Съдържание

IAO 2015, Теория, Група β — Задача 4. Constellation of White Leopard

Оригинал на условието (PDF)Съобщи за грешка

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

According to an ancient legend of Middle Volga there was a constellation called White Leopard (Белый Барс – Pardus Album) in the sky in the very past, in which the number of stars were exactly equal to the number of letters in the Greek alphabet, and the stars had magnitudes $\alpha$ PaA – $+0.10^{m}$ , $\beta$ PaA – $+0.20^{m}$ , $\gamma$ PaA – $+0.30^{m}$ , $\delta$ PaA – $+0.40^{m}$ , and so on with adding $0.10^{m}$ till $\omega$ PaA. Calculate the total magnitude of the stars of this constellation.

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

There are 24 letters in the Greek alphabet, so take the 24 magnitudes from +0.10 to +2.40.

The source first jokes: $0.10+0.20+\cdots+2.40=12\cdot2.50=30$ magnitudes — “Sorry, a joke.”

Really we must add to the first star's flux the other 23 fluxes, decreasing exponentially with pitch

10^{0.4\cdot0.10}\approx1.09648.

The infinite sum can be found by a geometric progression. The source prints

\Phi(1\text{–}\infty)=\Phi(1)+\Phi(\infty)/1.09648,

\Phi(1\text{–}\infty)=\Phi(\infty)/1.04713=\Phi(1),

\Phi(1\text{–}\infty)=\frac{\Phi(1)}{1-1/1.04713}=\Phi(1)\cdot11.37.

Source discrepancies: the middle line is printed as an equality chain, and the subsequent denominator uses 1.04713 instead of the preceding pitch 1.09648. The printed multiplier 11.37 follows the latter pitch.

The magnitude of the infinite series would be

m(1\text{–}\infty)=0.10-2.5\log11.37\approx-2.54.

There are only 24 stars. Subtract the flux of the analogous infinite series starting at the 25th star, magnitude 2.50. Every star in this second series is 2.40 magnitudes weaker; so is the series as a whole:

m(25\text{–}\infty)=2.50-2.5\log11.37\approx-0.14.

The source jokes again about subtracting magnitudes: $-2.54-(-0.14)=-2.40$ — “Sorry, a joke once again.” Fluxes must instead be subtracted. The second flux is $10^{0.4\cdot2.4}\approx9.12$ times weaker:

\Phi(25\text{–}\infty)=\frac{\Phi(1\text{–}\infty)}{10^{0.96}}=\frac{\Phi(1)\cdot11.37}{9.12}\approx\Phi(1)\cdot1.246,

\Phi(1\text{–}24)=\Phi(1\text{–}\infty)-\Phi(25\text{–}\infty)=\Phi(1)(11.37-1.246)=\Phi(1)\cdot10.12.

Hence

\boxed{m(1\text{–}24)=0.10-2.5\log10.12\approx-2.41.}

Attention! The second joke happens to give almost the right number, by pure coincidence. Not any solution giving the same numerical value is right; pay attention when checking.

Other approaches. Calculate each star's flux, add the fluxes, and recover the total apparent magnitude. A proper calculation gives exactly the same answer (easy to check in Excel), but is lengthy without a computer or programmable calculator.

A very rough linear approximation uses the “average” magnitude $(0.10+2.40)/2=1.25$ , giving

1.25-2.5\log24\approx1.25-3.45=-2.20.

The source says this is very rough and not suitable as a workable solution, but “a little better than nothing”.

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.

Светлина и астрономически измервания · Звездни величини

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