Съдържание

IAO 2011, Теория, Групи α и β — α-1. Наблюдение на звезда

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Всички материали от IAO 2011

XVI Международная астрономическая олимпиада / XVI International Astronomy Olympiad — Казахстан, Алма-Ата, 22 – 30. IX. 2011 — Round: Theo · 22 септември 2011 г.

Обща информация

Възможно е не всички въпроси да имат верен отговор. Някои въпроси (може би главният въпрос или някой от подвъпросите) могат да нямат смисъл. В този случай, Вие трябва да напишете във Вашия отговор (на английски или руски език) "impossible situation-ситуация невозможна". Разбира се, отговорът трябва да бъде обяснен числено или логически. Данните от таблиците (данни за планети, звезди, константи и т.н.) могат да бъдат използвани за решаването на всяка задача. Отговорите "Да-Yes" или "Нет-No" трябва да бъдат написани на английски или руски език.

Source ownership and groups

The Bulgarian manuscript contains distinct α-1 and β-1 tasks, distinct α/β Planetarium versions, then shared αβ-3–5. Existing IDs p1–p5 are retained; the unrelated β-1 is now p6. The English original supplies the Jupiter photograph and complete reference tables absent from the Bulgarian manuscript. Bulgarian date 18 October 2009 is retained; English question and key say 19 October 2009.

Clipped official sketches

The five-page official key is a preliminary jury sketch. Its page tops are clipped on pages 2–5 in the actual PDF, also visible in fresh PyMuPDF rendering. Missing opening/continuation passages are explicitly marked, not reconstructed. The ray-diagram lower label is partly cut off.

Missing drawings and grading

No detailed grading sheet occurs in the year inventory. The introduction refers to correct student solutions worth 8 points. The requested Mars Bear drawing and promised Jupiter-moon sector/point drawings are absent from the supplied key.

Printed discrepancies

Retained: latitude intermediate −88°28′ versus final −88°23′; solar mass 1.989 × 10⁻³⁰ kg versus reference table 1989000 × 10²⁴ kg; Planetarium inequality R ≪ F; Mars sketch says sunset and discusses eccentricity although the question allows a circular orbit. Jupiter sketch uses “difference in declinations”, reports 1.5% accuracy and gives centre-distance rather than the requested surface distance. These are source statements, not newly calculated answers.

Условие

Проведени са наблюдения с невъоръжено око на 16-ти юни 2008, като е използвано Универсално време. Наблюдател забелязва, че една звезда преминава през зенита в $0^{h}18^{m}$ . В $8^{h}17^{m}$ нейната височина над хоризонта е била $87^{\circ}12'$ .

Намерете географската ширина на която са се провели наблюденията.

Отговори

Решение

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.

Latitude of observation

The difference is $7^{\mathrm h}59^{\mathrm m}$ between the two observations. It is almost exactly $1/3$ of a sidereal day (the exact value is $7^{\mathrm h}58^{\mathrm m}41^{\mathrm s}$ ). During this time the star moves $120^\circ$ of its daily path. And its zenith distance has changed by $\zeta=90^\circ-87^\circ12'=2^\circ48'$ . This value is small, so it is possible to work with the approximation of plane geometry. If $\beta$ is the angular distance from the star to the pole P, we may conclude (see figure) that during this time the star has moved from the zenith point Z to point Y, and changed the zenith distance by

\zeta=2\beta\times\sin(120^\circ/2)=1.732\times\beta,

therefore

\beta=\zeta/1.732=2^\circ48'/1.732=1^\circ37'.

Hence, the declination of the star is equal to

\delta=90^\circ-\beta=90^\circ-1^\circ37'=88^\circ23'.

Accordingly, this star culminates at zenith only at latitude $\varphi=\delta$ , that is the latitude also equals $88^\circ23'$ .

But it is not a complete solution. However, the value of $88^\circ23'$ is only the absolute value of the unknown value. In principle, both the values: $+88^\circ23'$ (Northern latitude, the star is near the North Pole) and $-88^\circ28'$ (Southern latitude, the star near the South Pole) can meet the condition of the more general problem. However, according to the condition of our problem the observations were made with the naked eye on 16 June, when polar day appears at such northern latitudes. Therefore, correct answer is only one: $-88^\circ23'$ ( $88^\circ23'$ South latitude).

Transcription note: the intermediate southern value is printed $-88^\circ28'$ ; the calculation and final answer print $-88^\circ23'$ . Both are retained.

Small polar-circle geometry with centre P, radii beta, zenith Z, point Y and central angle 120 degrees.

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: bgtheo.pdf · официални решения: theosol.pdf · допълнителни данни: alphatheo.pdf · допълнителни данни: betatheo.pdf · допълнителни данни: data_1.png · допълнителни данни: data_2.png