IOAA 2024, data-analysis — Задача 1. D1. Photometric comparison of surveys (75 points)

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

Условие

1. The Data Analysis competition will be 3 hours in duration and is marked out of a total of 150 points.

2. There are Detailed Worksheets for carrying out detailed work / rough work. On each of the Detailed Worksheets, please fill in

• Student Code
• Question no.
• Page no. and total number of pages.

3. Start each problem on a new page of the Detailed Worksheets. Please write only on the printed side of the sheet. Do not use the reverse side. If you have written something on any sheet that you do not want to be marked, cross it out.

4. Graph Paper is required for your solutions. On each Graph Paper sheet, please fill in

• Student Code
• Question no.
• Graph no. and total number of graph paper sheets used.

5. There is a summary Answer Sheet with your student ID code for your final answers.

6. Please remember that the graders may not understand your language. As far as possible, write your solutions only using mathematical expressions and numbers. If it is necessary to explain something in words, please use short phrases (if possible in English).

7. You are not allowed to leave your exam desk without permission. If you need any assistance (malfunctioning calculator, need to visit a restroom, need more Detailed Worksheets, etc.), please put up your hand to signal the invigilator.

8. The beginning and end of the competition will be indicated by a long sound signal. Additionally, there will be a short sound signal fifteen minutes before the end of the competition (before the final long sound signal).

9. At the end of the competition you must stop writing immediately. Sort and put your Summary Answer Sheets, Graph Papers, and Detailed Worksheets in one stack. Put all other papers in another stack. You are not allowed to take any sheet of paper out of the examination area.

10. Wait at your table until your envelope is collected. Once all envelopes are collected, your student guide will escort you out of the competition room.

You are an astronomer working with large photometric surveys, such as the Sloan Digital Sky Survey (SDSS) and the Dark Energy Survey (DES), both of which have your host, Observatório Nacional, as a participant. SDSS used a 2.5 m telescope in Apache Point, USA, during the 2000s, and DES used a 4 m telescope in Cerro Tololo, Chile, from 2013 to 2019. Even though they mostly covered different hemispheres of the sky, they had an equatorial region in common known as Stripe 82 that you can use to compare and calibrate the photometry of different data sets, like SDSS and DES.

The following tables containing object positions and magnitudes from Stripe 82 were downloaded for analysis. However, due to a file system corruption on the computer, the file names were scrambled, and now you cannot tell which table belongs to which survey.

Tables 1 and 2 appear next to each other below, with an identification number for each source, its equatorial coordinates, and its magnitude in the $g$-band ($m_g$) with its error (err $m_g$).

Table 1:

$ID_1$	RA (deg)	Dec (deg)	$m_g$ (mag)	err $m_g$ (mag)
1	0.047255	0.000406	21.7649	0.0120
2	0.064741	0.021568	21.1111	0.0067
3	0.064911	0.026395	21.3931	0.0084
4	0.098343	0.054871	21.3934	0.0088
5	0.022256	0.039129	21.9933	0.0157
6	0.006188	0.066928	21.5490	0.0088
7	0.083945	0.074259	21.9395	0.0126
8	0.076715	0.079496	21.4808	0.0089
9	0.057422	0.076006	21.8897	0.0127
10	0.024412	0.087688	21.8341	0.0126
11	0.044723	0.091782	21.8868	0.0172
12	0.071089	0.091053	21.4390	0.0098

Table 2:

$ID_2$	RA (deg)	Dec (deg)	$m_g$ (mag)	err $m_g$ (mag)
1	0.006167	0.066874	21.9020	0.0576
2	0.018660	0.007450	21.8039	0.0529
3	0.047853	0.061487	21.3007	0.0418
4	0.050870	0.015659	21.1678	0.0388
5	0.051270	0.020812	21.2524	0.0401
6	0.057414	0.075999	21.8884	0.0578
7	0.064745	0.021583	21.3634	0.0422
8	0.064910	0.026419	21.6428	0.0488
9	0.071102	0.091058	21.9259	0.0751
10	0.074946	0.002792	21.3258	0.0410
11	0.076709	0.079474	21.5303	0.0476
12	0.092635	0.077395	21.6995	0.0513
13	0.098343	0.054854	21.6542	0.0499
14	0.099332	0.093711	21.8802	0.0577

(a) From these tables, which survey (SDSS or DES) is Table 1 and which is Table 2? Assume that both surveys are equivalent regarding detector response, exposure times, and site characteristics. [5 т.]

(b) Using the data in the table, plot the magnitude ($m_g$) on the $x$-axis (linear scale) and the error in magnitude (err $m_g$) on the $y$-axis (logarithmic scale) using the semi-log paper marked as Graph 1. Estimate the angular coefficient A (slope) and linear coefficient B ($y$-axis intercept) for each dataset. There is no need to calculate the associated errors. [35 т.]

(c) The Signal to Noise ratio ($S/N$) is approximately the inverse of the error in the magnitude, $S/N \approx 1/(\mathrm{err}\ m_g)$. Using the linear fit calculated in the previous part, what is the $S/N$ reached for each survey at a magnitude of $m_g = 21.5\ \mathrm{mag}$? [5 т.]

(d) An object in Table 1 that is within 1 arcsecond of an object in Table 2 can be considered to be the same object. By looking at the RA and Dec of the objects in both tables, identify the objects in common and write down a new table with the matching IDs, $ID_1$ and $ID_2$. [15 т.]

(e) Using the matched table from part (d), plot the $g$-band magnitude of each survey against the other, Table 1 on $x$-axis, and Table 2 on $y$-axis using the millimetre (linear) paper marked as Graph 2. Draw on error bars for each point in both horizontal and vertical directions, using values double err $m_g$ (known as a $2\sigma$ uncertainty). From your graph, identify the stars that would be suitable for photometric calibration between the two surveys and write down their correspondings IDs from Table 1. [15 т.]

Решение

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