Notes on the convection in the ATLAS9 model atmospheres

Transcription

1 Astron. Astrophys. 318, (1997) ASTRONOMY AND ASTROPHYSICS Notes on the convection in the ATLAS9 model atmospheres F. Castelli 1, R.G. Gratton 2, and R.L. Kurucz 3 1 CNR-Gruppo Nazionale Astronomia and Osservatorio Astronomico, Via G.Tiepolo 11, I Trieste, Italy 2 Osservatorio Astronomico di Padova, Vicolo dell Osservatorio 5, I Padova, Italy 3 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA Received 9 April 1996 / Accepted 20 June 1996 Abstract. The mixing-length theory for the convection, as it is used in the ATLAS9 code (Kurucz, 1993a), is summarized and discussed. We investigated the effect of the modification called approximate overshooting on the model structure of the Sun and of stars with T eff included between 4000 K and 8500 K, log g included between 2.5 and 4.5, and metallicities [M/H]=0.0 and [M/H]= 3.0. We found that the Kurucz solar model (SUNK94) with the overshooting option switched on reproduces more observations than that without overshooting. In the H γ and H β regions no solar model is able to reproduce the level of the true continuum deduced from high-resolution observations absolutely calibrated. At 486 nm the computed continuum is about 6.6% higher than that inferred from the observed spectrum. We found that the largest effect of the approximate overshooting on the model structure occurs for models with T eff >6250 K and it decreases with decreasing gravity. The differences in (b y), (B V), and (V K) indices computed from models with the overshooting option switched on and off, correspond to T eff differences which may amount up to 180 K, 100 K, 60 K respectively. The differences in T eff from Balmer profiles may amount up to 340 K and they occur also for T eff < 6250 K down to about 5000 K. The c 1 index yields gravity differences log g as a function of log g which, for each T eff, grow to a maximum value. The maximum log g decreases with increasing temperatures and ranges, for solar metallicity, from 0.7 dex at log g=0.5 and T eff =5500 K to 0.2 dex at log g=4.5 and T eff =8000 K. This behaviour does not change for [M/H] = 3.0. Comparisons with the observations indicate that model parameters derived with different methods are more consistent when the overshooting option is switched off (NOVER models), except for the Sun. In particular for Procyon, T eff and log g from NOVER models are closer to the parameters derived from model independent methods than are T eff and log g derived from the Kurucz (1995) grids. However, no model is able to explain the whole observed spectrum of either the Sun or Procyon with a unique T eff, regardless of whether the overshooting option is switched on or off. Independently of the convection option, the largest differences Send offprint requests to: F. Castelli in T eff derived with different methods are of the order of 200 K for Procyon and 150 K for the Sun. Key words: stars: atmospheres stars: fundamental parameters stars: individual: Procyon Sun: general convection 1. Introduction Stellar observations are very often interpreted by means of the stellar model atmospheres, so that it is very important to know which is the physics used for the model calculations. In particular, cool stars are affected by convection and the more the stars are metal deficient the more the convection plays an important role in their atmospheres. Because Kurucz model atmospheres, together with the corresponding fluxes and colours, are widely used in several fields of the astrophysical research, we estimated it worthwhile to discuss how convection is handled in the ATLAS9 codes stored on the CD-ROM No. 13 (Kurucz, 1993a) and in the successive update versions, and which were used for computing the grids of models, fluxes, and colours stored on the CD-ROMs No. 13 (Kurucz, 1993a), No. 19 (Kurucz, 1994), and successive releases (Kurucz, 1995). The basic convection theory in ATLAS9 is the mixinglength theory as described in Kurucz (1970), but some modifications have been added after the year 1970, the most important being related to (1) the optical thickness of the convective elements, (2) the horizontal averaging of the opacity, and (3) the inclusion of an approximate overshooting. The first two modifications are discussed in Lester, Lane, and Kurucz (1982). The first one was incorporated in the several ATLAS versions successive to ATLAS5 (Kurucz, 1970), while the second one was finally included in ATLAS9. In this paper we recall the formulas on which convection in ATLAS9 is based, and we discuss in particular the third modification and its effects on the (B V), (V K), (b y), c 1 color indices and on the Balmer profiles. We also discuss the ATLAS9 model for the Sun. Finally, Procyon (HR 2943 = HD 61421=α CMi, F5 IV-V) is used as example to show how

2 842 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres the different convection options in the models affect the value of the effective temperature T eff, derived from the several methods which compare observed and computed quantities. 2. Convection in ATLAS9 We briefly summarize the mixing-length formulation and its modifications and indicate the values of the free parameters adopted for computing the grids of the ATLAS9 models (Kurucz, 1993a, 1994, 1995). The complete description of the mixing-length theory can be found in several papers and textbooks. For our summary we refer mostly to Vitense (1953), Cox & Giuli (1968), and Mihalas (1970) The mixing-length approach (ML) and the first ATLAS modification: γ thin The basic idea of the mixing-length theory is that the convective energy is carried out by upward and downward hot and cool bubbles, which travel for some distance L, the mixing-length, before dissolving. Convection may occur when the density of the bubble decreases at least as rapidly as the density of the surrounding medium. The Schwarzschild surface, where the condition is no longer fulfilled, represents the border of the convection zone in the atmosphere. Observations of the solar granulation have shown that this border does not correspond to the real situation in stars. The mixing-length L is a free parameter having such values that the mixing-length to the pressure scale height ratio L/H p is usually included between 0.5 and 2.0, for both models of internal structure and atmosphere. In addition to the value of the mixing-length there are other free parameters that have to be considered when the convective flux, the convective velocity, and the efficiency parameter are computed. We indicate which are the values adopted for these constants in the Kurucz (1993, 1994, 1995) ATLAS9 models. In the framework of the mixing-length theory the convective flux can be expressed as (cf. Mihalas (1970), (6-274)): 4πH conv =k L/H p ρ c p T v( - ), (1) where the value of the free-parameter k is 0.5 if only upward or only downward bubbles are considered, while it is 1 if both upward and downward elements are taken into account. The convective velocity is (cf. Mihalas (1970), (6-280)): v= k1gq( ) H p L/2, (2) where k 1 is a free parameter which takes into account the work which is dissipated by the frictional forces. If half work is supposed to be dissipated k 1 =0.5. Finally, the efficiency parameter has the form (cf. Mihalas (1970), (6-286)): γ=γ thick +γ thin = ρcp 8σT 3 2+τ 2 e 2τ e v, (3) where: and γ thick = 3 8 γ thin = ρc pv σt 3 k Ross ρ V A, (4) 2c pρv 16κ RossσT 3 ρvl v. (5) The ratio V/A of the volume to the surface area of the convective element is a further arbitrary parameter which has to be fixed. For a sphere of diameter L, or a cube of side L, or a cylinder of diameter and height L, it is L/6. In these cases: γ thick = ρcpkrossρl 16σT 3 v. (6) With the notations: ω=τ e =k Ross ρl and y=1/2, we recover the Henyey et al. (1965) formula for a linear temperature distribution inside the bubble: γ thick = cpρ 8σT 3 vωy. (7) The grid of models from ATLAS9 were all computed with L/H p =1.25. As it is discussed in Sect. 4, this value was fixed by Kurucz from the comparison of the observed and computed solar irradiance. The free parameters in the formulas (1), (2), and (4) have the values: k=0.5, k 1 =0.5 and V/A=L/6. The efficiency factor is that given by formula (3), which includes γ thin. The addition of γ thin to γ thick is the first modification of the mixing-length convection as was described by Kurucz (1970) for ATLAS5. A summary of the values assumed by the free parameters in the successive ATLAS versions is given in Castelli (1996) The second modification of ML in ATLAS9: the horizontally averaged opacity (HAO) In the convective layers, the Rosseland opacity κ Ross (j), corresponding to the temperature T(j) of the j layer is replaced by the opacity κ Ross (j), defined by 1 κ = Ross f κ + (1 f ) Ross(T+ T) κ, (8) Ross(T T) namely κ Ross (j) is obtained by averaging the opacities of the hot and cold convective elements crossing the j layer. The numbers of these elements fix the value of the new free parameter f. ATLAS9 assumes f=1/2, which implies an equal number of hot and cold elements. This modification of the standard ML was introduced in model atmospheres by Lester, Lane and Kurucz (1982) and it was taken from a work of Deupree (1979), who pointed out how, in this way, the convective flux in the upper part of a stellar envelope is very close to the flux obtained from a two-dimensional stellar model.

3 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres 843 Because Lester et al. (1982) added also a variable mixinglength in their computations, the results from their paper can not be directly compared with the ATLAS9 results, when the third modification, namely the overshooting option, is switched off. We have tried to reproduce with ATLAS9 the convective fluxes computed by Deupree (1979) for an internal structure model with parameters L=50L, M=0.575M, T eff =6050 K, X=0.7, Z=0.001, and L/H p =1.0. Fig. 1a shows the variation of F conv /F total with temperature for both the ML (full line) and ML + HAO (dashed line) approaches in the case of an atmospheric model with parameters T eff =6050 K, log g=2.5, [M/H]= 1, L/H p =1.0, which correspond approximately to those used by Deupree (1979). Fig. 1b compares the corresponding curves as they are plotted in Fig. 1 of Deupree s (1979) paper. ATLAS9 does not yield the large difference between the two convective fluxes shown by Deupree (1979). In ATLAS9, the horizontally average opacity acts in the sense of decreasing the convective flux. However, the values f=1/2 in formula (8) and k=1/2 in formula (1) are inconsistent, because the former implies the presence of both upward and downward convective elements, while the latter implies only upward (or only downward) moving bubbles. Differences between the solar disk center intensity I c computed with the standard ML and with ML+HAO occur in the nm region. I c from the model with ML+HAO convection is larger than I c from the model with the only ML convection less than 1%. I c from the model with ML+HAO convection and k=1 in formula (1) (both upward and downward moving bubbles) is lower than I c from the model with the only ML convection and k=1/2 (only upward or only downward moving bubbles) less than 0.8%. The conclusion is that the effect of the HAO modification is nearly negligible, at least in the solar case The third modification of ML in ATLAS9: the approximate overshooting (AO) The meaning of approximate overshooting was explained by Kurucz (1992). He assumes that the center of a bubble stops at the top of the convection zone so that there is convective flux one bubble radius above the convection zone. That flux is found by computing the convective flux in the normal way and then smoothing it over a bubble diameter. In practice, for an atmosphere with N layers, the overshooting flux F over (j) at the j level is the ML convective flux F conv (j) averaged over a prefixed 2 H(j) atmospheric thickness, where H(j) is the geometrical height of the j level in the atmosphere. If H p is the pressure scale height, the thickness H(j) is defined by: H(j)=min (W H p (j)/2, H(N)-H(j), H(j)-H(1)), (9) where W is a weight which controls the amount of smoothing and N indicates the last layer in the atmosphere (corresponding to τ Ross =100 for the models of the grids). In the ATLAS9 code stored on CD-ROM No.13 (Kurucz, 1993a), W was defined as: Fig. 1a and b. Comparison of convective fluxes computed with the standard mixing-length theory (ML)(full line) and the mixing-length theory modified for the horizontally averaged opacity (ML+HAO) (dashed line): a from ATLAS9 model; b Deupree s (1979) results based on an internal structure model W= Fconv(N) F tot, (10) where F tot =F rad +F conv. If, owing to numerical instabilities, F conv becomes larger than F tot, W is assumed to be equal 1. This definition of W led to several discontinuities in the color indices from models with effective temperatures included between 7500 K and 6000 K. Color indices stored on CD-ROM No.13 (Kurucz, 1993a) and CD-ROM No.19 (Kurucz, 1994) are affected by such a shortcoming. For instance, North et al. (1994) discussed the discontinuities in the case of the Geneva photometric system. The discontinuities were caused by the different way how convection was computed in models having the convective zone entirely lying inside the atmosphere (as i.e. in a 7500,4.0 model) and in models having the convective zone extending beyond the bottom of the atmosphere (as i.e in a 7500,4.5 model). Owing to the definition of the weight W, in the first case, the convective flux was computed without overshooting, while in the second case it was computed with overshooting. We replaced the (10) with: W= max(fconv(j)) F tot. (11) This new definition of W allowed to eliminate the disturbing discontinuities in the color indices and it has been adopted in a 1995 ATLAS9 version. The overshooting convective flux is therefore: F over (j)= 1 2 H(j) H(j)+ H(j) H(j) H(j) F conv (j)h(j)dh. (12)

4 844 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres Namely: F over (j)= 1 H(j) H(j) 0 2 H(j) [ H(j)+ H(j) 0 F conv (j)h(j)dh F conv (j)h(j)dh]. (13) At each layer, the final convective flux is the maximum between F conv (j) and F over (j). This last assumption makes the final convective flux greater than both the smoothed flux and the ML+HAO flux. Namely, in the convective atmosphere, the local flux F conv is replaced (when F conv <F over ) by the non local F over flux, which is the average of the local flux F conv over a H p (or shorter) height. As consequence, some positive convective flux is always present above the Schwarzschild layer. The modification of the approximate overshooting is different from the physical overshooting as it is usually defined (i.e. Renzini, 1987; Kippenhahn & Weigert, 1990), because the approximate overshooting assumes a zero velocity of the mean convective element at the Schwarzschild surface and a positive convective flux above it. Physical overshooting implies a non zero velocity at the Schwarzschild surface and a negative convective flux above it. The effect of the overshooting modification on the emerging radiation is larger than that yielded by a simply increasing of the L/H p parameter in the ML and it will be discussed in the next sections. 3. Codes and models All the convective models used in this paper were computed with the ATLAS9 version of June We will indicate with K95 the models computed with the overshooting option switched on, and with NOVER those with the overshooting option switched off. The K95 models differ from those stored on the CD-ROM 13 (Kurucz, 1993a) (K93) for the way how the approximate overshooting is handled as we described in Sect Furthermore, among the the K93 convective models, only those with solar metallicity were computed for 72 instead of 64 layers. These models are based on an improved solution of transfer equation, which is able to remove the jump between the temperatures of the first and second layers. Many wavelengths are optically thick at the first or second depth in the atmosphere in cool models with 64 optical depths. Increasing the number of layers toward the top makes the calculations for the upper layers much more accurate. If no chromospheric temperature rise is present, fluxes from the upper layers of models with 72 depths are much more reliable. In the K95 grids the larger number of layers and the use of the improved solution of the transfer equation was extended also to the models with non-solar metallicities. As far as the Kurucz solar model is concerned, we refer in this paper to that computed on January 1994 (SUNK94), which could be different from the solar model stored on the CD-ROM 13 distributed before Because the improvements related with the overshooting option do not affect the solar model, the SUNK94 model does not change when the 1995 version of the ATLAS9 is used. The SUNK94 model is listed in Table 1. In the computation of Balmer profiles we used both the BALMER9 and SYNTHE codes of Kurucz (1993a, 1993b). Both codes have subroutines from D. Peterson which account for Stark, van der Waals, Doppler, natural, and resonance broadening of the hydrogen lines. For H α,h β,h γ, and H δ, Stark profiles folded with the Doppler profiles are taken from Vidal, Cooper, and Smith (1973) (VCS) tables. The damping constants of the resonance profiles are computed according to Ali & Griem (1965, 1966), while the damping constants of the natural profiles are computed from the transition probablities A ij. The BALMER9 code computes the first four Balmer lines normalized to the flux at ± 100 Å from the line center. The SYNTHE code computes a synthetic spectrum including all the known lines lying in the studied region. Both BALMER9 and SYNTHE codes are consistent with the ATLAS9 code, in the sense that same input data and same numerical approaches for solving the basic equations are used in all the codes. There are some differences in the opacities computations. For instance, no molecular continous opacities have been still inserted in SYNTHE. 4. The solar model and the value of the mixing-length Owing to the availability of absolute values for the solar central disk radiation and for the irradiance, the Sun is usually adopted to test the model predictions for cool stars. The variation of the solar opacity with frequency allows looking to different physical depths in the atmosphere, so that disk-center absolute intensity and irradiance can be used over an extended range of wavelengths in order to test the run of the temperature versus optical depth in a computed solar model. In the Sun, the peak of the emitted radiation as a function of wavelength is in the range from 410 to 510 nm, so that this is one of the best suited regions for getting information on the deepest layers where convection occurs. In the next subsections we compare some observed quantities, such as the disk central intensity, the irradiance, the limbdarkening curves, (U B), (B V), (b y) color indices, Balmer profiles, and the Mg I line at nm, with the quantities computed from solar models differing only in the way that ML convection is handled. We will show that the SUNK94 model yields the closest agreement with the observations. The only exception are Balmer profiles which would require a T eff K higher than that of the Sun in order to fit the observed profiles Comparison between observed and computed disk central intensities and irradiances The parameters of the SUNK94 model are T eff =5777 K, log g=4.4377, microturbulent velocity ξ=1.5 km s 1 for the line blanketing, and abundances from Anders & Grevesse (1989).

5 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres 845 Fig. 2a e. Comparison between observed (dashed line) and computed (full line) intensities I λ (0) from the center of the solar disk. Observations are from Neckel & Labs (1984). a and b Computed intensities correspond to L/H p=1.25 and L/H p=2.0 respectively. In both cases solar models are computed with the overshooting option switched off. c I λ (0) is from the SUNK94 model (L/H p=1.25, overshooting option switched on). d I λ (0) is from a solar model computed with the only ML, L/H p=2.0, and k=1 in formula (1); e I λ (0) is from the solar model No. 743 of the Kurucz (1979b) grid computed with ATLAS6

6 846 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres Table 1. The Kurucz solar model (SUNK94) TEFF LOG G SOLAR MODEL WITH SDSC OPACITY VTURB 1.5 KM/S L/H 1.25 ELECTRON ROSSELAND HEIGHT ROSSELAND FRACTION RADIATIVE PER CENT FLUX RHOX TEMP PRESSURE NUMBER DENSITY MEAN (KM) DEPTH CONV FLUX ACCELERATION ERROR DERIV E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E We adopted these same data for all the solar models we computed for this paper. We use elemental abundances relative to the total atomic number density. On this scale, the hydrogen and helium solar abundances are N H /N tot =0.911 and N He /N tot = Observed low-resolution spectra are those from Neckel & Labs (1984) for the visible region and from Labs et al. (1987) for the ultraviolet region. The spectra for the visible region, from to nm, consist of absolute 20 Å integrals for both the central intensity I λ (0) and the disk-averaged radiation (mean intensity) F(λ). We associated to the central wavelength of each 20 Å interval the values of I λ (0) and F(λ) averaged over 20 Å. F(λ) was converted to irradiance in order to be consistent with the ultraviolet data. In fact, Labs et al. (1987) tabulated irradiances, averaged over each passband, from 200 to 358 nm. These solar spectra are the same data used by Kurucz (1992). The comparison of the observed I λ (0) with that computed with the ML convection modified only for the horizontally averaged opacity (Sect. 2.2) has shown that the computed I λ (0) is larger than the observed one in the nm region; the discrepancy slightly decreases with increasing L/H p. It goes from about 7% for L/H p =1.25 to about 5% for L/H p =2.0 (Fig. 2a and 2b). This small variation of I λ (0) with the mixing-length parameter shows that the emerging radiation is weakly affected by the specific value assumed for the L/H p parameter. The approximate overshooting (Sect. 2.3) is much more effective in reducing the computed I λ (0) than the simple increasing of the L/H p value. Kurucz (1992) assumed L/H p =1.25 as giving the best fit between the observed and computed irradiances when convection is represented by the ML theory modified for both the horizontally averaged opacity and the approximate overshooting. Actually I λ (0) yielded by the SUNK94 model (L/H p =1.25) is generally larger than the observed one by about 3% in the nm region (Fig. 2c). A mixing-length value L/H p =2.0 would still improve the fit.

7 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres 847 Fig. 3. a Comparison between observed (dashed line) and computed (full line) solar irradiances. Observations are from Neckel & Labs (1984) for the visible region and from Labs et al. (1987) for the ultraviolet region. Solar model is SUNK94 given in Table 1. b Comparison between irradiances computed from models which differ only for the overshooting option. It is on in the SUNK94 model (full line) and off in the SUN- NOVERC125 model (dashed line) Because the mixing-length theory is far from being a rigorous theory, it is easy to obtain for the Sun the same results yielded by the SUNK94 model by dropping the two modifications to the mixing-length and by changing some of the free parameters. For instance, the same difference of about 3% between the observed and computed disk central intensity can be obtained by assuming k=1 in the flux formula (1) and L/H p =2.0 (Fig. 2d). Finally, Fig. 2e shows the solar central intensity I λ (0) computed from the solar model No. 743 of the old ATLAS6 grid (Kurucz, 1979b). For these ATLAS6 models the mixing-length parameter was assumed to be L/H p =1.0. The comparison of Fig. 2e with Fig. 2c points out the big improvement attained by the SUNK94 model. When the observed irradiance is compared with that computed from the SUNK94 model the result is the same as that already shown by Kurucz (1992) and obtained with a previous solar model (SUNK92). Observed and computed solar irradiances agree very well in the whole nm region (Fig. 3a). The SUNK94 model differs from the SUNK92 model mostly for the number of layers (72 instead of 64) which extend toward lower optical depths, for a better treatment of the radiation emerging from the uppermost layers, and for a few changes in some opacities routines, as that for H. Irradiances from the SUNK94 model and from a model with the overshooting option switched off and L/H p =1.25 (SUNNOVERC125), differ each from the other not more than 2.2% in the nm region (Fig. 3b). For a mixing-length parameter L/H p =2.0 the model without overshooting (SUN- NOVERC20) yields an irradiance which differs not more than 1% from that yielded by the SUNK94 model. The SUNK94 irradiance fits better the observations than the SUNNOVER irradiances. In conclusion, when the SUNNOVERC125 model is used, the difference between the observed and computed irradiances is smaller than that between the observed and computed central-disk intensities (Fig. 2a-2b) 4.2. The limb-darkening curves Observational profiles of the continuum solar limb darkening at a number of wavelengths from to nm and from to nm can be derived from the tables of Pierce & Slaughter (1977) and Pierce, Slaughter & Weinberger (1977) respectively. Blackwell et al. (1995) compared the observed limbdarkening curves with those predicted by the SUNK92 model. They concluded that for λ>500 nm the observed I λ (cosθ)/i λ (0) are larger than the computed ones, while for wavelengths shorter than 500 nm computations fit observations very well. We obtained the same results from the SUNK94 model. However, Fig. 4a shows that the agreement between observation and computations is not too bad up to nearly 600 nm. The difference increases in the whole wavelength range from 400 to 2400 nm

8 848 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres Fig. 4a and b. Comparison between observed (points) and computed (full line) solar limb-darkening curves I λ (cosθ)/i λ (0). Observations are from Pierce & Slaughter (1977) and Pierce, Slaughter & Weinberger (1977). Computed limb-darkening curves are from models which differ only for the overshooting option. a it is on (SUNK94 model) b it is off (SUNNOVERC125 model). The different curves correspond to different values of cosθ Table 2. Observed and computed color indices for the Sun Color Observed Computed Computed indices SUNK94 SUNNOVERC125 (U B) ± (B V) ± ± (b y) 0.406± ± Neckel (1994) 2 Schmidt-Kaler (1982) 3 Gray (1992) 4 Edvardsson et al. (1993) when SUNNOVERC125 or SUNNOVERC20 models are used to compute the solar intensities I λ (cosθ) (Fig. 4b). A fact to be considered when observed limb-darkening curves are compared with the computations is that the line opacity is negligible only for few wavelengths, as we can infer from the analysis of high-resolution spectra (Kurucz et al, 1984). This implies that the continuum windows selected by Pierce & Slaughter (1977) and Pierce et al. (1977) may not always correspond to the real continuum at several wavelengths Color indices Table 2 compares (U B), (B V), and (b y) observed color indices with those derived from the SUNK94 and the SUN- NOVERC125 models. Owing to the uncertainty in the solar colors and to the small differences between the SUNK94 and the SUNNOVERC125 colors we cannot state which model has to be preferred The Balmer profiles Comparison of Balmer profiles from BALMER9 with those from SYNTHE has shown that the metallic lines do not affect the shape of the wings of the H α and H β profiles. The violet wing of H γ predicted by the synthetic spectrum is a little bit broader than that predicted by BALMER9, owing to the presence of a strong Fe I line at nm. Observed high-resolution spectra were taken from the Solar Flux Atlas of Kurucz et al. (1984). We lowered the continuum

9 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres 849 by 1% in the H β region, because a close analysis of the way the continuum was drawn has shown that the continuum could have been drawn too high in this spectral range. Fig. 5 compares the normalized observed profiles with BALMER9 profiles computed both with the SUNK94 model (thin line) and the SUNNOVERC125 model (thick line) respectively. The SUNK94 model yields computed wings weaker than the observed ones for all H α,h β, and H γ profiles. The SUNNOVERC125 model reproduces very well all the observed Balmer lines. Fuhrmann et al. (1993, 1994) extensively discussed the comparison between observed and computed Balmer lines in cool stars. They showed that the wings of H α are independent of the value of the mixing-length parameter for stars with solar metallicity and for metal poor stars with T eff > 5500 K. Viceversa, the wings of H β and of the higher series members depend on it. Therefore they adopted H α wings to fix the effective temperature and the wings of H β to fix the value of L/H p. Finally they were able to compute the model (with T eff from H α and L/H p from H β ) which well reproduces the observed and computed H α and H β profiles at the same time. They derived L/H p =0.5 for the Sun. Our results for the Sun are different from those of Fuhrmann et al. (1993). The SUNNOVERC125 model, in which the convection is similar to that adopted by Fuhrmann et al. (1993), well reproduces all the profiles without any need of changing the L/H p value. When ATLAS9 models with overshooting are used, T eff must be increased up to 5875 K or 5900 K to fit the Balmer profiles, but also in this case all the profiles are well reproduced by the same L/H p =1.25. By the way, the mixinglength parameter should be 1.25 instead of 0.5, as we have derived from the comparison of observed and computed energy distributions. Our results for the Sun are different from those of Fuhrmann et al. (1993) probably because a different solar model has been used by us and by Fuhrmann et al. (1993) when computing the profiles. In particular, the solar model used by Fuhrmann et al. (1993) is based on continuum and line opacities different from those adopted in the SUNK94 and SUNNOVERC125 models. Also, the ML convection is different. We already showed in Fig. 2e the effect on the disk-center intensity of different ATLAS versions. By using our same codes and input data, but different observations, Van t Veer & Mégessier (1996) found for the Sun results very similar to those derived by Fuhrmann et al. (1993) and therefore different from the ours. We also found, as Van t Veer & Mégessier (1996), that H α computed from the SUNK94 model is weaker than the observed one. However, we derive an effective temperature about 100 K higher than that of the Sun, while Van t Veer& Mégessier (1996) require for the Sun 6200 K in order to fit at best both H α and H β profiles. We found that the same model, regardless of whether the overshooting option is switched on or off, fits both H α and H β profiles. In contrast with us, Van t Veer & Mégessier (1996) showed that the SUNNOVERC125 model (L/H p =1.25) well fits H α, but yields too weak computed wings for H β, so that they lowerd L/H p Fig. 5. Comparison between the solar observed spectrum and computed Balmer profiles: from top to bottom: H α,h β,h γ normalized to the continuum level. Computed profiles are from the SUNK94 model (thin line) and from the SUNNOVERC125 model (thick line). The models differ for the overshooting option which is switched on and off respectively up to 0.5, in analogy with Fuhrmann et al. (1993). Therefore, they suggested to adopt for the Sun a SUNNOVERC05 model (L/H p =0.5), because they found that this solar model yields agreement between the observed and computed profiles for both H α and H β lines. This suggestion of Van t Veer & Mégessier (1996) of lowering L/H p up to 0.5 is inconsistent with the need of a value for L/H p equal to 1.25 or even larger in order to fit observed and computed energy distribution (Fig. 2a and 2b). We note that the mixing-length parameter should be fixed from both the energy distribution and wings of H β profiles. In fact, normalized Balmer profiles change with L/H p mostly for the effect of L/H p on the continuum level. Fig. 6 compares both absolute H β profiles and continuum levels corresponding to two solar models differing only for the value of L/H p, which is 0.5 and

10 850 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres Fig. 6. Comparison between H β profiles in absolute flux units computed from solar models which differ only for the value of the mixinglength parameter L/H p. Thin line is for L/H p=1.25 (SUNK94 model), thick line is for L/H p=0.5. The ordinate is the flux H λ in erg cm 2 sec 1 nm respectively. The level of the continuum increases when L/H p decreases and it is much more affected by the convection than the wings, which become only a little bit broader when L/H p decreases. Finally, because different L/H p yield different levels for the continua, but almost the same wings, it follows that when we normalize the absolute profiles to the continuum level, the total effect of decreasing the mixing-length is to yield wings remarkably broader than those of the absolute profiles, mostly because the continuum level has increased. A possible explanation for the disagreement between our results and those of Van t Veer & Mégessier (1996) could be the different way of placing the level for the continuum. To further investigate this hypothesis, we analyzed Balmer profiles in absolute flux units. We converted the normalized spectrum of the Kurucz et al. (1984) Flux Atlas in absolute flux units by using the irradiance values of the pseudo-continuum tabulated in Kurucz et al. (1984). It came out that absolute profiles were not helpful for clarifying the disagreement because the solar computed continuum results higher than that inferred from the observed spectrum, so that a direct comparison of the wings of the observed and computed profiles is impossible (Fig. 7). In the wings of H β the continuum from the SUNK94 model is about 6.6% higher than that deduced from the observed spectrum. The discrepancy decreases with increasing wavelength, so that the difference nearly disappears at H α. The SUNNOVERC125 model yields a still larger difference between the observed and computed continuum levels, because, as we already discussed above, the overshooting effect is to decrease the emergent radiation. The comparison of Fig. 6 with Fig. 7 shows that there are no reasonable mixing-length values able to decrease the com- Fig. 7. From top to bottom: comparison between observed (thick) and computed (thin) solar H α,h β, and H γ profiles in absolute flux units. Computed profiles are from the SUNK94 model. The computed continuum is also drawn on the figure. The ordinate is the flux H λ in erg cm 2 sec 1 nm 1 for H α and in erg cm 2 sec 1 nm 1 for H β and H γ puted continuum to the level of the observed one. We obtained the same results by replacing the profiles from the Kurucz et al. (1984) Atlas by those from the Neckel (1987) Atlas. The disagreement between the observed and computed continua does not change when profiles averaged over the disk are replaced by profiles from the disk-center. In conclusion, the discrepancy between our results and those of Van t Veer & Mégessier (1996) could be due to the different observations and/or to different continuum levels The Mg I b line Holweger (1967, 1979) pointed out that the strong Mg I line at nm is well suited to test the structure of the solar

11 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres The problem of the level of the continuum in the Sun Fig. 8. Comparison between solar observed (thick line) and computed (thin line) normalized Mg I profiles at nm. model atmosphere, because the profile emerges from several depths in the atmosphere and it is almost independent of microturbulent velocity and radiative damping. Therefore, once the log gf value, the van der Waals damping constant γ VW, and the Mg abundance are known, the comparison of the computed and observed profiles may give information on the quality of the model structure. Line data in the Kurucz (1993b) line lists for Mg I nm are: log gf = (from Anderson et al., 1967) log γ rad = 7.99 sec 1, and log γ VW = 7.12 sec 1. The Van der Waals damping constant is derived from the classical value log γ VW = 7.60 sec 1 scaled for the correction factor log C 6 =1.2 (Gigas, 1988). However, log γ VW = 7.12 sec 1 yields a computed profile everywhere broader than the observed one, so that we preferred to fix log γ VW = 7.3 from the comparison of the observed and computed profiles. This empirically derived value well agrees with log γ VW = 7.26 sec 1 obtained for T=10000 K from the line broadening cross-sections from Anstee & O Mara (1995). The temperature dependence of the Anstee & O Mara (1995) computed linewidths is T 0.38, while it is T 0.3 in Kurucz codes. Fig. 8 shows the comparison between the observed and computed profiles for log γ VW = 7.26 sec 1 from Anstee & O Mara (1995) and by assuming a temperature dependence equal to T 0.38 for it. The results do not change in an appreciable way when the temperature dependence is T The agreement between the observed and computed normalized profiles shows that, within the error limits for the Van der Waals constant, the structure of the SUNK94 solar model is correct. We showed in the previous section that the SUNK94 model yields a very good agreement between the observed and computed irradiances in absolute flux units, but that it yields a level for the true continuum higher than that deduced from the observations when high resolution spectra absolutely calibrated are analyzed. The discrepancy decreases with increasing wavelength, being on the order of 6.6% in the H β region and almost zero at H α. This surprising inconsistency between the good agreement of observed and computed irradiances and the bad agreement between observed and computed continua, in spite of the same Neckel and Labs (1984) absolute calibration for the data was used, could be explained, for instance, with the inadequacy of the local theory of the mixing-length for the convection. In the previous section we showed that the changing of the mixinglength parameter, or the dropping of the overshooting option, are not able to eliminate the difference between the observed and computed continua. Possibly, 2D hydrodynamical computations would change the model structure in such a way that any inconsistency would be cancelled. Another possibility to explain the inconsistency is either a too large computed local line blanketing in low - resolution spectra or a too low local computed line blanketing in high - resolution spectra. In fact, high - resolution spectra are computed only with lines arising from observed levels, while low - resolution spectra include also lines arising from predicted levels. A less local line opacity in low-resolution spectra would yield an irradiance higher than that predicted by the SUNK94 model, so that both the computed true continuum and the computed irradiace would be higher than the predicted ones. In fact, the more numerous are the lines in a given wavelength interval of a low resolution spectrum, the more the computed flux lowers, while the level of the local continuum does not change. We checked that the computed continuum is the same both in low- and high - resolution spectra, which are computed with a different number of lines. We then increased the line opacity for computing high - resolution spectra in the H β region by adding the lines arising from the predicted levels. The difference between the observed and computed spectra shown in the middle panel of Fig. 7 does not change, indicating that the flux discrepancy in high-resolution spectra does not depend on the number of lines. Finally, the continuum opacity could be underestimated. We investigated the influence of the opacity sources on the continuum level: H is the dominant opacity source in the H β region, but also the ultraviolet line blanketing affects the model temperature structure. Minor opacities sources are the b-f and f-f transitions of neutral metals, the electron scattering, and Rayleigh scattering from neutral hydrogen The H opacity In the ATLAS9 version used by us the bound-free and free-free cross-sections for H are those computed by Wishart (1979) and by Bell & Berrington (1987) respectively. They estimated the

12 852 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres bound-free cross-section by 10%. The level of the continuum in the Hβ region lowers by 1.4%, a too small quantity to account for the discrepancy of about 6.6% between the observed and computed continua in absolute flux units (Sect. 4, Fig. 7) The metal opacity The continuum opacity from metals lowers the ultraviolet flux shortward than 255 nm and increases slightly the visual flux. When the total metal opacity is dropped the level of continuum in the H β region decreases of about 0.9%, a too low value to justify the assumption that the discrepancy between the observed and computed continuum levels is caused by some bound-free or free-free metal cross-section computed too large The Rayleigh scattering from neutral hydrogen and the electron scattering Their effect on continuum level is fully negligible in the visible The line blanketing Fig. 9. Comparison between computed H β profiles for the Sun in absolute flux units. Upper plot: profiles computed from solar models with different iron abundances. Thin line is for log N Fe/N tot=-4.37 (SUNK94 model), thick line is for log N Fe/N tot=-4.53 (Holweger et al., 1995). Lower plot: profiles computed from solar models with a different amount of line blanketing. Thin line is for a line blanketing corresponding to the Anders & Grevesse (1989) chemical composition (SUNK94 model), thick line is for a line blanketing corrsponding to a 0.2 dex lower metallicity. The ordinate is the flux H λ in erg cm 2 sec 1 nm 1 errors to be on the order of 1%. We checked that the H opacity computed by the ATLAS routines is exactly that obtained by means of the analytical formulas yielded by John (1988), which were derived by fitting the Wishart and Bell & Berrington crosssections with polynomials. Therefore, the ATLAS9 code stored on the CD-ROM No. 13 appears to be a later version than that used by Blackwell & Lynas-Gray (1994), who stated that the temperature derived from Kurucz models increases of 0.38% at 5000 K when the calculations of H opacity from John (1988) are used. To estimate the effect of hypothetical errors in the computation of the H cross-sections, we artfully increased the H The effect of line blanketing on the model structure is so important that the shape of the whole energy distribution may change even for small changes of line blanketing in some limited wavelength regions. Abundances different from those of Anders & Grevesse (1989) for some elements (i.e. the iron) or different line data would yield different ODF s than those used for computing the SUNK94 model, so that the whole model structure would be modified. To test the effect of the iron abundance on the model structure we replaced ODF s computed for log(n Fe /N tot ) equal to dex (Anders & Grevesse, 1989) by ODF s computed for log(n Fe /N tot ) = 4.53 (Holweger et al., 1995). The 0.16 dex smaller iron abundance than that from Anders & Grevesse (1989) lowers the line blanketing and decreases the absolute flux by about 1.1% in the H β region (Fig. 9, upper plot,thick line). Therefore, the different abundance for iron decreases the gap between the observed and computed continua even if to a less extent than that required to explain to 6.6% discrepancy. Fig. 10 compares the emerging energy distribution and continuum predicted by the SUNK94 model (dashed line) with those predicted by the model with the lower iron abundance (full line). The difference between the two computed continua decreases longward of the Balmer discontinuity with increasing wavelength, in analogy with the difference between the SUNK94 continuum and that suggested by high-resolution observations. Although computed continua are different, the emergent fluxes do not differ in an appreciable way. To show further the importance of the line blanketing for the model structure and therefore for the prediction of lowand high-resolution spectra, we simulated the effect of a still lower line blanketing than that given by ODF s computed for log (N Fe /N tot )= We computed a model having continuum opacities corresponding to solar abundances, but line opacities corresponding to a metallicity 0.2 dex lower than the solar one.

13 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres 853 Fig. 10. Comparison between fluxes and continua from solar models computed with the two different iron abundances log N Fe/N tot=-4.37 (SUNK94 model, dashed line) and log N Fe/N tot=-4.53 (full line) Fig. 9 (lower plot) compares the H β from this model with H β from the SUNK94 model. The difference is very similar to that between the observed and computed profiles shown in Fig. 7. This model is only an experiment with no pretension of reproducing the real solar atmosphere. In fact, it yields a too low irradiance longward 580 nm and a too low continuum in the H α region. However, it indicates that the line blanketing could be the cause of the disagreement in high resolution spectra between the observed and computed solar true continua as well as the theory of the mixing-length adopted for the convection. 6. The convective flux as a function of T eff and log g Because the solar model with overshooting better fits the observations than that without overshooting, it has been extrapolated that models with overshooting must fit also the observations of stars with temperatures, gravities, and metallicities different from the solar ones. However, if opacity, rather than convection, is the cause of the too high solar computed energy distribution, overshooting could make the fit worse in other stars with different absorption lines and different convection than in the Sun. To have an idea of the importance of the convection and of the effect of the overshooting as a function of the model parameters, we compared, for different T eff, log g, and [M/H], the F conv /F total - log τ Ross relations for convective fluxes computed with the overshooting option switched both on and off. For a few models we added also the relation for the only mixing-length (ML) obtained by dropping both the Lester et al. (1982)(HAO) and the approximate overshooting (AO) modifications (Sects. 2.2 and 2.3). Fig. 11 shows the F conv /F total - and the T- log τ Ross relations for the Sun. The F conv /F total -log τ Ross relations from the SUNK94 model, the SUNNOVERC125 model, and a solar model without the HAO and AO modifications are compared in the upper plot. The T-log τ Ross relations from the SUNK94 and SUNNOVERC125 models are compared in the lower plot. At log τ Ross =0, the difference of F conv /F tot from the SUNK94 and the SUNNOVERC125 model is 0.087, namely it is just the value of F conv /F tot in the SUNK94 model, because no convective flux is predicted at log τ Ross =0 when the overshooting option is switched off. We already discussed in the previous sections the effects of the different convection options on the emerging radiation. Fig. 12 shows, on the left, the changes of the F conv /F total - log τ Ross relations as a function of T eff for log g=4 and solar metallicity. The corresponding T-log τ Ross relations are plotted on the right. The convective zone increases with decreasing T eff, but it is gradually shifted towards larger depths, so that the structure of the superficial layers is less and less affected by the convection when T eff decreases, until the convection zone rises again toward the upper layers for temperatures lower than 4500K, owing to the dissociation of H 2. The overshooting pushes a small fraction of the convective flux closer to the surface, but its effect is not linearly correlated with the amount of the convective flux, because the differences in the T-log τ Ross relations for models computed with and without overshooting increase up to a maximum for T eff included between 7500 K and 6500 K and then decreases toward the lower temperatures, up to disappear for models with T eff < 4500 K. Analogous plots drawn for different gravities, same T eff, and same [M/H] show that the convective flux decreases with decreasing gravity, owing to the decreasing density. At a given effective temperature and gravity, the convective flux increases with decreasing metallicity owing to the increasing gas pressure

14 854 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres Table 3. The parameters of models affected by convection (columns 1 and 2) and the parameters of models which show the largest difference at log τ Ross=0 between F conv/f tot computed for the overshooting option switched on and off respectively. Columns 1 and 3 are for [M/H]=0 and columns 1,4 are for [M/H]= 3 Convective Models Max[ F(conv) Ftot OVER - F(conv) at τ Ross=1 Ftot NOVER ] [M/H]=0 [M/H]= 3.0 log g T eff (K) T eff (K) T eff (K) T eff from colour indices (V K), (B V), and (b y) In the previous section we showed that some models have a different structure depending whether the overshooting option is switched on or off. In this section we investigate the effect of the different model structure on the V K, B V, and b y colour indices, which are often adopted for fixing effective temperatures for cool stars; furthermore, we will try to state whether the color indices from the overshooting models (COLK95) or those from the no-overshooting models (COLNOVER) give T eff closer to the values derived from the infrared flux method (IRFM), which is almost model independent The dependence of the synthetic colour indices on the convection Fig. 11. Upper plot: the ratio F conv/f tot as a function of log τ Ross in the Sun for: (a) the SUNK94 model (full line), (b) the standard ML theory without any modification (crosses), (c) the SUNNOVERC125 model (no overshooting )(dashed line). Lower plot: the T-log τ Ross relation for the (a) and (c) cases and decreasing electron pressure which cause a growth of the hydrogen ionization zone. Table 3 shows which models are affected by convection for gravities ranging from log g=5.0 to log g=1.0. Furthermore, for the metallicities [M/H]=0 and [M/H]= 3, it lists the models which show the largest difference, at τ ross =1, between the F conv /F tot computed with the overshooting option switched on and off respectively. The maximum effective temperature of models affected by the different convection options decreases with decreasing log g. We computed grids of synthetic colours UBV, uvby, and RIJKL from models having the overshooting option switched off, microturbulent velocity ξ=2 km s 1, and metallicities [M/H]=0.0 and [M/H]= 3.0. For each gravity, we derived T eff by interpolating in the COLNOVER grids for the (V K), (B V), and (b y) color indices of the COLK95 grids. In this way, we may estimate the effect of the convection on the effective temperatures derived from the color indices. The temperature differences T eff =T eff over -T eff nover as function of T eff for the (V K), (B V), and (b y) indices are shown for different gravities and solar metallicity in the upper panels of Fig The largest T eff differences are about 60 K, 100 K, and 170 K for the (V K), (B V), and (b y) indices respectively. They occur for T eff and log g between K and for (V K), K and for (B V), K and for b y. Temperatures from the color indices computed from the no overshooting models are lower than those from color indices computed from the overshooting models. For all the three indices the value of T eff weakly depends on gravity for T eff <6500 K. For T eff > 6500 K the effect of the convection increases with increasing gravity.

15 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres 855 Fig. 12. On the left: The ratio F conv/f tot as a function of log τ Ross for solar metallicity, log g=4, and different T eff ranging from 8500 K to 3500 K. On the right: the T- log τ Ross relations corresponding to the models on the right. The symbols are the same as in Fig. 11 The effect of metallicity on the temperature differences T eff =T eff over -T eff nover as a function of T eff for the (V K), (B V), and (b y) indices is shown in the lower panels of Fig The functions T eff vs. T eff for the two metallicities [M/H]=0.0 and [M/H]= 3.0 are compared for log g = 4.0. T eff for [M/H]=0.0 differs from T eff for [M/H]= 3.0 no more than 40 K, 100 K, and 60 K when the (V K), (B V), and (b y) indices are considered, respectively. In particular: for the (V K) index and decreasing metallicity, T eff decreases for T eff > 6500 K then, it increases for T eff 6500 K. For [M/H]= 3.0, the largest T eff is about 50 K, at 5750 K. For (B V) and decreasing metallicity, T eff increases for T eff 5500 K, while T eff does not change with metallicity for T eff <5500 K. For [M/H]= 3.0, the largest T eff is about 200 K, at 7250 K. For the (b y) index and decreasing metallicity, T eff slightly increases for T eff >5750 K. For [M/H]= 3.0 the largest T eff is about 190 K at K. For T eff included between 4500 K and 5750 K, T eff does not change with metallicity, then it increases for T eff 4500 K. However, results for the no overshooting models with T eff <4000Khavetobe

16 856 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres Fig. 13. Differences between T eff from the COLK95 color grids and T eff from the COLNOVER synthetic grids for the same (V K) index and the same gravity. Different curves in the upper plot are for different gravities, and the same metallicity [M/H]=0.0. The two curves in the lower plot are for the same gravity log g=4, and the two different metallicities [M/H]=0.0 and [M/H]= 3.0 Fig. 14. The same as Fig. 13, but for the (B V) index considered with caution, owing to some convergence problems in the computation of models with log g>3.0 In summary: temperatures derived from the (V K) indices are nearly independent of the different treatment of convection, whatever the metallicity; for decreasing metallicity from [M/H]=0.0 to [M/H]= 3.0 the largest temperature differences T eff derived from the (B V) index increase from about 100 K to 200 K, while for the (b y) index the largest T eff increase from about 170 K to 190 K. The overshooting option affects mostly the models with T eff > 5500 K. This analysis suggests that the difference in T eff derived from (B V)or(b y) color indices due to the overshooting option will never be larger than about 200 K for each gravity and each metallicity of the grids Comparison of T eff from the colour indices and from the infrared flux method (IRFM) Although the largest difference between temperatures derived from (V K), (B V), and (b y) indices computed using models with the overshooting option switched on and off is only 200 K, we tried to determine which kind of models yields T eff s closer to those derived from methods almost model independent, as the infrared flux method (IRFM). Blackwell & Lynas-Gray (1994) (BLG) derived T eff using the infrared flux method for a sample of stars with effective temperatures included between 4000 and 8500 K, gravities log g from 1.5 to 4.5 and metallicities in the range from [M/H]=+0.6 Fig. 15. The same as Fig. 13, but for the (b y) index and [M/H]= 0.5. Smalley & Dworetsky (1995) (SD) revised the effective temperatures derived by Code et al. (1976) from the measured angular diameter and measured total flux. We compared the BLG and SD temperatures for stars with T eff <9000 K with those we derived from the colour indices computed from models with the overshooting option switched on and off respectively. Effective temperatures obtained from the infrared flux method can be directly compared with T eff derived from color

17 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres 857 Fig. 16. Upper plot shows the (V K)-T eff relation for solar metallicity and different gravities. The two curves in the lower plot show differences between T eff derived from (V K) index for solar metallicity, [M/H]=0.0, and T eff derived from the same (V K) index, but for [M/H]= 0.5 or [M/H]=+0.5 respectively. Uncertainties in T eff related with uncertainties in gravity and metallicity can be estimated indices only if both methods are independent of gravity and metallicity or, viceversa, if the gravity and the metallicity of each star in the sample are well known quantities. The IRFM method depends on models, and therefore on T eff, log g, and [M/H], in that it makes use of the computed monocromatic flux Φ(T,g,λ,A) at a preselected infrared wavelength. To this purpose, BLG assumed for each star of their sample solar metallicity and an approximate log g. Kurucz (1992) model atmospheres computed for solar metallicity were used. Mégessier (1994) showed that, in the framework of the Kurucz models, IRFM temperatures differ from 74 K at 8000 K to 0 K at 6200 K for gravity differences of 0.5 dex and that IRFM temperatues differ no more than 60 K for metallicity differences of 0.5 dex. We investigated the effect of the gravity and metallicity on the (V K), (B V), and (b y) indices. The only difference with a similar analysis performed by King (1993) is the use made by us of improved models (Kurucz, 1995) for cool stars. The results are shown in Fig For gravity differences of ±0.5 dex, effective temperatures from the (V K) index show differences ranging from ± 100 K at T eff =8000 K to about 0KforT eff 6500 K. Therefore, for temperatures lower than 6500 K the effect of the gravity on the (V K) index becomes negligible. For log g = 4, metallicity differences of 0.5 dex yield differences in T eff ranging from a Fig. 17. The same as Fig. 16, but for (B V) maximum of 70 K at 8500 K to a minimum of0katt eff =5250 K. For T eff <5250 K, the differences in T eff increase with decreasing temperature. They are about 20 K at 4250 K and about 120 K at 3750 K (Fig. 16). For gravity differences of ±0.5 dex, effective temperatures from the (B V) index show differences ranging from ± 100 K at T eff =8000 K to about 0 K for 5250 K T eff 6250 K. For lower temperatures the largest difference is about 70 K at 4500 K and log g = 2.5. Therefore, for temperatures lower than 6250 K, the (B V) index is nearly independent of gravity. The (B V) index depends on metallicity more than the (V K) index. For log g = 4, metallicity differences of 0.5 dex yield differences in T eff on the order of 200 K for T eff included between 4750 K and 6750 K (Fig. 17). The (b y) index depends on gravity more than the (B V) index. For gravity differences of 0.5 dex, the temperature differences are nearly 0 K only for T eff included between 5500 K and 6500 K. For larger and lower temperatures T eff may be on the order of 120 K and 90 K respectively. The dependence of the (b y) index on metallicity is different from that of the (B V) index. For log g = 4, metallicity differences of 0.5 dex yield differences in T eff on the order of 180 K at T eff =5000 K. Then T eff decreases for lower and larger temperatures (Fig. 18). Summarizing, the (V K) index is nearly unaffected by both gravity and metallicity for the whole K range of temperature, the largest uncertainty in T eff being on the order of 100 K for T eff >6500 K. The (B V) index is nearly unaffected by gravity for T eff 6500 K, but it is affected by metallicity so that T eff may be on the order of 200 K. The (b y) index depends on both gravity and metallicity, except for the K region where it is gravity independent.

18 858 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres Fig. 18. The same as Fig. 16, but for (b y) The stars for which we compared T eff derived from color indices with T eff derived with the IRFM method are listed in Table 4. Since we have been not able to recover the gravity values left out by BLG in their Table 6, we used only the stars of their sample with a given value for the gravity or for which we found a value in the literature. For the seven stars that we extracted from the SD sample (indicated with an asterisk in Table 4) we adopted the gravity values provided by SD. For all the stars we adopted models with solar metallicity in agreement with the BLG analysis. The actual metallicity of each star taken from the literature is listed in column 10 of Table 4. Observed (V K) indices were taken from BLG, except for Procyon, for which the value given by Steffen (1985) was adopted. (B V) indices were taken from the Bright Star Catalog (Hoffleit, 1964), and b y indices were taken from the Hauck & Mermilliod (1990) catalog. Observed (b y), (B V), and (V K) indices are listed in columns 5, 6, and 7 respectively. The reddening A(V) (column 3) was taken from BLG. We dereddened the (V K) indices by means of the relation E(V K) = 2.747E(B V), where E(B V) was derived from R V = A(V)/E(B V) = 3.1 (Clementini et al., 1995). (B V) indices were dereddened by using the above relation for E(B V), and (b y) indices were dereddened by means of the relation E(b y) = 0.74E(B V) (Crawford & Mandwewala, 1976). For each star, the effective temperatures corresponding to the assumed gravity and to the observed (b y), (B V), and (V K) indices were derived by interpolation in the uvby, UBV, and RIJKL synthetic grids, respectively. These effective temperatures are listed in columns 11, 12, and 13 of Table 4. For each column there are two T eff, the first one was derived from the COLK95 grids, the second one, in parenthesis, from the COL- NOVER grids. For comparison, column 8 lists T eff derived by BLG or SD by using the IRFM method. Fig. 19 shows the difference between temperatures obtained with the IRFM method and the temperatures derived from the (V K), (B V), and (b y) respectively. The crosses correspond to T eff derived from the COLNOVER colors grids. The points, triangles, and squares indicate T eff derived from the COLK95 color grids. The squares indicate the SD stars, the points and the triangles indicate the BLS stars, where points are for stars with [M/H] 0.2 in absolute value and triangles are for stars with [M/H]>0.2 in absolute value. The line connecting each cross with the corresponding point, or square, or triangle is the difference T eff between the effective temperatures derived from the COLK95 and the COLNOVER grids. As predicted by Fig. 13, 14, and 15, this difference is small for (V K), and then increases for (B V), and it is still larger for (b y). For each index, T eff is smaller for T eff <6250 K than for T eff included between 6250 K and 7250 K. Then T eff decreases again for temperatures larger than 7500 K. Effective temperatures from the COLK95 grids are always larger than effective temperatures from the COLNOVER grids, except for the (V K) index and T eff 5750 K, where, however, the differences are negligible. Fig. 13 shows that the (V K) index is almost independent of both gravity and metallicity for T eff included between 4000 K and 6250 K. Therefore, (V K) indices of stars with T eff included in this range can be used to estimate the difference between T eff derived from models and T eff derived from the IRFM method. Fig. 19 shows that T eff derived from the (V K) indices is 38 K larger, on average, than T eff derived from the IRFM method by BLG. Fig. 19 suggests that T eff from the COLNOVER indices are, on average, closer to T eff from the IRFM method than T eff derived from the COLK95 indices. This is confirmed by the comparison of columns 2 and 3 and of columns 4 and 5 of Table 5, where the averages of the differences between T eff derived from color indices and T eff obtained with the IRFM method are listed. We computed two separated averages T eff for T eff <6250 K and for T eff >6250 K. For each region, the differences between T eff from color indices and T eff from IRFM were computed by using both the COLK95 and the COLNOVER indices. Finally, we investigated the consistency of T eff derived from the COLK95 and COLNOVER indices. Because the (V K) index is nearly idependent of convection and the (b y) index somewhat depends on it, the comparison of effective temperatures derived from the (V K) and (b y) indices, when

19 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres 859 Table 4. The program stars (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) HD BS A(V) V b y B V V K T eff (K) log g [Fe/H] T eff (b-y) T eff (B-V) T eff (V-K) Notes no. no. IRFM over (nover) over (nover) over (nover) (7030) 7072 (6976) 6989 (6944) (4367) 4417 (4422) (5971) 6157 (6110) 6052 (6049) (8271) 8226 (8219) 8023 (8001) (8178) 8080 (8070) 8137 (8121) (6153) 6246 (6188) 6264 (6252) (4387) 4396 (4385) 4422 (4428) (5496) 5645 (5613) 5427 (5430) (6358) 6447 (6377) 6439 (6421) (6046) 6055 (6052) (5647) 5777 (5744) 5786 (5787) (5994) 6104 (6049) 6044 (6038) (5215) 5331 (5310) 5199 (5202) (8188) 8148 (8131) (7128) 7110 (7054) 7040 (6996) (5859) 5936 (5893) 5935 (5933) (5040) 4768 (4775) (8231) 8226 (8219) 8156 (8145) (4892) 4868 (4873) < (7229) 7250 (7520) (8207) 8222 (8222) 7983 (7981) < (6069) 5913 (5889) (6661) 6790 (6693) 6628 (6596) (6681) 6846 (6753) 6657 (6634) (4879) 4968 (4939) 4889 (4894) (8113) 8168 (8155) 8015 (7980) (5533) 5674 (5632) 5533 (5535) (8385) 8449 (8446) 8292 (8286) (6360) 6510 (6450) 6479 (6466) (5937) 6096 (6038) 6105 (6096) (6124) 6245 (6186) 6195 (6185) (6181) 6324 (6258) 6196 (6186) (7912) 7879 (7856) 7852 (7810) (4377) 4284 (4290) (6877) 7101 (7012) 6387 (6811) (6809) 6937 (6844) 6710 (6685) (6924) 7203 (7113) 7074 (7029) (5010) 4994 (4998) (6629) 6741 (6675) 6601 (6585) (5033) 4884 (4889) (6081) 6320 (6252) 6227 (6216) (7291) 7279 (7245) 7189 (7150) (5148) 5047 (5052) (6807) 6892 (6799) 6751 (6725) (8045) 8101 (8093) (5378) 5551 (5524) 5615 (5617) (5064) 5039 (5045) (8459) 8369 (8363) 8229 (8221) (6364) 6566 (6488) 6842 (6465) (7128) 5038 (5009) 5030 (5035) (6811) 6988 (6907) 6837 (6811) (5233) 5458 (5428) 5496 (5499) (7947) 7972 (7956) (5133) 5241 (5212) 5143 (5148) (4770) 4860 (4837) 4799 (4804) (5051) 5108 (5078) 4975 (4979) (6294) 6490 (6430) (4730) 4733 (4738) (5085) 5129 (5095) 5150 (5152) (4950) 5031 (5036) (4739) 4704 (4774) 4690 (4695) (6254) 6409 (6344) 6232 (6222) (5069) 5066 (5071) (8860) 8734 (8734) (7400) 7432 (7341) 7246 (7197) (5852) 6020 (5964) 6082 (6073) 1 1. log g and [M/H] from Künzli et al. (1995) 2. log g and [M/H] from Blackwell & Lynas-Gray (1994) 3. log g and [M/H] from Bell & Gustafsson (1989) 4. log g from Smalley & Dworetsky (1995), [M/H] = 0.0 (assumed) *. Stars from the Smalley & Dworetsky (1995) sample

20 860 F. Castelli et al.: Notes on the convection in the ATLAS9 model atmospheres Fig. 20. Comparison between T eff derived from (V K) index with T eff derived from (b y) index for the stars of Table 4 having T eff included between 6000 K and 8000 K. In the upper and lower plots synthetic indices are from the COLK95 and the COLNOVER grids, respectively Fig. 19. Comparison between T eff derived from observed and computed (V K), (B V), (b y) indices and T eff derived by Blackwell & Lynas-Gray (1994)(BLG) and by Smalley & Dworetsky (1995) (SM) from the IRFM method. Crosses are for effective temperatures from COLNOVER indices; points, squares and triangles are for effective temperatures from COLK95 indices. The squares indicate stars from SD. Points and triangles indicate stars from BLG with [M/H] 0.2 and [M/H]>0.2 (in absolute value), respectively Table 5. Difference between T eff from color indices and T eff from IRFM T eff <6250 K T eff >6250 K T eff (K) T eff (K) T eff (K) T eff (K) K95 NOVER K95 NOVER (V K) (B V) (b y) both COLK95 and COLNOVER grids are used, should indicate which convection yields effective temperatures closer to each other. We compared T eff derived from (V K) indices with T eff derived from (b y) indices for the stars of Table 4, having T eff included between 6000 K and 8000 K, because the models of this region are mostly affected by convection. We derived from Fig. 20 that the mean deviations of the points from the diagonal are 145 and 99 when the COLK95 and the COLNOVER indices are used respecticvely. This difference is not very large, however it indicates that, on average, NOVER models yield T eff from color indices more consistent each with other than K95 models do. 8. Gravity log g from c 1 Strömgren index For cool stars, surface gravity may be derived from the Balmer discontinuity as measured by the c 1 Strömgren index. Fig. 21 shows that, for a given metallicity and a given T eff, the c 1 color index may be used to fix the surface gravity of stars with T eff included between 5500 K and 7500 K. The c 1 0 -T eff relation is