APAO 2014, Теория, Група β — Задача 4. Fourth star of Orion’s Belt

X Asian-Pacific Astronomy Olympiad — Theoretical round · Group β · 24 ноември 2014 г.

Бележка

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.

Parameters of orbits and physical characteristics

Parameters of orbits and physical characteristics of planets, Sun and Moon.

Body	Average distance to central body (astr. units)	Average distance (million km)	Sidereal (or analogous) period (tropical years)	Period (days)	Eccentricity
Sun	$1.6\cdot10^9$	$2.5\cdot10^{11}$	$2.2\cdot10^8$	$8\cdot10^{10}$
Mercury	0.387	57.9	0.241	87.969	0.206
Venus	0.723	108.2	0.615	224.701	0.007
Earth	1.000	149.6	1.000	365.256	0.017
Moon	0.00257	0.38440	0.0748	27.3217	0.055
Mars	1.524	227.9	1.880	686.980	0.093
Jupiter	5.204	778.6	11.862	4332.59	0.048
Saturn	9.584	1433.7	29.458	10759.20	0.054
Uranus	19.191	2871.0	84.015	30685.93	0.046
Neptune	30.071	4498.6	164.778	60187.64	0.008

Body	Equatorial diameter (km)	Mass ($10^{24}$ kg)	Average density (g/cm³)	Gravity at surface (m/s²)	Maximum magnitude from Earth**	Albedo
Sun	1392000	1989000	1.409		−26.8
Mercury	4879	0.3302	5.43	3.70	−2.2	0.06
Venus	12104	4.8690	5.24	8.87	−4.7	0.78
Earth	12756	5.9742	5.515	9.81		0.36
Moon	3475	0.0735	3.34	1.62	−12.7	0.07
Mars	6794	0.6419	3.94	3.71	−2.0	0.15
Jupiter	142984	1899.8	1.33	24.86	−2.7	0.66
Saturn	120536	568.50	0.70	10.41	0.7	0.68
Uranus	51118	86.625	1.30	8.44	5.5	0.74
Neptune	49532	102.78	1.76	11.20	7.8	0.58

**) For outer planets and Moon — in mean opposition.

Coordinates	Irkutsk	Listvyanka
Longitude $\lambda$ (E)	$104^\circ18'$	$104^\circ53'$
Latitude $\varphi$ (N)	$52^\circ17'$	$51^\circ51'$
Distance to the center of the Earth	$R_I=6365.3$ km	$R_L=6365.4$ km
Timezone	UT+8	UT+8

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	68
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
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 a sphere	$S=4\pi R^2$
$\pi$	3.14159265
e	2.71828183
Golden ratio, $\varphi$	1.61803399

Условие

As you know, there is a Penguin among animals-observers, and he is a great joker. The Penguin decided that something is missing for artistic perception of Orion, and one must “put” a fourth star on the Babr’s picture of Orion’s Belt. For this, he found a photoflash lamp and, sitting somewhere in the field of view of the camera during the exposure, flashed once (sent one light pulse) to the camera during the exposure. Estimate at what distance from the Babr the Penguin was, if the shining of the “fourth star of Orion’s Belt” approximately corresponds to that of the three others in the picture. When photographing, the Babr-observer used an exposure time of the order of 10 seconds.

Take into account the parameters of the flash pulse. The photographic conditions (sensitivity of CCD, diaphragm) while photographing objects with the flash at the distance of 1 meters (from both flash and camera) should be the same as while photographing the same objects on a sunny day with an exposure time of about 1/1000 second.

The solution has to include a picture with artistic images of the Penguin and his flash lamp.

Решение

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Оригинал в Архива: betatheo.pdf · допълнителни данни: data1.pdf · допълнителни данни: data2.pdf