IAO 2017, Практически тур, Група β — Задача 6. Transiting extra-solar planet.

XXII International Astronomy Olympiad — Practical round, Group β · 10 т.

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Languages of the solutions. (For Practical and also for Observational rounds).

No text in any language is permitted to use in solutions. All the solution must be written using only drawing pictures, plotting graph, writing formulae, numerical values and standard international astronomy symbols (like ♂, ♎, α UMa, M31, 5ᵐ). Numerical values must be written using standard symbols but not symbols of National alphabet. Jury of the Practical round will not take into account any text in any language in solution.

For example, the names of constellations, written in English, Russian or any other language will be not taken into account (should be only UMa, Ori, etc.).

Some constants and formulae — archive reference

Archive reference sheet: Some constants and formulae, explicitly headed Round Theo, groups α and β. It is retained here as an attributed archive reference; it is not claimed to have been handed out during the practical round.

Quantity	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
Plank 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
Laboratory wavelength of Hα (Å)	6562.81
Tropical year length, $T$ (days)	365.242199
Sidereal year length, $T$ (days)	365.25636
Anomalistic year length, $T$ (days)	365.259636
Nodal period of lunar orbit (years)	~18.6
Standard atmosphere (Pa)	101325
Visible light extinction by the terrestrial atmosphere in zenith (minimum)	19%, $0.23^{\mathrm m}$
Height of homogeneous atmosphere (m)	7991
Refractive index of water for 20°C, $n$	1.334
Moment of inertia of a solid ball	$I=\frac25MR^2$
Moment of inertia of sphere	$I=\frac23MR^2$
Volume of a 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

Условие

The transit of an extra-solar planet (TrES-3b) across a star (TrES-3) was observed by the 1-m telescope of Weihai Observatory (WHO) on April 16, 2009, using the broad-band V filter and a PI CCD. The photometric data (selected part from the original one) are given in the Table 6 (see separate sheet), the time is in Julian days, and $m_V$ is the V-band magnitude.

6.1 Plot the transit light curve (magnitude vs time), determine the moment of mid-transit (the deepest transit) and write result (in Julian days) in form of «$T_{\mathrm{mid}}=\ldots$», estimate the duration of the transit (in days) in form of «$T_d=\ldots$», determine the depth of the transit (in magnitudes) in form of «$D=\ldots$». [5 т.]

6.2 Give the formula and calculate the radius of the extra-solar planet TrES-3b $R_p$ (in units of the radius of Jupiter: $R_{\mathrm{Jupiter}}$, write the answer in form of «$R_p=\ldots$»). The radius of the star (TrES-3) is known to be 0.813 radius of the Sun, $R_s=0.813R_{\mathrm{sun}}$. [2 т.]

6.3 Estimate the approximate value of the angle $i$, in unit of degrees (°) between the axis of this planet’s orbiting and the line of sight of the observer. Write the answer in form of «$i\approx\ldots$». [1 т.]

6.4 The next mid-transit moment after the above transit event was: JD = 2454939.578. The mass of the star (TrES-3) is known to be 0.924 mass of the Sun, $M_s=0.924M_{\mathrm{sun}}$. Calculate the orbital period of TrES-3b (in units of days, in form of «$P=\ldots$»), and the semi-major axis of the orbit (in units of au, in form of «$a=\ldots$»). [2 т.]

Table for problem 6

Parameters of orbits and physical characteristics of Sun, Earth, Moon and Jupiter

**) In mean opposition.

Отговори

Решение

Теми и трудност

Трудност: неоценена.

Екзопланети и среди · Небесна механика · Откриване на екзопланети · Закони на Кеплер

Предпоставки: Разширена олимпиадна подготовка

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Оригинал в Архива: betapract.pdf · официални решения: practsol.pdf · допълнителни данни: instructions.pdf · допълнителни данни: data.pdf · допълнителни данни: hrd.pdf