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3 Computed and empirical ${{\vec T}_{\sf eff}}$ vs.  $({\vec V-K})$  relations for dwarfs of solar metallicity. The effect of the interstellar extinction

Any determination of $T_{\rm eff}$  from a colour index depends crucially on how well we know the interstellar extinction AV, which is required for de-reddening (V-K). In this section we discuss the effect of the interstellar extinction on the determinations of  $T_{\rm eff}$  from (V-K)  for dwarfs of solar metallicity and compare $T_{\rm eff}$  from models with the $T_{\rm eff}$  from other determinations.

E(V-K)  was determined from the relation

E(B-V)  = E(V-K) /2.76 (Mathis 1999).

3.1 T $_\mathsfsl{eff}$  from $(\mathsfsl{V-K})$  for bright stars with "reference'' $\mathsfsl{ T_{eff}}$

The effective temperature is obtained most directly from the Stefan-Boltzmann equation which relates $T_{\rm eff}$  to the angular diameter $\theta$ and the total integrated flux at the earth ( $f_{\oplus}$) of a star:

$\displaystyle f_{\oplus} = \frac{1}{4} \sigma\theta^{2}T_{\rm eff}^{4} .$      

The number of stars for which angular diameters are available and from which "reference'' $T_{\rm eff}$  can be obtained is quite limited. A compilation of these "reference'' $T_{\rm eff}$ has been given by Smalley & Dworetsky (1995)[*]; these include several stars with luminosity class V and IV-V and spectral types A and F. The "reference'' $T_{\rm eff}$  for these stars are given in Table 4 (Col. 4) and are compared with those derived from the RIJKL grids (Col. 5) using $(V-K)_{\rm0}$  taken from Di B98 (Col. 8). We adopted the metallicity listed in Col. 7 and the gravity given in Col. 6. The agreement is generally good; the mean difference between the $T_{\rm eff}$ from the synthetic colors and the "reference'' $T_{\rm eff}$  is $-21\pm50$ K.


Table 4: Data for bright luminosity class V stars with reference $T_{\rm eff}$.

$\theta$a $f_{\oplus}$b $T_{\rm eff}$ $\log g$b [M/H] $(V-K)_{\rm0}$a E(B-V)
  10-3( $\hbox{$^{\prime\prime}$ }$) 10-6 erg cm-2 s-1 Reference Modelc       Pol.d (Di B98)
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

$\alpha $ CMa$^{\rm 1}$
$5.92\pm0.09$ $112.899\pm7.106$ $9916\pm174$ 10 033 4.33 +0.5e -0.099 $0.0018\pm0.0013$ 0.006
$\alpha $ Lyr$^{\rm 2}$ $3.24\pm0.07$ $29.737\pm1.820$ $9602\pm180$ 9558 3.95 -0.5e -0.001 $0.0005\pm0.0009$ 0.002
$\epsilon$ Sgr$^{\rm 3}$ $1.44\pm0.06$ $5.436\pm0.316$ $9418\pm337$ 9240 4.5 0.0 0.047 $0.0006\pm0.0008$ 0.012
$\beta$ Leo$^{\rm 4}$ $1.33\pm0.10$ $3.644\pm0.197$ $8867\pm355$ 8761 4.1 0.0 0.140 $0.0019\pm0.0016$ 0.004
$\alpha $ PsA$^{\rm 5}$ $2.10\pm0.14$ $8.638\pm0.459$ $8756\pm315$ 8760 4.2 0.0 0.144 $0.0008\pm0.0004$ 0.002
$\alpha $ CMi$^{\rm 6}$ $5.51\pm0.05$ $18.638\pm0.868$ $6551\pm82$ 6634 4.06 0.0 1.010 $0.0005\pm0.0005 $ 0.000

$^{\rm 1}$ BS 2491, $^{\rm 2}$ BS 7001, $^{\rm 3}$ BS 6879, $^{\rm 4}$ BS 4534, $^{\rm 5}$ BS 8728,  $^{\rm 6}$ BS 2943.
a Di Benedetto (1998) (Di B98).   b Smalley & Dworetsky (1995).   c  $T_{\rm eff}$ from RIJKL grids using (V-K) from Col. 8.
d Derived from polarization (see Appendix).   e Qiu et al. (2001).


Table 5: Data for F dwarfs that lie within $\sim $50 pc.

HD Dist. $T_{\rm eff}$ a (V-K)a $(V-K)_{\rm0}$a E(B-V)a E(B-V)b $(V-K)_{\rm0}$
    (pc.) (K)         (adopted)
(1) (2) (3) (4) (5) (6) (7) (8) (9)

3268 38 6 087 1.295 1.268 0.010 0.0040 1.284
14594 19445 39 6 014 1.340 1.311 0.010 0.0018 1.335
28644 40832 50 6 551 1.038 1.001 0.013 0.0034 1.029
54383 96574 50 6 113 1.288 1.251 0.013 0.0015 1.284
60098 107213 50 6 303 1.174 1.138 0.013 0.0022 1.168
76568 139798 36 6 756 0.924 0.897 0.010 0.0016 0.920
81800 151044 30 6 061 1.305 1.284 0.008 0.0043 1.293
91058 171620 52 6 129 1.281 1.243 0.014 0.0015 1.277
95492 182807 28 6 105 1.277 1.257 0.007 0.0027 1.270
98946 191096 52 6 783 0.922 0.885 0.014 0.0025 0.915
$\textstyle \parbox{13cm}{
$^{a}$ ~~~~Taken from Di B98. \\
$^{b}$ ~~~~Derived from polarization (see Appendix~A). \\ }$

The two stars with the largest departures from the model predictions are $\alpha $ CMa and $\epsilon$ Sgr. We see from Cols. 9 and 10 in Table 4 that these stars have the largest differences between the E(B-V) estimated by two different methods; this suggests that the correction of (V-K)  for the interstellar extinction may be a significant source of uncertainty in determining $T_{\rm eff}$. Column 10 is derived from the E(V-K)  correction taken from Di B98, who assumes Av = 0.8 mag kpc-1 (or E(B-V) = 0.25 mag kpc-1). Column 9 gives E(B-V)  derived from polarization data, as described in Appendix A. The E(B-V)  derived from the polarizations (Col. 9) are essentially zero while those assumed by Di B98 tend to be larger.

3.2 $\mathsfsl{ T_{eff}}$  from $(\mathsfsl{V-K})$ for stars nearer than 50 pc.

Evidence is given in the Appendix A that the extinction for stars within 50 pc is generally quite low (E(B-V) = 0.0025). There are twelve F stars that are nearer than 50 pc (mean distance 25 pc)[*] whose $T_{\rm eff}$  have been determined by both Di B98 and AAMR96. The mean difference between their estimates (Di B98 minus AAMR96) is very small ($14\pm10$ K). Thus these mean extinction corrections applied by Di B98 and AAMR96 (E(B-V) = 0.006 and 0.000 respectively) are too small to have a large effect on their calculated $T_{\rm eff}$.

We now consider another group of F stars that are closer than 52 pc (Table 5) for which we were able to derive E(B-V) from their polarizations (as explained in Appendix A). The E(B-V) of these stars is sufficently small so that the $T_{\rm eff}$ derived by Di B98 should be little affected by his adopted E(B-V). We have taken the (V-K) from Di B98 and corrected it by the extinction derived from the polarization (Table 5, Col. 8). These $T_{\rm eff}$  and $(V-K)_{\rm0}$  are plotted in Fig. 3; the case where the extinction is that used by Di B98 is shown by filled circles and the case where the extinction is derived from the polarization by open circles. The latter case agrees better with the colour vs. temperature relation given by the synthetic colors. Figure 3 shows that $T_{\rm eff}$  from the synthetic (V-K)  differs by less than 100 K from that given by the empirical relations taken from Di B98. However, the $T_{\rm eff}$  from the models is systematically larger than the empirical $T_{\rm eff}$. The discrepancy between the $T_{\rm eff}$  from the models and that obtained by BL98 using the Infrared flux method is somewhat larger. It is also interesting that the difference between the two empirical $T_{\rm eff}$ vs. (V-K)  relations given by BL98 and DiB98 increases with decreasing temperature.

\par\includegraphics[width=8cm,clip]{3596f3.eps} \end{figure} Figure 3: The ordinate is $(V-K)_{\rm0}$  and the abscissa is $T_{\rm eff}$ (K). The filled circles are ISO standards within 52 pc (for details see Table 6 and text). The filled circles correspond to the extinction used by Di B98 and the open circles to the extinction derived from the polarization. The lines show the relations derived from the RIJKL synthetic grid for solar metallicity and $\log g$ = 4.0 (full line) and 4.5 (dashed line). The dotted line is an empirical relation given by BL98.

Bearing in mind the uncertainties in the extinction, we conclude that the relation between $T_{\rm eff}$  and $(V-K)_{\rm0}$  given by the models is consistent within 100 K with the best data that we have at this time for dwarf stars of solar metallicity.

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