2 Derivation of the calibration

To derive the calibration we used the sample of stars used by Beers et al. (1999) for the calibration of the KPindex. For all these stars UBV photometry and HP2 and KP indices are available. As calibrators we selected the stars with low reddening from Beers et al. (i.e. $E(B-V)\le 0.01$). To make this criterion more stringent and thus to select a sample of truly unreddened stars, we searched the Hauck & Mermilliod (1998) $uvby\beta$ catalog, through the interface of the General Catalog of Photometric Data (GCPD, Mermilliod et al. 1996) to find out stars of the Beers et al. sample with $uvby\beta$ photometry. We then applied the Schuster & Nissen calibration to these data to obtain $E(b-y)\sim 0.74~E(B-V)$. Our final criterion was that both the estimate of E(B-V) of Beers et al. and the one derived from Strömgren photometry were less than or equal to 0.01 mag.

   

Table 1: Basic data for the calibrators

STAR	Typea	[Fe/H]	(b-y)	m1	c1	$\beta$
BD +18$^\circ$1479	D	-0.46	0.440	0.258	0.256	2.560
CD -24$^\circ$17504b	D	-3.70	0.317	0.039	0.301	2.596
G8-16	D	-1.59	0.322	0.071	0.292	2.595
G11-36	D	-0.68	0.384	0.139	0.255	2.579
G11-44	D	-2.07	0.334	0.054	0.263	2.597
G11-45	D	-0.01	0.431	0.249	0.359	2.602
G12-21	D	-1.32	0.339	0.090	0.284	2.592
G13-35	D	-1.63	0.331	0.060	0.285	2.594
G14-23	D	-0.27	0.370	0.150	0.316	2.590
G14-26	D	-0.20	0.387	0.181	0.349	2.590
G15-6	D	-0.65	0.436	0.210	0.253	2.561
G15-17	D	-0.39	0.475	0.315	0.246	2.553
G17-21	D	-0.66	0.367	0.125	0.306	2.588
G17-30	D	-0.48	0.396	0.164	0.303	2.574
G24-15	D	-1.10	0.342	0.092	0.273	2.597
G37-26	D	-1.93	0.351	0.058	0.208	2.584
G44-6	D	-0.54	0.397	0.160	0.259	2.566
G44-30	D	-0.89	0.427	0.177	0.200	2.557
G54-21	D	-0.03	0.387	0.195	0.319	2.602
G57-11	D	0.03	0.412	0.230	0.328	2.593
G58-23	D	-0.97	0.407	0.138	0.219	2.565
G58-25	D	-1.41	0.344	0.079	0.258	2.589
G58-41	D	-0.33	0.371	0.150	0.344	2.591
G59-1	D	-1.02	0.424	0.186	0.204	2.567
G59-24	D	-2.42	0.332	0.054	0.225	2.593
G60-48	D	-1.63	0.365	0.074	0.189	2.583
G62-44	D	-0.58	0.473	0.295	0.248	2.554
G62-52	D	-1.28	0.430	0.167	0.190	2.553
G63-46	D	-0.91	0.382	0.139	0.283	2.584
G65-47	D	-0.35	0.405	0.188	0.281	2.584
G66-15	D	-0.20	0.405	0.204	0.319	2.582
G80-15	D	-0.78	0.365	0.126	0.272	2.586
G88-32	D	-2.36	0.309	0.051	0.357	2.591
G97-43	D	-0.49	0.463	0.256	0.277	2.554
G113-24	D	-0.49	0.383	0.146	0.301	2.588
G114-26	D	-1.78	0.349	0.076	0.250	2.594
G119-64	D	-1.42	0.319	0.073	0.319	2.600
G121-12	D	-0.92	0.350	0.094	0.287	2.592
G160-3	D	-0.14	0.414	0.217	0.339	2.584
G162-16	D	-0.53	0.396	0.167	0.323	2.586
G162-51	D	-0.52	0.377	0.145	0.305	2.583
G162-68	D	-0.54	0.425	0.202	0.206	2.565
G165-39	D	-2.05	0.309	0.055	0.354	2.602
G200-62	D	-0.45	0.479	0.304	0.273	2.552
G229-34	D	-0.50	0.403	0.190	0.334	2.579
G271-34	D	-0.68	0.386	0.155	0.262	2.576
HD 693	TO	-0.38	0.328	0.130	0.405	2.621
HD 3567	SG	-1.29	0.328	0.089	0.330	2.600
HD 6461	G	-0.93	0.500	0.199	0.434	2.554
HD 16031	D	-1.71	0.323	0.069	0.302	2.606
HD 20010	SG	-0.27	0.339	0.156	0.411	2.624
HD 34328	D	-1.61	0.365	0.063	0.205	2.569
HD 76932	D	-0.99	0.359	0.119	0.298	2.584
HD 90508	D	-0.23	0.397	0.175	0.298	2.578
HD 105590	HB	-0.17	0.414	0.229	0.321	2.591
HD 113083	D	-1.04	0.367	0.122	0.254	2.587
HD 114762	D	-0.70	0.365	0.125	0.297	2.588
HD 134169	SG	-0.85	0.370	0.119	0.312	2.582
HD 184499	D	-0.58	0.390	0.143	0.314	2.578
HD 193901	D	-1.08	0.381	0.103	0.217	2.573
HD 200580	D	-0.75	0.364	0.149	0.305	2.599
HD 201889	D	-0.92	0.388	0.148	0.281	2.577
HD 201891	D	-1.13	0.358	0.104	0.262	2.590
HD 205156	D	-0.57	0.398	0.161	0.256	2.573
HD 219617	D	-1.31	0.344	0.078	0.246	2.597

^a D = dwarf; TO = turn-off; SG = subgiant; G = giant; HB = horizontal branch.
^b Not used in the calibration, because trimmed out after the first pass.

Out of the sample of Beers et al. (1999) 65 stars satisfy our criterion and they are reported in Table 1, the star name is given in Col. (1), Col. (2) gives the star type, according to Beers et al. (1999). Column (3) is [Fe/H] and Cols. (4)-(7) provide the Strömgren photometry extracted from the Hauck & Mermilliod (1998) catalogue. We performed a $\chi^2$ fit on this sample of stars for several functional forms. We computed the rms of the fit and we then discarded those stars whose residual was greater than $\rm 2.5\ \times$ rms. This was aimed at further cleaning the sample by rejecting stars which are either reddened or whose colours or line indices are affected by larger errors. Quite interesting only one star, CD $-24\hbox {$^\circ $ }$ 17504, was discarded, whichever functional form was used. This star is also the most metal-poor of the sample, in fact the only one below $\rm [Fe/H]=-3.00$. Thus our fits were all performed on a sample totalling 64 stars.

We began with the assumption that both (B-V) and HP2strongly depend on temperature; we therefore fit a linear relation

$$(B-V) = x_1 + x_2 \log HP2$$

and obtained a fit with $\chi^2 = 273.8$. We next began to add other terms in $\log KP$ and (U-B) and checked the significance of the new term through the F test (Bevington & Robinson 1992, p. 208). Linear terms in both $\log KP$ and (U-B) are highly significant (the probability corresponding to the observed $F_\chi$is $6.5\ 10^{-7}$ for $\log KP$ and $7.0 \ 10^{-13}$ for (U-B)). On the other hand further quadratic terms are not significant, the most significant being a term in $(\log KP)^2$, with a probability of the observed $F_\chi \approx 5.5 \ 10 ^{-2}$, i.e. a significance of $\sim 95 \%$. We therefore conclude that the best functional form is that with only linear terms.

 

$$(B-V)_0\! =\! x_1\! +\! x_2 \log HP2\! +\! x_3 \log KP\! +\! x_4 (U-B)_0.$$ (1)

Our best fit parameters are given in Table 2, together with their formal errors derived from the covariance matrix.

   

Table 2: Best fit parameters

	param.	error
x1	0.5734	0.0004
x2	-0.2759	0.0006
x3	0.1040	0.0003
x4	0.2676	0.0006

The rms of the fit was 0.0153 mag and $\chi^2 \approx 67.7$, i.e. $\chi^2_{60}= 1.13$, which indicates a good fit. In Fig. 1, panel a) we show a plot of (B-V)versus the right hand side of Eq. (1). Panels b) and c) display the residuals as a function of metallicity and (B-V) colour, respectively.

The range of validity of the calibration is fixed by the properties of the calibrator stars. In our case these are:

$$\begin{array}{lllll} \phantom{-}0.375 & \le & (B-V)\hfill & \... ... 9.790\\ -0.245 & \le & (U-B)\hfill & \le & 0.350. \end{array}$$

The mean metallicity of the calibrators is $\rm [Fe/H] \sim -0.9$. The metallicity range is $\rm -2.42\le [Fe/H] \le +0.03$. In practice we expect that our calibration is applicable to stars of intermediate metallicity. We note that the mean metallicity of the calibrators of Schuster & Nissen is $\rm [Fe/H]=-0.50$, while their metallicity range is $\rm -2.49\le [Fe/H] \le +0.22$. Thus there is an almost perfect overlap of the metallicity domains where the two calibrations are derived, with the Schuster & Nissen calibration extending towards slightly higher metallicities.

Figure 1: a) The right hand side of Eq. (1) for the calibrators as a function of (B-V). The star CD $-24\hbox {$^\circ $ }$ 17504 appears to be an outlier and has not been used in the derivation of the calibration, and it is not drawn in panels b) and c), b) the residuals (B-V)-FIT as a function of metallicity, c) same as panel b) but as a function of (B-V)

A matter of concern is wether the luminosity (gravity) dependence of the calibration is properly captured by the (U-B) colour. Inspection of Table 1 reveals that, although we did not impose any selection criterion on luminosity, most of our stars are dwarfs, only one is a giant, three sub-giants and one horizontal branch. This is a result of imposing a very tight criterion on reddening for the calibrators: all low-reddening stars are nearby and therefore most are dwarfs. So formally our calibration is valid only for dwarfs. In Fig. 2 we show the residuals of the fit as a box plot in the $\left ( (b-y)_0,c0\right )$ plane. There appears to be no obvious trend of the residuals with the luminosity of the stars, we may therefore expect that the calibration is in fact equally applicable to dwarfs and giants, as is the Schuster & Nissen calibration.

Figure 2: The residuals (B-V)-FIT in the $\left ( (b-y)_0,c0\right )$ plane. The stars have been divided into 10 bins 0.01 mag wide. The size of the symbol is largest for the stars in the bins $0.04< \vert(B-V)-{\rm FIT}\vert\le 0.05$ mag and decreases to the minimum size for stars in the bins with $\vert(B-V)-{\rm FIT}\vert\le 0.01$, as indicated in the scale plot shown below the figure. Negative residuals are shown with crossed squares, while positive residuals are shown with open squares. The star CD $-24\hbox {$^\circ $ }$ 17504 is drawn with an asterisk. The solid line represents the locus of points with $\log g=3.5$ for $\rm [Fe/H]=-1.0$. The dashed line is the same but for $\rm [Fe/H]=-2.5$