5 ${{\vec T}_{\sf eff}}$ from the 2MASS color indices and comparison with other determinations

In this section we take a number of recent determinations of $T_{\rm eff}$ and compare them with those obtained from colours determined from the 2MASS magnitudes after transformation to the Bessell-Brett system using Eqs. (5)-(7) in Sect. 4. In making comparisons with other determinations of $T_{\rm eff}$  we have chosen cases in which the correction for interstellar extinction is not a major source of uncertainty (in general, E(B-V) $\leq$ 0.05 mag).

5.1 $\mathsfsl{ T_{eff}}$ for Hyades dwarfs

Figure 5: Hyades members whose $T_{\rm eff}$  is taken from de Bruijne et al. (2001). The colours (ordinates) were determined from 2MASS magnitudes using the transformations in Sect. 4. The full and dashed lines show the computed $T_{\rm eff}$ vs. colour relations from Table 3 for $\log g=4.0$ and $\log g=4.5$, respectively.

The Hyades cluster is sufficiently close (mean distance $\sim$45 pc) that we can assume that E(B-V) = 0.0. In their Hipparcos study of this cluster, de Bruijne et al. (2001) have given $T_{\rm eff}$  for main sequence stars ( $4.33\leq\log g\leq4.36$) which are based on two recent calibrations of the $T_{\rm eff}$ versus (B-V) relation: (1) Bessell et al. (1998) in combination with Alonso et al. (1996) and (2) Lejeune et al. (1997, 1998). We have used the eleven hottest of these Hyades stars ( $T_{\rm eff}>6500$ K) for which 2MASS magnitudes are available. Table 6 lists the stars and their $T_{\rm eff}$  and $\log g$  according to de Bruijne et al. (2001). Their V magnitudes were taken from the Hipparcos Input Catalogue (Turon et al. 1992) and the 2MASS magnitudes were transformed to the Bessell-Brett system with Eqs. (5)-(7). The quoted errors of both the V and the 2MASS magnitudes were used to determine the errors of the colours. These data, given in Table 6, are compared in Fig. 5 with the computed $T_{\rm eff}$ vs. colour relations for $\log g$ = 4.0 and 4.5 that are given in Table 3; the agreement is generally satisfactory. The mean difference between the observed and the synthetic colours for the temperatures adopted by de Bruijne et al. are $+0.002\pm0.013$, $-0.009\pm0.014$ and $+0.010\pm0.009$ for $(V-J)_{\rm0}$, $(V-H)_{\rm0}$ and $(V-K)_{\rm0}$ respectively. The mean differences between the $T_{\rm eff}$ given by de Bruijne et al. and those derived from the synthetic and the observed colours are $-5\pm33$ K, $+23\pm36$ K and $-26\pm23$ K for $(V-J)_{\rm0}$, $(V-H)_{\rm0}$ and $(V-K)_{\rm0}$  respectively.

5.2 $\mathsfsl{ T_{eff}}$ for nearby Blue Horizontal Branch stars

2MASS magnitudes are available for thirteen out of the twenty nine nearby BHB stars that were discussed by Kinman et al. (2000) (KCCBHV); these thirteen stars are listed in Table 7.

The observed dereddened colour indices $(V-J)_{\rm BB0}$, $(V-H)_{\rm BB0}$  and $(V-K)_{\rm BB0}$  given in Cols. 7, 9, and 11 were obtained from the observed 2MASS magnitudes (Cols. 3-5) using the the transformation equation given in Sect. 4, the reddening E(B-V)  given in Col. 6 and the following reddening relations obtained from Mathis (1999) for A_v = 3.1 E(B-V): E(V-J) = 2.23E(B-V), E(V-H) = 2.55E(B-V) and E(V-K) = 2.76E(B-V).

We used Table 2 to derive $T_{\rm eff}$  from the dereddened colour indices. According to KCCBHV the abundances of the $\alpha$-elements are enhanced by about 0.4 dex over the iron in these stars. Since the red colour-indices of these stars have a weak dependance on their metallicity (Fig. 2), we adopted the synthetic colours of Table 2 which were computed from non-$\alpha$-enhanced models. We also adopted the metallicities listed in Col. 2 of Table 7; these are close to those obtained by KCCBHV.

In Fig. 6 we compare the $T_{\rm eff}$  derived from $(V-J)_{\rm BB0}$, $(V-H)_{\rm BB0}$  and $(V-K)_{\rm BB0}$  (using Table 2) with the $T_{\rm eff}$  from the literature taken from KCCBHV, Adelman & Philip (1990 1994 and 1996) and Gray et al. (1996). All these temperatures are summarized in Table 13 of KCCBHV. If T_J is the effective temperature derived from Table 2 for a BHB star of known $(V-J)_{\rm0}$  and [M/H], then we define the difference $\Delta T_{J}$ as the $T_{\rm eff}$ for the BHB star given in the literature minus T_J. The differences $\Delta T_{H}$  and $\Delta T_{K}$  are defined similarly. These differences are shown plotted against T_JHK  in Fig. 6, where T_JHK  is the weighted mean of T_J, T_H and T_K. In the case of the KCCBHV temperatures, these differences are shown by filled circles and the mean values of $\Delta T_{J}$, $\Delta T_{H}$  and $\Delta T_{K}$ are are $+58\pm44$ K, $+110\pm45$ K and $+79\pm40$ K respectively. The corresponding rms deviations are 153 K, 157 K and 137 K. The error bars of $\Delta T_{J}$, $\Delta T_{H}$ and $\Delta T_{K}$ in Fig. 6 take into account the quoted errors of $T_{\rm eff}$ given by KCCBHV and the quoted photometric errors of the 2MASS observations.

The open circles in Fig. 6 are similarly derived from the $T_{\rm eff}$  given by Adelman & Philip (1990, 1994, 1996) and Gray et al. (1996). In this case, the mean values of $\Delta T_{J}$, $\Delta T_{H}$ and $\Delta T_{K}$ are $-122\pm95$ K, $-25\pm119$ K and $-66\pm92$ K respectively and the corresponding rms deviations are 286 K, 356 K and 277 K. When $T_{\rm eff}$ from Castelli & Cacciari (2001, hereafter CC) are considered the mean values of $\Delta T_{J}$, $\Delta T_{H}$ and $\Delta T_{K}$ are $+90\pm50$ K, $+158\pm50$ K and $+129\pm37$ K respectively and the corresponding rms deviations are 167 K, 165 K and 121 K.

Table 8 compares T_JHK with the $T_{\rm eff}$  from KCCBHV (which is based mostly on optical data) and those from CC (based on IUE ultraviolet energy distributions). The mean of the differences (Col. 5) between the KCCBHV $T_{\rm eff}$  and the weighted mean T_JHK is $+93\pm37$ K with an rms deviation of 127 K. The mean of the differences (Col. 6) between the CC $T_{\rm eff}$  and the weighted mean T_JHK is $+138\pm37$ K with an rms deviation of 122 K.

These BHB stars are at distances of several hundred parsecs and at various galactic latitudes, so the uncertainty in their E(B-V)  is at least 0.01 mag. This corresponds to an uncertainty of about 50 K at 7500 K and 180 K at 9000 K in the derived temperatures. Bearing this in mind, the agreement between previously derived values of $T_{\rm eff}$  for BHB stars and those derived from the 2MASS data seems satisfactory.

5.3 $\mathsfsl{ T_{eff}}$ for blue metal-poor stars ( $\mathsfsl{12\leq{V}\leq14}$)

The blue metal-poor (BMP) stars were originally defined by Preston et al. (1994) as having $0.15\leq(B-V)\leq0.36$, $\log g$ $\approx$ 4 and [Fe/H] <-1. They are presumed to be the same as the "Class A'' stars found by Kinman et al. (1994). Preston & Sneden (2000) have obtained echelle spectra of sixty-two of their BMP stars and shown that a high proportion are single-line binaries and likely to be blue stragglers; only 44 of their sample have [Fe/H] <-1. Preston & Sneden derived a preliminary effective temperature from a $T_{\rm eff}$  vs. (B-V), [Fe/H] relation and then adjusted it so as to minimize the variation of the calculated abundance with respect to excitation potential. We picked five of their hottest stars for which 2MASS data are currently available; they are listed in Table 9 and also in Table 10 which gives their $\log g$, [Fe/H] and V from Preston & Sneden (2000). We used the procedure described in Sect. 5.2 to obtain T_J, T_H, T_K  for each star from the 2MASS data using Table 3 and thus derived T_JHK. The colours of these stars were de-reddened using the E(B-V) of SFD (Table 9, Col. 6).

 

 

Table 7: Temperatures T_J, T_H, and T_K for BHB stars derived from Table 2 and 2MASS colours.

HD/BD	Va	$J_{\rm 2M}$	$H_{\rm 2M}$	$K_{\rm 2M}$	E(B-V)	$(V-J)_{\rm BB0}$	TJ	$(V-H)_{\rm BB0}$	TH	$(V-K)_{\rm BB0}$	TK
	[M/H]
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)
2857	9.99	9.495	9.351	9.305	0.022	0.385	7665	0.567	7459	0.580	7467
	-1.5	$\pm$0.030	$\pm$0.025	$\pm$0.033			$\pm$92		$\pm$50		$\pm$70
14829	10.31	10.120	10.044	10.033	0.018	0.088	9001	0.200	8553	0.179	8673
	-2.0	$\pm$0.029	$\pm$0.026	$\pm$0.028			$\pm$190		$\pm$100		$\pm$113
60778	9.10	8.742	8.629	8.651	0.028	0.241	8179	0.387	7908	0.331	8103
	-1.5	$\pm$0.028	$\pm$0.043	$\pm$0.027			$\pm$110		$\pm$115		$\pm$80
74721	8.71	8.537	8.526	8.507	0.012	0.093	8945	0.140	8792	0.129	8876
	-1.5	$\pm$0.044	$\pm$0.034	$\pm$0.029			$\pm$275		$\pm$150		$\pm$138
86986	8.00	7.540	7.515	7.485	0.022	0.354	7777	0.413	7838	0.410	7884
	-1.5	$\pm$0.030	$\pm$0.028	$\pm$0.039			$\pm$95		$\pm$72		$\pm$100
87047	9.72	9.321	9.273	9.229	0.006	0.333	7855	0.421	7828	0.435	7831
	-2.0	$\pm$0.033	$\pm$0.056	$\pm$0.041			$\pm$109		$\pm$140		$\pm$102
109995	7.63	7.295	7.275	7.262	0.010	0.257	8134	0.314	8131	0.296	8222
	-1.5	$\pm$0.028	$\pm$0.022	$\pm$0.021			$\pm$107		$\pm$65		$\pm$62
130095	8.13	7.828	7.856	7.818	0.072	0.084	9002	0.072	9171	0.067	9210
	-1.5	$\pm$0.038	$\pm$0.070	$\pm$0.034			$\pm$250		$\pm$450		$\pm$195
167105	8.97	8.736	8.738	8.725	0.024	0.121	8777	0.151	8741	0.131	8869
	-1.5	$\pm$0.025	$\pm$0.029	$\pm$0.020			$\pm$140		$\pm$130		$\pm$96
202759	9.09	8.431	8.347	8.221	0.072	0.437	7522	0.543	7526	0.626	7390
	-2.0	$\pm$0.036	$\pm$0.063	$\pm$0.050			$\pm$100		$\pm$140		$\pm$101
252940	9.10	8.455	8.370	8.301	0.048	0.476	7401	0.590	7412	0.621	7384
	-1.5	$\pm$0.045	$\pm$0.031	$\pm$0.046			$\pm$122		$\pm$65		$\pm$93
+25 2602	10.12	9.826	9.855	9.829	0.008	0.221	8289	0.229	8428	0.225	8475
	-2.0	$\pm$0.032	$\pm$0.030	$\pm$0.031			$\pm$130		$\pm$110		$\pm$106
+42 2309	10.77	10.554	10.583	10.498	0.013	0.131	8715	0.139	8795	0.193	8594
	-1.5	$\pm$0.034	$\pm$0.036	$\pm$0.022			$\pm$190		$\pm$170		$\pm$90

$\parbox{16.7cm}{ $^{a}$\space Data taken from KCCBHV.}$

Figure 6: Differences between the $T_{\rm eff}$ of BHB stars given in the literature and those derived from 2MASS data and the $T_{\rm eff}$-colours relations for BHB stars given in Table 2. For further details see the text in Sect 5.2.

Table 10 compares the T_JHK  temperature with those from Preston & Sneden (2000) and with those from Wilhelm et al. (1999) who also observed the same BMP stars using UBV photometry and low resolution spectra.

The mean difference between the Preston and Sneden $T_{\rm eff}$ and our T_JHK  is $-63 \pm119$ K and the rms deviation of these differences is 237 K. The mean difference between the $T_{\rm eff}$  of Wilhelm et al. and our T_JHK is $+137 \pm157$ K; the rms deviation of these differences is 314 K. The mean difference between the Preston & Sneden $T_{\rm eff}$  and those of Wilhelm et al. is $-200\pm168$ K and the rms deviation of these differences is 335 K.

The agreement between the T_JHK  and these previous $T_{\rm eff}$  is satisfactory, if we take into account their binary nature and that they are probably all photometric variables. Their V-amplitudes (when known) are given in the footnotes to Table 10. Our use of non-simultaneous optical and infrared magnitudes will clearly produce errors in our temperatures and we have tried to take these and other photometric errors into account in calculating the errors for our T_JHK. We have not taken into account any other errors such as those in our estimated E(B-V). The random errors of both our $T_{\rm eff}$  and those of Preston & Sneden are probably about 150 K, while for Wilhem et al. they are probably $\sim$300 K. The systematic difference between our temperatures and those of Preston & Sneden is not significant.

5.4 $\mathsfsl{ T_{eff}}$ for BMP and BHB stars observed by Wilhelm et al. (1999) in field BS 15621

Wilhelm et al. (1999) have given $T_{\rm eff}$ for large numbers of both BMP and BHB stars. We have chosen one of their fields (BS 15621) for which both E(B-V) is low and 2MASS data are available. We evaluated T_J, T_H, T_K and the weighted T_JHK  for the BMP and BHB stars in this field as described in Sect. 5.2; the results are shown in Table 11. Our adopted $\log g$  and metallicity [M/H] are given in Table 11, Col. 2. The adopted E(B-V)  (Table 11, Col. 6) are taken from SFD and do not differ greatly from those assumed by Wilhelm et al. (Table 12, Col. 5). We interpolated in Table 3 for the BMP stars and in Table 2 for the BHB stars. Wilhelm et al. give an uncertain [Fe/H] of 0.0 for the BHB star BS 15621-0039. We interpolated in Table 3 for this star, since a metallicity ${\rm [M/H]}=0.0$ is not available for BHB stars in Table 2. If we had used Table 2 and ${\rm [M/H]}=-1.0$, the resulting temperature would have been 137 K higher. Our results are given in Table 11 (Cols. 8, 10, 12) and in Table 12 (Col. 6). The errors quoted for T_J, T_H and T_K  are derived from the errors in the photometry while those quoted for the weighted T_JHK are derived from the errors of T_J, T_H and T_K.

Table 12 compares our T_JHK with the $T_{\rm eff}$  from Wilhelm et al. (1999). The difference $\Delta T$ between the $T_{\rm eff}$  of Wilhelm et al. and our T_JHK is shown plotted against T_JHK in Fig. 7. The errors for $\Delta T$ in this plot assume an error of 300 K for the $T_{\rm eff}$  of Wilhelm et al.. The mean difference $<\Delta T>$ for the BMP stars is $+249\pm74$ (K) which is comparable with the difference found for the other BMP star data of Wilhelm et al. (1999) and which we discussed in Sect. 5.3.

Figure 7: Difference $\Delta$T between Wilhelm et al. $T_{\rm eff}$  and TJHK given in Table 13. Filled circles are BHB stars and open circles are BMP stars.

 

 

Table 8: Comparison of $T_{\rm eff}$  from 2MASS colours for BHB stars with the parameters from KCCBHV^a  and CC^b.

HD/BD	TJHK/$\log g$/[M/H]	$T_{\rm eff}$/$\log g$/[Fe/H]	$T_{\rm eff}$/$\log g$/[M/H]	$\Delta$ $T_{\rm eff}$  from	$\Delta$ $T_{\rm eff}$  from
	This paper	KCCBHVa	CCb	KCCBHVa	CCb
2857	$7495\pm37$/3.00/-1.5	7550/3.00/-1.73	7600/2.8/-1.75a	+55	+105
14829	$8659\pm70$/3.20/-2.0	8900/3.20/-2.39	8900/3.1/-2.5a	+241	+241
60778	$8076\pm56$/3.10/-1.5	8050/3.10/-1.49	8250/2.9/-1.50a	-26	+174
74721	$8850\pm95$/3.30/-1.5	8900/3.30/-1.42	8800/3.2/-1.50a	+50	-50
86986	$7833\pm50$/3.20/-1.5	7950/3.20/-1.81	8100/2.8/-1.75a	+117	+267
87047	$7839\pm66$/3.10/-2.0	7850/3.10/-2.47	7900/2.8/-2.50a	+11	+61
109995	$8172\pm41$/3.10/-1.5	8500/3.10-1.72	8500/3.0/-1.75a	+328	+328
130095	$9135\pm145$/3.30/-1.5	9000/3.30/-1.87	9100/3.2/-1.75a	-135	-35
167105	$8813\pm68$/3.30/-1.5	9050/3.30/-1.56	9000/3.1/-1.50a	+237	+187
202759	$7471\pm63$/3.05/-2.0	7500/3.05/-2.16	7500/2.8/-2.00a	+29	+29
252940	$7403\pm49$/2.95/-1.5	7550/2.95/-1.77	7650/2.7/-1.75a	+147	+247
+25 2602	$8410\pm66$/3.20/-2.0	8410/3.17/-1.98	$\cdots$	0	$\cdots$
+42 2309	$8649\pm73$/3.20/-1.5	8800/3.20/-1.63	8750/3.0/-1.75a	+151	+101

$\parbox{13.8cm}{ $^{a}$\space Kinman et~al. (\cite{kin00}). \\ $^{b}$\space Castelli \& Cacciari (\cite{cck}). \\ }$

 

 

Table 9: Temperatures T_J, T_H, and T_K for blue metal poor stars derived from Table 3 and 2MASS colours.

ID	$\log g$	$J_{\rm 2M}$	$H_{\rm 2M}$	$K_{\rm 2M}$	E(B-V)	$(V-J)_{\rm BB0}$	TJ	$(V-H)_{\rm BB0}$	TH	$(V-K)_{\rm BB0}$	TK
CS-	[M/H]
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)
22871-040	4.2	12.147	12.041	12.001	0.101	0.289	8185	0.406	8047	0.396	8100
	-1.5	$\pm$0.026	$\pm$0.031	$\pm$0.031		$\pm$0.030	$\pm$105	$\pm$0.034	$\pm$90	$\pm$0.034	$\pm$90
29497-017	4.2	13.571	13.435	13.459	0.016	0.495	7512	0.668	7416	0.613	7558
	-1.0	$\pm$0.032	$\pm$0.037	$\pm$0.044		$\pm$0.035	$\pm$100	$\pm$0.040	$\pm$85	$\pm$0.046	$\pm$100
22966-043	3.7	13.011	12.852	12.825	0.017	0.451	7605	0.649	7425	0.644	7464
	-2.0	$\pm$0.032	$\pm$0.030	$\pm$0.037		$\pm$0.068	$\pm$190	$\pm$0.067	$\pm$140	$\pm$0.070	$\pm$150
29497-030	4.2	11.960	11.760	11.730	0.016	0.592	7291	0.832	7140	0.831	7163
	-2.0	$\pm$0.031	$\pm$0.031	$\pm$0.029		$\pm$0.034	$\pm$95	$\pm$0.034	$\pm$70	$\pm$0.033	$\pm$65
29499-057	4.5	13.356	13.315	13.226	0.023	0.384	7954	0.462	7993	0.517	7882
	-2.0	$\pm$0.033	$\pm$0.030	$\pm$0.041		$\pm$0.045	$\pm$139	$\pm$0.042	$\pm$106	$\pm$0.051	$\pm$118

 

 

Table 10: Comparison of $T_{\rm eff}$  for blue metal poor stars from various sources.

ID	Preston & Snedena					Wilhelm et al.b				This paper
CS-	V	$T_{\rm eff}$	$\log g$	[Fe/H]		$T_{\rm eff}$	$\log g$	[Fe/H]		$T_{\rm JHK}$
		(K)				(K)				(K)
(1)	(2)	(3)	(4)	(5)		(6)	(7)	(8)		(9)
22871-040c	$12.72\pm0.015$	7880	4.2	-1.66		7722	3.7	-2.1		$8103\pm54$
29497-017d	$14.16\pm0.015$	7500	4.2	-1.19		7768	4.9	-0.5		$7486\pm54$
22966-043e	$13.56\pm0.060$	7300	3.7	-1.96		7577	4.1	-1.4		$7480\pm90$
29497-030	$12.65\pm0.015$	7500	4.2	-2.16		7426	3.9	-2.5		$7180\pm43$
29499-057f	$13.85\pm0.030$	7700	4.2	-2.33		8386	4.3	-2.9		$7946\pm69$

^a Data taken from Preston & Sneden (2000).
^b Data taken from Wilhelm et al. (1999).
^c Light amplitude $\Delta V\sim0.01$ mag (Preston & Landolt 1999).
^d Light amplitude $\Delta V<0.01$ mag (Preston & Landolt 1999).
^e Light amplitude $\Delta V= 0.12$ mag (Preston & Landolt 1999).
^f Light amplitude $\Delta V= 0.04$ mag (Preston & Landolt 1999).

 

 

Table 11: Temperatures T_J, T_H, T_K for a sample of BMP and BHB stars $^{\dagger }$  in BS 15621 by using Tables 2, 3 and 2MASS colours.

BS 15621	$\log g$a	$J_{\rm 2M}$	$H_{\rm 2M}$	$K_{\rm 2M}$	E(B-V)	$(V-J)_{\rm BB0}$	TJ	$(V-H)_{\rm BB0}$	TH	$(V-K)_{\rm BB0}$	TK
Va	[M/H]
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)
-0002	3.8	11.421	11.236	11.207	0.018	0.318	7981	0.542	7620	0.539	7657
11.84	-1.0	$\pm$0.034	$\pm$0.030	$\pm$0.029		$\pm$0.034	$\pm$110	$\pm$0.030	$\pm$67	$\pm$0.029	$\pm$64
-0012	3.8	11.765	11.654	11.628	0.017	0.408	7700.	0.557	7587	0.551	7631
12.27	-1.0	$\pm$0.034	$\pm$0.029	$\pm$0.034		$\pm$0.034	$\pm$100	$\pm$0.029	$\pm$65	$\pm$0.034	$\pm$75
-0022	4.3	12.211	12.087	12.047	0.022	0.560	7295	0.721	7267	0.728	7277
12.88	-0.5	$\pm$0.036	$\pm$0.033	$\pm$0.031		$\pm$0.036	$\pm$103	$\pm$0.033	$\pm$60	$\pm$0.031	$\pm$65
-0037	3.7	11.053	10.973	10.950	0.037	0.447	7569	0.557	7569	0.544	7630
11.64	-1.0	$\pm$0.035	$\pm$0.036	$\pm$0.029		$\pm$0.035	$\pm$100	$\pm$0.036	$\pm$80	$\pm$0.029	$\pm$65
-0040	4.2	13.327	13.159	13.193	0.050	0.313	8071	0.508	7769	0.435	7968
13.81	-1.0	$\pm$0.030	$\pm$0.031	$\pm$0.034		$\pm$0.030	$\pm$112	$\pm$0.031	$\pm$73	$\pm$0.034	$\pm$85
-0041	4.0	12.120	11.937	11.946	0.026	0.362	7835	0.581	7523	0.538	7649
12.60	-0.5	$\pm$0.034	$\pm$0.040	$\pm$0.041		$\pm$0.039	$\pm$122	$\pm$0.034	$\pm$75	$\pm$0.040	$\pm$90
-0048	3.0	11.640	11.531	11.531	0.020	0.457	7470	0.602	7387	0.570	7494
12.20	0.0	$\pm$0.036	$\pm$0.057	$\pm$0.028		$\pm$0.036	$\pm$110	$\pm$0.057	$\pm$123	$\pm$0.028	$\pm$60
-0072	3.8	12.221	12.085	12.046	0.030	0.482	7400	0.653	7277	0.657	7307
12.83	0.0	$\pm$0.031	$\pm$0.033	$\pm$0.047		$\pm$0.031	$\pm$87	$\pm$0.033	$\pm$70	$\pm$0.047	$\pm$100
${\bf -0009}$	2.9	13.265	13.104	13.069	0.022	0.555	7177	0.754	7072	0.756	7106
13.93	-1.5	$\pm$0.032	$\pm$0.034	$\pm$0.037		$\pm$0.032	$\pm$83	$\pm$0.034	$\pm$65	$\pm$0.037	$\pm$70
${\bf -0015}$	3.0	11.841	11.715	11.693	0.020	0.325	7856	0.488	7640	0.478	7700
12.27	-2.0	$\pm$0.034	$\pm$0.035	$\pm$0.049		$\pm$0.034	$\pm$114	$\pm$0.035	$\pm$81	$\pm$0.049	$\pm$120
${\bf -0025}$	3.3	15.031	14.928	14.929	0.023	0.030	9479	0.167	8654	0.134	8839
15.17	-2.0	$\pm$0.047	$\pm$0.071	$\pm$0.110		$\pm$0.047	$\pm$460	$\pm$0.071	$\pm$390	$\pm$0.110	$\pm$710
${\bf -0031}$	2.9	13.800	13.561	13.517	0.030	0.560	7194	0.837	6951	0.846	6970
14.49	-2.0	$\pm$0.033	$\pm$0.040	$\pm$0.052		$\pm$0.033	$\pm$85	$\pm$0.040	$\pm$73	$\pm$0.052	$\pm$100
${\bf -0032}$	3.3	14.592	14.713	14.699	0.031	0.147	8603	0.052	9368	0.031	9493
14.86	-2.0	$\pm$0.040	$\pm$0.068	$\pm$0.106		$\pm$0.040	$\pm$220	$\pm$0.068	$\pm$624	$\pm$0.106	$\pm$950
${\bf -0039}$	3.3	13.369	13.360	13.256	0.049	0.064	8938	0.099	8824	0.165	8509
13.60	0.0	$\pm$0.031	$\pm$0.035	$\pm$0.038		$\pm$0.031	$\pm$220	$\pm$0.035	$\pm$200	$\pm$0.038	$\pm$180
${\bf -0043}$	3.4	14.251	14.234	14.214	0.039	0.016	9633	0.061	9306	0.044	9404
14.41	-2.0	$\pm$0.040	$\pm$0.050	$\pm$0.068		$\pm$0.040	$\pm$410	$\pm$0.050	$\pm$430	$\pm$0.068	$\pm$500

$^{\dagger }$ The ID of BHB stars are shown in boldface.
^a Adopted values for interpolation in Tables 2 and 3.

In the case of the BHB stars the mean difference $<\Delta T>$ is $- 504 \pm211$ (K) and $\Delta T$ becomes increasingly negative as T_JHK  increases. We note that in the case of the four BHB stars whose T_JHK exceeds 8000 K, the $\log g$ assumed by Wilhelm et al. (1999) are significantly less than those predicted by Eq. (9) for the T_JHK.

An inspection of the $T_{\rm eff}$ that Wilhem et al. derive for BHB stars show that they are cooler than might be expected; thus 9% are less than 7000 K, 50% between 7000 and 8000 K, 36% between 8000 and 9000 K and 5% greater than 9000 K. Also, the $(B-V)_{\rm0}$ of a few of the coolest of these stars suggests that they may be RR Lyrae stars or even (e.g. CS 16027-0049 with $(B-V)_{\rm0}$ = 0.54 and $T_{\rm eff}$  = 6200 K) that they belong to to the red horizontal branch. We would expect the BHB stars to have $T_{\rm eff}$ that range from 7600 K corresponding to $(B-V)_{\rm0}$ $\sim+0.20$ at the blue end of the instability strip to temperatures greater than 10 000 K. It therefore seems that systematic errors may be present in the $T_{\rm eff}$  of their whole BHB sample. A further investigation of this is published elsewhere (Kinman & Miller 2002) and shows that the trend shown for the BHB stars in Fig. 7 is present in a much larger sample and is related to the difference between the $\log g$ used by Wilhelm et al. (1999) and that predicted from T_JHK and Eq. (2).

5.5 The BHB stars in the globular cluster M 13

Peterson et al. (1995) have determined $T_{\rm eff}$  for BHB stars in the globular cluster M 13. 2MASS data are available for these stars but the errors in the 2MASS $K_{\rm s}$ magnitudes are too large for reliable colours to be derived from them. We used the V magnitudes of Cudworth & Monet (1979) and E(B-V) from SFD and assumed [Fe/H] = -1.5 to derive $T_{\rm eff}$ using Table 2. These $T_{\rm eff}$ are compared with those of Peterson et al. in Table 13. The errors for T_J and T_H were derived by assuming an error of 0.03 mag in V and the quoted errors for the 2MASS magnitudes. The difference between the $T_{\rm eff}$  of Peterson et al. and the mean of T_J and T_H is given as $\Delta$ $T_{\rm eff}$  in Col. 9. Its mean value $< \Delta T_{\rm eff}>~ = +152\pm154$ and the rms deviation of these differences is 344 K. Considering the faintness of these stars and consequently the relatively large errors in the colours and derived temperatures, this agreement is satisfactory. As Peterson et al. point out, there is significant uncertainty in the V-magnitudes of these stars which could produce a systematic error in the resulting temperatures. There is no indication, however, of differences $\Delta$ $T_{\rm eff}$  as large as those found for the field BHB stars observed by Wilhelm et al.

 

 

Table 12: Comparison of $T_{\rm eff}$  from 2MASS colours for BMP and BHB stars in field BS 15621 with $T_{\rm eff}$  from Wilhelm et al. (1999).

ID $^{\dagger }$	Wilhelm et al.					2MASS
15621-	$T_{\rm eff}$ (K)	$\log g$	[Fe/H]	E(B-V)		TJHK (K)
(1)	(2)	(3)	(4)	(5)		(6)
0002	8390	3.8	-1.2	0.01		$7691\pm43$
0012	7778	3.8	-0.9	0.00		$7624\pm44$
0022	7521	4.3	-0.6	0.01		$7275\pm41$
0037	7875	3.7	-1.1	0.04		$7598\pm45$
0040	8131	4.2	-0.9	0.04		$7896\pm50$
0041	7759	4.0	-0.4	0.01		$7622\pm52$
0048	7520	3.8	0.0	0.02		$7473\pm48$
0072	7517	3.8	-0.1	0.01		$7321\pm48$
0009	7093	2.9	-1.4	0.01		$7110\pm41$
0015	7511	2.9	-2.2	0.00		$7710\pm58$
0025	8254	2.9	-2.6:	0.01		$8975\pm274$
0031	7034	3.0	-1.9:	0.01		$7034\pm48$
0032	8271	2.9	-2.2:	0.01		$8724\pm203$
0039	8066	2.6	0.0:	0.04		$8728\pm114$
0043	7984	2.8	-3.0:	0.03		$9458\pm255$

$\parbox{9.1cm}{ $^{\dagger}$ ~The ID of BHB stars are shown in boldface. }$

 

 

Table 13: $T_{\rm eff}$  for BHB stars in M 13.

ID	r $^{\ddagger}$	$V_{\rm0}$	$(B-V)_{\rm0}$	E(B-V)	TJ	TH	$T_{\rm eff}$ (Peterson et al. $^{\dagger }$)	$\Delta$ $T_{\rm eff}$
	(arcsec)				(K)	(K)	(K)	(K)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
I-64	214	14.94	0.093	0.017	$7720\pm160$	$7800\pm160$	7970	+210
IV-83	225	15.02	0.114	0.016	$8385\pm250$	$8330\pm310$	8962	+604
II-68	233	14.93	0.031	0.019	$8680\pm340$	$8605\pm385$	8595	-48
J 52	395	15.03	0.084	0.016	$8650\pm275$	$8650\pm420$	8244	-406
J 11	398	14.94	0.054	0.016	$7520\pm150$	$7450\pm135$	7784	+299
SA 368	414	15.08	0.042	0.018	$8515\pm270$	$8150\pm260$	8586	+254

$\parbox{15cm}{ $^{\ddagger}$ ~Radial di... ...\\ $^{\dagger}$ ~From Table~\ref{tab6} of Peterson et~al. (\cite{pet95}).\\ }$