4 The intrinsic colours of giant stars

In this section we present the intrinsic colours of giant stars. Tables 5, 6 and 7 have been obtained by means of the plain numerical inversion of the fits derived in the preceding sections, and the direct use of the photometric database of the stars of the sample. We have checked that the quotients between temperatures derived by applying the different calibrations are consistent within the limit of accuracy of the temperatures and photometry used in the present programme.

The derivation of colours in the range of low temperatures ( $T_{{\rm eff}}\leq$ 4500 K) is uncertain because the gradient $\vert\Delta T_{{\rm eff}}/\Delta{\rm (colour)}\vert$ has in general small values (i.e. a minor variation in temperature implies a large variation in colour), and the size of temperature errorbars of stars in this range is large. The effect of this point is clearly illustrated by the case of (R-I). This colour can be derived either from Eq. (7), or combining Eqs. (5) and (6). Discrepancies are found between both approaches.

 

 

Table 5: Intrinsic broad band colours of giant stars in the system of Johnson for metallicities 0.2, and 0. Columns 2-11 (U-V), (B-V), (V-R), (V-I), (R-I), (V-K), (J-K), (J-H), (I-K), (V-L). Column 12. Bolometric correction to V. The values in brackets are close to the range of validity of calibrations and therefore less reliable

$T_{{\rm eff}}$ (K)	(U-V)	(B-V)	(V-R)	(V-I)	(R-I)	(V-K)	(J-K)	(J-H)	(I-K)	(V-L)	BC(V)
[Fe/H] = +0.2
3500	--	(1.930)	(1.685)	(2.800)	--	--	(1.250)	(0.975)	(1.955)	--	--
3750	--	(1.705)	(1.365)	(2.210)	(1.040)	(4.125)	(1.080)	(0.855)	(1.715)	(4.230)	(-1.425)
4000	3.315	1.515	1.155	1.865	0.825	3.450	0.935	0.745	1.505	3.560	-0.957
4250	2.885	1.345	1.000	1.625	0.695	2.995	0.815	0.660	1.325	3.100	-0.687
4500	2.475	1.190	0.875	1.435	0.600	2.640	0.715	0.585	1.170	2.735	-0.495
4750	2.090	1.055	0.770	1.275	0.520	2.355	0.625	0.515	1.035	2.435	-0.351
5000	1.720	0.935	0.680	1.135	0.460	2.115	0.550	0.450	0.915	2.170	-0.259
5250	1.320	0.825	0.605	1.015	0.405	1.890	0.480	0.400	0.805	1.940	-0.186
5500	0.995	0.725	0.540	0.910	0.360	1.690	0.420	0.350	0.705	1.735	-0.131
5750	0.800	0.625	0.485	0.815	0.320	1.505	0.365	0.300	0.620	1.550	-0.089
6000	0.650	0.550	0.430	0.725	0.285	1.335	0.315	0.265	0.540	1.380	-0.057
6250	0.530	0.475	0.385	0.645	0.250	1.175	0.270	0.225	0.465	1.220	-0.032
6500	0.425	0.410	0.345	0.575	0.225	1.030	0.235	0.190	0.400	1.075	-0.014
6750	0.400	0.350	0.305	0.505	0.195	0.895	0.195	0.160	0.340	0.950	-0.001
7000	0.380	0.295	0.270	0.440	0.175	0.770	0.160	0.135	0.280	0.815	+0.007
7250	(0.360)	0.245	0.240	0.380	0.150	0.650	0.130	0.105	0.230	0.695	+0.011
7500	(0.340)	0.200	0.210	0.325	0.135	0.540	0.100	0.080	0.180	0.585	+0.013
7750	(0.320)	(0.160)	0.185	0.275	0.115	0.440	0.070	0.060	0.135	0.480	+0.011
8000	--	(0.120)	(0.160)	(0.225)	(0.100)	(0.345)	(0.050)	(0.040)	(0.090)	(0.385)	(+0.007)
[Fe/H] = 0.0
3500	--	(1.925)	(1.660)	(2.800)	--	--	(1.250)	(0.980)	(1.955)	--	--
3750	--	(1.695)	(1.350)	(2.210)	(1.005)	(4.095)	(1.080)	(0.860)	(1.715)	(4.230)	(-1.411)
4000	3.240	1.495	1.145	1.865	0.815	3.435	0.935	0.755	1.505	3.560	-0.947
4250	2.790	1.320	0.990	1.625	0.690	2.980	0.815	0.670	1.325	3.100	-0.680
4500	2.365	1.165	0.865	1.435	0.595	2.630	0.715	0.590	1.170	2.735	-0.492
4750	1.955	1.030	0.760	1.275	0.520	2.345	0.625	0.520	1.035	2.435	-0.351
5000	1.565	0.905	0.680	1.135	0.460	2.105	0.550	0.460	0.915	2.170	-0.260
5250	1.160	0.790	0.600	1.015	0.405	1.885	0.480	0.405	0.805	1.940	-0.189
5500	0.910	0.690	0.535	0.910	0.360	1.685	0.420	0.355	0.705	1.735	-0.135
5750	0.730	0.605	0.480	0.815	0.320	1.500	0.365	0.310	0.620	1.550	-0.094
6000	0.590	0.530	0.425	0.725	0.285	1.335	0.315	0.270	0.540	1.380	-0.063
6250	0.475	0.460	0.380	0.645	0.255	1.180	0.270	0.235	0.465	1.220	-0.040
6500	0.375	0.395	0.340	0.575	0.225	1.035	0.235	0.200	0.400	1.075	-0.023
6750	0.320	0.335	0.300	0.505	0.200	0.900	0.195	0.170	0.340	0.950	-0.012
7000	0.300	0.285	0.265	0.440	0.175	0.775	0.160	0.140	0.280	0.815	-0.005
7250	(0.280)	0.235	0.235	0.380	0.155	0.660	0.130	0.110	0.230	0.695	-0.001
7500	(0.260)	0.190	0.205	0.325	0.135	0.550	0.100	0.090	0.180	0.585	-0.001
7750	(0.250)	(0.145)	0.180	0.275	0.120	0.445	0.070	0.065	0.135	0.480	-0.003
8000	--	(0.110)	(0.155)	(0.225)	(0.100)	(0.350)	(0.050)	(0.045)	(0.090)	(0.385)	(-0.008)

 

 

Table 6: Intrinsic broad band colours of giant stars in the system of Johnson for metallicities -1, -2 and -3. Columns 2-11 (U-V), (B-V), (V-R), (V-I), (R-I), (V-K), (J-K), (J-H), (I-K), (V-L). Column 12. Bolometric correction to V. The values in brackets are less reliable

$T_{{\rm eff}}$ (K)	(U-V)	(B-V)	(V-R)	(V-I)	(R-I)	(V-K)	(J-K)	(J-H)	(I-K)	(V-L)	BC(V)
[Fe/H] = -1.0
3500	--	(1.925)	(1.570)	--	--	--	--	--	--	--	--
3750	(3.505)	(1.665)	(1.300)	(2.210)	(0.925)	(3.985)	(1.080)	(0.875)	--	--	(-1.350)
4000	2.935	1.440	1.105	1.865	0.775	3.365	0.935	0.770	1.505	--	-0.905
4250	2.380	1.240	0.960	1.625	0.670	2.930	0.815	0.685	1.325	--	-0.657
4500	1.830	1.065	0.840	1.435	0.590	2.590	0.715	0.605	1.170	--	-0.486
4750	1.355	0.905	0.740	1.275	0.520	2.315	0.625	0.535	1.035	--	-0.358
5000	1.035	0.775	0.660	1.135	0.465	2.085	0.550	0.475	0.915	--	-0.273
5250	0.800	0.695	0.585	1.015	0.415	1.875	0.480	0.420	0.805	--	-0.209
5500	0.625	0.605	0.520	0.910	0.375	1.680	0.420	0.375	0.705	--	-0.162
5750	0.485	0.530	0.465	0.815	0.335	1.505	0.365	0.330	0.620	--	-0.128
6000	0.370	0.460	(0.415)	(0.725)	0.305	1.345	0.315	0.290	0.540	--	-0.104
6250	(0.270)	0.400	--	--	0.275	1.195	0.270	0.250	(0.465)	--	-0.087
6500	--	(0.340)	--	--	(0.245)	1.060	0.235	0.220	--	--	-0.076
6750	--	--	--	--	--	(0.930)	--	--	--	--	(-0.070)
[Fe/H] = -2.0
3750	--	--	--	--	--	--	--	--	--	--	(-1.306)
4000	2.830	(1.450)	(1.090)	1.865	(0.765)	(3.340)	(0.935)	(0.765)	(1.505)	--	-0.881
4250	2.130	1.220	0.950	1.625	0.670	2.915	0.815	0.675	1.325	--	-0.651
4500	1.430	1.015	0.835	1.425	0.595	2.580	0.715	0.600	1.170	--	-0.497
4750	1.085	0.835	0.735	1.275	0.535	2.310	0.625	0.530	1.035	--	-0.379
5000	0.820	0.710	0.655	1.135	0.480	2.085	0.550	0.470	0.915	--	-0.296
5250	0.625	0.645	0.580	1.015	0.435	1.885	0.480	0.415	0.805	--	-0.240
5500	0.475	0.565	0.520	0.910	0.395	1.700	0.420	0.370	0.705	--	-0.200
5750	(0.350)	0.495	(0.465)	(0.815)	0.360	1.535	0.365	0.325	(0.620)	--	-0.173
6000	--	0.435	--	--	--	1.380	0.315	0.285	--	--	-0.155
6250	--	0.375	--	--	--	1.235	0.270	(0.250)	--	--	-0.144
6500	--	(0.325)	--	--	--	(1.105)	(0.235)	(0.215)	--	--	(-0.139)
[Fe/H] = -3.0
4000	--	--	--	--	--	--	--	--	--	--	(-0.874)
4250	--	--	--	1.625	--	(2.930)	(0.815)	0.645	(1.325)	--	-0.662
4500	(1.320)	(1.050)	0.845	1.435	0.615	2.600	0.715	0.570	1.170	--	-0.525
4750	0.975	0.815	0.750	1.275	0.560	2.330	0.625	0.500	1.035	--	-0.415
5000	0.735	0.710	0.670	1.135	0.505	2.110	0.550	0.445	0.915	--	-0.330
5250	0.560	0.630	0.595	1.015	0.465	1.920	0.480	0.390	0.805	--	-0.281
5500	--	0.555	(0.535)	(0.910)	0.425	1.740	0.420	(0.340)	--	--	(-0.249)
5750	--	(0.495)	--	--	(0.390)	--	--	(0.300)	--	--	--

In Figs. 18a-d, we show the comparison between intrinsic colours of Table 2 and the empirical calibration of von Braun et al. (1998) fixing (V-K). Differences for (B-V) and (U-B) (colours which are more affected by blanketing effects) are stronger, while a better agreement is found for infrared colours (V-I) and (J-K). In Figs. 18c,d, we show also the comparison between intrinsic colours of Table 2 and the intrinsic colours of Bessell & Brett (1988) for solar metallicity stars. A fairly good agreement is seen for (J-K), however differences in (V-I) are around 0.10 mag for $(V-I)~\raisebox{-0.6ex}{$\stackrel{\textstyle >}{\sim}$ }~3$.

The effects of gravity on broad band colours and effective temperatures are in general of the order of observational errors, remaining thus concealed when considering individual stars. However, the average relations derived in the above sections are accurate enough to permit us to address this question. For this purpose, we show in Fig. 19 the comparison between colour: [Fe/H]: $T_{{\rm eff}}$ relations of giant stars derived in the above sections and those corresponding to main sequence stars (Paper III).

A noteworthy feature of the present relation $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(B-V)$ is that its global shape slightly differs from that corresponding to dwarf stars -classes V-VI- calibration (Fig. 19a), so that for a fixed (B-V), giants' temperatures are higher than dwarfs' in the range $(B-V)~\raisebox{-0.6ex}{$\stackrel{\textstyle >}{\sim}$ }~0.7$ and smaller in the range $(B-V)~\raisebox{-0.6ex}{$\stackrel{\textstyle <}{\sim}$ }~0.7$( $\Delta T_{{\rm eff}}\approx$ 300 K at (B-V)=0.2, $\Delta T_{{\rm eff}}\approx$ 0 K at (B-V)=0.7 and $\Delta T_{{\rm eff}}\approx -200$ K at (B-V)=1.2). The probable reason for this behaviour can be understood taking into account the variation of Paschen's continuum, and the TiO bands at 4954 Å and 5167 Å with gravity and temperature in the ranges considered. For 8000 K $\raisebox{-0.6ex}{$\stackrel{\textstyle >}{\sim}$ }~T_{{\rm eff}}~\raisebox{-0.6ex}{$\stackrel{\textstyle >}{\sim}$ }~T_{\odot}$ the shape of Paschen's continuum is dominated by the bound-free absorption of atomic hydrogen and the H^- ion. The wavelength dependence of the former is stronger, and its contribution becomes increasingly more important towards higher temperatures. Furthermore, the H bound-free opacity does not depend on the electron pressure, whereas the H^- bound-free absorption does. For a given $T_{{\rm eff}}$, the lower the atmospheric gravity the smaller the electron pressure, making the H^- bound-free contribution to decrease, and the continuum slope to become steeper. For the range $T_{\odot}~\raisebox{-0.6ex}{$\stackrel{\textstyle >}{\sim}$ }~T_{{\rm eff}}~\raisebox{-0.6ex}{$\stackrel{\textstyle >}{\sim}$ }~4000$ K, the true continuum is not very sensitive to gravity variations, but the strong TiO bands integrated within the V filter become more prominent with larger gravities (increasing the value of V magnitude). In conclusion, the combination of both mechanisms results in the observed effect: for a fixed temperature, giants have bluer (B-V) colours in the range $T_{{\rm eff}}~\raisebox{-0.6ex}{$\stackrel{\textstyle >}{\sim}$ }~5500$ K, and redder ones in the $T_{{\rm eff}}~\raisebox{-0.6ex}{$\stackrel{\textstyle <}{\sim}$ }$ 5500 K ( $\Delta(B-V)\approx$ 0.05 mag at $T_{{\rm eff}}=7000$ K, $\Delta(B-V)\approx$ 0.0 mag at $T_{{\rm eff}}=5500$ K, and $\Delta(B-V)\approx -0.15$ mag at $T_{{\rm eff}}=4000$ K).

The relation $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(R-I)$ for giants stars has no significant differences with that of dwarfs (Fig. 19b). Hence (R-I) is a temperature indicator free of surface gravity effects.

The effect of gravity on the relation $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(V-I)$ is also appreciable as in the case of (B-V) (Fig. 19c). In the whole range of temperatures considered, (V-I) colours of dwarf stars are bluer than those of giant stars of the same temperature ( $\Delta(V-I)\approx$ 0.1 mag at $T_{{\rm eff}}=7000$ K, $\Delta(V-I)\approx$ 0.04 mag at $T_{{\rm eff}}=5000$ K, and $\Delta(V-I)\approx$ 0.15 mag at $T_{{\rm eff}}=4000$ K). Conversely, temperatures of dwarfs are greater at fixed colour ( $\Delta T_{{\rm eff}}\approx$ 280 K at (V-I)=0.5, $\Delta T_{{\rm eff}}\approx$ 100 K at (V-I)=1 and $\Delta T_{{\rm eff}}\approx$ 120 K at (V-I)=2). The effect is again, probably related to the behaviour of opacity with gravity in V and I bands. The effect of gravity, although not shown in Fig. 19, is also appreciable in the case of (V-R), (J-H) and (J-K).

Finally, the present relation $T_{{\rm eff}}{:}~{\rm [Fe/H]}=\!{:}~(V-K)$for giant stars differs slightly from that of dwarfs (Fig. 19d): Over 6000 K dwarfs' temperatures are slightly higher than giants' ($\sim$ 50 K at $(V-K)\approx 0.2$), conversely (V-K) is redder for dwarfs ($\sim$ 0.03 mag). Under 4000 K dwarfs' temperatures are smaller ($\sim$ 100 K at $(V-K)\approx 4$), conversely (V-K) is bluer for dwarfs ($\sim$ 0.1 mag at 3900 K). In the range 6000 K $\raisebox{-0.6ex}{$\stackrel{\textstyle >}{\sim}$ }~ T_{{\rm eff}}~\raisebox{-0.6ex}{$\stackrel{\textstyle >}{\sim}$ }$ 4000 K dwarf and giant temperatures and colours are indistinguishable. The small size of the effect confirms that (V-K) is a temperature indicator which has only marginal dependence on stellar surface gravity.

 

 

Table 7: Intrinsic Strömgren colours of giant stars

	[Fe/H] = 0.20		[Fe/H] = 0.00		[Fe/H] = -1.00		[Fe/H] = -2.00		[Fe/H] = -2.50
$T_{{\rm eff}}$	(u-b)	(b-y)	(u-b)	(b-y)	(u-b)	(b-y)	(u-b)	(b-y)	(u-b)	(b-y)
8000	--	(0.070)	(1.56)	(0.065)	--	--	--	--	--	--
7750	--	0.100	1.54	0.095	--	--	--	--	--	--
7500	--	0.130	1.50	0.125	--	--	--	--	--	--
7250	--	0.160	1.48	0.155	--	--	--	--	--	--
7000	--	0.195	1.46	0.185	--	--	--	--	--	--
6750	--	0.230	1.44	0.225	--	--	--	--	--	--
6500	--	0.270	1.42	0.260	--	(0.240)	--	--	--	--
6250	--	0.310	1.43	0.300	--	(0.275)	--	(0.280)	--	--
6000	--	0.355	1.47	0.345	--	0.315	--	(0.315)	--	--
5750	(1.605)	0.400	1.54	0.390	--	0.360	--	0.355	--	--
5500	1.825	0.455	1.72	0.440	--	0.405	--	0.400	--	0.405
5250	2.070	0.510	1.965	0.500	(1.54)	0.460	--	0.450	--	0.450
5000	2.345	0.580	2.24	0.560	1.815	0.515	(1.555)	0.500	(1.47)	0.500
4750	2.650	0.645	2.545	0.630	2.125	0.575	1.870	0.560	1.79	0.580
4500	3.000	0.725	2.895	0.710	2.475	0.665	2.225	0.665	2.155	0.690
4250	3.405	0.820	3.295	0.810	2.875	0.770	2.630	0.790	2.575	(0.835)
4000	--	0.945	(3.765)	0.935	3.340	(0.910)	3.105	(0.965)	(3.06)	--
3750	--	--	--	(1.105)	(3.890)	--	(3.665)	--	--	--