IOAA 2024, data-analysis — Задача 2. D2. Shapley Hypothesis (75 points)

Data Analysis Examination, English (Official) — IOAA 2024, Rio de Janeiro, Brazil · 17 август 2024 г. · 75 т.

Условие

Globular clusters are one of the oldest components of galaxies. About a century ago, Harlow Shapley studied the distribution of globular clusters in the Milky Way in order to determine the distance from the Sun to the Galactic Centre, with the hypothesis that globular clusters were symmetrically distributed around the Galactic Centre. The table below shows the positions and distance moduli of a few known globular clusters in the Milky Way. The first three columns in the table show the cluster name, galactic longitude ($l$), and galactic latitude ($b$). The fourth column shows the distance modulus (i.e. the difference between the apparent and absolute magnitude), for which the values are extinction-corrected. Based on the data in the table:

Name	$l$ (degrees)	$b$ (degrees)	Distance modulus (mag)
NGC 6522	1.025	-3.926	14.3
NGC 6401	3.450	3.980	14.4
NGC 6342	4.898	9.725	14.5
NGC 6553	5.253	-3.029	13.6
NGC 6440	7.729	3.801	14.6
Ter 12	8.358	-2.101	13.6
VW-CL160	10.151	0.302	14.2
2MASS-GC01	10.471	0.100	12.6
NGC 6517	19.225	6.762	14.8
NGC 6402	21.324	14.804	14.8
NGC 6712	25.354	-4.318	14.3
NGC 6426	28.087	16.234	16.6
NGC 5466	42.150	73.592	16.0
NGC 7089	53.371	-35.770	15.3
NGC 288	151.285	-89.380	14.8
NGC 2298	245.629	-16.006	15.0
NGC 4590	299.626	36.051	15.1
NGC 4372	300.993	-9.884	13.8
NGC 362	301.533	-46.247	14.7
BH 140	303.171	-4.307	13.4
NGC 5927	326.604	4.860	14.6
Patchick 126	340.381	-3.826	14.5
NGC 5897	342.946	30.294	15.5
NGC 6380	350.182	-3.422	14.9
Djor 1	356.675	-2.484	15.0

(a) Calculate the distance (in parsecs) of each globular cluster from the Sun as well as their Cartesian coordinates ($x$, $y$, $z$). The $x$-axis points to the Galactic Centre and the $y$-axis points in the direction of galactic rotation. The system is right-handed. [25 т.]

(b) From the given data, estimate the distance from the Sun to the centre of the distribution of globular clusters and its uncertainty. [15 т.]

(c) To test the validity of Shapley's hypothesis that globular clusters are symmetrically distributed around the Galactic Centre, make histograms with five bins (i.e. sort the data and divide them into five equally-sized intervals) for each of the distributions in the $x$, $y$ and $z$ directions. Mark the value of the quartiles ($Q_1, Q_2, Q_3$) of the three distributions with solid lines on the histograms.

Hint: The three quartiles divide the sorted sample into four sections, each containing 25% of the data, with the second and third sections representing the interquartile range. [30 т.]

(d) Using the quartiles, calculate the symmetry factor value for the three distributions as given by:

$\Phi_x = \frac{\left|Q_{1,x}+Q_{3,x}-2Q_{2,x}\right|}{Q_{3,x}-Q_{1,x}}\ ,\quad \Phi_y = \frac{\left|Q_{1,y}+Q_{3,y}-2Q_{2,y}\right|}{Q_{3,y}-Q_{1,y}}\ ,\quad \Phi_z = \frac{\left|Q_{1,z}+Q_{3,z}-2Q_{2,z}\right|}{Q_{3,z}-Q_{1,z}}$

Classify the three distributions in the $x$, $y$, and $z$ directions based on their calculated symmetry factor values, according to the table shown below. Hence, on the answer sheet, write True (T) if the analysed sample follows Shapley's hypothesis or False (F) otherwise.

Symmetry factor value	Symmetry type
$0.0 \le \Phi \le 0.1$	symmetrical
$0.1 < \Phi \le 0.2$	quasi-symmetrical
$\Phi > 0.2$	asymmetrical

Решение

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