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International Astronomy Olympiad 2012 (Problems) IAO Problems 2012
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    АСТРОНОМИЧЕСКОЕ   ОБЩЕСТВО  EURO  – ASIAN ASTRONOMICAL SOCIETY Round    Theo   Group    язык   English language XVII Международная астрономическая олимпиада   XVII International Astronomy Olympiad Корея , Кванджу  16  –   24. X. 2012 Gwangju, Korea   Theoretical round. Problems to solve General note.  Maybe not all problems have correct questions. Some questions (maybe the main question of the problem, maybe one of the subquestions) may make no real sense. In this case you have to write in your answer (in English or Russian): « impossible situation  –    ситуация   невозможна » . Of course, this answer has to be explained numerically or logically. Data from the tables  (Planetary data, stars, constants, etc.) may be used for solving every problem. The answers « Да -Yes » or « Нет -No » ha ve to be written in English or Russian. 1. Transit of Venus.  Recently, on June 6, 2012, an infrequent astronomical phenomenon, transit of Venus across the solar disc, took place. The next transit of Venus will take place only in 2117. Calculate the date of that transit. (Answer without calculations will not be considered even as a partial solution.) 2.   Transit of Pseudovenus.  Recently, on June 6, 2012, an infrequent astronomical phenomenon, transit of Venus across the solar disc took place. Suppose somebody did not understand the phenomenon and ascribed it not to transit of real Venus but of some moon, which we name Pseudovenus, rotating around the Earth in a circular orbit. Find the radius of the orbit of Pseudovenus and diameter of this sky body. Effects due to axial rotating of the Earth should not be taken into account. 3. Old persons ’  star.  There is ancient legend in Korea that says, if you managed to see the “Old  person s’   star” thrice , you are lucky person and will live a long life. The “Old persons’   star”, now known as Canopus, was seen brighter and better in past times, but even now sometimes one can see this star in Korea. Estimate approximately what visible stellar magnitude Canopus may have when observing it from the southern coast of Jeju island (Korea) in the most favorable conditions. The territory of the island is located at latitudes between 33 °1 2' N and 33°3 4' N and longitudes between 126°09' E and 126°57' E. Take from the tables and recollect for yourself the necessary additional information. 4. Stars on Mars.  As you know, last year the Polar Bear (whom you have already met in the texts of many International Astronomy Olympiads) arrived to Mars for astronomical observations. Nowadays his friend Penguin also made a fascinating journey to Mars. At the same instant of time, the Bear and the Penguin observe stars in zenith and see Canopus and Sirius respectively. Estimate roughly, what is the distance (measured on the Martian surface) between the animals? At what height above the horizon does the Bear observe Sirius? The solution has to include a picture with an image of the Bear and the Penguin on Mars. Necessary sizes or angular sizes should be in the picture. Recollect for yourself the necessary information about the Polar Bear and Penguin. 5.   Venus and Earth.  At what maximum distance from the Venus ecliptic the Earth can be visible in the sky from Venus (actually, from a point outside the Venus atmosphere)? Orbits of the planets may be considered circular. 6. Parallaxes.  In our part of the Galaxy the mean distance between the stars is about 6 light years. Assume that an interferometer can measure parallaxes with an error of ±0 .001 arc second. How many stars of our Galaxy could have their parallax determined by this interferometer? Корея , Кванджу  2012 Gwangju, Korea    АСТРОНОМИЧЕСКОЕ   ОБЩЕСТВО  EURO  – ASIAN ASTRONOMICAL SOCIETY Round    Theo   Group    язык   English language XVII Международная астрономическая олимпиада   XVII International Astronomy Olympiad Корея , Кванджу  16  –   24. X. 2012 Gwangju, Korea   Theoretical round. Problems to solve General note.  Maybe not all problems have correct questions. Some questions (maybe the main question of the problem, maybe one of the subquestions) may make no real sense. In this case you have to write in your answer (in English or Russian): « impossible situation  –    ситуация   невозможна » . Of course, this answer has to be explained numerically or logically. Data from the tables  (Planetary data, stars, constants, etc.) may be used for solving every problem. The answers « Да -Yes » or « Нет -No » ha ve to be written in English or Russian. 1. Transit of Venus.  Recently, on June 6, 2012, an infrequent astronomical phenomenon, transit of Venus across the solar disc, took place. The next transit of Venus will take place only in 2117. Calculate the date of that transit. (Answer without calculations will not be considered even as a  partial solution.) 2.   Transit of Pseudovenus.  Recently, on June 6, 2012, an infrequent astronomical phenomenon, transit of Venus across the solar disc took place. Suppose somebody did not understand the  phenomenon and ascribed it not to transit of real Venus but of some moon, which we name Pseudovenus, rotating around the Earth in a circular orbit. Find the radius of the orbit of Pseudovenus and diameter of this sky body. Effects due to axial rotating of the Earth should not be taken into account. 3. Old persons ’  star.  There is ancient legend in Korea that says, if you managed to see the “Old  person s’   star” thrice , you are lucky person and will live a long life. The “Old persons’   star”, now known as Canopus, was seen brighter and better in past times, but even now sometimes one can see this star in Korea. Estimate approximately what visible stellar magnitude Canopus may have when observing it from the southern coast of Jeju island (Korea) in the most favorable conditions. The territory of the island is located at latitudes between 33 °1 2' N and 33°3 4' N and longitudes  between 126°09' E and 126°57' E. Take from the tables and recollect for yourself the necessary additional information. 4. Altair.  Estimate the density of the star Altair. 5.   Venus and Earth.  At what maximum distance from the Venus ecliptic the Earth can be visible at the sky from Venus (actually, from a point outside the Venus atmosphere)? Orbits of the planets may be considered circular. Estimate the stellar magnitude of the Earth in this situation. 6. Remote galaxy.  Astronomers have discovered a distant galaxy that in the Earth's sky, at the first glance looks like   Eridani, the same in colour, but 1000 times less in intensity. It appears, however, that this galaxy is composed only of stars similar to the Sun in physical characteristics. Find the number of stars in the galaxy. Корея , Кванджу  2012 Gwangju, Korea    АСТРОНОМИЧЕСКОЕ   ОБЩЕСТВО  EURO  – ASIAN ASTRONOMICAL SOCIETY Round    Theo   Group       язык    Русский   language язык   English language XVII Международная астрономическая олимпиада   XVII International Astronomy Olympiad Корея, Кванджу 16 –   24. X. 2012 Gwangju, Korea   Элементы орбит.   Физические характеристики некоторых планет, Луны , Солнца   и Эриды  Parameters of orbits. Physical characteristics of some planets, Moon, Sun and Eris Небесное тело,   Среднее    расстояние от   центрального тела   Сидерический  ( или аналогичный ) период обращения   На - клон   орби - Экс - цен - триси - Эквато -  риальн.   диаметр   Масса   Сред - няя плот - Ускор.своб. пад.   На - клон Макс.   блеск,   вид. с   Аль - планета   в   астр.   ед.   в    млн.   км   в   тропич.   годах   в   средних   сутках   ты , i    тет , е   км   10  24    кг   ность    г  /  см 3    у   пов .  м  /  с 2    оси   Земли  **)   бедо  Body, Average distance to central body Sidereal period (or analogous) Orbital inclin- Ec-centri- Equat. diameter Mass Av. den- Grav. accelr.   Axial Max. magn. Al-  planet in astr. units   in 10  6 km   in tropical years in days ation, i   city е   km    10  24   kg    sity g/cm  3   at surf. m/s  2    tilt From Earth **)    bedo Солнце  Sun 1,6·10 9   2,5·10 11   2,2·10 8   8·10 10  1392000 1989000 1,409 -26,74 m   Меркурий  Mercury 0,387 57,9 0,241 87,969 7,00 °  0,206 4 879 0,3302 5,43 3,70 0,01°  0,06 Венера  Venus 0,723 108,2 0,615 224,7007 3,40 0,007 12 104 4,8690   5,24 8,87 177,36 0,78 Земля  Earth 1,000 149,6 1,000 365,2564 0,00 0,017 12 756 5,9742   5,515 9,81 23,44 0,36 Луна  Moon 0,00257 0,38440 0,0748 27,3217 5,15 0,055 3 475 0,0735   3,34 1,62 6,7 -12,7 m  0,07 Марс  Mars 1,524 227,9 1,880 686,98 1,85 0,093 6 794 0,6419   3,94 3,71 25,19 -2,0 m  0,15 Юпитер  Jupiter 5,204 778,6 11,862 4 332,59 1,30 0,048 142 984 1899,8 1,33 24,86 3,13 -2,7 m  0,66 Сатурн  Saturn 9,584 1433,7 29,458 10 759,20 2,48 0,054 120 536 568,50 0,70 10,41 26,73 0,7 m  0,68 Эрида  Eris 68,05 205 029 43,82 0,435 2 326 0,0167   2,52 0,7 0,96 **) Для Луны –    в среднем противостоянии.  **) For Moon  –   in mean opposition. Корея , Кванджу  2012 Gwangju, Korea    АСТРОНОМИЧЕСКОЕ   ОБЩЕСТВО  EURO  – ASIAN ASTRONOMICAL SOCIETY Round    Theo   Group       язык    Русский   language язык   English language XVII Международная астрономическая олимпиада   XVII International Astronomy Olympiad Корея, Кванджу 16 –   24. X. 2012 Gwangju, Korea   Некоторые константы и формулы  Some constants and formulae Скорость света в вакууме , c (м/с)   299 792 458 Speed of light in vacuum, c (m/s)   Гравитационная постоянная , G (Н·м 2 / кг 2 )   6.67 4·10 -11   Constant of gravitation, G (N·m 2 /kg 2 )   Солнечная   постоянная , A ( Вт/м 2 )   1367 Solar constant, A (W/m 2 )   Параметр   Хаббла , H 0  ( км / с / МПк )   среднее   значение   диапазон   значений   71 50-100 mean value diapason of values   Hubble parameter, H 0  (km/s/Mpc)   Постоянная Планка , h (Дж·с)   6.626·10 -34  Plank constant, h (J·s)   Заряд электрона , e (Кл)   1.602·10 -19  Charge of electron, e (C)   Масса   электрона , m e  ( кг)   9.109 ·10 -31  Mass of electron, m e  (kg)   Соотношение   масс   протона   и   электрона   1836.15 Proton-to-electron ratio   Постоянная   Фарадея , F ( Кл / моль )   96 485 Faraday constant, F (C/mol)   Магнитная постоянная ,  0   (Гн/м)   1.257·10 -6  Magnetic constant,  0  (H/m)   Универсальная газовая постоянная , R (Дж/моль /K)   8.314 Universal gas constant, R (J/mol/K)   Постоянная   Больцмана , k (Дж/K)   1.381·10 -23  Boltzmann constant, k (J/K)   Постоянная Стефана - Больцмана ,   ( Вт / м 2 /K  4 )   5.670 ·10 -8  Stefan-Boltzmann constant,   (W/m 2 /K  4 )   Константа   смещения Вина , b (м·K)   0.002897 Wien’s displacement constant, b (m·K)   Лабораторная   длина   волны  H α   (Å)  6562.81 Laboratory wavelength of H α   (Å)   Длина тропического года , T ( сут ) 365.242199 Tropical year length, T (days) Стандартная атмосфера (Па)   101 325 Standard atmosphere (Pa)   Ослабление   видимого   света   земной   атмосферой   в   зените  ( минимально )   19%, 0.23 m  Visible light extinction by the terrestrial atmosphere in zenith (minimum)   Показатель   преломления   воды   при   20°C, n 1.334 Refractive index of water for 20°C, n Момент инерции шара  I = 2 / 5  MR  2  Moment of inertia of a solid ball Площадь   сферы  S = 4  R  2  Area of sphere    3.14159265    e 2.71828183 e Золотое сечение ,   1.61803399 Golden ratio,    Корея , Кванджу  2012 Gwangju, Korea
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