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IAO 2017, Теория, α+β — αβ-3. Mirror and plate.

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Всички материали от IAO 2017

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

There is a system, consisting of a plane mirror (100% reflection) and a plane glass plate that transmits K% of the light and reflects the other (100-K)% (this characteristics is the same for the light coming from both the directions). The mirror and plate are parallel and infinite on their right side. An observer is placed at the point O (marked by *) and sees a star with a magnitude $2^{\mathrm m}$ within a cone around direction «A». What magnitude(s) star(s) (if any) would the observer see in a cone near direction «B» that is symmetrical to «A»? Calculate numerical value for K = 50% and K = 5% for each such a star (if any visible) or explain why no stars are visible.

Mirror and plate, observer O and directions A and B

Решение

The observer sees the combined light from parallel rays A1, A2, A3, A4,.. in the direction B. Every beam AN once passed through the plate (losing K%), then successively reflected from the mirror (without losing intensity) and from plates (each time losing (100-K)% of the light), with N reflections from mirrors and N-1 reflections from the plate.

As the star is at a very large distance, all the rays AN are coming from the mentioned star $2^{\mathrm m}$ . Therefore, the observer sees in the direction “B” a single star, the intensity of which we need to calculate as the sum of the series:

P_0k+P_0(1-k)k+P_0(1-k)^2k+P_0(1-k)^3k+P_0(1-k)^4k+P_0(1-k)^5k+\ldots

where $P_0$ is the flow of light from the star $2^{\mathrm m}$ , and k is the fraction of the transmitted light from the plate (i.e., k = K/100).

The sum of the series $1+(1-k)+(1-k)^2+(1-k)^3+(1-k)^4+(1-k)^5+\ldots$ is equal to 1/k. Accordingly, the sum of our series is

P_0k(1+(1-k)+(1-k)^2+(1-k)^3+(1-k)^4+(1-k)^5+\ldots)=P_0k\,1/k=P_0.

Paradoxically, the fact! Regardless of the value of K the observer would see the only one star in the direction “B”, and its magnitude is $2^{\mathrm m}$ .

So the numerical answer in the both cases: $2^{\mathrm m}$ .

Basic criteria

Basic criteria	Points
Correct picture of reflections (or logic explanation)	1.5
Issue the only one star visible	2
Understanding about series, calculation intensity of star B as sum of series	2
Issue the result is not depend on K	1
Final answer	1.5

Оригинал в Архива: theo.pdf · официални решения: theosol.pdf · допълнителни данни: instructions.pdf · допълнителни данни: data.pdf · допълнителни данни: hrd.pdf