APAO 2015, Теория, Група β — Задача 2. Umbra and penumbra

XI Asian-Pacific Astronomy Olympiad — Theoretical round · Group β · 13 ноември 2015 г.

Parameters of orbits and physical characteristics

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

Body	Average distance to central body (astr. units)	Average distance (million km)	Sidereal (or analogous) period (tropical years)	Period (days)	Eccentricity
Sun	$1.6\cdot10^9$	$2.5\cdot10^{11}$	$2.2\cdot10^8$	$8\cdot10^{10}$
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
Eris	68.05			205029	0.435

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
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
Eris	2326	0.0167	2.52	0.7		0.96

**) 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	68
Hubble parameter, range 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 mass 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
Inclination of ecliptic to the celestial equatorial plane, $\varepsilon$ (epoch 2000.0)	$23^\circ26'21''.45$
Tropical year length, $T$ (days)	365.242199
Nodal period of lunar orbit (years)	−18.6
Standard atmosphere (Pa)	101325
Laboratory wavelength of Hα (Å)	6562.81
Wavelengths of visible light, $\lambda$ (Å), mean value	5526
Wavelengths of visible light, range of values (Å)	4000–7600
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, on clear sunny days, objects usually give rise to umbras and penumbras. Nowadays, the angle of the ecliptic plane to the celestial equator is $\varepsilon_0=23^\circ26.2'$ and is reduced by about $\upsilon=0.78'$ per century. Calculate in what century begins the period or in what century the period has ended when from the exactly vertical column located in Ashulia

2.1 umbra from the Sun

2.2 penumbra from the Sun

will disappear (or disappeared) at some sunny days of the year.

Ashulia’s coordinates are $23^\circ53.6'$ N, $90^\circ19.8'$ E.

Draw pictures explaining your solution.

Решение

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