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

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Всички материали от IAO 2015

XX International Astronomy Olympiad. Theoretical round. Group β · 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

Official sketches for solutions; for jury work only. The given sketches are not full; team leaders have to give more detailed explanations to students. Correct solutions in students' papers (enough for 8 points) may be shorter.

Jury note. Evaluate the solution in essence, not by formal existence of mentioned sentences or formulae. Their formal presence is not necessary to give the respective points. Give points if the following steps de facto use these positions.

The first source page prints these notes twice, under both “Sketches for solutions” and “Basic criteria”. No item-by-item numerical grading breakdown appears in the archived five images. The running headers on images 2–4 print “XX IAO 2014”, while their footers and the cover identify 2015.

Solution

3.1. The corresponding scheme is supplied separately in the archive.

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

The angles are

\alpha=\arccos\frac{A_V}{(1+e)A_M}=\arccos\frac{0.723}{1.093\cdot1.524}=64.3^\circ,

\beta=\arccos\frac{A_V}{A_E}=\arccos\frac{0.723}{1}=43.7^\circ.

The source prints the following chain for the third angle:

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

Source discrepancy: the two arccos expressions repeat the preceding β calculation, while 72° belongs to $180^\circ-\alpha-\beta$ .

3.3. Mars is at aphelion, opposite its perihelion position. Mars at perihelion may be in great opposition for Earth observers in late August (August 25). Direction 1 (Mars→Sun) therefore corresponds to Sun→Earth on August 25. Direction 2 corresponds to Sun→Earth 72° earlier, or 71 days earlier, June 15. By the sky map this direction is in Ophiuchus, near the Sagittarius border.

3.2. The situation is thus in mid-June. The Sun shines for the Bear at the North Pole, while the Moon, on the opposite side of the celestial sphere during eclipse, can be observed by the Penguin at the South Pole. Answer: B− P+.

3.4. The first situation was near equinox, the second in mid-June. Both require the Moon near its orbital nodes. The nodes move in the opposite direction with a period of 18.6 years. The minimum interval can occur if the line of nodes moves from the end-September/end-March line to the mid-June/mid-December line, about 28% of a period:

18.6\cdot0.28=5.2\ \text{years}.

Thus the minimum is about 5 years and almost 2–3 months. In 5 years and exactly 3 months the number of synodic lunar periods is

(365.25\cdot5.25)/29.53=64.94.

For our situations it should be an integer and a half:

64.5\cdot29.53=1904.7\ \text{days}.

The source checks against March 20, 2015 (total solar eclipse near the North Pole and equinox). 1904.7 days later is June 5–6, 2020. It says that a lunar eclipse, “although not full but partial”, will occur on June 5 evening, with the Moon in Sagittarius. This is the source's approximate check, explicitly not the total lunar eclipse assumed in the question; no stronger agreement is claimed.

Parameters of orbits and physical characteristics; local coordinates

Body	Average distance to central body (AU)	Average distance to central body (10⁶ km)	Sidereal (or analogous) period (tropical years)	Sidereal (or analogous) period (days)	Eccentricity	Equatorial diameter (km)	Mass (10²⁴ kg)	Density (g/cm³)	Gravity (m/s²)	Maximum magnitude	Albedo
Sun	$1.6\cdot10^9$	$2.5\cdot10^{11}$	$2.2\cdot10^8$	$8\cdot10^{10}$		1392000	1989000	1.409		−26.8
Mercury	0.387	57.9	0.241	87.969	0.206	4879	0.3302	5.43	3.70	−2.2	0.06
Venus	0.723	108.2	0.615	224.701	0.007	12104	4.8690	5.24	8.87	−4.7	0.78
Earth	1.000	149.6	1.000	365.256	0.017	12756	5.9742	5.515	9.81		0.36
Moon	0.00257	0.38440	0.0748	27.3217	0.055	3475	0.0735	3.34	1.62	−12.7	0.07
Mars	1.524	227.9	1.880	686.980	0.093	6794	0.6419	3.94	3.71	−2.0	0.15
Jupiter	5.204	778.6	11.862	4332.59	0.048	142984	1899.8	1.33	24.86	−2.7	0.66
Saturn	9.584	1433.7	29.458	10759.20	0.054	120536	568.50	0.70	10.41	0.7	0.68
Uranus	19.191	2871.0	84.015	30685.93	0.046	51118	86.625	1.30	8.44	5.5	0.74
Neptune	30.071	4498.6	164.778	60187.64	0.008	49532	102.78	1.76	11.20	7.8	0.58

For outer planets and Moon — in mean opposition. Blank cells are blank in the source.

Coordinates	Observatory	Kazan kremlin	Petrovskoye
Longitude E	48°49′	49°06′	49°06′
Latitude N	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

Теми и трудност

Трудност: неоценена.

Слънчева система · Планетни конфигурации · Затъмнения и покрития

Оригинал в Архива: theo_beta.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