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Условие Отговори Решение

OWAO 2024, Теория — Задача 7. Yet Another Pinwheel

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ΩWAO 2024 ◇ Theoretical Round ◇ Problems · 20 т.

Указания

The theoretical round is 5 hours long and accounts for 40 % of the total grade.
The maximum score for each problem is 20 points. The round total is 180 points.
There are Answer Sheets to write down detailed solutions.

On each Answer Sheet, please fill in

Student code
Problem number
Page number and total number of pages in the submitted work

Start each problem on a separate Answer Sheet.
Please write only on the printed side of the sheet. Do not write on the reverse side.
Your work will be scanned. Please use a pen so that the writing is clearly visible on the scan. Duplicate pencil lines with a pen.
Please remember that the jury may not understand your language. It is expected that solutions are written in plain English. Use mathematical expressions, numbers and drawings.
If you wrote something that you do not want to be marked, just cross it out like this.
Draft papers will not be graded.
Some problems are quite difficult. Remember that partial solutions are also scored.
A list of constants is given on page 4.

Good luck!

Условие

The radial surface-brightness profile of disc galaxies is overall well described by an exponential function:

I(R) = I_0 \exp\left(-\frac{R}{h}\right),

where $R$ is the distance from the center of the galaxy, $h$ is the radial scale length and $I_0$ is the central surface brightness measured in $L_\odot/\mathrm{pc}^2$ .

In the spectrum of the observed face-on disc galaxy (the axis of the galaxy coincides with the line of sight), the center of the $\mathrm{H}_\alpha$ line is observed at a wavelength of 6670 Å. The apparent magnitude of the galaxy is $14^{\mathrm{m}}$ and $h$ is assumed to be 3 kpc.

a) Estimate the effective radius $R_e$ of the galaxy. $R_e$ is the radius within which 50 % of the total light of a galaxy is emitted.

b) Estimate the central surface brightness $I_0$ .

c) Estimate the baryonic mass of the galaxy.

Отговори

Решение

a) First, we derive the formula for the total luminosity of the galaxy. It is necessary to integrate the surface brightness on the radius throughout the visible disk of the galaxy. When transforming the area integration to radius integration, a multiplier of $2\pi R$ appears: due to circular symmetry, the apparent disk of the galaxy can be divided into rings with a width of $dR$ and an area of $2\pi R\,dR$ with constant surface brightness, and the luminosities from such rings can be summed:

L = \int\limits_S I(R)\,dR = \int\limits_0^{+\infty} I(R) \cdot 2\pi R\,dR = 2\pi I_0 \int\limits_0^{+\infty} Re^{-\frac{R}{h}}\,dR\,;

\int Re^{-\frac{R}{h}}\,dR \overset{\tau \equiv \frac{R}{h}}{=\!=\!=} h^2 \int \tau e^{-\tau}\,d\tau = h^2\left(-\tau e^{-\tau} + \int e^{-\tau}\,d\tau\right) = -h^2(\tau + 1)e^{-\tau} + \mathrm{const}

\Longrightarrow\quad L = \left.-2\pi I_0 \cdot h^2\left(\frac{R}{h} + 1\right)e^{-\frac{R}{h}}\right|_{h=0}^{+\infty} = 2\pi I_0 h^2.

Next, we should find the value of the radius $R_e$ within which the half of the total luminosity, or $\pi I_0 h^2$ , is emitted:

\pi I_0 h^2 = \int\limits_0^{R_e} I(R) \cdot 2\pi R\,dR = \left.-2\pi I_0 h^2\left(\frac{R}{h} + 1\right)e^{-\frac{R}{h}}\right|_{h=0}^{R_e}

\Longrightarrow\quad \left(\frac{R_e}{h} + 1\right)e^{-\frac{R_e}{h}} \equiv (\tau_e + 1)\,e^{-\tau_e} = \frac{1}{2}.

For physical reasons (due to non-decreasing luminosity with increasing radius), the equation

f(\tau) = \frac{1}{2} - (\tau + 1)\,e^{-\tau} = 0

should have only one root $\tau = \tau_e$ . Beyond a few radial scale lengths, the surface brightness becomes extremely low, so we can choose a smaller interval for searching the root.

We look for the root on the interval $[0 : 10]$ using the bisection method:

$\tau$	0	10	5	2	1	1.5	1.7
$f(\tau)$	$-0.50$	0.50	0.46	0.09	$-0.23$	$-0.06$	0.007

Therefore, $R_e \approx 1.7h \approx 5$ kpc.

b) Let us determine the luminosity of the galaxy. To do this, we should estimate the distance $D$ to the galaxy. Knowing the value of the displacement of the center of the spectral line, we can estimate the galaxy redshift and apply Hubble’s law:

\begin{aligned} z &= \frac{v}{c} = \frac{\lambda_{\mathrm{obs}} - \lambda_0}{\lambda_0} = \frac{6670\ \text{Å} - 6563\ \text{Å}}{6563\ \text{Å}} = 0.016;\\ V &= cz = 3.0 \cdot 10^5\ \mathrm{km/s} \times 0.016 = 4.9 \cdot 10^3\ \mathrm{km/s};\\ D &= \frac{V}{H_0} = \frac{4.9 \cdot 10^3\ \mathrm{km/s}}{70\ \mathrm{(km/s)/Mpc}} = 70\ \mathrm{Mpc}. \end{aligned}

The redshift is quite small, and its effect on luminosity is negligible. So we can calculate the absolute magnitude of the galaxy to retrieve its luminosity and central surface brightness:

M = m + 5 - 5\lg r = 14 + 5 - 5\lg(7 \cdot 10^7) = -20.2^{\mathrm{m}};

L = 10^{-0.4(M - M_\odot)}L_\odot = 9.6 \cdot 10^9\,L_\odot

\Longrightarrow\quad I_0 = \frac{L}{2\pi h^2} = \frac{9.6 \cdot 10^9 L_\odot}{2\pi \times (3 \cdot 10^3\ \mathrm{pc})^2} = 1.7 \cdot 10^2\,L_\odot/\mathrm{pc}^2.

c) It is necessary to estimate the baryonic mass from the known galaxy luminosity. The galaxy is spiral and fainter than the Milky Way ( $M_{\mathrm{MW}} \approx -21.3^{\mathrm{m}}$ ). We don’t know much about this galaxy, but we can assume that the parameters of the galaxy are proportional to the parameters of the Milky Way.

The luminosity of the galaxy is $L = 10^{0.4(-21.3+20.2)}L_{\mathrm{MW}} \approx 0.4L_{\mathrm{MW}}$ .

Assuming the mass-to-light ratio of the galaxy and the Milky Way are the same, we obtain the total mass of the galaxy $\sim 0.4\mathfrak{M}_{\mathrm{MW}} \sim 10^{12}\mathfrak{M}_\odot \times 0.4 = 4 \cdot 10^{11}\mathfrak{M}_\odot$ . Also we recall that the baryonic mass in the Milky Way is about 10 % of the total mass. Then the baryonic mass of the galaxy is $4 \cdot 10^{10}M_\odot$ .

Another way is to use $L/\mathfrak{M} \sim L_\odot/\mathfrak{M}_\odot$ . Thus, $\mathfrak{M} \sim 10^{10}\,\mathfrak{M}_\odot$ (rough but effective).

Marking Scheme:

Question (a)
1. Expression of the total luminosity using an integral; $L = 2\pi I_0 h^2$ — 2 pt. + 2 pt.
2. Expression for $R_e$ ; simplified equation for $R_e$ or $\tau_e$ — 1 pt. + 2 pt.
3. Estimation of the root of $f(\tau) = 0$ — 3 pt. If the root of $I(R) = I_0/2$ is estimated instead of $R_e$ , the total score for Question (a) cannot exceed 1 pt.
Question (b) 4. Estimation of the distance to the galaxy — 2 pt. 5. Estimation of the luminosity — 2 pt. 6. Estimation of the central surface brightness — 3 pt.

Question (c) — 3 pt.

If the presence of dark matter is not taken into account or an incorrect mass-to-light ratio is used, no more than 2 pt. can be awarded for Question (c).

Константи

Universal
Speed of light	$c = 3.00 \cdot 10^8\ \mathrm{m/s}$
Planck constant	$h = 6.63 \cdot 10^{-34}\ \mathrm{J \cdot s}$
Boltzmann constant	$k_B = 1.38 \cdot 10^{-23}\ \mathrm{J/K}$
Gas constant	$\mathfrak{R} = 8.314\ \mathrm{J/(mol \cdot K)}$
Proton mass	$m_p = 1.673 \cdot 10^{-27}\ \mathrm{kg}$
Astronomical
Astronomical unit	$1\ \mathrm{au} = 149.6 \cdot 10^6\ \mathrm{km}$
Parsec	$1\ \mathrm{pc} = 206\,265\ \mathrm{au}$
Hubble constant	$H_0 = 70\ \mathrm{(km/s)/Mpc}$
Earth
Radius	$R_\oplus = 6371\ \mathrm{km}$
Mass	$\mathfrak{M}_\oplus = 5.97 \cdot 10^{24}\ \mathrm{kg}$
Obliquity of ecliptic	$\varepsilon = 23.4^{\circ}$
Surface gravity	$g = 9.81\ \mathrm{m/s^2}$
Orbital period	$T_\oplus = 365.26^{\mathrm{d}}$
Orbital eccentricity	$e_\oplus = 0.0167$
Hydrogen spectrum
Lyman L $\alpha$	$\lambda_{\mathrm{L}\alpha} = 1215.7$ Å
Balmer H $\alpha$	$\lambda_{\mathrm{H}\alpha} = 6562.8$ Å
Jupiter
Radius	$R_J = 6.99 \cdot 10^4\ \mathrm{km}$
Mass	$\mathfrak{M}_J = 1.90 \cdot 10^{27}\ \mathrm{kg}$
Orbital radius	$a_J = 5.20\ \mathrm{au}$
Orbital period	$T_J = 11.86\ \mathrm{yr}$
Sun
Radius	$R_\odot = 6.96 \cdot 10^5\ \mathrm{km}$
Mass	$\mathfrak{M}_\odot = 1.99 \cdot 10^{30}\ \mathrm{kg}$
Absolute magnitude	$M_\odot = 4.74^{\mathrm{m}}$ (bol.)
Effective temperature	$T_\odot = 5.8 \cdot 10^3\ \mathrm{K}$
Luminosity	$L_\odot = 3.828 \cdot 10^{26}\ \mathrm{W}$
Emission constants
Stefan–Boltzmann	$\sigma = 5.67 \cdot 10^{-8}\ \mathrm{(W/m^2)/K^4}$
Wien’s displacement	$b = 2898\ \mathrm{\mu m \cdot K}$
UBV system	Mean wavelengths
U band	$\lambda_U = 364\ \mathrm{nm}$
B band	$\lambda_B = 442\ \mathrm{nm}$
V band	$\lambda_V = 540\ \mathrm{nm}$

Number of Siriuses in the Solar system $\geqslant 1$

Оригинал в Архива: OWAO-2024-prob-T.pdf · официални решения: OWAO-2024-sol-T.pdf