Съдържание

IAO 2012, Теория — Задача 2. αβ-2. Пасаж на Псевдо-Венера

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. · 8 т.

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

Неотдавна, на 6 юни 2012 г. беше наблюдавано уникално астрономическо явление - преминаване на Венера по диска на Слънцето. Да предположим, че някой не е разбрал в какво се състои явлението и е приписал преминаването по слънчевия диск не на истинската Венера, а на някакво небесно тяло - спътник, да го наречем Псевдо-Венера, което се движи около Земята по кръгова орбита. Намерете радиуса на орбитата на Псевдо-Венера и диаметъра на това небесно тяло. Ефектите, свързани с въртенето на Земята около нейната ос да не се отчитат.

Отговори

Решение

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

Official solution sketch

Visible motion of Venus depends on the synodic motion of the planet. During the Venus transit we see just synodic velocity of Venus. The synodic angular speed of Venus is:

\omega=\omega_V-\omega_E,

where $\omega_V$ and $\omega_E$ are angular sidereal velocities of Venus and Earth respectively. If $R_V$ and $R_E$ are the radii of the orbits, the velocity of Venus in this system is,

V=\omega\cdot R_V=(\omega_V-\omega_E)\cdot R_V,

and its visible angular speed on the Earth’s sky,

u=\frac V{R_E-R_V}=\frac{(\omega_V-\omega_E)\cdot R_V}{R_E-R_V}.

But the synodic motion of the Pseudovenus has the direction opposite to that of planet’s rotation around the Sun. So the relation between the synodic $u$ and sidereal $\omega_P$ angular velocities of Pseudovenus will be

u=\omega_P+\omega_E

(opposite to formula $u=\omega_P-\omega_E$ for the same direction). So

\omega_P=u-\omega_E,

\omega_P=(\omega_V-\omega_E)\cdot R_V/(R_E-R_V)-\omega_E,

\begin{aligned}\omega_P&=\omega_V\cdot R_V/(R_E-R_V)-\omega_E\cdot(R_V/(R_E-R_V)+1)\\&=\omega_V\cdot R_V/(R_E-R_V)-\omega_E\cdot R_E/(R_E-R_V)\\&=(\omega_V\cdot R_V-\omega_E\cdot R_E)/(R_E-R_V)\\&=2\pi\cdot(R_V/T_V-R_E/T_E)/(R_E-R_V).\end{aligned}

For a body rotating around the Earth (mass $M$ ) in a circular orbit we may write,

\omega^2R=GM/R^2,

R^3=GM_E/\omega^2,

R_P=[GM_E(R_E-R_V)^2/4\pi^2\cdot(R_V/T_V-R_E/T_E)^2]^{1/3}.

Calculations give us,

R_P=2.92\cdot10^9\text{ m}=2.92\text{ mln. km}

(approx. 7.60 radii of lunar orbit).

There is also an other way to get the result after we have $\omega_P$ . It is to compare the motion of Pseudovenus with the motion of Moon. Since according to III Kepler law, $T$ is proportional to $R^{3/2}$ , $\omega$ is proportional to $R^{-3/2}$ ,

(\omega_P/\omega_M)=(R_P/R_M)^{-3/2},

R_P=R_M\cdot(\omega_M/\omega_P)^{2/3},

R_P=R_M\cdot(2\pi/T_M\omega_P)^{2/3}.

And calculations give us,

R_P=7.62R_M=2.93\text{ mln. km}.

Note for jury. Additional point may be given for the students who note that this distance is out of the Hill sphere for the Earth and so this situation is impossible.

We may find the size (diameter) of Pseudovenus using this distance $R_P$ and the angular size of the object that is equal to:

\alpha=r_V/(R_E-R_V),

where $r_V$ is the diameter of the real Venus.

r_P=\alpha\cdot R_P=r_V\cdot R_P/(R_E-R_V).

Calculations give us,

r_P=8.55\cdot10^5\text{ m}\approx850\text{ km}

(approx. 0.25 diameters of the Moon).

Basic criteria for examiners

Understanding that synodic motion should be used — 1 pt.

Formulae and result for synodic motion — 3 pt.

Using Gravitation law or III Kepler law, formulae and result for Pseudovenus orbit radius — 2 pt.

Note for jury. Additional point may be given for the students who note that this distance is out of the Hill sphere for the Earth and so this situation is impossible.

Calculation of Pseudovenus size (diameter) — 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