Съдържание

IOAA 2018, Теория — Задача 7. Sunspot

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IOAA 2018

Theoretical Competition · 25 т.

Указания

The theoretical competition will be 5 hours in duration and is marked out of a total of 300 points.
There are Answer Sheets for carrying out detailed work/rough work. On each Answer Sheet, please fill in
- Student Code
- Question No.
- Page no. and total number of pages.
Start each problem on a separate Answer Sheet. Please write only on the printed side of the sheet. Do not use the reverse side. If you have written something on any sheet which you do not want to be marked, cross it out.
Please remember that the graders may not understand your language. As far as possible, write your solutions only using mathematical expressions and numbers. If it is necessary to explain something in words, please use short phrases (if possible in English).
You are not allowed to leave your working desk without permission. If you need any assistance (malfunctioning calculator, need to visit a restroom, need more Answer Sheets, etc.), please put up your hand to signal the proctor.
The beginning and end of the competition will be indicated by a long sound signal. Additionally, there will be a short sound signal fifteen minutes before the end of the competition (before the final long sound signal).
At the end of the competition you must stop writing immediately. Sort and put your sheets in separate stacks,
- (a) Stack 1: answer sheets of part 1
- (b) Stack 2: answer sheets of part 2
- (c) Stack 3: answer sheets of part 3
- (d) Stack 4: question papers and paper sheets that you do not want to be graded.
Wait at your table until your envelope is collected. Once all envelopes are collected, your student guide will escort you out of the competition room.
A list of constants for this competition is given on the next page.

Point distribution of this exam

Problem number	Points
T1	10
T2	10
T3	10
T4	10
T5	10
T6	25
T7	25
T8	25
T9	25
T10	75
T11	75
Total	300

Table of constants

Mass ( $M_{\oplus}$ )	$5.98 \times 10^{24}$ kg	Earth
Radius ( $R_{\oplus}$ )	$6.38 \times 10^{6}$ m
Acceleration of gravity ( $g$ )	9.81 ms $^{-2}$
Obliquity of Ecliptic	23°27′
Length of Tropical Year	365.2422 mean solar days
Length of Sidereal Year	365.2564 mean solar days
Albedo	0.39
Mass ( $M$ ☾)	$7.35 \times 10^{22}$ kg	Moon
Radius ( $R$ ☾)	$1.74 \times 10^{6}$ m
Mean Earth-Moon distance	$3.84 \times 10^{8}$ m
Orbital inclination with the Ecliptic	5.14°
Albedo	0.14
Apparent magnitude (mean full moon)	−12.74
Mass ( $M_{\odot}$ )	$1.99 \times 10^{30}$ kg	Sun
Radius ( $R_{\odot}$ )	$6.96 \times 10^{8}$ m
Luminosity ( $L_{\odot}$ )	$3.83 \times 10^{26}$ W
Absolute Magnitude	4.80 mag
Surface Temperature	5772 K
Angular diameter at Earth	30′
Orbital velocity in Galaxy	220 kms $^{-1}$
Distance from Galactic center	8.5 kpc
1 au	$1.50 \times 10^{11}$ m	Physical constants
1 pc	206265 au
Gravitational constant ( $G$ )	$6.67 \times 10^{-11}$ Nm $^{2}$ kg $^{-2}$
Planck constant ( $h$ )	$6.62 \times 10^{-34}$ Js
Boltzmann constant ( $k_{\mathrm{B}}$ )	$1.38 \times 10^{-23}$ JK $^{-1}$
Stefan-Boltzmann constant ( $\sigma$ )	$5.67 \times 10^{-8}$ Wm $^{-2}$ K $^{-4}$
Hubble constant ( $H_0$ )	67.8 kms $^{-1}$ Mpc $^{-1}$
Speed of light in vacuum ( $c$ )	299792458 ms $^{-1}$
Magnetic Permeability of free space ( $\mu_0$ )	$4\pi \times 10^{-7}$ Hm $^{-1}$
1 Jansky (Jy)	$10^{-26}$ W m $^{-2}$ Hz $^{-1}$

Rayleigh-Jeans law is given by $B_\nu = \frac{2k_BT}{c^2}\nu^2$ , which is the power emitted per unit emitting area, per steradian, per unit frequency.

Условие

Magnetic fields are important in the physics of stars and sunspots. As an approximation, we can model the photosphere of the Sun consisting of a plasma, which can be simply treated as a single component ideal gas, and a magnetic field (B), which has an associated magnetic pressure $p_B = \frac{B^2}{2\mu_0}$ . It behaves like any other physical pressure except that it is carried by the magnetic field rather than by the kinetic energy of particles.

Assume that the number density of particles in the photosphere is constant everywhere, but the magnetic field inside the sunspot (B $_{\mathrm{in}}$ =0.1T) is much stronger than outside (B $_{\mathrm{out}}$ =5×10 $^{-3}$ T). From the blackbody spectrum, the temperature inside the sunspot is T $_{\mathrm{in}}$ ~4000K, while the temperature outside is T $_{\mathrm{out}}$ ~6000K (which is why the sunspot looks darker). For the sunspot to be stable, the inside must be in equilibrium with the outside.

(a) Estimate the number density of plasma particles in the solar photosphere.

(b) Compare your answer with an estimate of the number density of particles in the atmosphere at the surface of the Earth.

Отговори

Решение

Оригинал в Архива: Theory Problems.pdf · официални решения: Theory Solutions.pdf