Съдържание

IOAA 2014, Теория, предварителен вариант — Задача 13. Stars with Romanian names

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

Two Romanian astronomers Ovidiu Tercu and Alex Dumitriu from The Astronomical Observatory of the Museum Complex of Natural Sciences in Galati Romania, recently discovered – in September 2013- two variable stars. They used for that a telescope with the main mirror diameter of 40 cm and a SBIG STL-6303e CCD camera.

With the accord of the AAVSO (American Association of Variable Stars Observers), the two stars have now Romanian names: Galati V1 and respectively Galati V2. The two stars are circumpolar, located in Cassiopeia and respectively in Andromeda constellation. The two stars are visible above the horizon, form the territory of Romania, all over the year. The galactic coordinates of the two stars are: $\mathrm{Galati\ V1}\left(G_1 = 114.371^{\circ};\ g_1 = -11.35^{\circ}\right)$ and $\mathrm{Galati\ V2}\left(G_2 = 113.266^{\circ};\ g_2 = -16.177^{\circ}\right)$ .

Another star, discovered by the Romanian astronomer Nicolas Sanduleak, has also a Romanian name – Sanduleak -69° 202; it explodes as the supernova SN 1987. This star was localized in the Dorado constellation from the Large Magellan Cloud, by the coordinates:

\alpha = 5^{\mathrm{h}}35^{\mathrm{min}}28{,}03^{\mathrm{s}};\ \delta = -69^{\circ}16'11{,}79'';\ G = 279{,}7^{\circ};\ g = -31{,}9^{\circ}.

Estimate the angular distance between the stars Galati V1 and Galati V2.

Отговори

Решение

Problem 13. Marking scheme Stars with Romanian names

Correct geometrical calculations — 5 points
Correct calculations — 5 points

In the figure bellow the two stars $\sigma_1$ and $\sigma_2$ , are located using the galactic coordinates $\left(G_1;g_1\right)$ and respectively $\left(G_2;g_2\right)$ .on the geocentric celestial sphere. The spherical triangles $\sigma_1\mathrm{A}\sigma_2$ $\left(G_1;g_1\right)$ and respectively may be considered rectangular plane triangles because the angles $\Delta G = G_2 - G_1$ $\left(G_1;g_1\right)$ and respectively $\Delta g = g_2 - g_1$ are very small

Thus:

\sigma_1\sigma_2 = \sqrt{\left(\sigma_1\mathrm{A}\right)^2 + \left(\sigma_2\mathrm{A}\right)^2},

or:

\sigma_1\sigma_2 = \sqrt{\left(\sigma_1\mathrm{B}\right)^2 + \left(\sigma_2\mathrm{B}\right)^2},

Celestial hemisphere above the dashed galactic equator (points F′, F, centre O, the node Γ labelled “Nodul Galactic ascendending node of the North Pole”), with the Galactic North Pole J at the top; two parallels of galactic latitude with centres O₂ and O₁ and radii r₂ and r₁; the stars σ₂ and σ₁ as black dots, the points A and B completing the small triangles; marked angles Δg, Δφ, g₁, g₂, G₁, G₂ and ΔG, and the radius r from O. — Fig.

\sigma_1\mathrm{A} = r\cdot\Delta g;

\sigma_2\mathrm{A} = r_2\cdot\Delta G = r\cdot\cos g_2\cdot\Delta G;

\sigma_1\sigma_2 = r\cdot\Delta\varphi,

Where $\Delta\varphi$ is the angular distance between two stars

r\cdot\Delta\varphi = \sqrt{\left(r\cdot\Delta g\right)^2 + \left(r\cdot\cos g_2\cdot\Delta G\right)^2};

\Delta\varphi = \sqrt{\left(\Delta g\right)^2 + \left(\cos g_2\cdot\Delta G\right)^2};

\sigma_1\mathrm{B} = r_1\cdot\Delta G = r\cdot\cos g_1\cdot\Delta G;

\Delta\varphi = \sqrt{\left(\cos g_1\cdot\Delta G\right)^2 + \left(\Delta g\right)^2};

\left(G_1 = 114.371^{\circ};\ g_1 = -11.35^{\circ}\right);\ \left(G_2 = 113.266^{\circ};\ g_2 = -16.177^{\circ}\right);

\Delta G = G_2 - G_1 = -1{,}105^{\circ};\ \Delta g = g_2 - g_1 = -4{,}827^{\circ};

\cos g_1 = 0{,}98;\ \cos g_2 = 0{,}96;

\Delta\varphi = \sqrt{\left(-4{,}827^{\circ}\right)^2 + \left(0{,}96\right)^2\cdot\left(-1{,}105^{\circ}\right)^2} \approx 4{,}942^{\circ};

\Delta\varphi = \sqrt{\left(0{,}98\right)^2\cdot\left(-1{,}105^{\circ}\right)^2 + \left(-4{,}827^{\circ}\right)^2} \approx 4{,}946^{\circ},

The angular distance between Galați V1 and Galați V2.

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