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IOAA 2015, Анализ на данни — Задача 2. Problem 2

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Всички материали от IOAA 2015

International Olympiad on Astronomy and Astrophysics — Data Analysis Problem (Central Java - Indonesia, 2015)

Условие

BVRIJHKLMN photometry of 2 stars from the constellation Cassiopeia is given in Table 5. For both stars it is believed that their light is affected by extinction by diffuse Interstellar Medium (ISM) only. Assuming that the observation is done from outside the atmosphere.

Table 5 BVRIJHKLMN photometry of two stars in Cassiopeia

Star	MK class	$\dfrac{B}{\mathrm{mag}}$	$\dfrac{V}{\mathrm{mag}}$	$\dfrac{R}{\mathrm{mag}}$	$\dfrac{I}{\mathrm{mag}}$	$\dfrac{J}{\mathrm{mag}}$	$\dfrac{H}{\mathrm{mag}}$	$\dfrac{K}{\mathrm{mag}}$	$\dfrac{L}{\mathrm{mag}}$	$\dfrac{M}{\mathrm{mag}}$	$\dfrac{N}{\mathrm{mag}}$
HD 4817	K3Iab	8.08	6.18	4.73	3.64	2.76	1.86	1.54	1.32	1.59	-
HD 11092	K4II	8.66	6.57	-	-	3.10	2.14	1.63	1.41	1.65	1.44

Table 6 $(B-V)_0$ intrinsic colours for selected sp. types and luminosity classes

	$\dfrac{(B-V)_0}{\mathrm{mag}}$
	II	Iab / Ia
F0	-	0.15
G0	0.73	0.82
K0	1.06	1.18
K3	1.40	1.42
K4	1.42	1.50

Table 7 Infrared intrinsic colours for selected sp. types of supergiant stars

	$\dfrac{(V-R)_0}{\mathrm{mag}}$	$\dfrac{(V-I)_0}{\mathrm{mag}}$	$\dfrac{(V-J)_0}{\mathrm{mag}}$	$\dfrac{(V-H)_0}{\mathrm{mag}}$	$\dfrac{(V-K)_0}{\mathrm{mag}}$	$\dfrac{(V-L)_0}{\mathrm{mag}}$	$\dfrac{(V-M)_0}{\mathrm{mag}}$	$\dfrac{(V-N)_0}{\mathrm{mag}}$
F0	0.20	0.31	0.36	0.51	0.60	0.64	0.65	0.82
G0	0.55	0.90	1.14	1.52	1.71	1.72	1.72	1.98
K0	0.95	1.59	2.01	2.64	2.80	2.87	2.79	3.14
K3	1.13	1.96	2.41	3.14	3.25	3.39	3.25	3.63
K4	1.20	2.13	2.59	3.37	3.44	3.62	3.46	3.84

Table 8 Infrared intrinsic colours for selected sp. types of giant stars

	$\dfrac{(V-R)_0}{\mathrm{mag}}$	$\dfrac{(V-I)_0}{\mathrm{mag}}$	$\dfrac{(V-J)_0}{\mathrm{mag}}$	$\dfrac{(V-H)_0}{\mathrm{mag}}$	$\dfrac{(V-K)_0}{\mathrm{mag}}$	$\dfrac{(V-L)_0}{\mathrm{mag}}$	$\dfrac{(V-M)_0}{\mathrm{mag}}$	$\dfrac{(V-N)_0}{\mathrm{mag}}$
K0	0.60	1.03	1.23	1.72	1.94	1.97	1.90	1.92
K3	0.86	1.39	1.84	2.40	2.69	2.82	2.70	2.73
K4	0.96	1.61	2.16	2.77	3.05	3.22	3.08	3.02

Period–luminosity diagram: mean apparent magnitude $\langle R\rangle$ (21.5–24.5, reversed axis) versus log Period(days) (1.2–2.0); data points with a few outliers (open circles), a solid least-squares line and two dashed boundary lines. — Fig. 2 $\langle R\rangle$ is the mean apparent magnitude in filter R

Table 9 Effective wavelengths of selected photometric filters

Filter	B	V	R	I	J	H	K	L	M	N
$\lambda_F$ /nm	450	555	670	870	1200	1620	2200	3500	5000	9000

Scatter plot with a fitted solid line and two dashed lines of mean apparent magnitude ⟨I⟩ (21.0–24.0, reversed axis) versus log Period(days) (1.2–2.0); filled circles follow the trend, a few open circles are outliers. — Fig. 3 ⟨I⟩ is the mean apparent magnitude in filter I

a) Using the data given in Tables 5 to 9, plot $E_{\mathrm{X-V}}/E_{\mathrm{B-V}}$ as a function of $1/\lambda_{\mathrm{X}}$ for filters B, V, R, I, J, H, K, L, M, N for both stars. Fit approximate curves by eye (in particular, note that $E_{\mathrm{X-V}}/E_{\mathrm{B-V}} \sim const.$ as $1/\lambda_{\mathrm{X}} \rightarrow 0$ ). X is each band in the photometric system.

b) Using the graphs obtained in a), estimate $R_{\mathrm{V}}$ for each star.

R_{\mathrm{V}} = \frac{A_{\mathrm{V}}}{E_{\mathrm{B-V}}}

( $A_{\mathrm{V}}$ is the absorption in V, and $E_{\mathrm{B-V}}$ is the color excess)

Now apply these results in order to derive a distance estimate for IC 342, a spiral galaxy in Cassiopeia obscured by Milky Way. You should assume that the properties of the ISM in IC 342 are similar to those of the ISM in our Galaxy.

c) Using the period-magnitude diagrams for 20 Cepheids from IC 342 (Figures 2 and 3) and assuming the period-luminosity relations:

\langle M_{\mathrm{R}} \rangle = -2.91\left(\log\left(\frac{P}{\mathrm{day}}\right) - 1\right) - 4.04 \quad \mathrm{and} \quad \langle M_{\mathrm{I}} \rangle = -3.00\left(\log\left(\frac{P}{\mathrm{day}}\right) - 1\right) - 4.06

where $\langle M_{\mathrm{R}} \rangle$ and $\langle M_{\mathrm{I}} \rangle$ are the mean absolute magnitudes in filters R and I, find $A_{\mathrm{R}}$ for objects in IC 342. Find the distance to IC 342.

Решение

a) Using the data from tables 5 to 9 and recalling that MK class II corresponds to giant stars while MK classes Ia and Iab correspond to supergiants, we easily obtain tables 10 to 12 and hence figures containing the plots of $E_{X-V}/E_{B-V}$ against $1/\lambda_X$ for both stars.

Table 10 (10 points)

Star	$\frac{B-V}{\text{mag}}$	$\frac{V-V}{\text{mag}}$	$\frac{R-V}{\text{mag}}$	$\frac{I-V}{\text{mag}}$	$\frac{J-V}{\text{mag}}$	$\frac{H-V}{\text{mag}}$	$\frac{K-V}{\text{mag}}$	$\frac{L-V}{\text{mag}}$	$\frac{M-V}{\text{mag}}$	$\frac{N-V}{\text{mag}}$
HD 4817	1.9	0	-1.45	-2.54	-3.42	-4.32	-4.64	-4.86	-4.59	-
HD 11092	2.09	0	-	-	-3.47	-4.43	-4.94	-5.16	-4.92	-5.13

Table 11 (10 points)

Star	$\frac{(B-V)_0}{\text{mag}}$	$\frac{(V-V)_0}{\text{mag}}$	$\frac{(R-V)_0}{\text{mag}}$	$\frac{(I-V)_0}{\text{mag}}$	$\frac{(J-V)_0}{\text{mag}}$	$\frac{(H-V)_0}{\text{mag}}$	$\frac{(K-V)_0}{\text{mag}}$	$\frac{(L-V)_0}{\text{mag}}$	$\frac{(M-V)_0}{\text{mag}}$	$\frac{(N-V)_0}{\text{mag}}$
HD 4817	1.42	0	-1.13	-1.96	-2.41	-3.14	-3.25	-3.39	-3.25	-3.63
HD 11092	1.42	0	-0.96	-1.61	-2.16	-2.77	-3.05	-3.22	-3.08	-3.02

Table 12 (10 points)

Star	$\frac{E_{B-V}}{E_{B-V}}$	$\frac{E_{V-V}}{E_{B-V}}$	$\frac{E_{R-V}}{E_{B-V}}$	$\frac{E_{I-V}}{E_{B-V}}$	$\frac{E_{J-V}}{E_{B-V}}$	$\frac{E_{H-V}}{E_{B-V}}$	$\frac{E_{K-V}}{E_{B-V}}$	$\frac{E_{L-V}}{E_{B-V}}$	$\frac{E_{M-V}}{E_{B-V}}$	$\frac{E_{N-V}}{E_{B-V}}$
HD 4817	1.00	0.00	-0.67	-1.21	-2.10	-2.46	-2.90	-3.06	-2.79	-
HD 11092	1.00	0.00	-	-	-1.96	-2.48	-2.82	-2.90	-2.75	-3.15

(10 points) — plot for HD 4817; (10 points) — plot for HD 11092.

b) We have that $E_{\lambda-V} = A_\lambda - A_V \to -A_V$ as $\lambda \to \infty$ , so the plot of $E_{X-V}/E_{B-V}$ against $1/\lambda_X$ intersects the vertical axis at

\frac{-A_V}{E_{B-V}} = -R_V.

(10 points)

Hence $R_V$ can be read off as minus the intersection of the plot with vertical axis. For the two stars we get Table 13 (the intersection points are obtained by fitting a curve to guide the eye, noting that as $1/\lambda \to 0$ , the curve becomes flat, as hinted at above).

Table 13 (10 points)

Star	$R_V$
HD 4817	3.1
HD 11092	3.0

Next,

R_{r,r-i} = \frac{A_r}{E_{r-i}} = \left(\frac{E_{r-V}}{E_{B-V}} + \frac{A_V}{E_{B-V}}\right)\frac{E_{B-V}}{E_{r-i}} = \frac{\frac{E_{r-V}}{E_{B-V}} + R_V}{\frac{E_{r-V}}{E_{B-V}} - \frac{E_{i-V}}{E_{B-V}}},

where $E_{r-V}/E_{B-V}$ and $E_{i-V}/E_{B-V}$ can be read off from the plot (again, by fitting a curve to guide the eye). Hence we get Table 14.

Table 14 (10 points)

Star	$R_r$
HD 4817	3.7
HD 11092	3.6

Hence we take the expected values to be $R_V \approx 3.1$ and $R_r \approx 3.7$ . Note that the first value agrees with the widely used ratio of the total to selective extinction in filters B and V.

c) First let us find the apparent distance moduli $\mu_{r,i}$ in filters r and i. Reading off the fitted values e.g. at $\log(P/\text{day}) = 1.6$ from figures 2 and 3 and substituting into the period-luminosity relations, we find $\mu_r = 29.0$ mag and $\mu_i = 28.6$ mag, so $E_{r-i} = A_r - A_i = \mu_r - \mu_i = 0.4$ mag and so $A_r \approx 3.7 E_{r-i} = 1.5$ mag. Hence the unreddened distance modulus is $\mu_0 = \mu_r - A_r = 27.5$ mag and so we estimate the distance to IC 342 to be 3.2 Mpc. (20 points)

Scatter plot titled HD 4817: E(X-V)/E(B-V) on vertical axis from -3.500 to 1.500, 1/λ (1/nm) on horizontal axis from 0 to 0.0025; points descend from 1.0 at 0.0022 through 0 at ~0.0018 down to about -3.06 near 0.00025. — Plot of E(X-V)/E(B-V) against 1/λ (1/nm) for HD 4817

Scatter plot titled HD 11092: E(X-V)/E(B-V) on vertical axis from -3.500 to 1.500, 1/λ (1/nm) on horizontal axis from 0 to 0.0025; points descend from 1.0 at 0.0022 through 0 at ~0.0018 down to about -3.15 near 0.0002. — Plot of E(X-V)/E(B-V) against 1/λ (1/nm) for HD 11092

Оригинал в Архива: IOAA 2015 Data Analysis Problems ver 20150730_1717.pdf · официални решения: DataAnalysis_sol.pdf