Съдържание

OWAO 2022, Практически тур — Задача 2. RR Lyrae

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Всички материали от OWAO 2022

2022 ◇ Practical Round ◇ Problems

Условие

In the field of the globular star cluster M4, observed by the GAIA space telescope, a number of RR Lyrae variables were found. The equatorial coordinates of the center of M4 are $\alpha_0 = 16^{\mathrm{h}}\,23^{\mathrm{m}}\,35.22^{\mathrm{s}},\ \delta_0 = -26^{\circ}\,31'\,32.7''$ . You are given data on the RR Lyrae variables:

the equatorial coordinates $(\alpha, \delta)$ of objects,
the parallax $\varpi$ in milliarcseconds,
the components $(\mu_\alpha\cos\delta, \mu_\delta)$ of their proper motion,
the period of pulsation,
the average magnitudes in $G$ , $G_{BP}$ and $G_{RP}$ bands,
the average radial velocity $\langle V_r\rangle$ ,
the amplitude $X_G$ of pulsation in $G$ band and its error $\Delta X_G$ ,
the absorption $A_G$ in $G$ band.

a) Determine which stars do not belong to the cluster.

b) Find the period-luminosity relation for RR Lyrae variables of the M4 cluster.

c) Estimate the distance to the cluster.

d) Estimate the mass of the cluster.

Source ID	$\alpha$ , $^{\circ}$	$\delta$ , $^{\circ}$	$\varpi$	$\mu_\alpha\cos\delta$	$\mu_\delta$	$P$	$\langle G\rangle$	$\langle G_{BP}\rangle$	$\langle G_{RP}\rangle$	$\langle V_r\rangle$	$X_G$	$\Delta X_G$	$A_G$
Name			mas	mas/yr	mas/yr	days	mag	mag	mag	km/s	mag	mag	mag
60...496	245.7225	$-27.4102$	0.303	$-6.089$	$-2.476$	0.767	14.27	14.81	13.55		0.57	0.02	1.21
60...168	245.8310	$-26.6659$	0.570	$-12.015$	$-19.382$	0.507	13.00	13.45	12.43	72.38	0.89	0.07	0.84
...	...	...	...	...	...	...	...	...	...	...	...	...	...

The full data table can be found in M4_0.csv file.

Отговори

Решение

a) Let us calculate mean values and standard deviations of the kinematic parameters:

Parameter	Mean	St. dev.
$\varpi$ , mas	0.542	0.116
$\mu_\alpha\cos\delta$ , mas/yr	$-11.155$	4.596
$\mu_\delta$ , mas/yr	$-17.715$	4.000
$\langle V_r\rangle$ , km/s	69.35	8.62

We should discard the stars most deviating from the mean by 3 or all 4 parameters:

ID	$\Delta\varpi/\sigma_{\varpi}$	$\Delta(\mu_\alpha\cos\delta)/\sigma_{\mu_\alpha}$	$\Delta\mu_\delta/\sigma_{\mu_\delta}$	$\Delta V_r/\sigma_{V_r}$
6045280135437770496	2.06	1.10	3.81	—
6045651495491644928	2.20	4.32	0.45	0.90
6045438018442520576	3.57	1.03	2.43	3.24

Then we calculate absolute magnitudes taking extinction in the G band into account:

M_G = m_G + 5 + 5\log\varpi - A_G.

Now we can continue the selection process. Since the stars in a globular cluster have approximately the same age, the period of pulsation and the absolute magnitude should also be quite similar:

ID	$\Delta\varpi/\sigma_{\varpi}$	$\Delta(\mu_\alpha\cos\delta)/\sigma_{\mu_\alpha}$	$\Delta\mu_\delta/\sigma_{\mu_\delta}$	$\Delta V_r/\sigma_{V_r}$	$\Delta M_G/\sigma_{M_G}$	$\Delta P/\sigma_P$
6045502305516138368	1.83	0.50	0.83	1.32	2.59	1.09
6045488767778755456	2.48	0.46	1.51	0.19	0.42	0.40
6045466055990618496	0.84	3.28	0.92	-	1.46	3.53

After the second round of selection, there is no parameter with a deviation greater than $3\sigma$ .

b) The parameters of the period-luminosity relation $\lg P$ – $M_G$ are obtained using least squares:

M_G = -1.657\lg P\,[\mathrm{d}] + 0.289.

c) The distance to the cluster is estimated from the mean parallax:

R = \frac{1\ \mathrm{pc}}{\varpi\,['']} = \frac{1\ \mathrm{kpc}\cdot\mathrm{mas}}{0.561\ \mathrm{mas}} = 1.78\ \mathrm{kpc} \approx 1.8\ \mathrm{kpc}.

d) To estimate the mass of a cluster, it is necessary to determine the velocities of the stars relative to the center of the cluster. As a first approximation,

\vec{v}_i = \vec{V}_i - \vec{V}_{\mathrm{GC}},

where

\vec{V}_i \simeq V_{r,i}\cdot\vec{e}_r + \mu_{\alpha,i}\cos\delta_i\cdot R\cdot\vec{e}_\alpha + \mu_{\delta,i}\cdot R\cdot\vec{e}_\delta,

\vec{V}_{\mathrm{GC}} \simeq \left\langle \vec{V}_i \right\rangle.

According to the virial theorem,

\left\langle v_i^2 \right\rangle \sim \frac{GM_{M4}}{r_{\mathrm{GC}}}.

The radius of the cluster may be estimated as

r_{\mathrm{GC}} \simeq R\cdot\max_i\left\{\sqrt{(\alpha_i - \alpha_0)^2\cos\delta_0 + (\delta_i - \delta_0)^2}\right\}.

Finally, the mass of the cluster

M_{M4} \sim \frac{\langle v_i^2\rangle\cdot r_{\mathrm{GC}}}{G} \sim \frac{(2.4\ \mathrm{km/s})^2\times 90\ \mathrm{pc}}{G} \sim 10^5 M_{\odot}.

Marking Scheme:

Star selection — 6 pt:
- using 4 parameters with 2 iterations — full points.
- only parallax is used — 2 pt.
- 2 iteration with parallax or parallax and some parameters — 4 pt.
Coefficient — 6 pt. In particular, correct coefficient with incorrect sampling — 3 pt.
Distance to the cluster — 2 pt.
Mass of the cluster (any appropriate method) — 6 pt. In particular,
- Speed of stars relative to the center of the cluster is ignored — 3 pt.
- Inadequate final answer — $\leqslant 1$ pt.

Оригинал в Архива: OWAO_2022_Practical_Problems.pdf · официални решения: OWAO_2022_Practical_Solutions.pdf