Съдържание

IAO 2012, Теория — Задача 9. β-6. Далечна галактика

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IAO 2012

XVII Международная астрономическая олимпиада — XVII International Astronomy Olympiad, Корея, Кванджу, 16 – 24. X. 2012, Gwangju, Korea. Round: Theo. Group: α, β. Задачи за теоретичния тур · 16 октомври 2012 г.

Общи забележки

Общи забележки: Не е изключено не във всички задачи въпросите да са поставени коректно. В такива случаи трябва да се напише в отговора (на руски или на английски) „ситуация невозможная - impossible situation“. Разбира се, отговорът трябва да бъде подкрепен с пресмятания или логически разсъждения.

Данните от таблиците (Слънчева система, звезди, константи) могат да се използват във всяка задача.

Отговорите „Да - Yes“ или „Нет - No“ трябва да бъдат написани на руски или на английски.

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.

Printed date

The transit key prints “December 31, 2017” in an intermediate line, while its calculations and final answer concern 2117.

Pseudovenus notation

The orbital-radius formula retains the source’s linear slash and multiplication notation; the Moon-comparison formula likewise retains the printed placement of T_M and omega_P. The key explicitly permits a bonus for noting that the resulting orbit is outside Earth’s Hill sphere.

Mars discrepancy and absent answer drawing

The Mars key calculates about 2130 km but prints ~2300 km in its final answer. The requested Bear-and-Penguin drawing earns one point, but no completed drawing is supplied in the official key.

Galaxy arithmetic and rubric

The galaxy key prints 3.74+7.5=10.24 and M=−28.2, followed by a −0.2 correction and approximately 20 trillion stars; these printed inconsistent values are retained. Its listed criteria total 9 points (3+3+1+2), unlike the general note about 8 points, so the problem total remains unspecified.

Reference editions

The native planetary table follows the English PDF reference page (Mercury inclination 7.00°). The separately archived data_1.png instead prints 7.01°. Both source files were read; the discrepancy is retained here.

Source ownership

The Bulgarian manuscript’s nine IDs and shared αβ / distinct α and β tasks are retained. Full English sketches, both criterion sheets, all four standalone reference images and both English question editions were read. No Bulgarian official key is present in the full year inventory. The supplied sketches explicitly are not complete pedagogical solutions; no missing derivations are invented.

Условие

Астрономите са открили далечна галактика, която на земното небе на пръв поглед много прилича на $\varepsilon$ Еридан - същата по цвят, но 1000 пъти по-слаба по интензивност. Изяснило се, обаче, че тази галактика се състои предимно от звезди, по физически характеристики същите като Слънцето. Намерете броя звезди в галактиката.

Отговори

Решение

Официални скици на решенията и критерии за оценяване, на оригиналния английски език.

Official solution sketch

The fact that the galaxy, consisting of yellow stars like the Sun, looks like the orange star ε Eridani indicates redshift. $z=(\lambda_1-\lambda_0)/\lambda_0$ . According to Wien’s displacement law $\lambda T=\mathrm{const}$ .

$\lambda_1T_1=\lambda_0T_0$ . Therefore, $z=(T_0-T_1)/T_1$ .

The approximate temperatures of the stars may be found from the Hertzsprung-Russell diagram,

T_0\approx5800\text{ K},\quad T_1\approx4900\text{ K}.

Hence $z\approx(5800-4900)/4900\approx0.18$ .

The galaxy is receding from us with a speed $V=zc$ . $V=0.18\cdot300000$ km/s $=54000$ km/s.

According to Hubble’s law, $V=R\cdot H$ , $R=V/H=54000\text{ km/s}/71\text{ km/s/Mpc}\approx760\text{ Mpc}$ .

Thus, the galaxy is located at the distance 760 Mpc, and the fact that it looks like ε Eridani ( $3^m.74$ ), weakened in brightness by factor 1000 ( $3^m.74+7^m.5=10^m.24$ ) allow us to conclude that (excluding the effect of the redshift) the absolute magnitude of the galaxy is:

M=m-5^m\cdot\log(R/10\text{ pc})=3^m.74+7^m.5-5^m\cdot\log(76000000)\approx-28^m.2.

Redshift leads to the fact that every photon coming to us loses some of the energy, $E=h\nu=hc/\lambda$ ,

$\Delta E=h\Delta\nu=hc(1/\lambda_0-1/\lambda_1)$ , $\Delta E/E=\Delta T/T_0\approx0.16$ , $E_1/E_0=T_1/T_0\approx0.84$ . This change the magnitude by

\Delta m=-2^m.5\cdot\lg(0.84)\approx0^m.2.

Thus, taking into account the effect of redshift, absolute magnitude of the galaxy is

M=-28^m.2+-0^m.2\approx-28^m.4.

The absolute magnitude of the Sun is $4^m.8$ , the difference is $\Delta M=33^m.2$ .

Using this value we can conclude that the total number of stars is $10^{\Delta M/2.5}\approx10^{13.28}\approx1.9\cdot10^{13}$ or about 20 trillion stars.

Basic criteria for examiners

Understanding about redshift, taking temperatures from Hertzsprung-Russell diagram, finding the receding with a speed, and the distance to the galaxy according Hubble law — 3 pt

Calculating necessary difference in stellar magnitudes — 3 pt

Taking into account effect of red shift that gives correction $\Delta m\approx0^m.2$ — 1 pt

Final result of number of stars — 2 pt

Оригинал в Архива: bgtheo.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 · допълнителни данни: alphatheo.pdf · допълнителни данни: betatheo.pdf · допълнителни данни: data_1.png · допълнителни данни: data_2.png · допълнителни данни: data_3.png · допълнителни данни: data_4.png