Съдържание

IOAA 2018, Теория — Задача 8. A Possible Dark Matter Deficient Galaxy

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

Earlier this year, a team of astronomers reported their discovery of a galaxy with much less dark matter than the galaxy evolution model predicted (van Dokkum et al. 2018, Nature). This galaxy, named NGC 1052-DF2, is located close to the elliptical galaxy NGC 1052 (D=20Mpc from the Sun) in the sky. The shape of NGC 1052-DF2 resembles an ellipse with semi major axis (a) of 22.6" and $\frac{b}{a} = 0.85$ . Half of the total light from the galaxy comes from within this ellipse and the mean surface brightness within the ellipse is about 24.7 mag arcsec $^{-2}$ .

(a) Calculate the total apparent magnitude of this galaxy.

(b) The team suggested the galaxy is a companion of NGC 1052. Determine the total mass of stars in NGC 1052-DF2, assuming it has a mass to light ratio $\left(\frac{M/M_{\odot}}{L/L_{\odot}}\right)$ of 2.0.

(c) The team identified 10 globular clusters in NGC 1052-DF2 with a mean galactocentric distance of 78.4". They also measured the velocity dispersion of these clusters to be not more than 8.4 km/s. Estimate the dynamical mass of this galaxy. For simplicity, assume the mass distribution in the galaxies is uniform and is spherically symmetric.

(d) This discovery was challenged by other groups (Kroupa et al., Nature, 2018, Truijlo et al., MNRAS, 2018), who claimed that NGC 1052-DF2 is not a satellite of NGC 1052, and it is located at a much smaller distance to us. Show why a smaller distance would weaken the assertion of the dark matter deficiency in NGC 1052-DF2.

Отговори

Решение

(1) Total area with in the ellipse (actually, half-light ellipse) of NGC 1052-DF2 is:

$A_{DF2} = \pi ab = 0.85\pi \times 22.6^{''2} = 1363.9 arcsec^2$ (2 points)

Magnitude of the part of the galaxy within the ellipse:

$m_{ell} = SB - 2.5 \times log(A_{DF2}) = 24.7 - 2.5 \times log(1363.9) = 16.9$ (2 points)

Total magnitude of the whole galaxy:

$m_{gal} = m_{ell} - 2.5 \times log(2) = 16.1$ (2 points)

(2) Absolute mag of the galaxy:

$M_{0,gal} = m_{gal} + 5 - 5 \times log(D) = -15.4$ (2 points)

Convert to solar luminosity:

$\frac{L_{gal}}{L_{\odot}} = 10^{-0.4(M_{0,gal}-M_{0,\odot})} = 1.2 \times 10^8$ (2 points)

Thus, the total stellar mass should be:

$\frac{M_{gal}}{M_{\odot}} = \frac{L_{gal}}{L_{\odot}} \times \frac{M}{L} = 2.4 \times 10^8$ (2 points)

(3) Mean galactocentric distance of globular clusters:

$r_{gc} = \frac{\theta D}{206265} = 7.6$ kpc (2 points)

Using viral theorem:

$\langle 2K \rangle + \langle U \rangle = 0$ ,

$\langle K \rangle = \frac{1}{2}M\langle v^2 \rangle = \frac{1}{2}M\sigma^2$ ,

$\langle U \rangle = \frac{3}{5}\frac{GM^2}{r}$ (4 points)

Thus the maximal dynamical mass of this system should be:

$M_{dyn} = \frac{5r_{gc}\sigma^2}{3G} = 2.1 \times 10^8M_{\odot}$ , even less than the total stellar mass within this radius. (3 points)

(4) From Question 2 and 3, we know:

$M_{star} \propto L_{gal} \propto F_{gal} \times D^2$

However, $M_{dym} \propto D$ (2 points)

Thus

$M_{dym}/M_{star} \propto 1/D$ (2 points)

If the distance measured is smaller by a certain factor, the stellar mass to dynamical mass ratio would increase by the same factor, thus the dark matter in NGC 1052-DF2 would not be as deficient as Dragonfly team claimed.

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