2 The grids of synthetic colors

The synthetic grids of VJHK colours used in this paper are based on the ATLAS9 Kurucz models computed by Castelli with the overshooting option for the convection switched off (NOVER grids, Castelli et al. 1997). When (V-K)  colours in the Johnson (J) system are considered, the NOVER RIJKL grids available at http://kurucz.harvard.edu were used, while when (V-J), (V-H), and (V-K)  colours in the Bessell-Brett (BB) homogenized system are considered, the BCP NOVER grid of colours were used. We recall that for (V-K), the conversion from the J system to the BB system is (Bessell & Brett 1988):

$$(V-K)_{\rm J} = 1.007[(V-K)_{\rm BB}-0.01].$$ (1)

We used the BCP colours to generate two tables of synthetic indices $(V-J)_{\rm BB}$, $(V-H)_{\rm BB}$, and $(V-K)_{\rm BB}$  vs.  $T_{\rm eff}$. Table 2 gives $T_{\rm eff}$ vs. colour relations specifically for BHB stars in the interval 7000 to 10 500 K for ${\rm [M/H]}=-1.0$, -1.5, and -2.0. The $T_{\rm eff}$ vs. colour relations were set up by assuming that $\log g$  satisfies the empirical relation:

$$\log g = 4.375\log T_{\rm eff} - 13.967$$ (2)

which we derived from the field BHB stars studied by Kinman et al. (2000). This relation is compatible with the data for cluster BHB stars with $T_{\rm eff}$ $\la$ 10 000 K discussed by Moehler (2001). The colours versus $T_{\rm eff}$  relations of Table 2 were obtained by interpolating in the synthetic indices for a given $T_{\rm eff}$  and the specific $\log g$ derived from the above linear relation.

Table 3 gives $T_{\rm eff}$  for $(V-J)_{\rm BB}$, $(V-H)_{\rm BB}$, and $(V-K)_{\rm BB}$ in the interval 6500 to 10 500 K for metallicities ${\rm [M/H]}=0.0$, -1.0, -1.5 and -2.0. This table is in two parts; the first for $\log g$ = 4.0 and the second for $\log g$ = 4.5. In both Tables 2 and 3 the step in $T_{\rm eff}$  is 10 K.

Figures 1 and 2 show the effect of gravity and the effect of the metallicity respectively on the relations $T_{\rm eff}$ vs.  $(V-J)_{\rm BB}$, $T_{\rm eff}$ vs.  $(V-H)_{\rm BB}$, and $T_{\rm eff}$ vs.  $(V-K)_{\rm BB}$. Table 1 shows the effect of gravity on the $T_{\rm eff}$ vs.  $(V-K)_{\rm BB}$ relation indicating that it is at a maximum at 8000 K with $\Delta$ $T_{\rm eff}$ = 123 K for $\Delta$$\log g$ = 1.0. Figure 1 shows that this effect is of the same order as for the $T_{\rm eff}$ vs.  $(V-J)_{\rm BB}$ and $T_{\rm eff}$ vs.  $(V-H)_{\rm BB}$ relations. The effect of errors in the metallicity on the $T_{\rm eff}$ vs. colour relations are not larger than those caused by gravity. Table 1 shows that, for (V-K), the largest difference in $T_{\rm eff}$  produced by a [M/H] change of 1.0 is about 100 K. Figure 2 shows that this behaviour is similar for all the three colour indices.

Figure 1: Computed $T_{\rm eff}$ vs. colour relations for different $\log g$  and ${\rm [M/H]}=0.0$.

Figure 2: Computed $T_{\rm eff}$ vs. colour relations for different metallicities and $\log g$ = 4.0.

Houdashelt et al. (2000) have used updated MARC-SSG models to obtain colours on the Johnson-Glass system for stars with 4000 K $\leq$  $T_{\rm eff}$ $\leq$ 6500 K. Their hottest model (6500 K) is somewhat cooler than the temperatures with which we are concerned in this paper. Nevertheless, it may be interesting to note that at this temperature, for $\log g$ = 4.0, ${\rm [Fe/H]}=0.0$, their relation gives a V-K = 1.047 compared with 1.079 for the BCP colors. Thus, at 6500 K, their model gives a $T_{\rm eff}$  which is $\sim$56 K cooler than that given by BCP. Such systematic differences do not seem unreasonable considering the use of different atmospheric models and the different calibrations for the colours.