2024

Two surveys of Stripe 82

Round: Data analysis
Points: 75
Level: Bronze
Taught in: Section D10

Adapted from IOAA 2024 (Rio de Janeiro), data-analysis problem 1. We have restated it in our own words and kept the original data. The solution was last checked on 20 October 2026.

The Sloan Digital Sky Survey (SDSS) used a 2.5 m telescope at Apache Point in the United States during the 2000s, and the Dark Energy Survey (DES) used a 4 m telescope at Cerro Tololo in Chile from 2013 to 2019. They mostly covered different parts of the sky, but both observed an equatorial strip called Stripe 82, which can be used to compare and calibrate their photometry. The two tables below list stars in Stripe 82, each with an identification number, its right ascension and declination, and its magnitude $m_{g}$ in the g band with its uncertainty $σ_{g}$ . The names of the two files were lost, so we do not know which table came from which survey.

Table 1

ID1	RA (°)	Dec (°)	$m_{g}$	$σ_{g}$
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

ID2	RA (°)	Dec (°)	$m_{g}$	$σ_{g}$
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) Which table comes from SDSS and which from DES? Take the two surveys to be alike in their detectors, exposure times and sites. (5 points)

(b) For each table, plot $σ_{g}$ on a logarithmic scale against $m_{g}$ on an ordinary scale, and find the slope $A$ and intercept $B$ of the straight line $\log σ_{g} = A m_{g} + B$ . The uncertainties of $A$ and $B$ are not needed. (35 points)

(c) The signal-to-noise ratio is about $1 / σ_{g}$ . From the lines of part (b), what signal-to-noise ratio does each survey reach at a magnitude of 21.5? (5 points)

(d) A star of Table 1 that lies within 1″ of a star of Table 2 can be taken to be the same star. Find the stars that appear in both tables and list their pairs of identification numbers. (15 points)

(e) For the stars in both tables, plot the magnitude from Table 2 against the magnitude from Table 1 on ordinary graph paper, with error bars of twice $σ_{g}$ in both directions. Which stars would be suitable for calibrating the photometry of one survey against the other? Give their numbers in Table 1. (15 points)

I solved it without the solution

At the same magnitudes, the uncertainties of Table 1 run from 0.0067 to 0.0172, about 0.011 on average, and those of Table 2 from 0.0388 to 0.0751, about 0.050 on average. The 4 m telescope of DES collects 2.56 times as much light as the 2.5 m telescope of SDSS, so it measures the same star with the smaller uncertainty. Table 1 comes from DES, and Table 2 from SDSS. With everything else equal, the signal-to-noise ratio of a faint star grows in proportion to the diameter of the telescope, so the uncertainties should differ by a factor of about 1.6, while the tables differ by about 4.5. The sign of the difference answers part (a), and its size shows that the two surveys differ in more than their mirrors.
The plotted points of each table lie close to a straight line. The least-squares line of (D.6) through the points $(m_{g}, \log σ_{g})$ has a slope of $A = 0.278$ for SDSS and $A = 0.406$ for DES. The intercept $B$ is the value of the line at $m_{g} = 0$ , far off the graph, so it is found from the slope and the centre of the points. For SDSS the centre is at $m_{g} = 21.596$ and $\log σ_{g} = - 1.307$ , so $B = - 1.307 - 0.278 \times 21.596 = - 7.31$ . For DES the centre is at $m_{g} = 21.640$ and $\log σ_{g} = - 1.967$ , and with the slope to one more figure, $B = - 1.967 - 0.4063 \times 21.640 = - 10.76$ . A best line drawn by hand gives values close to these, and the official solution accepted any within 10% of them.
At $m_{g} = 21.5$ the SDSS line gives $\log σ_{g} = - 1.333$ , so $σ_{g} = 0.046$ and the signal-to-noise ratio is about 22. The DES line gives $σ_{g} = 0.0095$ and a signal-to-noise ratio of about 106. The SDSS slope lies between 0.2 and 0.4, and the DES slope of 0.406 is as close to the 0.4 of a sky-limited star as the scatter of its points allows, as the physical picture expects. The larger telescope reaches the higher signal-to-noise ratio.
All the stars lie within 0.1° of the celestial equator, where $\cos δ$ differs from 1 by less than two parts in a million, so the separation of two stars on the sky is $\sqrt{Δ α^{2} + Δ δ^{2}}$ , and 1″ is 0.000278°. Comparing each star of Table 1 with the stars of Table 2 near its right ascension gives seven pairs closer than 1″. As (ID1, ID2) they are (2, 7), (3, 8), (4, 13), (6, 1), (8, 11), (9, 6) and (12, 9), all within 0.21″ of each other. The next closest pair, (3, 7), is 17″ apart.
The differences between the magnitudes of Table 2 and Table 1 for the stars 2, 3, 4, 6, 8, 9 and 12 of Table 1 are 0.25, 0.25, 0.26, 0.35, 0.05, 0.00 and 0.49. The error bars of each difference, twice the two uncertainties combined in quadrature by (S.16), are 0.09 to 0.15 long either side.
Stars 8 and 9 lie on the line where the two magnitudes are equal, within their error bars. Stars 2, 3 and 4 lie together about 0.25 above that line, and star 6, at 0.35, agrees with them within its error bar of 0.12. Star 12, at 0.49, lies away from both groups. A calibration measures the offset between the magnitudes of two surveys, so the useful stars are those that agree with one another up to one offset.

AnswerTable 1 comes from DES and Table 2 from SDSS. The lines are $A = 0.278$ , $B = - 7.31$ for SDSS and $A = 0.406$ , $B = - 10.76$ for DES, which reach signal-to-noise ratios of about 22 and 106 at magnitude 21.5. Seven stars appear in both tables, and stars 8 and 9 of Table 1 agree between the surveys with no offset, while stars 2, 3, 4 and 6 agree with an offset of about 0.28 magnitudes, their mean difference. The official solution accepted either group, 8 and 9 for a calibration with no offset, or 2, 3, 4 and 6 for one with an offset.

This problem also appears among the olympiad problems of Section D10, Problems for Chapter D, which teaches the ideas it uses. Equation and figure numbers refer to the course.