Съдържание

Условие Таблица 1. Данные о Гиадах / Table 1. Hyades data Таблица 2. Данные о NGC 2682 / Table 2. NGC 2682 data Диаграмма Герцшпрунга-Рассела / Hertzsprung-Russell diagram Решение

IAO 2012, Практически тур, Група β — Задача 8. Clusters

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IAO 2012

IAO2012 — Practical round (β) · 10 т.

Practical round. Problems to solve

XVII International Astronomy Olympiad. Gwangju, Korea,16–24 X 2012. Group β.

Note. If you find somewhere in the problems an impossible situation, write in English «impossible situation».

Бележки от източника

The official Fireball solution explicitly acknowledges an incorrectly plotted observing point II. It gives A/B latitudes36.47°/36.10° in7.2, then36.49°/36.11° in7.4 and its final table. These printed values are retained.

Missing plotting form

The original blank Cluster graph papers(A) and(B) referred to in8.1/8.2 are absent from the complete same-year source inventory; the available paper.png is the Fireball geographic grid.

Printed solution details

The official Clusters solution says NGC2686 in its8.4 explanation and NGC2682 elsewhere; both are retained. Its Hyades magnitude formula and stated answer0.75 are preserved without recalculation.

Shared source pack

The six official practical solution images and grading sheet cover both group editions; this edition includes only its matching problem8. Four theory-labelled reference sheets were inspected. The orbital-properties table supports Moon, and the Hertzsprung-Russell diagram is explicitly mentioned by the Clusters solution. The constants and individual-star tables supply reference context, not extra practical tasks. The geographic grid belongs to Fireball.

Условие

Using the moving cluster method, the Hyades cluster is known to be 45 pc away. This open cluster is important as a standard candle, because we can use it to determine the distances of other clusters. However, the interstellar medium absorbs light making a star appear fainter and redder, which is called the interstellar extinction $A_V$ and reddening $E_{(B-V)}$ , both measured in stellar magnitudes. The true distance modulus can be computed using the relation

m-M=5\log d-5+A_V.

The empirical relation between $A_V$ and $E_{(B-V)}$ is

A_V=3\cdot E_{(B-V)}.

In Tables I and II, you are provided with photometric data of the stars of the two open clusters, Hyades and NGC 2682.

8.1 Make the colour-magnitude diagrams of the Hyades cluster and NGC 2682 using the provided scaled marked graph paper (A). In the diagrams, draw the main sequence line of each cluster.

8.2 Plot the colour-colour diagrams of the Hyades cluster and NGC 2682 using the provided scaled marked graph paper (B).

8.3 Assuming that the interstellar reddening of Hyades cluster is negligible, derive the interstellar reddening, $E_{(B-V)}$ of NGC 2682.

8.4 Determine the distance to NGC 2682.

8.5 Find the absolute magnitude and colour index (B-V) of the main sequence turn-off star in each cluster, approximately.

8.6 Which cluster is older? (Write in English «Hyades» or «NGC 2682».)

Таблица 1. Данные о Гиадах / Table 1. Hyades data

$m_V$	(B-V)	(U-B)
7.78	+0.62	+0.16
7.14	+0.51	+0.05
8.46	+0.72	+0.31
7.47	+0.57	+0.08
4.22	+0.14	+0.12
6.02	+0.34	+0.04
5.13	+0.21	+0.12
9.99	+1.06	+0.95
6.62	+0.42	-0.01
5.65	+0.28	+0.08
3.61	+0.99	+0.84
4.80	+0.16	+0.12
3.85	+0.96	+0.74
4.27	+0.12	+0.11
9.05	+0.84	+0.53
8.06	+0.64	+0.17

Таблица 2. Данные о NGC 2682 / Table 2. NGC 2682 data

$m_V$	(B-V)	(U-B)
12.80	+0.79	+0.27
12.67	+0.68	+0.19
12.93	+0.93	+0.59
15.64	+0.89	+0.53
15.19	+0.80	+0.30
17.33	+1.19	+0.97
12.16	+1.02	+0.81
12.22	+0.42	+0.03
13.66	+0.55	+0.03
12.55	+0.41	+0.03
14.00	+0.61	+0.11
16.38	+1.00	+0.70
14.96	+0.76	+0.28
14.23	+0.64	+0.12
13.14	+0.45	+0.01
13.25	+0.52	+0.01

Диаграмма Герцшпрунга-Рассела / Hertzsprung-Russell diagram

Official reference Hertzsprung-Russell diagram with spectral classes, temperature, absolute magnitudes and named stars

Решение

All necessary data may be taken from tables and the Hertzsprung-Russell diagram.

8.1

Color-Magnitude Diagrams (blue dots and line for Hyades cluster, red dots and line for NGC2682).

Official colour-magnitude diagram and fitted sequences for both clusters

8.2

Color-Color Diagram (blue dots and line for Hyades cluster, red dots and line for NGC2682):

Official U-B versus B-V colour-colour diagram for both clusters

8.3

On the color-color diagram (problem 8-2), the reddening line is $E(U-B)/E(B-V)=0.67$ . However, data points for two clusters are not sufficient to get such detailed reddening, we accept the difference (B-V) between two main-sequence lines on the color-color diagram as reddening E(B-V), which is $E(B-V)\sim0.06$ (+- 0.02 acceptable).

8.4

On the color-magnitude diagrams in Problem 8-1, the magnitude difference $\Delta m_V$ between the main sequence line of Hyades and the main sequence line of NGC 2686, which is $\Delta m_V=-6.3$ (±0.2 acceptable).

m_V(\mathrm{Hyades})-m_V(\mathrm{NGC}\ 2682)=5\log45-5\log d-3\cdot E(B-V),

-6.3=5\log45-5\log d-0.18,

$d=741$ pc (660 pc - 851 pc : acceptable).

8.5

Find the absolute magnitude and colour index (B-V) of the main sequence turn-off star in each cluster, approximately.

On the color-magnitude diagram of Hyades, MT (main sequence turn off point) is $(B-V)\sim0.12$ (acceptable ±0.02) and $m_V\sim4.3$ (acceptable ±0.2).

m_V-M_V=5\cdot\log45-5-0.18\Longrightarrow M_V=0.75\quad(\text{acceptable }\pm0.2).

Ans: $(B-V)=(B-V)_0=0.12$ , $M_V=0.75$ .

On the color-magnitude diagram of NGC2682, MT (main sequence turn off point) is $(B-V)\sim0.42$ (±0.02 acceptable) and $m_V=12.6$ (±0.2 acceptable).

Intrinsic color of the turn off star is $(B-V)_0=(B-V)-E(B-V)=0.36$ (acceptable : 0.32 - 0.40)

\Longrightarrow m_V-M_V=5\cdot\log d-5+A\Longrightarrow M_V=3.41\quad(\text{acceptable }3.09-3.73).

Ans: $(B-V)=0.42$ , $(B-V)_0=0.36$ , $M_V=3.41$

8.6

NGC2682.

Practical round. Basic grading criteria

Part	Points	Criteria
8.1	2.0	correct plot with main sequence line : 2.0 point, plots without main sequence line : deduction of 0.1 point.
8.2	2.0	correct plot with different marks for each cluster : 2.0 point, plots without identifying each cluster : deduction of 0.1 point.
8.3	2.0	E(B-V) ~ 0.06 ±0.02, get 2.0 point (±0.03, get 1.5 point; ±0.04, get 1.0 point; ±0.05, get 0.5 point).
8.4	2.0	ΔmV ~ -6.3 ±0.2 ➜ distance ~741 pc. Correct method to get distance with wrong ΔmV values : deduction of 0.2 point
8.5	1.8	only (B-V) value for main sequence turn off stars of each cluster (Hyades and NGC 2682) : get 0.9 point, only absolute magnitude of it get 0.9.
8.6	0.2	NGC2682.

Теми и трудност

Трудност: неоценена.

Звезди · Светлина и астрономически измервания · Диаграма на Херцшпрунг–Ръсел · Междузвездно поглъщане · Определяне на разстояния

Предпоставки: Разширена олимпиадна подготовка

Оригинал в Архива: betapract.pdf · официални решения: practsol1.png · допълнителни данни: practsol2.png · допълнителни данни: practsol3.png · допълнителни данни: practsol4.png · допълнителни данни: practsol5.png · допълнителни данни: practsol6.png · допълнителни данни: pract_cr.png · допълнителни данни: paper.png · допълнителни данни: data_1.png · допълнителни данни: data_2.png · допълнителни данни: data_3.png · допълнителни данни: data_4.png