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IAO 2011, Теория, Група α — Задача 2. Planetarium

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Всички материали от IAO 2011

XVI International Astronomy Olympiad. Almaty, Kazakhstan, 22 – 30. IX. 2011. Round Theo, Group α. Theoretical round. Problems to solve · 22 септември 2011 г. · 8 т.

Внимание: Бележка към темата

The shared five-page official solution is a preliminary jury sketch. Tops of scanned pages 2–5 are truncated in the source itself (MediaBox equals CropBox); missing passages are marked explicitly. No detailed grading rubric is present in the year inventory; the solution introduction says correct student solutions are worth 8 points. The Bear drawing for problem 3 and the promised sector/point drawings for problem 5 are not supplied. The ray diagram for problem 4 has a partly clipped lower label. Printed slips, including the solar-mass exponent, latitude inconsistency and planetarium inequality, are retained with notes where relevant.

Бележка

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» has to be written in English or Russian.

Условие

Classical devices "planetaria" are arranged so that each group of stars is projected on a dome by a small optical system. Foils with hole-stars of the corresponding sizes are often used as these "slides" of constellations which are projected on the dome, so that most of the light is blocked by the foil (the black sky appears), and only light through the holes are transmitted (so stars appear). For example, images of $0^{\mathrm{m}}$ stars on the foil have the size $l_0 = 0.1\ \mathrm{mm}$ , and stars up to $6^{\mathrm{m}}$ are demonstrated, the focal length of the projecting system is $f = 25\ \mathrm{cm}$ , and the device has 16 separate projecting systems for every hemisphere. The dome of the planetarium of the observatory “Bobek” has a diameter of $2R = 10\ \mathrm{m}$ .

Let's suppose that all slides have been removed for replacement by more perfect ones, and the whole light began to be projected on the dome. What would the total stellar magnitude of the illuminated dome be (the artificial gray sky)? Would it be possible to read a newspaper in such an illuminance?

The answer has to include a list of the necessary parameters with formulae and numerical values.

Отговори

Решение

Внимание: Непълно решение

The top of the source scan is cut off; missing opening/continuation lines are marked. All surviving supplied calculation is transcribed.

Theoretical round. Sketches for solutions. For preliminary work of Jury ONLY.

Note for jury and team leaders. The proposed sketches are not full; the team leaders have to give more detailed explanations for students. But the correct solutions in the students’ papers (enough for 8 pts) may be shorter.

Dome illumination

Source gap: the top of the scanned page is cut off. The surviving calculation begins with the following displayed formula; the missing opening is not reconstructed.

L=\Gamma\cdot l_0,

where $\Gamma$ is the magnification coefficient of the projecting system, $\Gamma=R/r$ , $r$ is the distance from the hole in the foil to the projecting lens. By the lens formula, $1/R+1/r=1/F$ , but since $R\ll F$ , one may use $\Gamma\approx R/F$ .

L=\Gamma\cdot l_0\approx(R/F)\cdot l_0=20\cdot0.1\,\mathrm{mm}=2\,\mathrm{mm},

Accordingly, the area of the illuminated surface of the dome which gives the star $0^{\mathrm m}$ is equal to

S_{m0}=\pi L^2/4\approx3.14\,\mathrm{mm}^2.

After slides-foils have been removed the whole dome will be illuminated, i.e. a hemisphere of radius $R=5\,\mathrm m$ will shine, the area of shining surface is

S=2\pi R^2\approx157\,\mathrm m^2.

Thus, the shining area will be

K=S/S_{m0}=\frac{2\pi R^2}{\pi L^2/4}=8(R/L)^2=8(F/l_0)^2=50000000.

In terms of stellar magnitudes the difference is

\Delta m=2.5^{\mathrm m}\lg K=2.5^{\mathrm m}\lg(8(F/l_0)^2)\approx19.25^{\mathrm m}.

Thus, the stellar magnitude of the dome will be $0^{\mathrm m}-19.25^{\mathrm m}=-19.25^{\mathrm m}$ . It gives an illuminance 1000 times less, than the illuminance outdoors at clear sunny day, and in 400 times larger than illuminance from full Moon. It corresponds approximately to indoors illuminance at daytime. Of course, it is possible to read without any problem.

We should note that the solution does not require the values of limiting stellar magnitude $6^{\mathrm m}$ , the number of projecting systems for the hemisphere and the diameter of the dome of the planetarium.

Transcription note: the source's inequality $R\ll F$ is retained, although its numerical approximation uses the opposite scale relation.

Parameters of orbits. Physical characteristics of Sun, some planets, Moon and Galilean moons of Jupiter

Элементы орбит. Физические характеристики Солнца, некоторых планет, Луны и Галилеевых спутников Юпитера

Parameters of orbits. Physical characteristics of Sun, some planets, Moon and Galilean moons of Jupiter

Небесное тело, планета / Body, planet	Среднее расстояние от центрального тела, в астр. ед. / Average distance to central body, in astr. units	Среднее расстояние от центрального тела, в млн. км / Average distance to central body, in mln. km	Сидерический (или аналогичный) период обращения, в тропич. годах / Sidereal (or analogous) period, in troph. years	Сидерический (или аналогичный) период обращения, в средних сутках / Sidereal (or analogous) period, in days	Эксцентриситет, e / Eccentricity, e	Экваториальн. диаметр, км / Equat. diameter, km	Масса, $10^{24}$ кг / Mass, $10^{24}$ kg	Средняя плотность, г/см³ / Av. density, g/cm³	Ускор. своб. пад. у пов., м/с² / Grav. acceler. at surf., m/s²	Наклон оси / Axial tilt	Альбедо / Albedo
Солнце Sun	$1{,}6\cdot 10^{9}$	$2{,}5\cdot 10^{11}$	$2{,}2\cdot 10^{8}$	$8\cdot 10^{10}$		1392000	1989000	1,409
Меркурий Mercury	0,387	57,9	0,241	87,97	0,206	4 879	0,3302	5,43	3,70	0,01°	0,06
Венера Venus	0,723	108,2	0,615	224,70	0,007	12 104	4,8690	5,24	8,87	177,36	0,78
Земля Earth	1,000	149,6	1,000	365,26	0,017	12 756	5,9742	5,515	9,81	23,44	0,36
Луна Moon	0,00257	0,38440	0,0748	27,3217	0,055	3 475	0,0735	3,34	1,62	6,7	0,07
Марс Mars	1,524	227,9	1,880	686,98	0,093	6 794	0,6419	3,94	3,71	25,19	0,15
Юпитер Jupiter	5,204	778,6	11,862	4 332,59	0,048	142 984	1899,8	1,33	24,86	3,13	0,66
Сатурн Saturn	9,584	1433,7	29,458	10 759,20	0,054	120 536	568,50	0,70	10,41	26,73	0,68

Спутник / Moon	Среднее расстояние от центра планеты, в астр. ед. / Average distance to planet center, in astr. units	Среднее расстояние от центра планеты, в тыс. км / Average distance to planet center, in thous. km	Сидерич. период обращения в средних сутках / Sidereal period in days	Наклонение орбиты (°) / Orbit inclination (°)	Эксцентриситет, e / Eccentricity, e	Диаметр, км / Diameter, km	Масса, $10^{21}$ кг / Mass, $10^{21}$ kg	Средняя плотность, г/см³ / Av. density, g/cm³	Ускор. своб. пад. у пов., м/с² / Grav. acceler. at surf., m/s²	Макс. блеск, вид. с Земли ) / Max. magn. From Earth )	Геом Альбедо / Geom Albedo
Ио Io	0,00282	421,70	1,769137	0,050°	0,0041	3 643	89,31	3,53	1,80	$5{,}02^{m}$	0,63
Европа Europa	0,00449	671,03	3,551181	0,471°	0,0094	3 122	48,00	3,01	1,32	$5{,}29^{m}$	0,67
Ганимед Ganymede	0,00716	1070,41	7,154553	0,204°	0,0011	5 262	148,19	1,94	1,43	$4{,}61^{m}$	0,43
Каллисто Callisto	0,01259	1882,71	16,689018	0,205°	0,0074	4 821	107,59	1,83	1,23	$5{,}65^{m}$	0,20

*) В среднем противостоянии. *) In mean opposition.

Some constants and formulae

Некоторые константы и формулы

Some constants and formulae

Константа	Значение / Value	Constant
Скорость света в вакууме, c (м/с)	299 792 458	Speed of light in vacuum, c (m/s)
Гравитационная постоянная, G (Н·м²/кг²)	$6.674\cdot 10^{-11}$	Constant of gravitation, G (N·m²/kg²)
Солнечная постоянная, A (Вт/м²)	1367	Solar constant, A (W/m²)
Параметр Хаббла, H₀ (км/с/МПк), среднее значение	71	mean value, Hubble parameter, H₀ (km/s/Mpc)
Параметр Хаббла, H₀ (км/с/МПк), диапазон значений	50-100	diapason of values, Hubble parameter, H₀ (km/s/Mpc)
Постоянная Планка, h (Дж·с)	$6.626\cdot 10^{-34}$	Plank constant, h (J·s)
Заряд электрона, e (Кл)	$1.602\cdot 10^{-19}$	Charge of electron, e (C)
Масса электрона, mₑ (кг)	$9.109\cdot 10^{-31}$	Mass of electron, mₑ (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, μ₀ (Гн/м)	$1.257\cdot 10^{-6}$	Magnetic constant, μ₀ (H/m)
Универсальная газовая постоянная, R (Дж/моль/K)	8.314	Universal gas constant, R (J/mol/K)
Постоянная Больцмана, k (Дж/K)	$1.381\cdot 10^{-23}$	Boltzmann constant, k (J/K)
Стандартная атмосфера (Па)	101 325	Standard atmosphere (Pa)
Постоянная Стефана-Больцмана, σ (Вт/м²/K⁴)	$5.670\cdot 10^{-8}$	Stefan-Boltzmann constant, σ (W/m²/K⁴)
Константа смещения Вина, b (м·K)	0.002897	Wien’s displacement constant, b (m·K)
Лабораторная длина волны Hα (Å)	6562.81	Laboratory wavelength of Hα (Å)
Длина тропического года, T (сут)	365.242199	Tropical year length, T (days)
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	$S = 4\pi R^{2}$	Area of sphere
π	3.14159265	π
e	2.71828183	e
Золотое сечение, φ	1.61803399	Golden ratio, φ

Оригинал в Архива: alphatheo.pdf · официални решения: theosol.pdf