IOAA 2019, theory — Задача 13. Distance to the Coma galaxy cluster

Theory problem sheet · 40 т.

Условие

The Coma galaxy cluster (Abell 1656) has an angular diameter on the sky of about 100 arcminutes, and contains more than 1000 individual galaxies, most of which are dwarf and giant ellipticals orbiting the common center of mass of the cluster in approximately circular orbits. The table below lists the measured radial velocities of a few individual cluster member galaxies.

No.	$v_{\mathrm{r}}$ (km/s)	No.	$v_{\mathrm{r}}$ (km/s)	No.	$v_{\mathrm{r}}$ (km/s)	No.	$v_{\mathrm{r}}$ (km/s)
1	6001	6	7116	11	7156	16	7111
2	7666	7	7004	12	7522	17	8292
3	6624	8	4476	13	7948	18	5358
4	5952	9	6954	14	4951	19	4957
5	5596	10	8953	15	7797	20	7183

The formula for standard deviation:

$\sigma = \sqrt{\frac{\sum_{j=1}^{n}(x_i - \bar{x})^2}{n-1}}$

a) Derive the distance of the cluster from the mean radial velocity of the galaxies listed in the table. [8 т.]

b) Estimate the physical diameter of the cluster (in Mpc). [4 т.]

c) The virial theorem states that if the galaxy cluster is in dynamic equilibrium, then the mean kinetic energy, $\langle K\rangle$, and the mean gravitational potential energy, $\langle U\rangle$, are related by

$-2\langle K\rangle = \langle U\rangle$

assuming the Coma cluster is spherical.

For simplicity, assume that each galaxy has approximately the same mass, $m$.

Use the virial theorem to prove that, in this case, the cluster mass $M$ (also called as the virial mass) can be expressed as

$M = \frac{5R}{G}\,\sigma_{\mathrm{r}}^2$

where $\sigma_{\mathrm{r}}^2$ is the velocity dispersion of the cluster. [10 т.]

d) Using the data in the table, estimate the virial mass of the Coma cluster in solar masses. [12 т.]

e) The total luminosity of the Coma cluster (in solar luminosity, $L_\odot$) is $L \approx 5\times10^{12}L_\odot$. Calculate the mass-luminosity ratio of the cluster in solar mass per solar luminosity units. [2 т.]

f) Which of the following statements is true (more than one answer is possible)? Mark the letter of your answer with × on the answer sheet.

(A) The mass-luminosity ratio of the Coma cluster is much higher than that of a typical spiral galaxy, like the Milky Way. (B) The mass-luminosity ratio of the Coma cluster is similar to that of a typical spiral galaxy. (C) The mass-luminosity ratio of the Coma cluster is much less than that of a typical spiral galaxy. (D) The Coma cluster contains much more dark matter than a typical spiral galaxy. (E) The Coma cluster contains much less dark matter than a typical spiral galaxy. [4 т.]

Решение

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Оригинал в Архива: TH-problem-final.pdf · официални решения: TH-solution-final.pdf