IAO 2017, Теория, α+β — α-4. Eclipse in USA.
XXII International Astronomy Olympiad — Theoretical round, α and β · 27 октомври 2017 г. · 8 т.
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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» have to be written in English or Russian.
Do not forget that in clean copy you should write solutions only on even (left) pages. The odd (right) pages will be used by your team leader to translate solutions into English and also to emphasize your thoughts in solutions.
| EVEN PAGE / ЧЁТНАЯ СТРАНИЦА | ODD PAGE / НЕЧЁТНАЯ СТРАНИЦА |
|---|---|
| Текст решения. Text of solution. Текст решения. Text of solution. Текст решения. Text of solution. | × |
Ещё раз прочитайте перед началом работы
Общее замечание. Не исключено, что не во всех задачах вопросы поставлены корректно. Некоторые вопросы (возможно, главный вопрос задачи, возможно – подвопрос) могут не иметь смысла. В этом случае следует написать в ответе (по-русски или по-английски): «ситуация невозможна – impossible situation». Естественно, ответ должен быть подкреплён вычислениями или логическими рассуждениями.
Данные из таблиц (Солнечная система, звёзды, константы) могут быть использованы в любой задаче.
Ответы «Да-Yes» или «Нет-No» должны быть написаны по-русски или по-английски.
Не забудьте, что в чистовике вы должны писать решения только на чётных (левых) страницах. Нечётные (правые) страницы предназначены для того, чтобы руководитель вашей команды перевёл решения на английский язык и подчеркнул основные мысли решения.
Некоторые константы и формулы / Some constants and formulae
| Величина / Quantity | Значение |
|---|---|
| Скорость света в вакууме, c (м/с) / Speed of light in vacuum | 299 792 458 |
| Гравитационная постоянная, G (Н·м²/кг²) / Constant of gravitation | 6.674·10⁻¹¹ |
| Солнечная постоянная, A (Вт/м²) / Solar constant | 1367 |
| Параметр Хаббла, H₀ (км/с/МПк) / Hubble parameter, mean; range | 68; 50–100 |
| Постоянная Планка, h (Дж·с) / Planck constant | 6.626·10⁻³⁴ |
| Заряд электрона, e (Кл) / Charge of electron | 1.602·10⁻¹⁹ |
| Масса электрона, mₑ (кг) / Mass of electron | 9.109·10⁻³¹ |
| Соотношение масс протона и электрона / Proton-to-electron mass ratio | 1836.15 |
| Постоянная Фарадея, F (Кл/моль) / Faraday constant | 96 485 |
| Магнитная постоянная, μ₀ (Гн/м) / Magnetic constant | 1.257·10⁻⁶ |
| Универсальная газовая постоянная, R (Дж/моль/К) / Universal gas constant | 8.314 |
| Постоянная Больцмана, k (Дж/К) / Boltzmann constant | 1.381·10⁻²³ |
| Постоянная Стефана-Больцмана, σ (Вт/м²/К⁴) / Stefan-Boltzmann constant | 5.670·10⁻⁸ |
| Константа смещения Вина, b (м·К) / Wien’s displacement constant | 0.002897 |
| Лабораторная длина волны Hα (Å) / Laboratory wavelength | 6562.81 |
| Длина тропического года, T (сут) / Tropical year length | 365.242199 |
| Длина сидерического года, T (сут) / Sidereal year length | 365.25636 |
| Длина аномалистического года, T (сут) / Anomalistic year length | 365.259636 |
| Период обращения узлов лунной орбиты (лет) / Nodal period of lunar orbit | -18.6 |
| Стандартная атмосфера (Па) / Standard atmosphere | 101 325 |
| Ослабление видимого света слоем 1 атмосферы (минимально) / Visible light extinction in zenith | 19%, 0.23ᵐ |
| Высота однородной атмосферы (м) / Height of homogeneous atmosphere | 7991 |
| Показатель преломления воды при 20°C, n / Refractive index of water | 1.334 |
| Момент инерции шара / Moment of inertia of a solid ball | I=²⁄₅MR² |
| Момент инерции сферы / Moment of inertia of sphere | I=²⁄₃MR² |
| Объём шара / Volume of a ball | V=⁴⁄₃πR³ |
| Площадь сферы / Area of sphere | S=4πR² |
| π | 3.14159265 |
| e | 2.71828183 |
| Золотое сечение, φ / Golden ratio | 1.61803399 |
Данные о некоторых звёздах / Data of some stars
| Звезда / Star | Designation | R.A. | DEC | p | Mag. | Temperature | Mass |
|---|---|---|---|---|---|---|---|
| Солнце / Sun | ☉ | 0ʰ–24ʰ | −23°26′–+23°26′ | 8″.794 | −26ᵐ.74 | 5777 K | 1 M☉ |
| Бетельгейзе / Betelgeuse | α Ori | 05ʰ55ᵐ10ˢ | 07°24′25″ | 0″.005 | 0ᵐ.5 | 3590 K | 11.6 M☉ |
| Вега / Vega | α Lyr | 18ʰ36ᵐ56ˢ | 38°47′01″ | 0″.130 | 0ᵐ.03 | 9600 K | 2.14 M☉ |
| Наос / Naos | ζ Pup | 08ʰ03ᵐ35ˢ | −40°00′12″ | 0″.003 | 2ᵐ.21 | 42000 K | 40 M☉ |
| Толиман / Toliman (Rigel Kent) | α Cen A | 14ʰ39ᵐ36ˢ | −60°50′07″ | 0″.747 | −0ᵐ.01 | 5810 K | 1.11 M☉ |
| Толиман / Toliman (Rigel Kent) | α Cen B | 14ʰ39ᵐ36ˢ | −60°50′07″ | 0″.747 | 1ᵐ.33 | 5260 K | 0.93 M☉ |
| Проксима Центавра / Proxima Centauri | V645 Cen, α Cen C | 14ʰ29ᵐ43ˢ | −62°40′46″ | 0″.769 | 11ᵐ.06 | 3040 K | 0.123 M☉ |
| Координаты / Coordinates | WHO Observatory | Xinglong Observatory |
|---|---|---|
| φ (N / с.ш.) | 37°32′ | 40°24′ |
| λ (E / в.д.) | 122°03′ | 117°35′ |
| Часовой пояс / Timezone | UT+08 | UT+08 |
Theoretical round. Sketches for solutions
Note. The given sketches are not full; the team leaders have to give more detailed explanations to students. But the correct solutions in the students’ papers (enough for 8 pts) may be shorter.
Note. Jury members should evaluate the student’s solutions in essence, and not by looking on formal existence the mentioned sentences or formulae. The formal presence of the mentioned positions in the text is not necessary to give the respective points. Points should be done if the following steps de facto using these positions.
Theoretical round. Basic criteria. For work of Jury
Note. Jury members should elaborate more detailed criteria, and also create criteria for other correct ways of the student’s solutions.
Source limitations
The original PDF and its embedded scans clip the opening text on solution pages 2, 4 and 6. Ellipses mark these unrecoverable passages; complete surviving derivations, answers and grading rows are retained. The double-star solution refers to a figure that is absent from the supplied packet.
Printed discrepancies
The answer key uses 1.34ᵐ for α Centauri B while its data table gives 1.33ᵐ. The summit key invokes brown dwarfs despite the stated A–M main-sequence restriction, and calls an isosceles triangle equilateral. Its printed sin β/2 expressions are preserved. The mirror key says losing K% on transmission despite the statement saying transmitting K%. The eclipse key prints V₃ = V₂ − V_E and L_NS for the along-track width, while nearby prose uses Vφ and L_WE; these are not silently changed. The printed β-2 grading allocations and subtotals are retained.
Условие
A little more than two months ago, on August 21, 2017 from West to East of the USA territory a total solar eclipse was observed. The middle line of the track of the eclipse is shown schematically on the map. Below are four pictures taken by the Bulgarian-Russian group of observers who were in state of Oregon, through which the Moon’s shadow ran as it shown in Fig.α4. (see separate sheet). The local time (UT −07) of the shots is indicated below each image. By using the data and making calculations, estimate:
4.1.
altitude of the Sun in Oregon at the moment of totality;
4.2.
time t (in seconds) between the positions 1 and 2 of the shadow as marked at Fig.α4.
Решение
4.1.
Let us analyze the ellipse of the Moon’s shadow on the Earth's surface to calculate the altitude h of the Sun. If the diameter of the Moon's shadow near the Earth is denoted by D, the lengths of these axes equal to:
in the azimuth direction, where the Moon and the Sun are: ,
in the direction perpendicular to this azimuth: .
Thus, from these two equations: .
Measuring lengths of the axes of the shadow “2” on the given map, we get:
Answer: about 42°.
Note. In reality, the height of the Sun during the totality phase in Oregon was from 39° (in the West part) to 45° (in the East part), and in the selected for measurement place “2” it was 41.9°.
4.2.
The size of the eclipsing shadow along the central line of the path is the speed of the shadow multiplied by the duration of the total phase, and the difference in the positions of shadows “1” and “2” is the speed of the shadow multiplied by the time that we need to find.
The duration of the totality can be obtained by analyzing pictures 2 and 4. Both of them demonstrate the so-called "diamond ring", a unique phenomenon, lasting a few seconds only. It is seen that images of approximately symmetrical in shape, and were shorted about 3-4 seconds before the second contact and after 3-4 seconds of the third one. In addition, from the fact that the points of the second and the third contacts are almost symmetric, it can be concluded that the Bulgarian-Russian group observed the central eclipse and, therefore, was located about in the middle of the totality path. Thus, it is possible to obtain the totality duration at the middle of the path by calculating the time difference of the dating of these images.
Note. The above few seconds before the second contact and after the third one is a very delicate moment. It should not be considered an error if a participant will not take it into account and use the value .
Making the necessary measurements on the map, we get:
Answer: about 180 seconds.
Note. In reality, 180 seconds is the interval between the positions of the shadow on the map.
Basic criteria
| Basic criteria | Points |
|---|---|
| 4.1., including | 3 |
| Understanding shadow ellipse parameters depend on the altitude | 0.5 |
| Formulae , , | 1 |
| Measurements of axis | 0.5 |
| Calculations and answer | 0.5 |
| Correct number of significant digits (not more than 2) | 0.5 |
| 4.2., including | 5 |
| Understanding to use times of 2 and 4 as duration | 1 |
| Calculation of duration using photos | 1 |
| Understanding the size of shadow along the path is Vτ | 0.5 |
| Ratio between the size of shadow along the path and distance between 1 and 2 | 0.5 |
| Measurements of size and distance | 0.5 |
| Calculations and answer | 1 |
| Correct number of significant digits (not more than 2) | 0.5 |
Оригинал в Архива: theo.pdf · официални решения: theosol.pdf · допълнителни данни: instructions.pdf · допълнителни данни: data.pdf · допълнителни данни: hrd.pdf