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IAO 2012, Теория, старша възраст — Задача 5. Venus and Earth

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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

At what maximum distance from the Venus ecliptic the Earth can be visible at the sky from Venus (actually, from a point outside the Venus atmosphere)? Orbits of the planets may be considered circular.

Estimate the stellar magnitude of the Earth in this situation.

Решение

α-5. Venus and Earth.

The maximum space distance from the Earth to the Venus ecliptic is (see figure)

H=R_\oplus\times\sin i,

where $R_\oplus$ is radius of the Earth orbit (1 AU), and $i$ is the orbital inclination of the terrestrial orbit to the plane of Venus ecliptic, which is evidently equal to the orbital inclination of the Venus orbit to the plane of the Earth’s ecliptic, i.e. $3.4°$ .

H=1\text{ AU}\times\sin3.4°=0.0593\text{ AU},

Geometry of Earth and Venus relative to the Venus ecliptic

In the Venetian sky, the Earth in such position can be visible at the maximum distance at the configuration of opposition, when the distance between Venus and Earth is minimal and equals to

L=R_\oplus-R_V=1\text{ AU}-0.723\text{ AU}=0.277\text{ AU}

(all angles are small and we do not take into account the inclination in these calculations). The maximal distance in the sky (angle $\beta$ ) may be found from the equation

H=L\times\tan\beta,

\operatorname{tg}\beta=H/L=0.0593\text{ AU}/0.277\text{ AU}=0{,}214,

\beta=12{,}1°.

β-5. Venus and Earth.

First part of the solution, see solution of problem α-5.

To estimate stellar magnitude of the Earth visible from vicinities of Venus in opposition we may compare it with Mars visible from the Earth in opposition (below $\alpha$ is albedo, $D$ are diameters of the bodies and $R$ are distances, indices E, V, M and S correspond to Earth, Venus, Mars and Sun).

The flux to Venus from Earth

F_V\sim\alpha_E\cdot D_E^2\cdot(1/R_{V-E})^2\cdot(1/R_{S-E})^2.

The flux to Earth from Mars

F_M\sim\alpha_M\cdot D_M^2\cdot(1/R_{E-M})^2\cdot(1/R_{S-M})^2.

The ratio of fluxes

F_E/F_M=(\alpha_E/\alpha_M)\cdot(D_E^2/D_M^2)\cdot(R_{E-M}\cdot R_{S-M})^2/(R_{V-E}\cdot R_{S-E})^2.

Taking the necessary values from the table of Solar system, we may calculate:

F_E/F_M=(0.36/0.15)\cdot(12756/6794)^2\cdot(0.524\cdot1.524)^2/(0.277\cdot1)^2.

F_E/F_M\approx70.

So the Earth visible from vicinities of Venus at opposition is brighter than Mars visible from Earth at opposition, and the difference in stellar magnitudes is equal

\Delta m=-2^m.5\cdot\lg(F_E/F_M)\approx-4^m.6.

Stellar magnitude of the Earth

m=-2^m.0+\Delta m\approx-6^m.6.

Of course it is not the only possible correct way for solution.

Basic criteria for examiners.

First part of the solution — 4 pt, (see criteria for problem α-5 and divide number of points by factor 2).

Second part of the solution — 4 pt, including:

Formula for flux — 2 pt.

(including 0.5 pt for each correct dependence on $\alpha$ , $D$ , $R$ (obs-planet), $R$ (sun-planet)).

Calculation of the ratio of fluxes — 1 pt.

Final result — 1 pt.

α-5 criteria referenced above.

Understanding the configuration, picture (may be not drawn but explained) — 3 pt.

Taking for solution correct values from tables — 1 pt.

Calculation of necessary distances — 2 pt.

Final result — 2 pt.

Теми и трудност

Трудност: неоценена.

Слънчева система · Планетни конфигурации

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