Съдържание

IOAA 2018, Теория — Задача 2. Distance

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IOAA 2018

Theoretical Competition · 10 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

An observer measured trigonometric parallaxes of stars in a star cluster. Due to random errors, the measured parallax values are distributed symmetrically around the expected value with standard deviation equal to 0.05 mas (milliarcsec). Assume there are no systematic errors and assume all stars in the said cluster have the same luminosity. It is known that the distance of this cluster from us is R= 5 kpc.

He gave the data table to 4 of his students (A, B, C and D) and they estimated the distance to the cluster in the following ways:

A. Convert each parallax measurement into distance and then find the average distance ( $R_A$ )

B. Take the average of all parallaxes first and then convert the average parallax into distance. ( $R_B$ )

C. Convert each parallax measurement into distance and then take the median distance value. ( $R_C$ )

D. Find the median value of the parallaxes and then convert the median value into distance. ( $R_D$ )

State if the following statements are true or false. In case a given mathematical relation is false, give the correct relation.

(l) If the i $^{\mathrm{th}}$ star gave the smallest value of parallax and the j $^{\mathrm{th}}$ star gave the highest value of parallax, in all likelihood $R_i-R > R-R_j$ [2 т.]

(m) $R_A = R$ (i.e. there is a high chance that the distance estimated by A fairly matches the true distance) [2 т.]

(n) $R_B = R$ (i.e. there is a high chance that the distance estimated by B fairly matches the true distance) [2 т.]

(o) $R_C < R$ (i.e. there is a high chance that the distance estimated by C will be systematically lower than the true distance) [2 т.]

(p) $R_D = R$ (i.e. there is a high chance that the distance estimated by D fairly matches the true distance) [2 т.]

Отговори

Решение

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