APAO 2014, Теория, Група β — Задача 6. ISS near Zenith

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.

BodyAverage distance to central body (astr. units)Average distance (million km)Sidereal (or analogous) period (tropical years)Period (days)Eccentricity
Sun1.6⋅1091.6\cdot10^92.5⋅10112.5\cdot10^{11}2.2⋅1082.2\cdot10^88⋅10108\cdot10^{10}
Mercury0.38757.90.24187.9690.206
Venus0.723108.20.615224.7010.007
Earth1.000149.61.000365.2560.017
Moon0.002570.384400.074827.32170.055
Mars1.524227.91.880686.9800.093
Jupiter5.204778.611.8624332.590.048
Saturn9.5841433.729.45810759.200.054
Uranus19.1912871.084.01530685.930.046
Neptune30.0714498.6164.77860187.640.008
BodyEquatorial diameter (km)Mass (102410^{24} kg)Average density (g/cm³)Gravity at surface (m/s²)Maximum magnitude from Earth**Albedo
Sun139200019890001.409−26.8
Mercury48790.33025.433.70−2.20.06
Venus121044.86905.248.87−4.70.78
Earth127565.97425.5159.810.36
Moon34750.07353.341.62−12.70.07
Mars67940.64193.943.71−2.00.15
Jupiter1429841899.81.3324.86−2.70.66
Saturn120536568.500.7010.410.70.68
Uranus5111886.6251.308.445.50.74
Neptune49532102.781.7611.207.80.58

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

CoordinatesIrkutskListvyanka
Longitude λ\lambda (E)104∘18′104^\circ18'104∘53′104^\circ53'
Latitude φ\varphi (N)52∘17′52^\circ17'51∘51′51^\circ51'
Distance to the center of the EarthRI=6365.3R_I=6365.3 kmRL=6365.4R_L=6365.4 km
TimezoneUT+8UT+8
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 parameter, H0H_0 (km/s/Mpc), mean value68
Hubble parameter, diapason of values50–100
Planck 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}
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 (Å)6562.81
Tropical year length, TT (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, nn1.334
Moment of inertia of a solid ballI=25MR2I=\frac25MR^2
Volume of a ballV=43πR3V=\frac43\pi R^3
Area of a sphereS=4πR2S=4\pi R^2
π\pi3.14159265
e2.71828183
Golden ratio, φ\varphi1.61803399

Условие

On the day of the beginning of the X APAO, November 24, 2014, at 05 h 01 min (Irkutsk time) the spacecraft Soyuz-TMA-15M was launched. At 10 h 48 min it was docked with the ISS. It is interesting to note that sometimes ISS flies very close to Zenith for an observer in Listvyanka. At what minimum zenith distance can ISS be seen from Listvyanka?

November 24, 2014, the main parameters of the orbit of the ISS are:

  • Inclination of the orbit: i=51.648∘i=51.648^\circ.
  • Mean motion (revolutions per day): n=15.5142n=15.5142.

The shape of the Earth can be considered as spherical (with radius equal to the distance from Listvyanka to the center of the Earth), and the ISS orbit as circular.

Note: It is not a problem to estimate. Precise calculations may be necessary to get correct result.

Решение

Внимание: Непълно решение

В архива няма официално решение на тази задача.

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