Съдържание

IOAA 2015, Теория, кратки задачи, английски — Задача 1

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IOAA 2015

SHORT PROBLEMS

Условие

Several exoplanets have been observed in the Gliese 876 system $(\mathrm{M_G} = 0.33 \pm 0.03\mathrm{M_\odot})$ as given in the following table,

Gliese System	Mass	Semi Major Axis (AU)
Gliese 876 b	$2.276\ \mathrm{M_J}$	0.2083
Gliese 876 c	$0.714\ \mathrm{M_J}$	0.1296
Gliese 876 d	$6.8\ \mathrm{M_\oplus}$	0.0208
Gliese 876 e	$15\ \mathrm{M_\oplus}$	0.334

where $\mathrm{M_\odot}$ is mass of Sun, $\mathrm{M_J}$ is mass of Jupiter ( $M_J = 1.89813\times10^{27}\ kg$ ), and $\mathrm{M_\oplus}$ is mass of Earth. Assume that all these planets revolve around Gliese 876 in the same direction. Two planets are said to be in resonant orbits if the synodic period of one planet with respect to the other planet is an integer multiple of the orbital period of the second planet.

Find if any of the exoplanets of Gliese 876 system may have resonant orbits.

Решение

Answer and Marking Scheme:

1 Determine planet periods

[50]

Synodic period of planet 1 and 2 (assume $P_1 < P_2$ ) is $\frac{1}{P_S} = \frac{1}{P_1} - \frac{1}{P_2} \;\rightarrow\; P_S = \frac{P_1P_2}{P_2 - P_1}$

Resonant orbit happens if

P_S = mP_1 \;\rightarrow\; \frac{P_1P_2}{P_2 - P_1} = mP_1 \;\rightarrow\; m = \frac{P_2}{P_2 - P_1} > 1,\ \mathrm{integer}

Using the data of the Gliesean system we have

Gliese System	Mass (kg)	Semi Major Axis (m)	Period (Earth, days) $P = 2\pi\sqrt{\frac{a^3}{GM_G}}$
Gliese 876 star	$6.64359\times10^{29}$
Gliese 876 b	$4.31938\times10^{27}$	$3.1161\times10^{10}$	60.07568
Gliese 876 c	$1.35564\times10^{27}$	$1.9388\times10^{10}$	29.48304
Gliese 876 d	$4.079\times10^{25}$	$3.1116\times10^{9}$	1.8956
Gliese 876 e	$8.7194\times10^{25}$	$5.0011\times10^{10}$	122.1432

2 Calculate resonant orbit for each planet

From the results above we see that

Gliese 876 c and Gliese 876 b have resonant orbit

m = \frac{P_b}{P_b - P_c} = 1.96 \approx 2

[25]

Gliese 876 b and Gliese 876 e have resonant orbit

m = \frac{P_e}{P_e - P_b} = 1.9679 \approx 2

[25]

13. Earlier marking-scheme variant

Two protons collide to produce deuteron, positron, and neutrino in the center of a star. Write down the reaction equation and calculate the minimum temperature needed for this reaction. Assume that the neutrino mass is negligible.

Answer

1 $p + p + \Delta E \rightarrow D + e^{+} + \nu_e$ [20]

2 $\Delta E = (m_e + m_D - 2m_p)c^2 = (m_e + (m_p + m_n) - 2m_p)c^2 = (m_e + m_n - m_p)c^2$ [20]

3 $\Delta E = (9.109\times10^{-31} + 1.675\times10^{-27} - 1.673\times10^{-27})\mathrm{kg}\,(2.99792\times10^{8}\mathrm{ms^{-1}})^2 = 2.6198\times10^{-13}\ J$ [20]

4 $\Delta E \sim k_BT \rightarrow T \sim \frac{\Delta E}{k_B}$ [20]

5 $T = \frac{2.6198\times10^{-13}\ J}{8.315\frac{J}{mol\ K}} = \frac{2.6198\times10^{-13}mol\ K}{8.315}\times6.02\times10^{23}\frac{1}{mol} = 1.894\times10^{10}\ K$ [20]

Оригинал в Архива: Short_problem_Final_EN.pdf · официални решения: Short Problem Solution-Marking Scheme 30.07.pdf · допълнителни данни: Short_Problem_Solution.docx