Съдържание

Earth Moon Sun Jupiter Saturn Physical constants Условие Решение

IOAA 2017, Теория — Задача 4. 21-cm HI galaxy survey

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IOAA 2017

IOAA 2017 — Theoretical Examination · 10 т.

Original Examination Instructions

The theoretical examination lasts for 5 hours and is worth a total of 300 marks.
Dedicated IOAA Summary Answer Sheets are provided for writing your answers. Enter the final answers into the appropriate boxes in the corresponding Summary Answer Sheet. On each Summary Answer Sheet, please fill in
- Student Code (Country Code and 1 digit)
There are Answer Sheets for carrying out detailed work/rough work. On each Answer Sheet, please fill in
- Student Code (Country Code and 1 digit)
- 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 evaluated, cross it out.
Use as many mathematical expressions as you think may help the evaluator to better understand your solutions. The evaluator may not understand your language. If it is necessary to explain something in words, please use short phrases (if possible in English).
You are not allowed to leave your work desk without permission. If you need any assistance (malfunctioning calculator, need to visit a restroom, need more Answer Sheets, etc.), please draw the attention of the proctor using the Help card.
The beginning and end of the examination will be indicated by a long sound signal. Additionally, there will be a buzzer sound, fifteen minutes before the end of the examination (before the final sound signal).
At the end of the examination you must stop writing immediately. Sort and put your sheets in separate stacks, a) Stack 1: Summary Answer Sheets, Answer Sheets of part 1
b) Stack 2: Summary Answer Sheets, Answer Sheets of part 2
c) Stack 3: Summary Answer Sheets, Answer Sheets of part 3
d) Stack 4: question papers and paper sheets 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 examination room.
A list of constants and a table of the mark distribution for this exam are given on the next two pages.

Original List of Constants

Earth

Quantity	Value
Mass ( $M_\oplus$ )	$5.98\times10^{24}$ kg
Radius ( $R_\oplus$ )	$6.38\times10^6$ m
Acceleration of gravity ( $g$ )	$9.8 \mathrm{m\cdot s^{-1}}$
Obliquity of Ecliptic	$23^\circ27'$
Length of Tropical Year	365.2422 mean solar days
Length of Sidereal Year	365.2564 mean solar days
Albedo	0.39

Moon

Quantity	Value
Mass (M with subscript ☾)	$7.35\times10^{22}$ kg
Radius (R with subscript ☾)	$1.74\times10^6$ m
Mean distance from Earth	$3.84\times10^8$ m
Orbital inclination with the Ecliptic	$5.14^\circ$
Albedo	0.14
Apparent magnitude (mean full moon)	−12.74

Sun

Quantity	Value
Mass ( $M_\odot$ )	$1.99\times10^{30}$ kg
Radius ( $R_\odot$ )	$6.96\times10^8$ m
Luminosity ( $L_\odot$ )	$3.83\times10^{26}$ W
Absolute Magnitude ( $\mathcal M_\odot$ )	4.80 mag
Angular diameter	0.5 degrees
Rotational velocity in the Galaxy	$220 \mathrm{km\,s^{-1}}$
Distance from Galactic centre	8.5 kpc

Jupiter

Quantity	Value
Mass	$1.89\times10^{27}$ kg
Orbital semi-major axis	5.20 au
Orbital period	11.86 year

Saturn

Quantity	Value
Mass	$5.68\times10^{26}$ kg
Orbital semi-major axis	9.58 au
Orbital period	29.45 year

Physical constants

Quantity	Value
1 au	$1.50\times10^{11}$ m
1 pc	206,265 au
Gravitational constant ( $G$ )	$6.67\times10^{-11} \mathrm{N\cdot m^2\cdot kg^{-2}}$
Planck constant ( $h$ )	$6.62\times10^{-34} \mathrm{J\cdot s}$
Boltzmann constant ( $k_B$ )	$1.38\times10^{-23} \mathrm{J\cdot K^{-1}}$
Stefan-Boltzmann constant ( $\sigma$ )	$5.67\times10^{-8} \mathrm{W\cdot m^{-2}\cdot K^{-4}}$
Hubble constant ( $H_0$ )	$67.8 \mathrm{km\,s^{-1}\,Mpc^{-1}}$
Speed of light in vacuum ( $c$ )	$299,792,458 \mathrm{m\cdot s^{-1}}$
Proton mass	$938.27 \mathrm{MeV\cdot c^{-2}}$
Deuterium mass	$1875.60 \mathrm{MeV\cdot c^{-2}}$
Neutron mass	$939.56 \mathrm{MeV\cdot c^{-2}}$
Helium-3 mass	$2808.30 \mathrm{MeV\cdot c^{-2}}$
Helium-4 mass	$3727.40 \mathrm{MeV\cdot c^{-2}}$

Original Mark Distribution

Problem number	Marks
T1	10
T2	10
T3	10
T4	10
T5	10
T6	15
T7	20
T8	20
T9	20
T10	25
T11	50
T12	40
T13	60
Total	300

Условие

A radio telescope is equipped with a receiver which can observe in a frequency range from 1.32 to 1.52 GHz. Its detection limit is 0.5 mJy per beam for a 1-minute integration time. In a galaxy survey, the luminosity of the HI spectral line of a typical target galaxy is $10^{28}$ W with a linewidth of 1 MHz. For a large beam, the HI emitting region from a far-away galaxy can be approximated as a point source. The HI spin-flip spectral line has a rest-frame frequency of 1.42 GHz.

What is the highest redshift, $z$ , of a typical HI galaxy that can be detected by a survey carried out with this radio telescope, using 1-minute integration time? You may assume in your calculation that the redshift is small and the non-relativistic approximation can be used. Note that $1 \mathrm{Jy}=10^{-26} \mathrm{W\,m^{-2}\,Hz^{-1}}$ .

Решение

Solution

Case 1: If we only consider the frequency range where HI spectral line can be detected by the receiver of this radio telescope,

z\approx\frac{f_0-f}{f_0}.

[1 Mark]

The lowest frequency, 1.32 GHz, can be used to observe the highest redshifted HI, at

z=\frac{1.42-1.32}{1.42}=0.0704.

[1 Mark]

Alternative solution: Above is the cosmological redshift formula in frequency conventionally used in radio astronomy. However, if the cosmological redshift in optical astronomy is used, student may use

z\approx\frac{f_0-f}f=\frac{1.42-1.32}{1.32}=0.0758

and the student should be awarded full mark for this part.

Case 2: Use the information given to calculate the redshift limit set by the flux limit due to furthest distance that typical HI galaxy can be detected by the telescope

for low $z$ , non-relativistic redshift

d=\frac{v_r}{H_0}=\frac{cz}{H_0}.

[1 Mark]

The HI spectral-line flux density for a galaxy with HI luminosity $L$ and line-width $\Delta f$ at distance $d$ is

\frac{L}{4\pi d^2}\times\frac1{\Delta f}\ ( \mathrm{W\,m^{-2}\,Hz^{-1}}).

[2 Mark]

Setting this to the detection limit and solve correctly, and taking special care of converting various units

S=\frac{LH_0^2}{4\pi\Delta fc^2z^2}\ge S_{\mathrm{lim}}=0.5\times10^{-26}.

[1 Mark]

Therefore,

z\le\frac{H_0}c\sqrt{\frac{L}{4\pi\Delta fS_{\mathrm{lim}}}},

[1 Mark]

Correctly substitute numerical values,

z\le\frac{67.8\ ( \mathrm{km^{-1}\,Mpc^{-1}})}{2.99\times10^5\ ( \mathrm{km\,s^{-1}})\times3.08\times10^{22}\ ( \mathrm{m\,Mpc^{-1}})}\sqrt{\frac{10^{28}\ \mathrm W}{4\pi\times10^6\ ( \mathrm{Hz})\times0.5\times10^{-29}\ ( \mathrm{W\,m^{-2}\,Hz^{-1}})}}.

[1 Mark]

z\le0.0929.

[1 Mark]

Conclusion: For typical HI galaxy, the highest redshift is limited by the receiver’s lowest frequency limit and we get $z_{\max}=0.0704$ (or 0.0758 for alternative solution given above). [1 Marks]

The last part is marked only if student calculate both instrumental factors. If students use more precise cosmological effects, they will be marked accordingly.

Source discrepancies retained: the first flux-limit equality prints $0.5\times10^{-26}$ , whereas the numerical substitution prints $0.5\times10^{-29}$ . The numerical Hubble-constant numerator prints units km⁻¹ Mpc⁻¹.

Бележки за изданието

The original question and key contain several differing expressions. The Galactic-arm question uses a flat curve beyond 5 kpc and its worked key uses 2 kpc. The HI survey key prints differing flux-threshold exponents and an unusual Hubble-constant unit. The Bethlehem key has differing multiplication and remainder values. The Galactic Outflow key heading says T2 while its full prompt matches T9. GOTO assigns a different pair of marks to parts a) and d) in the question and key. The Shipwreck key prints differing sidereal-time values in its main and alternative methods. Visible missing-font glyph squares and the Exomoon key’s grading dependency are preserved. The original constants table prints the acceleration unit as m·s^-1. These printed forms remain literal.

Оригинал в Архива: Theory.pdf · официални решения: Theory_Solution.pdf