Съдържание

IAO 2015, Теория, Група α — Задача 3. Close conjunction

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IAO 2015

XX International Astronomy Olympiad. Theoretical round. Group α · 15 октомври 2015 г.

Read once more before you start your work

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.

Ещё раз прочитайте перед началом работы.

Общее замечание. Не исключено, что не во всех задачах вопросы поставлены корректно. Некоторые вопросы (возможно, главный вопрос задачи, возможно – подвопрос) могут не иметь смысла. В этом случае следует написать в ответе (по-русски или по-английски): «ситуация невозможна – impossible situation». Естественно, ответ должен быть подкреплён вычислениями или логическими рассуждениями.

Данные из таблиц (Солнечная система, звёзды, константы) могут быть использованы в любой задаче.

Ответы «Да-Yes» или «Нет-No» должны быть написаны по-русски или по-английски.

Условие

Some time after the events described in the previous problem (nobody knows even the order of magnitude how long after – minutes, or hours, or days, or years ...), Venus at the point of eastern elongation came to close conjunction with Mars, which was located near the aphelion of its orbit. At the same time a total lunar eclipse occurred on the Earth.

3.1 Draw the corresponding scheme.

3.2 Explain, which animal (sitting at the same poles) may see this lunar eclipse. (At the very end of your explanation write as answer B+ or B− for the Bear and P+ or P− for the Penguin.) An artistic picture of the observations of the animals is welcome.

3.3 Calculate in which constellation the eclipsed Moon was observed.

3.4 Estimate minimal possible time passed from the situation of the previous problem to the situation of the current one.

Source sky map

Supplied sky map with constellation boundaries, stars and celestial coordinate grid

Отговори

Решение

αβ-3. Close conjunction.

The corresponding scheme is drawn right on the right (3.1.). The values of angles are:

Official conjunction diagram: Sun, Earth, Moon, Venus, Mars, angles α β γ and directions1 and2

\alpha = \arccos(A_V/(1+e)A_M) = \arccos(0.723/1.093\cdot1.524) = 64.3^\circ.

\beta = \arccos(A_V/A_E) = \arccos(0.723/1) = 43.7^\circ.

\gamma = 180^\circ - \alpha - \beta = \mathrm{arcos}(A_V/A_E) = \arccos(0.723/1) = 72^\circ.

Mars is in aphelion, which means it is opposite to the position of perihelion. We should remember that Mars in perihelion may be in great opposition for observers on the Earth in the late August (actually, August 25). So the current direction 1 (Mars→Sun) corresponds to the direction Sun→Earth for August 25. Direction 2 corresponds to the direction Sun→Earth 72° earlier, i.e. 71 day earlier, for June 15. By the sky map we may find that this direction is to a point in Ophiuchus constellation just near to the border with Sagittarius (3.3.).

The above conclusion means also that this situation was released in the middle of June. At this time the Sun is shining for the Bear at the North pole, and the Moon (that is in the opposite point of the celestial sphere at the moment of eclipse) may be observed by Penguin at the South pole. (3.2. B– P+)

3.4. To find the period from the situation from the previous problem to the current situation, we should take into account that the first situation was near equinox, and the second – in the middle of June. And both of them are possible only with the Moon is near its orbital nodes. The orbital nodes of the Moon move in opposite direction with period 18.6 years. That is the minimum possible time between the two situations may be in the case the line of nodes moved from end September – end March line to middle June – middle December line, that is about 28% of the period or 18.6×0.28 ≈ 5.2 years. So the minimum time is about 5 years and almost 2-3 months later. The number of synodic periods of the Moon in 5 years and exactly 3 months is (365.25×5.25)/29.53 = 64.94. For our situations this value should be an integer and a half. 64.5×29.53 = 1904.7.

Let us check. The situation very near the first condition was just on March 20, 2015 (total solar eclipse very close to North pole and just at the day of equinox). 1904.7 days later is June 5-6, 2020. Yes! The lunar eclipse (although not full but partial) will occur in the evening of June 5. And the Moon will be in the constellation of Sagittarius.

Parameters of orbits and physical characteristics of planets, Sun and Moon

Body, planet	Average distance to central body, in astr. units	Average distance to central body, in mln. km	Sidereal (or analogous) period, in tropical years	Sidereal (or analogous) period, in days	Eccentricity, e	Equat. diameter, km	Mass, $10^{24}$ kg	Av. density, g/cm³	Grav. acceler. at surf., m/s²	Max. magn. from Earth **)	Albedo
Sun	$1{,}6\cdot10^9$	$2{,}5\cdot10^{11}$	$2{,}2\cdot10^8$	$8\cdot10^{10}$		1392000	1989000	1,409		$-26{,}8^{\mathrm{m}}$
Mercury	0,387	57,9	0,241	87,969	0 ,206	4 879	0,3302	5,43	3,70	$-2{,}2^{\mathrm{m}}$	0,06
Venus	0,723	108,2	0,615	224,701	0,007	12 104	4,8690	5,24	8,87	$-4{,}7^{\mathrm{m}}$	0,78
Earth	1,000	149,6	1,000	365,256	0,017	12 756	5,9742	5,515	9,81		0,36
Moon	0,00257	0,38440	0,0748	27,3217	0,055	3 475	0,0735	3,34	1,62	$-12{,}7^{\mathrm{m}}$	0,07
Mars	1,524	227,9	1,880	686,980	0,093	6 794	0,6419	3,94	3,71	$-2{,}0^{\mathrm{m}}$	0,15
Jupiter	5,204	778,6	11,862	4 332,59	0,048	142 984	1899,8	1,33	24,86	$-2{,}7^{\mathrm{m}}$	0,66
Saturn	9,584	1433,7	29,458	10 759,20	0,054	120 536	568,50	0,70	10,41	$0{,}7^{\mathrm{m}}$	0,68
Uranus	19,191	2871,0	84,015	30 685,93	0,046	51 118	86,625	1,30	8,44	$5{,}5^{\mathrm{m}}$	0,74
Neptune	30,071	4498,6	164,778	60 187,64	0,008	49 532	102,78	1,76	11,20	$7{,}8^{\mathrm{m}}$	0,58

**) For outer planets and Moon – in mean opposition.

Coordinates	Observatory	Kazan kremlin	Petrovskoye
λ ( E / в.д. )	48° 49′	49° 06′	49° 06′
φ ( N / с.ш. )	55° 50′	55° 48′	55° 41′
Timezone	UT+3	UT+3	UT+3

Some constants and formulae

Constant or formula	Value
Speed of light in vacuum c (m/s)	299792458
Constant of gravitation G (N·m²/kg²)	$6.674\cdot10^{-11}$
Solar constant A (W/m²)	1367
Hubble parameter H₀ (km/s/Mpc), mean	68
Hubble parameter, range	50–100
Planck constant h (J·s)	$6.626\cdot10^{-34}$
Charge of electron e (C)	$1.602\cdot10^{-19}$
Mass of electron mₑ (kg)	$9.109\cdot10^{-31}$
Proton-to-electron ratio	1836.15
Faraday constant F (C/mol)	96485
Magnetic constant μ₀ (H/m)	$1.257\cdot10^{-6}$
Universal gas constant R (J/mol/K)	8.314
Boltzmann constant k (J/K)	$1.381\cdot10^{-23}$
Stefan–Boltzmann constant σ (W/m²/K⁴)	$5.670\cdot10^{-8}$
Wien displacement constant b (m·K)	0.002897
Laboratory wavelength Hα (Å)	6562.81
Tropical year length T (days)	365.242199
Nodal period of lunar orbit (years)	−18.6
Standard atmosphere (Pa)	101325
Visible light extinction by terrestrial atmosphere in zenith (minimum)	19%, 0.23 magnitudes
Height of homogeneous atmosphere (m)	7991
Refractive index of water for 20°C n	1.334
Moment of inertia of a solid ball	$I=\frac25MR^2$
Volume of a ball	$V=\frac43\pi R^3$
Area of sphere	$S=4\pi R^2$
π	3.14159265
e	2.71828183
Golden ratio φ	1.61803399

20 brightest stars, other stars and Greek alphabet

Name	Designation	RA	Dec	Parallax (arcsec)	Magnitude	Spectral class
Altair	α Aql	19h50m47s	08°52′06″	0.195	0.77	A7
Capella	α Aur	05h16m41s	45°59′53″	0.073	0.08	G5+G0
Arcturus	α Boo	14h15m38s	19°10′57″	0.089	−0.04ᵛ	K1
Canopus	α Car	06h23m57s	−52°41′45″	0.028	−0.72	F0
Toliman (Rigel Kent)	α Cen A; B	14h39m36s	−60°50′07″	0.747	−0.01; 1.33	G2; K1
Hadar	β Cen	14h03m49s	−60°22′23″	0.009	0.61	B1
Sirius	α CMa	06h45m09s	−16°42′58″	0.375	−1.46	A1
Procyon	α CMi	07h39m18s	05°13′30″	0.288	0.38	F5
Acrux	α Cru	12h26m36s	−63°05′57″	0.010	0.77	B0
Becrux	β Cru	12h47m43s	−59°41′20″	0.009	1.30	B0
Deneb	α Cyg	20h41m26s	45°16′49″	0.002	1.25	A2
Achernar	α Eri	01h37m43s	−57°14′12″	0.026	0.46	B3
Pollux	β Gem	07h45m19s	28°01′35″	0.097	1.14	K0
Vega	α Lyr	18h36m56s	38°47′01″	0.123	0.03	A0
Betelgeuse	α Ori	05h55m10s	07°24′25″	0.005	0.5ᵛ	M2
Rigel	β Ori	05h14m32s	−08°12′06″	0.013	0.12	B8
Fomalhaut	α PsA	22h57m39s	−29°37′20″	0.130	1.16	A3
Antares	α Sco	16h29m24s	−26°25′55″	0.024	0.96	M1+B4
Aldebaran	α Tau	04h35m55s	16°30′33″	0.048	0.85ᵛ	K5
Spica	α Vir	13h25m12s	−11°09′41″	0.023	0.98	B1

Some other stars

Name	Designation	RA	Dec	Parallax (arcsec)	Magnitude	Spectral class
Hamal	α Ari	02h07m10s	23°27′45″	0.050	2.01	K2
Polaris	α UMi	02h31m49s	89°15′51″	0.007	1.97ᵛ	F7
Kochab	β UMi	14h50m42s	74°09′20″	0.025	2.07	K4
Proxima Centauri	V645 Cen, α Cen C	14h29m43s	−62°40′46″	0.769	11.05	M5.5

Greek alphabet

Uppercase	Lowercase	Name
Α	α	alpha
Β	β	beta
Γ	γ	gamma
Δ	δ	delta
Ε	ε	epsilon
Ζ	ζ	zeta
Η	η	eta
Θ	θ	theta
Ι	ί	iota
Κ	κ	kappa
Λ	λ	lambda
Μ	μ	mu
Ν	ν	nu
Ξ	ξ	xi
Ο	ο	omicron
Π	π	pi
Ρ	ρ	rho
Σ	σ	sigma
Τ	τ	tau
Υ	υ	upsilon
Φ	φ	phi
Χ	χ	chi
Ψ	ψ	psi
Ω	ω	omega

Theoretical round. Sketches for solutions. For jury job ONLY

Theoretical round. Sketches for solutions. For jury job ONLY

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.

Note. Jury members should evaluate the student's solutions in essence, and not by looking on formal existence the mentioned sentences or formulae. The formal presence of the mentioned positions in the text is not necessary to give the respective points. Points should be done if the following steps de facto using these positions.

Theoretical round. Basic criteria. For work of Jury

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.

Теми и трудност

Трудност: неоценена.

Слънчева система · Планетни конфигурации · Затъмнения и покрития

Оригинал в Архива: theo_alpha.pdf · официални решения: theosol1.jpg · допълнителни данни: theosol2.jpg · допълнителни данни: theosol3.jpg · допълнителни данни: theosol4.jpg · допълнителни данни: theosol5.jpg · допълнителни данни: theo_instr.pdf · допълнителни данни: data_1.pdf · допълнителни данни: data_2.pdf · допълнителни данни: data_3.pdf · допълнителни данни: hrd.pdf · допълнителни данни: sky_map.pdf