IOAA 2024, Анализ на данни — Задача 2. D2. Shapley Hypothesis

(a)

The first step is to convert the extinction corrected distance moduli ($DM$) of all globular clusters to distance ($d$), in parsecs:

So, it follows to calculate the cartesian coordinates ($x,y,z$) of the globular clusters with respect to the Sun, using the galactic coordinates (longitude $l$ and latitude $b$). The $x$-axis points to the Galactic Centre, the $y$-axis points to the direction of galactic rotation and the $z$-axis is perpendicular to the galactic disk, and points in the direction antiparallel to the angular momentum. The conversion of coordinates should be done as follows:

The table below shows the calculated values of $d,x,y$, and $z$ for all globular clusters.

(b)

In order to estimate the distance from the Sun to the centre of the distribution and its uncertainty, we first need the mean values of each coordinate $\bar x,\bar y$ and $\bar z$, as well as the standard deviations in each axis, $\sigma_x,\sigma_y$ and $\sigma_z$. The calculations of the mean and standard deviation for the $x,y$, and $z$ coordinates are as follows:

Mean in the three axes:

Standard deviations in the three axes:

Consequently, we obtain the uncertainties associated to the mean in each axis:

Uncertainties of means in the three axes:

Finally, the distance $D$, in parsecs, from the Sun to the centre of the distribution of globular clusters is estimated as:

And, to estimate its uncertainty $\delta D$, we should perform an error propagation calculation:

(c)

The table below shows the calculated values of the quartiles for each distribution.

The expected histograms for each axis are as follows:

(d)

F (False). The analysed sample does not follow Shapley's hypothesis, because all distributions are asymmetrical. See table below for the values of calculated symmetry factors for each distribution.

Marking scheme — General Rules

These are general guidelines for the grading of all problems. All of the rules listed below refer to deductions on specific items in the marking scheme.

Marking scheme

(a) 20 points

• 2.5 points: Calculate the distance to each GC using distance modulus. Deduct $-\frac{2.5}{25}$ points for each wrong value.
• 2.5 points: Equate change of coordinates from polar to Cartesian coordinates.
• 15 points: Calculate $x,y$ and $z$ components of each GC. 5 points for each component. Deduct $-\frac{5}{25}$ points for each wrong value.

(b) 20 points

• 3 points: Calculate $\bar x,\bar y$ and $\bar z$. 1 point for each value.
• 6 points: Calculate $\sigma_{\bar x},\sigma_{\bar y}$ and $\sigma_{\bar z}$. 2 points for each value.
• 2 points: Write formula of $D$.
• 3 points: Reasonable method for $\sigma_D$ estimation.
• 2.5 point: Calculate numerical value of $D$.
• 3.5 points: Calculate numerical value of $\sigma_D$. If the numerical value appears reasonable but the student did not explain their method, deduct -1.5 points.

(c) 30 points

• 24 points: Correct histogram for $x,y$ and $z$. 8 points for each plot. Deduct 50% of the points for this criterion if the student uses the wrong method for constructing the histograms. Reminder: bin's size and first bin position.
• 3 points: Calculate $Q_{1x},Q_{2x},Q_{3x},Q_{1y},Q_{2y},Q_{3y},Q_{1z},Q_{2z}$ and $Q_{3z}$.
• 3 points: Represent $Q_{1x},Q_{2x},Q_{3x},Q_{1y},Q_{2y},Q_{3y},Q_{1z},Q_{2z}$ and $Q_{3z}$ on their respective histograms (1 point for each histogram).

(d) 5 points

• 3 points: Calculate $\phi_x,\phi_y$ and $\phi_z$ and evaluate their symmetry types (1 point for each value).
• 2 points: Correctly concluding that it is FALSE.