Съдържание

IAO 2012, Теория, старша възраст — Задача 3. Old persons’ star

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Всички материали от IAO 2012

XVII International Astronomy Olympiad — Theoretical round — Group β

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» have to be written in English or Russian.

Planetary data

Элементы орбит. Физические характеристики некоторых планет, Луны, Солнца и Эриды

Parameters of orbits. Physical characteristics of some planets, Moon, Sun and Eris

Небесное тело, планета / Body, planet	Среднее расстояние от центрального тела / Average distance to central body, in astr. units	in $10^6$ km	Сидерический (или аналогичный) период обращения / Sidereal period (or analogous), in tropical years	in days	Наклон орбиты / Orbital inclination, $i$	Эксцентриситет / Eccentricity, $e$	Экваториальн. диаметр / Equat. diameter, km	Масса / Mass, $10^{24}$ kg	Средняя плотность / Av. density, g/cm³	Ускор. своб. пад. у пов. / Grav. accelr. at surf., m/s²	Наклон оси / Axial tilt	Макс. блеск, вид. с Земли / Max. magn. From Earth **)	Альбедо / Albedo
Солнце / Sun	$1,6\cdot10^9$	$2,5\cdot10^{11}$	$2,2\cdot10^8$	$8\cdot10^{10}$			1392000	1989000	1,409			$-26,74^{\mathrm m}$
Меркурий / Mercury	0,387	57,9	0,241	87,969	$7,00^\circ$	0,206	4 879	0,3302	5,43	3,70	$0,01^\circ$		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	$-12,7^{\mathrm m}$	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	$-2,0^{\mathrm m}$	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	$-2,7^{\mathrm m}$	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	$0,7^{\mathrm m}$	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.

Константи

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·м²/кг²)	$6.674\cdot10^{-11}$	Constant of gravitation, G (N·m²/kg²)
Солнечная постоянная, A (Вт/м²)	1367	Solar constant, A (W/m²)
Параметр Хаббла, $H_0$ (км/с/МПк), среднее значение	71	mean value, Hubble parameter, $H_0$ (km/s/Mpc)
диапазон значений	50-100	diapason of values
Постоянная Планка, h (Дж·с)	$6.626\cdot10^{-34}$	Plank constant, h (J·s)
Заряд электрона, e (Кл)	$1.602\cdot10^{-19}$	Charge of electron, e (C)
Масса электрона, $m_e$ (кг)	$9.109\cdot10^{-31}$	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, $\mu_0$ (Гн/м)	$1.257\cdot10^{-6}$	Magnetic constant, $\mu_0$ (H/m)
Универсальная газовая постоянная, R (Дж/моль/К)	8.314	Universal gas constant, R (J/mol/K)
Постоянная Больцмана, k (Дж/К)	$1.381\cdot10^{-23}$	Boltzmann constant, k (J/K)
Постоянная Стефана-Больцмана, $\sigma$ (Вт/м²/К⁴)	$5.670\cdot10^{-8}$	Stefan-Boltzmann constant, $\sigma$ (W/m²/K⁴)
Константа смещения Вина, b (м·К)	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)
Стандартная атмосфера (Па)	101 325	Standard atmosphere (Pa)
Ослабление видимого света земной атмосферой в зените (минимально)	$19\%,\ 0.23^{\mathrm m}$	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	$I=\frac25 MR^2$	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

Data of some stars

Данные о некоторых звёздах / Data of some stars

			RA	DEC	$p$	$\mathbf m$	S C	масса / mass
Солнце	Sun	$\odot$	$0^{\mathrm h}-24^{\mathrm h}$	$-23^\circ26'\ +23^\circ26'$	$8''.794$	$-26^{\mathrm m}.74$	G2	$1\ M_\odot$
Альдебаран	Aldebaran	α Tau	$04^{\mathrm h}35^{\mathrm m}55^{\mathrm s}$	$16^\circ30'33''$	$0''.048$	$0^{\mathrm m}.85^{\mathrm v}$	K5	$2.5\ M_\odot$
Альтаир	Altair	α Aql	$19^{\mathrm h}50^{\mathrm m}47^{\mathrm s}$	$08^\circ52'06''$	$0''.195$	$0^{\mathrm m}.77$	A7	$1.7\ M_\odot$
Антарес	Antares	α Sco	$16^{\mathrm h}29^{\mathrm m}24^{\mathrm s}$	$-26^\circ25'55''$	$0''.006$	$0^{\mathrm m}.96$	M1+B4	$22.4\ M_\odot$
Арктур	Arcturus	α Boo	$14^{\mathrm h}15^{\mathrm m}40^{\mathrm s}$	$19^\circ10'57''$	$0''.089$	$-0^{\mathrm m}.04^{\mathrm v}$	K1	$1.1\ M_\odot$
Ахернар	Achernar	α Eri	$01^{\mathrm h}37^{\mathrm m}43^{\mathrm s}$	$-57^\circ14'12''$	$0''.026$	$0^{\mathrm m}.46$	B3
зв.Барнарда	Barnard’s star	Oph	$17^{\mathrm h}57^{\mathrm m}48^{\mathrm s}$	$04^\circ41'36''$	$0''.545$	$9^{\mathrm m}.54$	M4
Бетельгейзе	Betelgeuse	α Ori	$05^{\mathrm h}55^{\mathrm m}10^{\mathrm s}$	$07^\circ24'25''$	$0''.005$	$0^{\mathrm m}.5^{\mathrm v}$	M1
Вега	Vega	α Lyr	$18^{\mathrm h}36^{\mathrm m}56^{\mathrm s}$	$38^\circ47'01''$	$0''.129$	$0^{\mathrm m}.03$	A0
Денеб	Deneb	α Cyg	$20^{\mathrm h}41^{\mathrm m}26^{\mathrm s}$	$45^\circ16'49''$	$0''.002$	$1^{\mathrm m}.25$	A2
Канопус	Canopus	α Car	$06^{\mathrm h}23^{\mathrm m}57^{\mathrm s}$	$-52^\circ41'45''$	$0''.010$	$-0^{\mathrm m}.72$	F0
Капелла	Capella	α Aur	$05^{\mathrm h}16^{\mathrm m}41^{\mathrm s}$	$45^\circ59'53''$	$0''.073$	$0^{\mathrm m}.08$	G5+G0
Полярная	Polaris	α UMi	$02^{\mathrm h}31^{\mathrm m}49^{\mathrm s}$	$89^\circ15'51''$	$0''.0076$	$1^{\mathrm m}.97^{\mathrm v}$	F7
Процион	Procyon	α CMi	$07^{\mathrm h}39^{\mathrm m}18^{\mathrm s}$	$05^\circ13'30''$	$0''.288$	$0^{\mathrm m}.38$	F5
Ригель	Rigel	β Ori	$05^{\mathrm h}14^{\mathrm m}32^{\mathrm s}$	$-08^\circ12'06''$	$0''.013$	$0^{\mathrm m}.12$	B8
Сириус	Sirius	α CMa	$06^{\mathrm h}45^{\mathrm m}09^{\mathrm s}$	$-16^\circ42'58''$	$0''.375$	$-1^{\mathrm m}.46$	A1
Спика	Spica	α Vir	$13^{\mathrm h}25^{\mathrm m}12^{\mathrm s}$	$-11^\circ09'41''$	$0''.023$	$0^{\mathrm m}.98$	B1
Альфа Центавра	Alpha Centauri	α Cen	$14^{\mathrm h}39^{\mathrm m}36^{\mathrm s}$	$-60^\circ50'07''$	$0''.751$	$-0^{\mathrm m}.01$ ; $1^{\mathrm m}.33$	G2 ; K1	$2.0\ M_\odot$
Бета Центавра	Beta Centauri	β Cen	$14^{\mathrm h}03^{\mathrm m}49^{\mathrm s}$	$-60^\circ22'23''$	$0''.009$	$0^{\mathrm m}.61$	B1	$21\ M_\odot$
Эпсилон Эридана	Epsilon Eridani	ε Eri	$03^{\mathrm h}32^{\mathrm m}56^{\mathrm s}$	$-09^\circ27'30''$	$0''.311$	$3^{\mathrm m}.74$	K2	$0.82\ M_\odot$

Note for jury and team leaders

The proposed 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.

Basic criteria for examiners — 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.

Условие

There is ancient legend in Korea that says, if you managed to see the “Old persons’ 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 $33°12^{\prime}$ N and $33°34^{\prime}$ N and longitudes between $126°09^{\prime}$ E and $126°57^{\prime}$ E. Take from the tables and recollect for yourself the necessary additional information.

Решение

Canopus is a star of the southern sky; if it is visible in Korea, it should be close to the horizon. Absorption and scattering of light play an important role under such conditions of observations. Therefore, the most favorable conditions for the observation are at the southern point of the island Jeju, and Canopus is at upper culmination. The latitude of this point is the smallest of the given range, i.e. $33°12^{\prime}$ N. At this latitude Canopus culminates at the altitude

h=90°-\varphi+\delta=90°-33°12^{\prime}+-52°42^{\prime}=4°06^{\prime}.

Atmospheric absorption and scattering are significant for observations at such a low altitude. The depth of atmosphere which a beam of light from the star passes is $1/\sin h$ times larger than the depth that the light from a star located in zenith passes. $1/\sin4°06^{\prime}\approx14$ . (Otherwise, one may use the formal formula using the zenith angle $1/\cos z=1/\cos85°54^{\prime}\approx14$ .) It is known that under the most favorable conditions, the loss of light when it passes one atmosphere (due to absorption and scattering) is about 20% (or 19% as it is written in the supplement table), or (in magnitudes) $0^m.23$ . Since the magnitudes of stars in the table ( $m_0=-0^m.72$ for Canopus) are given with the zenith absorption and scattering ( $+0^m.23$ to the magnitude visible from space), the additional absorption of light at a height of $4°06^{\prime}$ will be $(1/\tan4°06^{\prime}-1)\approx13$ times more than in zenith. Thus,

\Delta m=0^m.23(1/\sin4°06^{\prime}-1)\approx3^m.0.

m_1=m_0+\Delta m=-0^m.72+3^m.0\approx2^m.3.

Note for jury. This problem is to estimate. So the exact values are not too important, especially knowledge of the exact value of 20% (15%–30% may be reasonable), or (in magnitudes) $0^m.23$ ( $0^m.15$ – $0^m.4$ may be reasonable). And the reasonable answers are from $1^m.5$ to $4^m$ .

Basic criteria for examiners.

Understanding that $\varphi=33°12^{\prime}$ should be used — 1 pt.

Calculation of the altitude on which Canopus culminates at — 2 pt.

Calculation of the depth of atmosphere to pass (like $1/\operatorname{tg}4°06^{\prime}\approx14$ ) — 2 pt.

Calculation of the loss of light in atmosphere in stellar magnitudes ( $\Delta m\approx\ldots$ or similar) — 2 pt.

Final result — 1 pt.

Теми и трудност

Трудност: неоценена.

Светлина и астрономически измервания · Междузвездно поглъщане

Предпоставки: Разширена олимпиадна подготовка

Оригинал в Архива: betatheo.pdf · официални решения: theosol1.png · допълнителни данни: theosol2.png · допълнителни данни: theosol3.png · допълнителни данни: theosol4.png · допълнителни данни: theo_cr_1.png · допълнителни данни: theo_cr_2.png