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IAO 2015, Теория, Група β — Задача 2. Eclipse on the Poles

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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

The White Bear and the Penguin from the previous International Astronomy Olympiads returned to their poles (North and South respectively), and decided to observe an annular solar eclipse. The Penguin was lucky to see an amazing picture: at the maximum phase of the eclipse the centres of both discs, solar and lunar, appeared just on the visible horizon. And what did the Bear observe at this time? Draw what the White Bear saw at that moment, and also contour by dotted line the true positions of the Sun and the Moon.

Assume that the Earth is spherical. The drawing should include an artistic picture with an image of the Bear on North Pole; necessary sizes or angular sizes should be pointed out in the picture. Recollect for yourself the necessary information about the animals.

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

It seems that since the points of observations are opposite, the situation should be absolutely symmetric: the White Bear should also observe half of the eclipsed solar disk on the horizon. But this is quite incorrect. Three circumstances must be taken into account.

Horizon depression. For any observer the physical horizon is slightly lowered. A person standing on a flat surface sees about 2.5′ depression. For Earth's radius $R$ and eye height $h$ , the horizon dip is

\arccos\frac{R}{R+h}\approx\frac{\sqrt{(R+h)^2-R^2}}R\approx\sqrt{\frac{2h}R}\quad\text{(radians)}.

Take an Imperial Penguin with eyes about 1 m above the surface: depression about 2′. The White Bear is sitting, with eyes also about 1 m high, so depression about 2′ (standing, more than 2 m and about 3′). If the Penguin sees the solar centre on its horizon, it is already 2′ below the mathematical horizon. At the North Pole it is 2′ above the mathematical horizon, or 4′ above the sitting Bear's physical horizon — a quarter of the solar radius. Horizon depression is the smallest effect here.

Atmospheric refraction. This varies slightly with the weather, averaging about 35′ at the horizon. When the solar centre is visible on the South Pole horizon, at the North Pole it can be seen well above the horizon. Refraction at the South Pole contributes 35′, and at the North Pole about 25′ (not 35′, since the Sun is higher). This is still not the principal effect.

Lunar parallax. An annular eclipse is visible only in a narrow band on Earth. The distance to the Moon is 384 thousand km. The source then calls 12730 km the Earth's “polar radius” and prints

12730/38400\approx0.0332\ \mathrm{rad}\approx1.9^\circ.

Source discrepancy: the printed 12730 km is a diameter, and the denominator 38400 lacks a zero relative to the preceding lunar distance. The printed numerical angular result is retained.

The Bear does not observe any solar eclipse, neither annular nor partial. The solar centre is about $4'+35'+25'=64'$ (about 1°) above the horizon. The lunar disk is approximately 1°54′ below the solar one, totally below the horizon. More precisely this is 1°54′ below the true solar position; including refraction, the true Moon is 2°19′ below the apparent Sun.

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 · Небесна сфера и сферична астрономия · Затъмнения и покрития · Атмосферни ефекти

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