responses shall be written legibly and unambiguously.
  7. Details on each page:
    • Your participant code.
    • DO NOT WRITE YOUR NAME ON YOUR ANSWER SCRIPT.
Section A

Each question is worth 2 marks.

Summary of answers:

Q1Q2Q3Q4Q5Q6
DCACBB

Условие

The luminosity distance dLd_L is defined in terms of the relationship between the absolute magnitude MM and the apparent magnitude mm of an astronomical object M=m−5(log⁡(dL−1))M = m - 5(\log(d_L - 1)), and the angular diameter distance dAd_A is defined in terms of an object’s physical size xx and the angular size θ\theta of the object as viewed from the Earth: dA=xθd_A = \frac{x}{\theta}. The comoving distance dCd_C is similar to the proper distance, but factors out the expansion of the universe, giving a distance that does not change in time – it is equal to proper distance dPd_P at present time.

(a) Show that the luminosity distance dL=dC(1+z)d_L = d_C(1 + z) and the angular diameter distance dA=dC1+zd_A = \frac{d_C}{1+z}. [3 т.]

(b) The intensity, or radiance, (the radiant flux per unit solid angle per unit projected area) of a distance light source changes with redshift Iz∝(1+z)mI_z \propto (1 + z)^m. Determine the exponent mm. [4 т.]

(c) In an Einstein-de Sitter Universe, the comoving distance is dC=2cH0(1−11+z)d_C = \frac{2c}{H_0}(1 - \frac{1}{\sqrt{1+z}}). An object is measured to have redshift z=1z = 1 and spans 102 kly across. Determine its angular diameter θ\theta as viewed from the Earth. [2 т.]

Отговори

Решение

Оригинал в Архива: SAO 2021 Question Paper.pdf · официални решения: SAO 2021 Compiled Solutions.pdf