The determination of an effective temperature $T_{\rm eff}$ is an essential preliminary to deriving the chemical abundances in a stellar atmosphere. If a moderately high resolution ( $\lambda$ / $\Delta\lambda \ga 15~000$ ) spectrum is available, several independent methods may be used to derive $T_{\rm eff}$ from the spectra, and their inter-agreement can be used to assess their accuracy (e.g. see Kinman et al. 2000, Table 7). For fainter stars, only a single broad-band colour such as (B-V) may be available to give an observational constraint on $T_{\rm eff}$ . The relation between (B-V) and $T_{\rm eff}$ has recently been discussed by Castelli (1999) for dwarfs and giants and also by Sekiguchi & Fukugita (2000, hereafter SF00) primarily for stars with $T_{\rm eff}$ cooler than 7000 K. For hotter stars, $(B-V)_{\rm0}$ becomes increasingly insensitive to $T_{\rm eff}$ and the (B-V) vs. $T_{\rm eff}$ relation is also quite sensitive to $\log g$ (see Table 1). Caution is needed therefore in the use of the (B-V) vs. $T_{\rm eff}$ relation for stars hotter than 7000 K; not only must $\log g$ be well determined but the accuracy of the method decreases rapidly with increasing temperature (see Table 1).

Table 1: Comparison of $(B-V)_{\rm0}$ and $(V-K)_{\rm0}$ colour vs. $T_{\rm eff}$ relations for various $T_{\rm eff}$ .
&nbs; Change in $T_{\rm eff}$ for

$T_{\rm eff}$ $(B-V)_{\rm0}$ relation for $(V-K)_{\rm0}$ relation for

colour change $\log g$ change colour change $\log g$ change [M/H] change

of 0.01 mag^a of 1.0^b of 0.01 mag^a of 1.0^b of 1.0^c

(1) (2) (3) (4) (5) (6)

7000 K 52 K 178 K 20 K 104 K 60 K

8000 K 59 K 488 K 27 K 123 K 50 K

9000 K 100 K 630 K 49 K 78 K 60 K

10 000 K 172 K 655 K 78 K 16 K 100 K

$\textstyle \parbox{13cm}{ $^{a}$ ~For $\log g= 4.0$\space and ${\rm [M/H]} = 0... ...$ ~From $\log g=4.0$\space and ${\rm [M/H]}=-1.0$\space to ${\rm [M/H]}=-2.0$ }$

**Table 1:** Comparison of $(B-V)_{\rm0}$ and $(V-K)_{\rm0}$ colour vs. $T_{\rm eff}$ relations for various $T_{\rm eff}$ .
&nbs;	Change in $T_{\rm eff}$ for
$T_{\rm eff}$	$(B-V)_{\rm0}$ relation for		$(V-K)_{\rm0}$ relation for
	colour change	$\log g$ change		colour change	$\log g$ change	[M/H] change
	of 0.01 mag^a	of 1.0^b		of 0.01 mag^a	of 1.0^b	of 1.0^c
(1)	(2)	(3)		(4)	(5)	(6)
7000 K	52 K	178 K		20 K	104 K	60 K
8000 K	59 K	488 K		27 K	123 K	50 K
9000 K	100 K	630 K		49 K	78 K	60 K
10 000 K	172 K	655 K		78 K	16 K	100 K

We therefore need another way to estimate $T_{\rm eff}$ which can be used to check that derived from (B-V). A particular application is for metal-poor A-type halo stars with $V\la 15$ . An extensive discussion of empirical $T_{\rm eff}$ calibrations has been given by Bessell et al. (1998). For earlier type stars, they prefer optical colour-indices to derive $T_{\rm eff}$ because "the lower precision of much (V-K) photometry (from independent observations of V and K magnitudes) produces larger uncertainties in the $T_{\rm eff}$ - colour relations''. The 2MASS sky survey provides near-IR magnitudes in the J, H, and $K_{\rm s}$ (K-short) wavebands for stars as faint as 15th magnitude and so in principle can provide another way to estimate $T_{\rm eff}$ providing a sufficiently accurate V-magnitude is available. Obviously the stars must also not be variable or be composite. In this paper we investigate how well the 2MASS magnitudes can be used to derive $T_{\rm eff}$ for fainter hot stars for which the use of (B-V) lacks accuracy.

In Sect. 2 we present the synthetic grids of colour indices used in this paper, which are based on the ATLAS9 (Kurucz 1993) models.

In Sect. 3 we compare the computed $T_{\rm eff}$ vs. $(V-K)_{\rm0}$ relation with the best-determined data for several nearby stars. This includes the "reference'' $T_{\rm eff}$ given by Smalley & Dworetsky (1995) and also the recent $(V-K)_{\rm0}$ and $T_{\rm eff}$ data published by Di Benedetto (1998, hereafter Di B98), Blackwell & Lynas-Gray (1998, hereafter BL98) and Alonso et al. (1996, hereafter AAMR96) for main-sequence stars of solar metallicity. The assumptions that these authors have made about the interstellar extinction affect both their $T_{\rm eff}$ and $(V-K)_{\rm0}$ .

In Sect. 4, we investigate the problem of transforming the 2MASS magnitudes to the Bessell-Brett (1988) homogenized system, so that they will be compatible with the $T_{\rm eff}$ vs. colour relations from Bessell et al. (1998) (hereafter BCP) that are computed in the same photometric system. Finally, in Sect. 5, we compare the $T_{\rm eff}$ that are obtained from $(V-J)_{\rm0}$ , $(V-H)_{\rm0}$ and $(V-K)_{\rm0}$ colours (using 2MASS data) with those obtained in previous investigations. We considered the hotter Hyades dwarfs extracted from the sample studied by de Bruijne et al. (2001) (Sect. 5.1); field blue horizontal branch (BHB) stars already studied by Kinman et al. (2000) in the optical region and by Castelli & Cacciari (2001) in the ultraviolet region (Sect. 5.2); a small number of blue metal-poor (BMP) stars taken from the sample studied by Preston & Sneden (2000) and Wilhelm et al. (1999) (Sect. 5.3); the BMP and BHB stars in the high-latitude field BS 15621 field among those studied by Wilhelm et al. (1999) (Sect. 5.4) and six of the outlying BHB stars in the globular cluster M 13 that were studied by Peterson et al. (1995) and for which reliable 2MASS data are available (Sect. 5.5).

1 Introduction