IOAA 2013, Анализ на данни — Задача 4. Question 4

(1)

The student should be able to import the ascii data of the txt file in the MS Excel spreadsheet application. (12 Points)

(2)

In order to calculate the angular distance $\varphi$ (next step), all coordinates should be converted into decimal degrees. The students should be able to convert the [hours, min, sec] and the [°, ′, ″] into decimal degrees, using. For the right ascension: $\alpha_{\mathrm{deg}}=h\times15+\frac{m}{60}+\frac{s}{3600}$ and for the declination: $\delta_{\mathrm{deg}}={}^\circ+\frac{{}^\prime}{60}+\frac{{}^{\prime\prime}}{3600}$. All coordinates are positive, so the student does not have to worry about checking the sign, which is not trivial. (12 Points)

(3)

Using the cosine law, $\varphi=\arccos(\sin\delta_1\times\sin\delta_2+\cos\delta_1\times\cos\delta_2\times\cos(a_1-a_2))$, the angular distance, $\varphi$, between each of the stars and the point of convergence should be calculated. (15 Points)

(4)

The student should easily calculate the proper motion of each star, by inserting the given equation, $\mu=\sqrt{(\mu_\alpha\cos\delta)^2+\mu_d^2}$, in the spreadsheet. (9 Points)

(5)

Again, inserting the given equation, $r_\mu=\frac{v_r\tan\varphi}{4.74047\mu}$ in the spreadsheet (remembering to divide the given values of $\mu$ by 1000 to get arcseconds), the student should be able to calculate the distance, $r_\mu$, of each star. Any star whose distance from the centre of the cluster is larger than 10 pc should be omitted from the following calculations. (12 Points)

(6)

The trigonometric parallax distance is given by $r_\pi=\frac1{\pi\times10^{-3}}$. This equation should be inserted in the spreadsheet. The result is in parsecs [pc]. (6 Points)

(7)

Using the statistical functions AVERAGE and STDEV of MS Excel, the student should easily calculate the average and the standard deviation of $r_\mu$ and $r_\pi$. (6 Points)

(8)

The method with the smaller standard deviation is, obviously more accurate. (3 Point)

(macros are allowed)

DataHyades.xlsx — archived worked values

Point of convergence: R.A. 6h7m, displayed 90.1167 degrees; Dec. 6°56′, displayed 6.9333 degrees. $r_0=46.3400$. Values below are transcribed from the archived workbook, without recalculation. The workbook and printed solution use an incorrect right-ascension conversion; the distance filter is also one-sided. These source errors affect the worked results.

DataHyades.xlsx — residuals

DataHyades.xlsx — without distant stars