IOAA 2018, Теория — Задача 9. Radio Galaxy

Theoretical Competition · 25 т.

Указания
  1. The theoretical competition will be 5 hours in duration and is marked out of a total of 300 points.
  2. 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.
  3. 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.
  4. 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).
  5. 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.
  6. 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).
  7. 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.
  8. 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.
  9. A list of constants for this competition is given on the next page.

Point distribution of this exam

Problem numberPoints
T110
T210
T310
T410
T510
T625
T725
T825
T925
T1075
T1175
Total300
Table of constants
Mass (M⊕M_{\oplus})5.98×10245.98 \times 10^{24} kgEarth
Radius (R⊕R_{\oplus})6.38×1066.38 \times 10^{6} m
Acceleration of gravity (gg)9.81 ms−2^{-2}
Obliquity of Ecliptic23°27′
Length of Tropical Year365.2422 mean solar days
Length of Sidereal Year365.2564 mean solar days
Albedo0.39
Mass (MM☾)7.35×10227.35 \times 10^{22} kgMoon
Radius (RR☾)1.74×1061.74 \times 10^{6} m
Mean Earth-Moon distance3.84×1083.84 \times 10^{8} m
Orbital inclination with the Ecliptic5.14°
Albedo0.14
Apparent magnitude (mean full moon)−12.74
Mass (M⊙M_{\odot})1.99×10301.99 \times 10^{30} kgSun
Radius (R⊙R_{\odot})6.96×1086.96 \times 10^{8} m
Luminosity (L⊙L_{\odot})3.83×10263.83 \times 10^{26} W
Absolute Magnitude4.80 mag
Surface Temperature5772 K
Angular diameter at Earth30′
Orbital velocity in Galaxy220 kms−1^{-1}
Distance from Galactic center8.5 kpc
1 au1.50×10111.50 \times 10^{11} mPhysical constants
1 pc206265 au
Gravitational constant (GG)6.67×10−116.67 \times 10^{-11} Nm2^{2}kg−2^{-2}
Planck constant (hh)6.62×10−346.62 \times 10^{-34} Js
Boltzmann constant (kBk_{\mathrm{B}})1.38×10−231.38 \times 10^{-23} JK−1^{-1}
Stefan-Boltzmann constant (σ\sigma)5.67×10−85.67 \times 10^{-8} Wm−2^{-2}K−4^{-4}
Hubble constant (H0H_0)67.8 kms−1^{-1}Mpc−1^{-1}
Speed of light in vacuum (cc)299792458 ms−1^{-1}
Magnetic Permeability of free space (μ0\mu_0)4π×10−74\pi \times 10^{-7} Hm−1^{-1}
1 Jansky (Jy)10−2610^{-26} W m−2^{-2}Hz−1^{-1}

Rayleigh-Jeans law is given by Bν=2kBTc2ν2B_\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 wants to use the Five-hundred-meter Aperture Spherical radio Telescope (FAST) in China to observe a radio galaxy at redshift of z=0.06z = 0.06. We assume that the radio source is compact compared to the beam size of the telescope at the observing frequencies, i.e., the source is point-like as seen through the telescope. To detect a point source with FAST, it must be sufficiently strong (bright) relative to the noise level (for single polarization observations), σ, which depends on the bandwidth, Δν\Delta\nu, and the integration time (the radio astronomy equivalent of exposure time), ti_i, as follows:

σ=2kBTsysAetiΔν\sigma = \frac{2k_BT_{sys}}{A_e\sqrt{t_i\Delta\nu}}

where TsysT_{sys} is the system temperature (about 150 K in the frequency range of 0.28 GHz – 0.56 GHz and 25 K in the frequency range of 1.05 GHz – 1.45 GHz), and Ae_e = 4.6x104^4 m2^2 is the effective area of the telescope taking into account the total efficiency of the instrument.

This radio galaxy has an observed continuum flux density of fν=2.5×10−3Jyf_\nu = 2.5 \times 10^{-3}Jy at an observing frequency of 0.4 GHz. The bandwidth Δν\Delta\nu for the continuum observation centered at 0.4 GHz is 2.8×108^8 Hz.

(a) In order to detect the continuum flux density at 0.4 GHz with a signal-to-noise ratio of 30 (a so-called 30σ detection), what is the required integration time, ti_i?

(b) We want to search for the neutral Hydrogen (HI) in the galaxy using 21cm absorption line. The HI 21cm line, with rest frame frequency of 1.4204 GHz. Calculate the observed frequency (υobs_{obs}) of the HI line for this galaxy.

(c) The radio continuum emission from this galaxy can be described by a power law fν∼ναf_\nu \sim \nu^\alpha, with a spectral index of α=−0.2\alpha = -0.2. Calculate the continuum flux density at νobs\nu_{obs} for this galaxy.

(d) The line width of the HI 21cm absorption line is 90 km/s. Calculate the line width in Hz at the observing frequency of νobs\nu_{obs}. According to Figure 1, the HI 21cm line absorbs 4% of the continuum flux density (on average) over the line width of 90 kms−1^{-1}. In order to detect the absorption line at ≥3σ in three consecutive 30 kms−1^{-1} channels, what is the required integration time?

Framed spectrum plot with tick marks on all sides and no axis numbers: a flat curve labelled 'Continuum level' dips into a single deep absorption line; a blue double arrow across the line is labelled '90 km s⁻¹', and a yellow vertical arrow spanning the depth is labelled '4% of the continuum level'.
Figure 1: Spectrum of the HI 21cm absorption relative to the continuum emission in the radio galaxy

Отговори

Решение

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