Съдържание

IAO 2014, Теория, Група β — Задача 2. Great oppositions

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IAO 2014

XIX International Astronomy Olympiad. Theoretical round. Group β · 12 октомври 2014 г. · 8 т.

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.

Условие

Now great oppositions of Mars happen every 15 or 17 years. To simplify colonization of Mars, to improve environmental conditions on it, and to increase the effect of the great oppositions, our civilization has conceived an ambitious project: to reduce by 6.0% the semi-major axis of this planet’s orbit, while keeping the eccentricity of the orbit unchanged. How often will the great oppositions of Mars occur in the case of the new orbit?

Решение

Parameters of orbits and physical characteristics

Body	Distance (AU)	Distance (10⁶ km)	Period (tropical years)	Period (days)	Inclination i (°)	Eccentricity	Equatorial diameter (km)	Mass (10²⁴ kg)	Density (g/cm³)	Gravity (m/s²)	Axial tilt (°)	Max. magnitude	Albedo
Sun	$1.6\cdot10^9$	$2.5\cdot10^{11}$	$2.2\cdot10^8$	$8\cdot10^{10}$			1392000	1989000	1.409			−26.74
Mercury	0.387	57.9	0.241	87.969	7.00	0.206	4879	0.3302	5.43	3.70	0.01		0.06
Venus	0.723	108.2	0.615	224.7007	3.40	0.007	12104	4.8690	5.24	8.87	177.36		0.78
Earth	1.000	149.6	1.000	365.2564	0.00	0.017	12756	5.9742	5.515	9.81	23.44		0.36
Moon	0.00257	0.38440	0.0748	27.3217	5.15	0.055	3475	0.07348	3.34	1.62	6.7	−12.7	0.07
Mars	1.524	227.9	1.880	686.98	1.85	0.093	6794	0.6419	3.94	3.71	25.19	−2.0	0.15
Jupiter	5.204	778.6	11.862	4332.59	1.30	0.048	142984	1899.8	1.33	24.86	3.13	−2.7	0.66
Saturn	9.584	1433.7	29.458	10759.20	2.48	0.054	120536	568.50	0.70	10.41	26.73	0.7	0.68
Eris	68.05			205029	43.82	0.435	2326	0.0167	2.52	0.7			0.96

Source footnote: for outer planets and Moon — in mean opposition. Blank cells are blank in the source.

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	71
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
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

Coordinates	Bishkek	Cholpon-Ata
Longitude E	74°34′	77°05′
Latitude N	42°52′	42°39′
Timezone	UT+6	UT+6

28 brightest stars and some other stars

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
Miaplacidus	β Car	09h13m12s	−69°43′02″	0.029	1.68	A2
Toliman (Rigel Kent)	α Cen	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
Adara	ε CMa	06h58m38s	−28°58′19″	0.008	1.50	B2
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
Gacrux	γ Cru	12h31m10s	−57°06′48″	0.037	1.60	M4
Deneb	α Cyg	20h41m26s	45°16′49″	0.002	1.25	A2
Achernar	α Eri	01h37m43s	−57°14′12″	0.026	0.46	B3
Castor	α Gem	07h34m36s	31°53′18″	0.065	1.58	A1+A2
Pollux	β Gem	07h45m19s	28°01′35″	0.097	1.14	K0
Regulus	α Leo	10h08m22s	11°58′02″	0.041	1.35	B7
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
Bellatrix	γ Ori	05h25m08s	06°20′59″	0.013	1.64	B2
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
Shaula	λ Sco	17h33m37s	−37°06′14″	0.005	1.62	B2
Aldebaran	α Tau	04h35m55s	16°30′33″	0.048	0.85ᵛ	K5
Elnath	β Tau	05h26m18s	28°36′27″	0.025	1.66	B7
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
Sheratan	β Ari	01h54m39s	20°48′30″	0.055	2.64	A5
Thuban	α Dra	14h04m23s	64°22′33″	0.011	3.65	A0
Rastaban	β Dra	17h30m26s	52°18′05″	0.009	2.79	G2
Eltamin	γ Dra	17h56m36s	51°29′20″	0.021	2.24	K5
Polaris	α UMi	02h31m49s	89°15′51″	0.007	1.97ᵛ	F7
Kochab	β UMi	14h50m42s	74°09′20″	0.025	2.07	K4
Pherkad	γ UMi	15h20m44s	71°50′02″	0.007	3.06ᵛ	A3
Yildun	δ UMi	17h32m19s	86°35′11″	0.019	4.35	A1
Proxima Centauri	V645 Cen, α Cen C	14h29m43s	−62°40′46″	0.769	11.05	M5.5

Оригинал в Архива: betatheo.pdf · официални решения: theosol.pdf · допълнителни данни: theocriteria.pdf · допълнителни данни: sombrero.pdf · допълнителни данни: constants.pdf · допълнителни данни: hrd.pdf

Съдържание

IAO 2014, Теория, Група β — Задача 2. Great oppositions

Условие

Решение

Solution

Official grading criteria