Съдържание

IAO 2012, Теория, старша възраст — Задача 2. Transit of Pseudovenus

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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

Recently, on June 6, 2012, an infrequent astronomical phenomenon, transit of Venus across the solar disc took place. Suppose somebody did not understand the phenomenon and ascribed it not to transit of real Venus but of some moon, which we name Pseudovenus, rotating around the Earth in a circular orbit. Find the radius of the orbit of Pseudovenus and diameter of this sky body. Effects due to axial rotating of the Earth should not be taken into account.

Решение

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.

Теми и трудност

Трудност: неоценена.

Небесна механика · Синодични периоди

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