Съдържание

IOAA 2014, Теория, предварителен вариант — Задача 11. Space – ship orbiting the Sun

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IOAA 2014

8th International Olympiad on Astronomy and Astrophysics, Suceava - Gura Humorului - August 2014 — THEORETICAL TEST, 15 short problems

Indications

The problems were elaborated concerning two aspects: a) To cover merely all the subjects from the syllabus;
b) The average time for solving the items is about 15 minutes per a short problem;
In your folder you will find out the following: a) Answer sheets
b) Draft sheets
c) The envelope with the subjects in English and the translated version of them in your mother tongue;
The solutions of the problems will be written down only on the answer sheets you receive on your desk. PLEASE WRITE ONLY ON THE PRINTED SIDE OF THE PAPER SHEET. DON’T USE THE REVERSE SIDE. The evaluator will not take into account what is written on the reverse of the answer sheet.
The draft sheets is for your own use to try calculation, write some numbers etc. BEWARE: These papers are not taken into account in evaluation, at the end of the test they will be collected separately . Everything you consider as part of the solutions have to be written on the answer sheets.
Each problem have to be started on a new distinct answer sheet.
On each answer sheet please fill in the designated boxes as follows: a) In PROBLEM NO. box write down only the number of the problem: i.e. 1 – 15 for short problems, 16 – 19 for long problems. Each sheet containing the solutions of a certain problem, should have in the box the number of the problem;
b) In Student ID – fill in your ID you will find on your envelope, consisted of 3 letters and 2 digits.
c) In page no. box you will fill in the number of page, starting from 1. We advise you to fill this boxes after you finish the test
We don’t understand your language, but the mathematic language is universal, so use as more relationships as you think that your solution will be better understand by the evaluator. If you want to explain in words we kindly ask you to use short English propositions.
Use the pen you find out on the desk. It is advisable to use a pencil for the sketches.
At the end of the test: a) Don’t forget to put in order your papers;
b) Put the answer sheets in the folder 1. Please verify that all the pages contain your ID, correct numbering of the problems and all pages are in the right order and numbered. This is an advantage of ease of understanding your solutions.
c) Verify with the assistant the correct number of answer sheets used fill in this number on the cover of the folder and sign.
d) Put the draft papers in the designated folder, Put the test papers back in the envelope.
e) Go to swim

GOOD LUCK !

Условие

A spherical space –ship orbits the Sun on a circular orbit, and spin around one of its axes. The temperature on the exterior surface of the ship is $T_N$ . Find out the apparent magnitude of the Sun and the angular diameter of the Sun as seen by the astronaut on board of the space – ship. The following values are known:, $T_S$ - the effective temperature of the Sun; $R_S$ - the radius of the Sun; $d_0$ - the Earth –Sun distance; $m_0$ - apparent magnitude of Sun measured from Earth; $R_N$ - the radius of the space –ship.

Отговори

Решение

Problem 11. Marking scheme Space – ship orbiting the Sun

Correct use of the formulas (1) for the apparent brightness — 3 points
Correct use of the formula (2) by using Pogson formula — 3 points
Correct solution and reasoning — 4 points

Detailed solution

According to the Stefan – Boltzmann law, the luminosity of the Sun is:

L_{sun} = Q_{sun}\cdot 4\pi R_{\mathrm{Sun}}^2 = \sigma T_{\mathrm{Sun}}^4\cdot 4\pi R_{\mathrm{Sun}}^2, \qquad (1)

At distance $d$ from the Sun , where the space ship is the energy which passes the unit of surface in an unit of time is:

\phi_{\mathrm{Sun,d}} = \frac{L_{\mathrm{S}}}{4\pi d^2} = \frac{\sigma T_{\mathrm{S}}^4\cdot 4\pi R_{\mathrm{S}}^2}{4\pi d^2}. \qquad (2)

The space ship receive through its entire surface, in the unit of time, the energy:

P_{received} = \frac{\sigma T_{\mathrm{Sun}}^4\cdot 4\pi R_{\mathrm{Sun}}^2}{4\pi d^2}\cdot\pi R_{ship}^2.

Corresponding to its temperature, $T_{\mathrm{N}}$ , according the Stefan - Boltzmann law, the emitted energy by starship through its hole surface in the unit of time :

P_{\mathrm{emis,N}} = \sigma T_{\mathrm{N}}^4\cdot 4\pi R_{\mathrm{N}}^2.

When the temperature stabilized at thermic equilibrium :

P_{received,\mathrm{N}} = P_{\mathrm{emis,N}}

\frac{\sigma T_{\mathrm{S}}^4\cdot 4\pi R_{\mathrm{S}}^2}{4\pi d^2}\cdot\pi R_{\mathrm{N}}^2 = \sigma T_{\mathrm{N}}^4\cdot 4\pi R_{\mathrm{N}}^2

the distance of orbiting the Sun of the space ship is:

d = \frac{T_{\mathrm{S}}^2R_{\mathrm{S}}}{2T_{\mathrm{N}}^2},

The angular diameter of the Sun as seen from the space ship :

\alpha/2 = \frac{R_{\mathrm{S}}}{d}

\alpha = \frac{2R_{\mathrm{S}}}{d} = 4\left(\frac{T_{\mathrm{N}}}{T_{\mathrm{S}}}\right)^2

According the Pogson formula written for Sun seen from Earth and space ship the following relation occurs:

\lg\frac{E_{\mathrm{S,Nava}}}{E_{\mathrm{S,P}}} = -0{,}4\left(m - m_0\right) \qquad (2)

2\cdot\lg\left(\frac{d_0}{d}\right) = -0{,}4\left(m - m_0\right)

The apparent magnitude of the Sun as seen from the space ship

m = m_0 - 5^{\mathrm{m}}\cdot\lg\frac{2d_0T_{\mathrm{N}}^2}{R_{\mathrm{S}}T_{\mathrm{S}}^2}

Оригинал в Архива: th_short_ver1.pdf · официални решения: th_short_ver1_sol.pdf