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OWAO 2025, Теория — Задача 6. Glimpse of GLIMPSE

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Всички материали от OWAO 2025

2025 ◇ Theoretical Round ◇ Problems

Внимание: Бележка към темата

All eight whole original tasks and 29 lettered parts, complete official worked solutions, alternatives and marking schemes are retained. Constants are native question page 5 and repeated key page 31, fully transcribed once as a shared reference. No paper/time/task/part total is printed, so no totals are inferred from marking criteria. Source mathematical/subscript/endpoint/equality discrepancies are preserved with attributed local notes. The separately printed cooler-star Criterion (x) remains inside original task 8; no extra problem is created.

Константи

Universal

Gravitational constant	$G=6.67\cdot10^{-11}\ \mathrm{m^3/(kg\cdot s^2)}$
Speed of light	$c=3.00\cdot10^8\ \mathrm{m/s}$
Planck constant	$h=6.63\cdot10^{-34}\ \mathrm{J\cdot s}$
Boltzmann constant	$k_B=1.38\cdot10^{-23}\ \mathrm{J/K}$
Gas constant	$\mathfrak R=8.314\ \mathrm{J/(mol\cdot K)}$
Proton mass	$m_p=1.673\cdot10^{-27}\ \mathrm{kg}$

Astronomical

Astronomical unit	$1\ \mathrm{au}=149.6\cdot10^6\ \mathrm{km}$
Parsec	$1\ \mathrm{pc}=206265\ \mathrm{au}$
Hubble constant	$H_0=70\ \mathrm{(km/s)/Mpc}$

Earth

Radius	$R_\oplus=6371\ \mathrm{km}$
Mass	$\mathfrak M_\oplus=5.97\cdot10^{24}\ \mathrm{kg}$
Obliquity of ecliptic	$\varepsilon=23.4^\circ$
Surface gravity	$g=9.81\ \mathrm{m/s^2}$
Orbital period	$T_\oplus=365.26\ \mathrm{days}$
Orbital eccentricity	$e_\oplus=0.0167$

Hydrogen spectrum

Lyman $\mathrm L\alpha$	$\lambda_{\mathrm L\alpha}=1215.7\ \text{Å}$
Balmer $\mathrm H\alpha$	$\lambda_{\mathrm H\alpha}=6562.8\ \text{Å}$

Jupiter

Radius	$R_J=6.99\cdot10^4\ \mathrm{km}$
Mass	$\mathfrak M_J=1.90\cdot10^{27}\ \mathrm{kg}$
Orbital radius	$a_J=5.20\ \mathrm{au}$
Orbital period	$T_J=11.86\ \mathrm{yr}$

Sun

Radius	$R_\odot=6.96\cdot10^5\ \mathrm{km}$
Mass	$\mathfrak M_\odot=1.99\cdot10^{30}\ \mathrm{kg}$
Absolute magnitude	$M_\odot=4.74^{\mathrm m}\ \text{(bol.)}$
Effective temperature	$T_\odot=5.8\cdot10^3\ \mathrm K$
Luminosity	$L_\odot=3.828\cdot10^{26}\ \mathrm W$
Color index	$(B-V)_\odot=+0.65^{\mathrm m}$

Emission constants

Stefan-Boltzmann	$\sigma=5.67\cdot10^{-8}\ \mathrm{W/(m^2\cdot K^4)}$
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	$\lambda_V=540\ \mathrm{nm}$
K band	$\lambda_K=2190\ \mathrm{nm}$

Условие

The cluster GLIMPSE-C01 is located in the Milky Way. The tip of the red giant branch (TRGB) corresponds to an apparent magnitude of $K=8.7^{\mathrm m}$ . Its absolute magnitude, assuming a metallicity of $[\mathrm{Fe/H}]=-1.5$ , is $M_K=-6.1^{\mathrm m}$ (Ivanov et al., 2005).

a) Assuming the extinction is $a_K=0.45\ \mathrm{mag/kpc}$ , estimate the distance to the cluster.

b) Estimate the total extinction $A_V$ in the $V$ band, given the cluster’s Galactic coordinates: latitude $b=0^\circ$ , longitude $l=31^\circ$ .

NOTE. According to Rieke & Lebofsky (1985), $a_V/a_K=9$ .

c) In reality, the extinction, even in the $K$ band, is uncertain: $a_K=0.45\pm0.08\ \mathrm{mag/kpc}$ . Find the uncertainty in the distance.

d) Is this cluster more likely a globular cluster or an open cluster? Explain.

Отговори

Решение

a) The relationship between the apparent magnitude, absolute magnitude, distance, and interstellar extinction is given by

K=M_K-5+5\lg(r/\mathrm{pc})+a_K\cdot(r/1000\ \mathrm{pc})\quad\Longrightarrow\quad5\lg(r/\mathrm{pc})+0.45\cdot(r/1000\ \mathrm{pc})=19.8^{\mathrm m}.

We solve this equation numerically using the bisection method. Since both terms on the left-hand side are monotonically increasing functions of $r$ , the equation has a single root. Define the function:

f(r)=5\lg(r/\mathrm{pc})+0.45\cdot(r/1000\ \mathrm{pc})-19.8.

We evaluate this function at several test points to bracket the root:

$r/\mathrm{pc}$	[1000	10000]	[3000	5000]	4000
$f(r)$	−4.35	4.70	−1.06	0.94	0.01

Therefore, the distance is approximately 4 kpc.

b) The problem provides the ratio of absorption coefficients in two bands. Given the previously determined distance and the absorption coefficient in the $K$ band, the total absorption in the $V$ band is:

A_V=a_V\cdot r=9a_K\cdot r=9\times0.45\ \mathrm{mag/kpc}\times4\ \mathrm{kpc}=16.2\ \mathrm{mag}.

c) The uncertainty in the extinction value is significant. Although this uncertainty is symmetric in magnitude space, it translates into an asymmetric range of possible distances. The most straightforward method to determine this range is to calculate the distance using the upper and lower bounds of the extinction coefficient:

5\lg(r_{\min}/\mathrm{pc})+0.53\cdot(r_{\min}/1000\ \mathrm{pc})=19.8^{\mathrm m}\quad\Longrightarrow\quad r_{\min}=3.7\cdot10^3\ \mathrm{pc}.

5\lg(r_{\max}/\mathrm{pc})+0.37\cdot(r_{\max}/1000\ \mathrm{pc})=19.8^{\mathrm m}\quad\Longrightarrow\quad r_{\max}=4.3\cdot10^3\ \mathrm{pc}.

Thus, accounting for rounding, the distance estimate is $4.0\pm0.3\ \mathrm{kpc}$ . In reality, a more precise calculation using better initial data would likely yield a slightly larger upper uncertainty, revealing an asymmetric error interval.

d) The specified condition of low metallicity immediately rules out a young open cluster. The high absolute magnitude of the tip of the red giant branch indicates the presence of highly luminous, evolved stars. This stellar population is more typical for a globular cluster or a very old open cluster — the latter being quite rare. The object’s location in the Galactic plane, combined with its longitude pointing toward the central region, is consistent with the known distribution of globular clusters, which can have such coordinates.

In conclusion, the object is most likely a globular cluster (which is consistent with its actual identity), although an old open cluster remains a remote theoretical possibility.

Marking Scheme:

Question (a) Distance to the cluster
1. Equation relating stellar magnitudes, distance, and absorption — 3 pt.
2. Number of possible solutions to the equation — 2 pt.
3. Correct answer — 2 pt.
Question (b) Total extinction in $V$ band
1. Relation for absorption in two bands — 2 pt.
2. Correct answer — 2 pt.
Question (c) Uncertainty in the distance
1. Relation for the uncertainty in distance, or stating the method of calculating the boundary values — 2 pt.
2. Correct answers — 1+1 pt.
Question (d) Cluster type
1. Correct reasoning concerning the cluster’s classification — 4 pt.
2. Correct binary answer — 1 pt.

Теми и трудност

Оценена трудност: 3/5.

Светлина и астрономически измервания · Галактики и космология · Междузвездно поглъщане · Звездни величини · Млечен път

Предпоставки: Гимназиална подготовка, Разширена олимпиадна подготовка

Оригинал в Архива: OWAO_2025_Theoretical_Problems.pdf · официални решения: OWAO_2025_Theoretical_Solutions.pdf