Съдържание

Условие Отговори Решение

IOAA 2014, Теория, окончателен вариант — Задача 8. Space – ship orbiting the Sun

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IOAA 2014

8th International Olympiad on Astronomy and Astrophysics — Theoretical Test, 12 short problems (final edition)

Указания

In your folder you will find the following items:

a. Answer sheets

b. Rough work sheets

c. The envelope with the problems The solutions of the problems should be written only on the answer sheets you receive. 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 side of the answer sheet.
The rough worksheets are for your own use for doing calculations, write some numbers etc. BEWARE: These sheets are not taken into account for the evaluation. At the end of the test they will be collected separately. Everything you consider as part of the solutions should be written on the answer sheets.
Each problem should be started on a separate answer sheet.
On each answer sheet please fill in the designated boxes as follows:

a. In the „PROBLEM NO.” box write down only the number of the problem: i.e. 1 – 12 for each short problems, 13 – 15 for each 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 that you will find on your envelope, consisting of 3 letters and 2 digits.

c. In the „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 language of Mathematics is universal, so, please, use as many mathematical expressions as you think that may help the evaluator to better understand your solutions. If you want to explain something in words we kindly ask you to use short phrases(if possible in English).
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:

d. Don’t forget to put your papers in order.

e. Put the answer sheets in 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 will help us to understanding your solutions.

f. Verify with the assistant the correct number of answer sheets used and fill in this number on the cover of the folder and sign it.

g. Put the draft papers in the designated folder. Put the test papers back in the envelope.

h. Go to swim

GOOD LUCK !

Marking scheme

ALL THE CORRECT SOLUTIONS WHICH ARE DIFFERENT FROM THE AUTHOR’S SOLUTION WILL BE EVALUATED AND MARKED ACORDINGLY TO THE MARKING SCHEME.

Условие

A spherical space –ship orbits the Sun on a circular orbit, and spin around an axis of rotation that is perpendicular to the orbital plane of the space-ship. The temperature on the exterior surface of the ship is $T_N$ . Assume the space -ship is a perfect black body and there is no activity inside it.

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 8. Marking scheme Space – ship orbiting the Sun

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,

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}.

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)

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\cdot\lg\frac{2d_0T_{\mathrm{N}}^2}{R_{\mathrm{S}}T_{\mathrm{S}}^2}

Оригинал в Архива: Short Theoretical.pdf · официални решения: short_final_sol.pdf · допълнителни данни: ioaa1.jpg · допълнителни данни: ioaa2.jpg · допълнителни данни: ioaa3.jpg · допълнителни данни: ioaa4.jpg · допълнителни данни: ioaa5.jpg · допълнителни данни: ioaa6.jpg · допълнителни данни: ioaa7.jpg