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Условие Отговори Решение

OWAO 2024, Теория — Задача 3. Planet on the Spot

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Всички материали от OWAO 2024

ΩWAO 2024 ◇ Theoretical Round ◇ Problems · 20 т.

Указания

The theoretical round is 5 hours long and accounts for 40 % of the total grade.
The maximum score for each problem is 20 points. The round total is 180 points.
There are Answer Sheets to write down detailed solutions.

On each Answer Sheet, please fill in

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Start each problem on a separate Answer Sheet.
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Draft papers will not be graded.
Some problems are quite difficult. Remember that partial solutions are also scored.
A list of constants is given on page 4.

Good luck!

Условие

There is a large round starspot with a temperature of 2900 K near the center of the disk of a far away star with a radius of $0.7R_\odot$ and a temperature of 3400 K. The decrease in brightness due to the presence of the spot turns out to be the same as during the central transit of a planet with a radius of $6 \cdot 10^4$ km. The axis of rotation of the star is perpendicular to the line of sight, the rotation period is 20 earth days.

a) Estimate the radius of the spot.

b) Estimate the duration of the central transit of the planet through the spot if the radius of the planet’s orbit is 0.6 au. The planet’s orbit lies in the plane of the star’s equator.

Отговори

Решение

a) First, we express how the amount of radiation from the star changes in the presence of a spot and at the time of the planet’s transit.

If there is a spot of the radius $R_s$ on the disk of the star, instead of some area of the surface of the star with a temperature of $\Theta$ , we see a spot with a temperature of $\Theta_s$ . Using the Stefan–Boltzmann law we get

\Delta E_s = \pi R_s^2\sigma\Theta^4 - \pi R_s^2\sigma\Theta_s^4 = \pi R_s^2\sigma\left(\Theta^4 - \Theta_s^4\right).

The planet covers a part of the disk of a star with a radius of $R_p$ , so $\Delta E_p = \pi R_p^2\sigma\Theta^4$ .

The changes in visible brightness are the same, therefore the amounts of energy lost in both events are the same:

\pi R_s^2\sigma\left(\Theta^4 - \Theta_s^4\right) = \pi R_p^2\sigma\Theta^4,

\Longrightarrow\quad R_s = R_p \cdot \frac{\Theta^2}{\sqrt{\Theta^4 - \Theta_s^4}} = 6 \cdot 10^4\ \mathrm{km} \times \frac{3400^2}{\sqrt{3400^4 - 2900^4}} = 8.7 \cdot 10^4\ \mathrm{km}.

Such large spots, although irregularly shaped, are formed on BY Draconis type stars.

b) Usually, the direction of the giant planet’s orbital motion coincides with the direction of the star’s rotation. Let us estimate the relative velocity of the spot and the planet.

The velocity of the spot is determined by the radius and rotation period of the star.

V_s = \frac{2\pi R}{T} = \frac{2\pi \times (0.7 \times 7 \cdot 10^8\ \mathrm{m})}{20 \times 86400\ \mathrm{s}} = 1.8 \cdot 10^3\ \mathrm{m/s}.

The temperature and the radius determine the M spectral type star. For such stars, we may take the estimation of the star mass at $0.6\,\mathfrak{M}_\odot$ . Then the orbital velocity of the planet is

V_p = \sqrt{\frac{G\mathfrak{M}}{a}} = \sqrt{\frac{0.6G\mathfrak{M}_\odot}{0.6a_\oplus}} = V_\oplus = \frac{2\pi a_\oplus}{T_\oplus} = 3 \cdot 10^1\ \mathrm{km/s}.

The mass estimation could also be obtained based on the mass-luminosity relation $L \propto \mathfrak{M}^4$ for the main sequence stars (but this ratio is rather rough for such low-mass stars). Since the temperature and the radius are known, the star luminosity can be estimated from the Stefan–Boltzmann law:

\begin{aligned} L &= 4\pi R^2\sigma\Theta^4 =\\ &= 4\pi \times (0.7 \times 7 \cdot 10^8\ \mathrm{m})^2 \cdot 5.67 \cdot 10^{-8}\ \mathrm{W/m^2/K^4} \times (3400\ \mathrm{K})^4 = 2.3 \cdot 10^{25}\ \mathrm{W} = 0.06L_\odot\,. \end{aligned}

Taking into account the mass-luminosity relation, we get $\mathfrak{M} = 0.5\mathfrak{M}_\odot\,$ .

The velocities of the planet and the spot differ by more than an order of magnitude, so for an estimate, we can consider the relative velocity equal to the velocity of the planet. From the first to the last contact, the planet has to cover the distance

D = 2R_p + 2R_s = 3 \cdot 10^5\ \mathrm{km}.

This will take a time interval

\Delta t = \frac{D}{V_p} = \frac{3 \cdot 10^5\ \mathrm{km}}{3 \cdot 10^1\ \mathrm{km/s}} = 1 \cdot 10^4\ \mathrm{s} \approx 3^{\mathrm{h}}.

Note that the transit time is very short, so we can ignore the change in the apparent shape of the spot when the star’s disk rotates.

Marking Scheme:

Starspot size estimation — 6 pt.
Planet orbital velocity — 5 pt. Acceptable range for the star’s mass is $[0.3; 0.8]\,\mathfrak{M}_\odot$
Velocity of the spot — 4 pt.
Transit duration — 5 pt.

Note. The problem text did not specify which configurations are considered as the beginning and the end of the transit. For example, no penalty is applied if the participant considers the time interval between inner contacts (from the second to the third):

\begin{aligned} \tilde D &= 2(R_s - R_p) = 5.4 \cdot 10^4\ \mathrm{km},\\ \Delta\tilde t &= \frac{\tilde D}{V_p} = 1.8 \cdot 10^3\ \mathrm{s} = 0.5^{\mathrm{h}}. \end{aligned}

Константи

Universal
Speed of light	$c = 3.00 \cdot 10^8\ \mathrm{m/s}$
Planck constant	$h = 6.63 \cdot 10^{-34}\ \mathrm{J \cdot s}$
Boltzmann constant	$k_B = 1.38 \cdot 10^{-23}\ \mathrm{J/K}$
Gas constant	$\mathfrak{R} = 8.314\ \mathrm{J/(mol \cdot K)}$
Proton mass	$m_p = 1.673 \cdot 10^{-27}\ \mathrm{kg}$
Astronomical
Astronomical unit	$1\ \mathrm{au} = 149.6 \cdot 10^6\ \mathrm{km}$
Parsec	$1\ \mathrm{pc} = 206\,265\ \mathrm{au}$
Hubble constant	$H_0 = 70\ \mathrm{(km/s)/Mpc}$
Earth
Radius	$R_\oplus = 6371\ \mathrm{km}$
Mass	$\mathfrak{M}_\oplus = 5.97 \cdot 10^{24}\ \mathrm{kg}$
Obliquity of ecliptic	$\varepsilon = 23.4^{\circ}$
Surface gravity	$g = 9.81\ \mathrm{m/s^2}$
Orbital period	$T_\oplus = 365.26^{\mathrm{d}}$
Orbital eccentricity	$e_\oplus = 0.0167$
Hydrogen spectrum
Lyman L $\alpha$	$\lambda_{\mathrm{L}\alpha} = 1215.7$ Å
Balmer H $\alpha$	$\lambda_{\mathrm{H}\alpha} = 6562.8$ Å
Jupiter
Radius	$R_J = 6.99 \cdot 10^4\ \mathrm{km}$
Mass	$\mathfrak{M}_J = 1.90 \cdot 10^{27}\ \mathrm{kg}$
Orbital radius	$a_J = 5.20\ \mathrm{au}$
Orbital period	$T_J = 11.86\ \mathrm{yr}$
Sun
Radius	$R_\odot = 6.96 \cdot 10^5\ \mathrm{km}$
Mass	$\mathfrak{M}_\odot = 1.99 \cdot 10^{30}\ \mathrm{kg}$
Absolute magnitude	$M_\odot = 4.74^{\mathrm{m}}$ (bol.)
Effective temperature	$T_\odot = 5.8 \cdot 10^3\ \mathrm{K}$
Luminosity	$L_\odot = 3.828 \cdot 10^{26}\ \mathrm{W}$
Emission constants
Stefan–Boltzmann	$\sigma = 5.67 \cdot 10^{-8}\ \mathrm{(W/m^2)/K^4}$
Wien’s displacement	$b = 2898\ \mathrm{\mu m \cdot K}$
UBV system	Mean wavelengths
U band	$\lambda_U = 364\ \mathrm{nm}$
B band	$\lambda_B = 442\ \mathrm{nm}$
V band	$\lambda_V = 540\ \mathrm{nm}$

Number of Siriuses in the Solar system $\geqslant 1$

Оригинал в Архива: OWAO-2024-prob-T.pdf · официални решения: OWAO-2024-sol-T.pdf