Съдържание

IAO 2013, Теория — Задача 3. α-3. Наблюдения от Глизе 581 g

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IAO 2013

XVIII Международная астрономическая олимпиада. XVIII International Astronomy Olympiad. Литва, Вильнюс, 6 – 14. IX. 2013, Vilnius, Lithuania. Round Theo · 6 септември 2013 г. · 8 т.

Внимание: Бележка към темата

Ръкописни български условия за групи α, β и общи αβ; решенията и критериите са на английски. В αβ-6 ръкописът дава 3400 km, а английският оригинал и ключът — 3400 pc. Източниковите разминавания са запазени и отбелязани.

Общо указание

Общо указание. Възможно е не във всички задачи въпросите да са поставени коректно. Някои въпроси (възможно е това да е основният въпрос в задачата, а може и да е някой подвъпрос) може да нямат смисъл. В такива случаи в отговора трябва да се напише (на руски или на английски): „ситуация невозможна - impossible situation“. Разбира се, отговорът трябва да е подкрепен с пресмятания или логически разсъждения.

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

Отговорите „Да - Yes“ или „Нет - No“ трябва да бъдат написани на руски или на английски.

Общо указание

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

Отговорите „Да - Yes“ или „Нет - No“ трябва да бъдат написани на руски или на английски.

Parameters of orbits and physical characteristics

The bilingual source sheet gives the following values. Blank cells reproduce blank source cells.

Body / planet	Average distance (AU)	Average distance ( $10^6$ km)	Sidereal period (tropical years)	Sidereal period (days)	Orbital inclination $i$	Eccentricity $e$	Equatorial diameter (km)	Mass ( $10^{24}$ kg)	Average density (g/cm³)	Gravity at surface (m/s²)	Axial tilt	Maximum magnitude from Earth	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	0.0735	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

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, $H_0$ (km/s/Mpc), mean value	71
Hubble parameter, diapason of values	50–100
Planck constant, h (J·s)	$6.626\cdot10^{-34}$
Charge of electron, e (C)	$1.602\cdot10^{-19}$
Mass of electron, $m_e$ (kg)	$9.109\cdot10^{-31}$
Proton-to-electron ratio	1836.15
Faraday constant, F (C/mol)	96485
Magnetic constant, $\mu_0$ (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, $\sigma$ (W/m²/K⁴)	$5.670\cdot10^{-8}$
Wien’s displacement constant, b (m·K)	0.002897
Laboratory wavelength of Hα (Å)	6562.81
Tropical year length, T (days)	365.242199
Standard atmosphere (Pa)	101325
Visible light extinction by the terrestrial atmosphere in zenith (minimum)	19%, 0.23 mag
Refractive index of water for 20°C, n	1.334
Moment of inertia of a solid ball	$I=\frac25MR^2$
Volume of ball	$V=\frac43\pi R^3$
Area of sphere	$S=4\pi R^2$
$\pi$	3.14159265
e	2.71828183
Golden ratio, $\varphi$	1.61803399

Data of some stars

Star	Designation	RA	DEC	Parallax $p$	Apparent magnitude $m$	Spectral class	Mass ( $\mathcal M_{\odot}$ )
Sun	⊙	0h–24h	−23°26′ to +23°26′	8.794″	vis −26.74; bol −26.82	G2	1
Proxima Centauri	V645 Cen, α Cen C	14h 29m 43s	−62°40′46″	0.769″	11.05	M5.5	0.123
Alpha Centauri A	α Cen A	14h 39m 37s	−60°50′02″	0.747″	−0.01	G2	1.1
Alpha Centauri B	α Cen B	14h 39m 35s	−60°50′14″		1.34	K1	0.9
Beta Centauri	β Cen	14h 03m 49s	−60°22′23″	0.009″	0.61	B1	21
Epsilon Eridani	ε Eri	03h 32m 56s	−09°27′30″	0.311″	3.74	K2	0.82
Gliese 581	HO Lib	15h 19m 27s	−07°43′20″	0.16″	vis 10.57; bol 8.0	M3V	0.31

The Alpha Centauri parallax is printed once beside the pair; the second cell is left blank here as in the source.

Historical units

Units of power

1 horse-power (hp) = 735.49875 W.

Local units of length in the end of XVIII century

1 arshin (arsh) = 0.711187 m
1 span (sp) = 1/4 arshin
1 vershok (vrsh) = 1/4 span
1 sajene (sj) = 3 arshin
1 verst (vrst) = 500 sajene

Условие

3.1. Каква ще бъде звездната величина на нашето Слънце при наблюдение от планетата Глизе 581 g? [5 т.]

3.2 Примерно в кое съзвездие ще се вижда оттам Слънцето? [3 т.]

Отговори

Решение

Официалното решение и критериите по-долу са на английски, както в източника.

Official English solution sketch. 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.

3.1. Apparent magnitude

The magnitude of the Sun, visible from any remote location is

m_{\odot}=m_0+5^{\mathrm m}\log(206265)-5^{\mathrm m}\lg p,

where $m_0$ is the apparent magnitude of the Sun visible from Earth, and p is the annual parallax of the location.

The magnitude of the Sun as seen from the system of Gliese 581 is

m_{\odot}=-26^{\mathrm m}.74+26^{\mathrm m}.57-5^{\mathrm m}\log0.16=-0^{\mathrm m}.17-(-3^{\mathrm m}.98)\approx3^{\mathrm m}.8.

(And it is not the only way of solution. The value can be found in many other ways, e.g., by formula $m_{\odot}=M-5^{\mathrm m}+5^{\mathrm m}\lg p$ ).

Source note: the double minus in the intermediate expression is written here as $-0^{\mathrm m}.17-(-3^{\mathrm m}.98)$ with parentheses for readability; the alternative formula above is retained as printed, including its plus sign before $5\lg p$ .

3.2. Approximate constellation

While observing from the planet Gliese 581 g, our Sun will be at a point opposite to where the system of Gliese 581 is seen from Earth. Because Gliese 581 is located in the zodiacal constellation of Libra, the sun will be approximately in the opposite zodiacal constellation, that is, in the constellation of Aries. (That solution is correct enough.)

Note: Using a sky-chart one can solve the problem more accurately. The point opposite to the location of the system Gliese 581 at the Earth’s sky, has the coordinates $03^{\mathrm h}\ 19^{\mathrm m}\ 27^{\mathrm s}$ and $07^\circ43^{\prime}20^{\prime\prime}$ . This point is located in the constellation of Taurus (Tau), very close to the borders of the constellations of Cetus (Cet) and Aries (Ari). As the students were not provided by a sky-chart, and question was “what is the approximate constellation”, the answer “approximately in the constellation, opposite to the constellation of Libra, that is in the constellation of Aries” is correct.

Official basic criteria

3.1. Totally 5 pt, including:

Formulae for magnitude of the Sun (any way solution) — 3 pt.
Correct calculations — 2 pt.

3.2. Totally 3 pt, including:

Understanding that Gliese 581 is located in the zodiacal constellation of Libra — 1 pt.
Sun will be at opposite point — 1 pt.
Answer: constellation of Aries — 1 pt.

Official solution sketches: 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.

Basic criteria: note for jury

Jury members should evaluate the students’ 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.

Оригинал в Архива: bgtheo.pdf · официални решения: theosol.pdf · допълнителни данни: theodata.pdf · допълнителни данни: theo_cr_1.pdf · допълнителни данни: theo_cr_2.pdf · допълнителни данни: theo_cr_3.pdf · допълнителни данни: constants.pdf · допълнителни данни: hrd.pdf · допълнителни данни: alphatheo.pdf · допълнителни данни: betatheo.pdf · допълнителни данни: theo_cr_1.png · допълнителни данни: theo_cr_2.png · допълнителни данни: theo_cr_3.png