Съдържание

IAO 2014, Теория, Група α — Задача 5. Архар

Оригинал на условието (изображение)Съобщи за грешка

Всички материали от IAO 2014

Задачи за теоретичния тур · 12 октомври 2014 г. · 8 т.

Обща забележка

Обща забележка. Възможно е не във всички задачи въпросите да са поставени коректно. Някои въпроси (възможно е да е главният въпрос на задачата или да е някой подвъпрос) може да нямат смисъл. В такъв случай в решението трябва да се напише на руски или на английски – „ситуация невозможна“ или „impossible situation“. Разбира се, решението трябва да се подкрепи с пресмятания или логически разсъждения.

Данните от таблиците (Слънчева система, звезди, константи) могат да се използват във всяка задача.

Отговорите „Да – Yes“ или „Нет – No“ трябва да се напишат на руски или на английски.

Условие

Архарът (киргизки планински овен), живеещ недалеч от Чолпон-Ата, се е заинтересувал от това, че на езерото Исък-Кул са пристигнали ученици от различни страни, които гледат в небето през нощта, и то не просто така, а чрез някакви тръби. Освен това, архарът е видял себе си на емблемата на това мероприятие, а също така е разбрал, че в негова чест е наречено едно от съзвездията на небето. Ще може ли архарът да види звездите от това съзвездие днес през нощта на небето? („Да-Yes“ или „Нет-No“). По кое време (по киргизко време) и на каква височина в течение на следващото денонощие ще кулминира най-ярката звезда от това съзвездие? Решението придружете с рисунка на архара, наблюдаващ на небето „своето“ съзвездие.

Емблемата с архара от оригиналния български лист

Решение

Theoretical round. Sketches for solutions

Argali means the constellation Aries. The Sun is in this constellation in late April – early May, now it is almost in the opposite position on the ecliptic, so Aries is perfectly visible at night now. The brightest star in the constellation is Hamal, α Ari, its right ascension is $02^{\mathrm h}07^{\mathrm m}$ . The upper culmination of a celestial body occurs when the sidereal time is equal to its right ascension. Sidereal time is equal to mean solar time at the date of autumnal equinox, and then begins to outstrip it with the speed of $1^{\mathrm d}/365.2422$ per day (about $3^{\mathrm m}56^{\mathrm s}$ per day). 23 days have passed between the night of September 22/23 (autumnal equinox in 2014) and October 15–16 (the next 24 hours). The difference between mean solar time and sidereal time is now:

24^{\mathrm h}00^{\mathrm m}\times 23/365.2422=1^{\mathrm h}31^{\mathrm m}.

So now sidereal time $02^{\mathrm h}07^{\mathrm m}$ corresponds to

02^{\mathrm h}07^{\mathrm m}-1^{\mathrm h}31^{\mathrm m}=0^{\mathrm h}36^{\mathrm m}

of mean solar time. Cholpon-Ata is located in 13° to the West of the meridian 90° (the standard meridian for UT+6 Timezone), so the mean solar time here is behind the local time UT+6 used in Kyrgyzstan by $13^{\circ}\times 4^{\mathrm m/\circ}=52^{\mathrm m}$ . So, to get the answer according to time UT+6, we have to add these 52 minutes, and we have:

0^{\mathrm h}36^{\mathrm m}+0^{\mathrm h}52^{\mathrm m}=1^{\mathrm h}28^{\mathrm m}

of Kyrgyz time. So, tonight it will be a favorable period for the mountain sheep to observe the constellation of Aries.

The altitude of α Ari in upper culmination may be found using the formula:

h_U=90^{\circ}-\varphi+\delta=90^{\circ}-42^{\circ}39'+23^{\circ}28'=70^{\circ}49'.

Also we should calculate the altitude of α Ari in lower culmination:

h_L=90^{\circ}-\varphi+\delta=-90^{\circ}+42^{\circ}39'+23^{\circ}28'=-23^{\circ}53'.

Theoretical round. Basic criteria

Solution: 7 pt in total, including:

Argali means constellation Aries — 1 pt.
Position of constellation in October, Aries now perfectly visible at night — 1 pt.
Understanding brightest star is Hamal, α Ari — 0.5 pt.
Correct calculation of time of upper culmination — 3 pt, including: difference with September 23 (or other reference point) — 1 pt; Cholpon-Ata 13° west of 90° meridian — 1 pt; correct use of RA and final result — 1 pt.
Correct altitude of upper culmination — 1 pt.
Correct altitude of lower culmination — 0.5 pt.

Artistic picture of Argali — 1 pt.

Официални решения и оценяване

Официалните скици на решенията и критериите са запазени на оригиналния английски език.

Note for jury and team leaders

The proposed sketches are not full; the team leaders have to give more detailed explanations for students. But the correct solutions in the students’ papers (enough for 8 pts) may be shorter.

Бележка

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.

Parameters of orbits. Physical characteristics of some planets, Moon, Sun and Eris

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.

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 in the sky. 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

Source discrepancy

the key prints 262.12° in 4.2; its printed β=48°33′ gives 262°54′ for 360°−2β. The printed angle and time are retained.

Source discrepancy

the lower-culmination line prints the same symbolic expression as the upper line, but substitutes −90°+φ+δ. The original mismatch is retained.

Source discrepancy

The printed individual grading allocations for Proxima Centauri sum to 9 points, although the general instructions describe each whole solution as sufficient for 8 points. Both the printed allocations and the 8-point whole-problem scale are retained.

Оригинал в Архива: bgtheo1.jpg · официални решения: theosol.pdf · допълнителни данни: bgtheo2.jpg · допълнителни данни: bgtheo3.jpg · допълнителни данни: bgtheo4.jpg · допълнителни данни: bgtheo5.jpg · допълнителни данни: bgtheo6.jpg · допълнителни данни: theocriteria.pdf · допълнителни данни: sombrero.pdf · допълнителни данни: constants.pdf · допълнителни данни: hrd.pdf