IAO 2009, Теория, Група α — Задача 3. Efficiency of eye.

Theoretical round. Problems to solve · 8 ноември 2009 г.

Бележка

Maybe not all problems have correct questions. Some questions (maybe the main question of the problem, maybe one of the subquestions) may have 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 table of planetary data may be used for solving every problem.

The answers «Да-Yes» or «Нет-No» has to be written in English or Russian.

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.

Parameters of orbits. Physical characteristics of some planets, Moon, Sun and Eris
Body, planetAverage distance (astr. units)Average distance (mln. km)Sidereal period (troph. years)Period (days)EccentricityEquat. diameter (km)Mass (102410^{24} kg)Av. density (g/cm³)Grav. acceler. (m/s²)Max. magn. from EarthAlbedo
Sun1.6⋅1091.6\cdot10^92.5⋅10112.5\cdot10^{11}2.2⋅1082.2\cdot10^88⋅10108\cdot10^{10}139200019890001.409−26.8m-26.8^{\mathrm m}
Mercury0.38757.90.24187.970.20648790.33025.433.700.06
Venus0.723108.20.615224.700.007121044.86905.248.870.78
Earth1.000149.61.000365.260.017127565.97425.5159.810.36
Moon0.002570.384400.074827.32170.05534750.07353.341.62−12.7m-12.7^{\mathrm m}0.07
Mars1.524227.91.880686.980.09367940.64193.943.71−2.0m-2.0^{\mathrm m}0.15
Jupiter5.204778.611.8624332.590.0481429841899.81.3324.86−2.7m-2.7^{\mathrm m}0.66
Saturn9.5841433.729.45810759.200.054120536568.500.7010.410.7m0.7^{\mathrm m}0.68
Eris68.012048620.43426000.01670.80.86

**) For Moon: in mean opposition.

Some constants and formulae
Constant or formulaValue
Speed of light in vacuum, cc (m/s)299792458
Constant of gravitation, GG (N·m²/kg²)6.674⋅10−116.674\cdot10^{-11}
Solar constant, AA (W/m²)1367
Hubble constant, H0H_0 (km/s/Mpc), mean value70
Hubble constant, diapason of values50–100
Plank constant, hh (J·s)6.626⋅10−346.626\cdot10^{-34}
Charge of electron, ee (C)1.602⋅10−191.602\cdot10^{-19}
Mass of electron, mem_e (kg)9.109⋅10−319.109\cdot10^{-31}
Proton-to-electron ratio1836.15
Faraday constant, FF (C/mol)96485
Magnetic constant, μ0\mu_0 (H/m)1.257⋅10−61.257\cdot10^{-6}
Universal gas constant, RR (J/mol/K)8.314
Boltzmann constant, kk (J/K)1.381⋅10−231.381\cdot10^{-23}
Standard atmosphere (Pa)101325
Stefan-Boltzmann constant, σ\sigma (W/m²/K⁴)5.670⋅10−85.670\cdot10^{-8}
Wien's displacement constant, bb (m·K)0.002897
Laboratory wavelength of Hα\alpha (Å)6563
Refractive index of water for20°C, nn1.334
Area of sphereS=4πR2S=4\pi R^2
π\pi3.14159265

Data of some stars

StarDesignationRADECmSC
Arcturusα\alpha Boo14h15m40s14^{\mathrm h}15^{\mathrm m}40^{\mathrm s}19∘10′57′′19^\circ10'57''-0.05K1
Vegaα\alpha Lyr18h36m56s18^{\mathrm h}36^{\mathrm m}56^{\mathrm s}38∘47′01′′38^\circ47'01''0.03A0
Denebα\alpha Cyg20h41m26s20^{\mathrm h}41^{\mathrm m}26^{\mathrm s}45∘16′49′′45^\circ16'49''1.25A2
Polarisα\alpha UMi02h31m51s02^{\mathrm h}31^{\mathrm m}51^{\mathrm s}89∘15′51′′89^\circ15'51''2.02F7
Siriusα\alpha CMa06h45m09s06^{\mathrm h}45^{\mathrm m}09^{\mathrm s}−16∘42′58′′-16^\circ42'58''-1.46A1
Бележки от източника

The original two-up official key photographs clip some right-hand line endings. Ellipses mark those unreadable endings; visible formulae and printed numerical values are retained. The reference table gives the Eris period as 204862 days, whereas the key uses 204852 days. Original inline slash notation is retained in the key. The photographed alpha masthead is cropped, but the original filename and distinct alpha tasks in the official key establish its group.

Условие

Estimate the theoretical limit of the maximum magnitude of stars that the human eye can see under very optimal conditions. Take into account that the eye's retina “remembers” the image about 1/7 of a second. A 0m0^{\mathrm m} star sends us about 101010^{10} photons/m2^2 every second.

Решение

Оригинал в Архива: alphatheo.png · допълнителни данни: theosol1.jpg · допълнителни данни: theosol2.jpg · допълнителни данни: constants.jpg