A&A 386, 286-295 (2002)
DOI: 10.1051/0004-6361:20020217
B. Caccin ^{1,2} - V. Penza ^{1} - M. T. Gomez ^{3}
1 - Dipartimento di Fisica, Università di Roma "Tor Vergata'',
via della Ricerca Scientifica 1, 00133 Roma, Italia
2 -
Consorzio Interuniversitario per la Fisica Spaziale (CIFS)
3 -
Osservatorio astronomico di Capodimonte, via Moiariello 16, 80131 Napoli, Italia
Received 22 November 2001 / Accepted 5 February 2002
Abstract
The ratios of spectral line depths are often used as indicators of the stellar effective temperature
.
In particular, Gray & Livingston (1997a) calibrated the temperature sensitivity of the ratios between the central depths of the line C I 538.032 nm and either the Fe I 537.958 or the Ti II 538.103, making use of observed spectra of several solar-like stars. The ultimate reason for choosing these lines was the subsequent application of their calibration to a long series of solar data, collected at Kitt Peak (in disk-integrated light) from 1978 to 1992, in order to get the
variation of the Sun during its 11-yr magnetic cycle (Gray & Livingston 1997b).
We propose a theoretical calibration that includes a careful treatment of convective transport and fits the stellar data very well , showing, at the same time, that the empirical calibration of Gray and Livingston incorporates in the
sensitivity an undesired dependence of line ratios on the surface gravities of the individual stars they used. A possible dependence of the calibration upon stellar rotation is also explored.
Key words: stars: late-type - stars: atmosphere - stars: fundamental parameters - Sun: activity - convection
The determination of the global parameters of a star is a key point in stellar physics. In particular, that of has a very long history, so that several different methods, based either on photometric or on spectroscopic data, are available. Each has its advantages and limitations, but errors of a few hundred degrees on are often encountered. Among them, those using spectral lines have always played an important role since the early times of stellar classification. With high-resolution spectra, differences of about K can be detected, although the scale is uncertain by at least ten times that amount, thus allowing to rank stars by and trace cyclic temperature changes (Gray & Johanson 1991). Usually, instead of looking at single line strengths, the central depth ratios or the equivalent width ratios of lines having different sensitivity to temperature variations are employed, as line ratios are less subject to instrumental and other systematic errors (e.g. Gray & Johanson 1991; Gray 1992b, 1994; Strassmeier & Schordan 2000; Kovtyukh & Gorlova 2000). Of course, different spectral lines and calibrations must be used for different temperature ranges and luminosity classes. The fundamental limits are likely set by intrinsic variations arising from surface features and rotational modulation, from variations during stellar magnetic cycles and other subtle effects that may occur in the atmospheres of the stars (Gray 1995). In a few favourable cases, errors of 1K on relative temperatures have been estimated, with inconsistencies of 50K between temperatures derived from colour indices and from spectral lines (Gray 1994).
Gray & Livingston (1997a, henceforth G&La) calibrated the temperature sensitivity C_{0}, defined through the following approximate linear relation
Suitable grids of theoretical photospheric models are currently available in the literature; in this work we used and modified a subset of Kurucz's models obtainable at http://cfaku5.harvard.edu/grids.html, that cover a wide range of the free parameters (effective temperature , surface gravity g, abundance [Fe/H] and microturbulence ). In all of the grid models, convective transport is treated according to the mixing-length theory (MLT), with a unique mixing-length ratio (where is the pressure scale height). However, two different series of models are available: the first (Kurucz 1994, henceforth Ku94) include the effects of overshooting, while in the second (Castelli et al. 1997, henceforth Ca97) the convection is treated, on the contrary, in the purely classical way (without overshooting). Actually (Kurucz 1992; Castelli et al. 1997), the introduction of overshooting in these models is not made by means of a really physical treatment of its effects, but only by computing the convective flux in the standard way and then smoothing it over a bubble diameter.
In order to explore the effects of changing the value of without recomputing a whole set of models every time, we have simply extended the radiative equilibrium part of Ca97 models, starting from the point where the convective instability begins ( ), with a gradient recalculated by us according to the MLT. In the same way we studied also the effects of using an alternative treatment of convection, extending the models with the Canuto-Mazzitelli theory (Canuto & Mazzitelli 1991, henceforth CMT). Here we describe very briefly the calculations we made in the two cases, with the same notations used by Canuto & Mazzitelli (1991).
In MLT the gradient throughout the convective zone is given by the solution of a cubic equation for the convective efficiency ,
defined as:
Figure 1: Comparison between a Ca97 model ( = 5750K, 4.5) and that recalculated by us with CMT (both values of are "canonical''). On the left we reported the gradients and on the right the temperature structures T(). | |
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In the CMT theory the approach to convection is rather different, since the whole energy spectrum of turbulence is taken into account. However there is a new relation that replaces Eq. (5), namely
For both classes of convective models,
is given through the parameter .
It's now commonly accepted that it is not correct to maintain constant the value of
along the main sequence, so we decided to consider in CMT a dependence of
on
and g, as suggested by Ludwig et al. (1999):
The use of such a relation for makes the models physically more consistent, however Eq. (11) was not intended to reproduce necessarily the temperature profile of the superadiabatic layers, therefore it may be not suitable, as noted by the authors themselves, to providing the optimum value for convective stellar atmospheres. In any way, we will presently see that the effect on of using either the MLT with constant or CMT with variable is rather small.
In Fig. 1 we show the effect of changing the convective theory (from MLT to CMT with "canonical'' values of ) on the temperature structure and on its gradient. As expected (Kiefer et al. 2000), in CMT the temperature gradient is steeper than in MLT (which in turn is steeper than that calculated in MLT by introducing the overshooting effects), but the two models begin to differ from each other only at , so that the effects cannot be too large.
In the following we will apply different kinds of models to a given set of stars, with given observed colours. Because the models that we have modified, in spite of their simplified treatment of convection (MLT), well reproduce the observed colours, we have checked, first of all, whether our passing to CMT alters very much the colours computed with given values of and g. To do that, we recalculated the UBV colours by means of available Kurucz's programs and adopting, as input data, the flux values corresponding to 1221 fixed wavelengths, calculated by ATLAS9 code (Kurucz 1993) running with our CMT models.
Figure 2: Colour differences obtained by changing the convective theory (from MLT with to CMT with ) in the Ca97 models are plotted vs. for two values of g. The differences are never large, but are greater for cooler stars. | |
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In Fig. 2 we plot vs.
the differences between the "new'' colours and those reported in Ca97 tables, for two values of .
Note that, although the differences are never very large, they are greater for cooler stars. The reason of that becomes clear if we look at Fig. 3, in which we plotted (on arbitrary scale) the difference between the temperature structure
in MLT and in CMT models for
5250 K and for
= 5750 K, superimposed to the temperature response function (Caccin et al. 1977) of the B-V colour, locating the layers where the temperature perturbation has the largest effect on the colour.
The expression of the response function of the emergent intensity is
Figure 3: Response function of (B-V) to a temperature perturbation at constant pressure (solid line), together with the difference (on arbitrary scale) between the temperature structure in Ca97 and in CMT models for = 5250K (dotted line) and for = 5750K (dashed line). | |
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The interpretation of Fig. 3 is immediate: where temperature perturbations have the largest effect on the emergent intensity (i.e. where RF is largest), the difference between the temperature structure in MLT and that in CMT models ( ) is larger for = 5250 K than for = 5750 K, so the effects on colours determination are more evident for the former. Notice also that the "effective'' sign of is different in the two cases and this changes the sign of .
However, the maximum error is 0.01 mag, which corresponds, according to Gray (1992a) calibration, to an uncertainty on of about 0.5% and cannot influence too much our choices of and g for the different stars.
In the same way, we don't expect a large effect on the colours calculation (hence on the determination) if we switch between models with or without overshooting: Castelli et al. (1997) found indeed that the largest effect of this change on the model structure occurs for 6500 K, that is hotter than any star we used (e.g. for K the difference between the B-V colours obtained with the two models corresponds to a difference in temperature of about 20 K). Moreover, in a subsequent paper (Castelli 1999) further comparisons with different synthetic grids of colors show again differences lower than 100 K.
STAR | log g | B-V | U-B | [F/H] | ||
Dra | 5432 | 5253 ^{(1)} | 4.50^{(1)} | 0.79^{(a)} | 0.38^{(a)} | -0.21^{(6)} |
Cet | 5441 | 5250 ^{(2)} | 4.50^{(2)} | 0.72^{(a)} | 0.21^{(a)} | -0.50^{(7)} |
51 Peg | 5831 | 5880 ^{(3)} | 4.37^{(3)} | 0.67^{(b)} | 0.20^{(b)} | +0.21^{(8)} |
SUN | 5791 | 5777 | 4.44 | 0.64^{(c)} | 0.18^{(c)} | +0.00 |
Per | 5970 | 5890 ^{(4)} | 4.19^{(4)} | 0.59^{(d)} | 0.12^{(d)} | +0.05^{(9)} |
And | 6081 | 6205 ^{(5)} | 4.00^{(5)} | 0.54^{(b)} | 0.06^{(b)} | +0.00^{(10)} |
Figure 4: Representative points of the stars in the ( , ) plane found in the literature. The bolded circles mark the values we used. The solid line represent the average relation between and g along the main sequence (Gray 1992a). | |
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STAR | log g | References | |
Dra | 5140 | 4.50 | Cayrel de Strobel & Bentolila (1983) |
5310 | 4.60 | Clegg et al. (1981)^{(6)} | |
5227 | 4.50 | Alonso et al. (1996) | |
5200 | 4.60 | Perrin et al. (1977) | |
5227 | 4.52 | Prieto & Lambert (2000) | |
5253 | 4.50 | Bell & Gustafsson (1989)^{(1)} | |
Cet | 5358 | 4.59 | Cayrel de Strobel & Bentolila (1983) |
5358 | 4.49 | Perrin et al. (1977) | |
5316 | 4.3 | Blackwell & Lynes-Gray (1994) | |
5250 | 4.7 | Arribas & Martínez-Roger (1988) | |
5300 | 4.4 | Flynn & Morell (1997)^{(7)} | |
5250 | 4.5 | Borges & Idiart (1995)^{(2)} | |
5330 | 4.30 | Tomkin & Lambert (1999) | |
51 Peg | 5728 | 4.27 | Cayrel de Strobel & Bentolila (1983) |
5793 | 4.33 | Fuhrmann et al. (1997) | |
5755 | 4.18 | Tomkin et al. (1995) | |
5880 | 4.37 | Gray R.O. et al. (2001)^{(3)} | |
5500 | 4.25 | Sadakane et al. (1999) | |
5755 | 4.18 | Cayrel de Strobel (1996) | |
5730 | 4.27 | Kobi & North (1990) | |
5795 | 4.41 | Gonzales et al. (2001)^{(8)} | |
5750 | 4.40 | Ford & Rasio (1999) | |
5669 | 4.27 | Prieto et al. (1999) | |
5740 | 3.76 | McWilliam (1990) | |
Per | 5929 | 4.10 | Cayrel de Strobel & Bentolila (1983)^{(9)} |
5890 | 4.19 | Nissen (1981) | |
5950 | 4.11 | Gray R.O. et al. (2001) | |
6000 | 4.0 | Borges & Idiart (1995) | |
5970 | 4.06 | Cayrel de Strobel (1996) | |
5930 | 4.09 | Kobi & North (1990) | |
5996 | 4.00 | Alonso et al. (1996) | |
5890 | 4.19 | Bell et al. (1994)^{(4)} | |
6088 | 4.5 | Castelli et al. (1997) | |
5929 | 4.09 | Perrin et al. (1977) | |
And | 6067 | 4.10 | Cayrel de Strobel & Bentolila (1983) |
6250 | 4.30 | Ford & Rasio (1999) | |
6155 | 4.00 | Alonso et al. (1999) | |
5998 | 4.00 | Perrin et al. (1977) | |
6205 | 4.0 | Blackwell & Lynes-Gray (1994)^{(5)} | |
6140 | 4.22 | Bell et al. (1994)^{(10)} | |
6212 | 4.17 | Tomkin et al. (1995) | |
6128 | 4.22 | Balachandran (1990) | |
6210 | 4.06 | Gray R.O. et al. (2001) | |
6125 | 3.98 | Prieto et al. (1999) |
Through their empirical calibration G&La obtained, for the two central-depth ratios C/Fe and C/Ti, the values C_{0}=346 and C_{0}=468 respectively.
Figure 5: Experimental values of C/Fe line-depth ratios (G&La), compared with our theoretical calculations. The straight lines are, in all cases, the best fit to the corresponding points. | |
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STAR | C/Fe | C/Ti | ||
exp. | theor. | exp. | theor. | |
Dra | 0.106 | 0.104 | 0.116 | 0.142 |
Cet | 0.103 | 0.105 | 0.135 | 0.143 |
51 Peg | 0.330 | 0.351 | 0.340 | 0.344 |
SUN | 0.273 | 0.279 | 0.293 | 0.298 |
Per | 0.457 | 0.400 | 0.418 | 0.372 |
And | 0.672 | 0.705 | 0.512 | 0.513 |
Line | lg( gf) | (km s^{-1}) | |||
CI 538.032 nm | -1.8 | 11.26 | 7.680 | 1 | 2 |
FeI 537.958 nm | -1.6 | 7.87 | 3.695 | 7 | 1 |
TiII 538.103 nm | -2.08 | 13.58 | 1.566 | 40 | 1 |
The same plot for C/Ti is shown in Fig. 6, where however the agreement is worse: the difference between the "experimental'' value of C_{0} (619) and the theoretical one (720) is 15% for CMT and reaches 25% for MLT (770). We note that the cause of the discrepancy can be attributed almost exclusively to an anomalous behavior of the coolest star, for which the reliability of our models might be less. In Table 3 we report explicitly, for the two lines, the experimental depth ratios obtained by G&La and the corresponding values calculated with our CMT models.
We have explored also the effects on C_{0} of changing the
value, finding a substantial insensitivity (3 to 4% for
changing from 1.25 to 2).
The corrisponding results obtained by using the Ku94 models provided values of C_{0} only slightly larger (2-5%) than the Ca97 ones.
Figure 6: Experimental values of C/Ti line-depth ratios (Gray 2001), compared with our theoretical calculations. The straight lines are, in all cases, the best fit to the corresponding points. | |
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Figure 7: Surface for the C/Fe line-depth ratio, obtained with a small grid of theoretical CMT models and [Fe/H] = 0. | |
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Figure 8: Experimental values of C/Fe equivalent width ratios (G&La), compared with our theoretical calculations. | |
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Figure 9: "Reconstructed'' experimental values of C/Ti equivalent width ratios (G&La), compared with our theoretical calculations. | |
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Even if we could already reproduce the G&La experimental data, to obtain a correct stellar calibration of the line-depth ratios vs. variations, we still have to take into account other paramenters that might play an important role in determining line profiles, like the stellar rotation, that produces a line broadening, but does not have any sensible effect on the equivalent widths (Stift & Strassmeier 1995). In this case, our calibration should remain, therefore, valid. From this point view the use of the equivalent widths in place of the line depths would be a better choice; unfortunately, as already said, the measurement of the former present, for our lines, larger uncertainties and, above all, their ratios are less sensible to variations (the values of C_{1} are larger than those of C_{0}, especially in C/Ti case). However, we will show that, even if the rotation modifies the values of the line depths and of their ratios, it has almost no effect on C_{0}, provided that the value of are small enough and the rotation is the same for all the stars of the sample. More sensible effects would however be present if the sample stars were characterized by a large dispersion in .
The literature data for G&La stars are reported in Table 5, where we see that for three stars (51 Peg, And and the Sun) there is not a considerable discrepancy among the data; for one (
Per) the difference from the catalogue value (Bernacca & Perinotto 1970) and those taken from more recent works is not negligible but not too large; finally for the first two stars (
Dra and
Cet) the data scattering is quite remarkable.
STAR | V sin i (km s^{-1}) | References |
Dra | 0.6 | Fekel (1997) |
0.8 | Gray (1984) | |
8 | Bernacca & Perinotto (1970) | |
Cet | 0 | Saar & Osten (1997) |
0.6 | Fekel (1997) | |
0.9 | Gray (1984) | |
2 | Simon & Landsman (1991) | |
2.4 | Soderblom (1981) | |
8 | Smith et al. (1983) | |
8 | Bernacca & Perinotto (1970) | |
51 Peg | 1.4 | Benz & Mayor (1984) |
1.7 | Soderblom (1983) | |
2 | Fuhrmann et al. (1997) | |
2 | Bernacca & Perinotto (1970) | |
2.3 | Queloz et al. (1998) | |
2.4 | Saar & Osten (1997) | |
2.6 | Melo et al. (2001) | |
2.8 | Mayor & Queloz (1995) | |
Per | 2.9 | Benz & Mayor (1984) |
3.5 | Soderblom (1983) | |
3.5 | Schrijver & Pols (1993) | |
8 | Bernacca & Perinotto (1970) | |
And | 9 | Gray (1986) |
9.2 | Soderblom (1983) | |
9.3 | Benz & Mayor (1984) | |
c | 11 | Bernacca & Perinotto (1970) |
SUN | 1.6 | Queloz et al. (1998) |
1.9 | Saar & Osten (1997) | |
1.7 | Simon & Landsman (1991) |
To calculate the normalized line profiles in a uniformly rotating star (
), we use the following expression
In Fig. 10 we reported the usual plots
vs.
where, for all stars, either
= 0 or
= 8 km s^{-1}; as anticipated, a "constant'' rotation has only the effect of shifting the straight line without changing C_{0} (the difference is 3%). This argument remains true also for the ratio C/Ti, that seems even less sensible (the difference is less than 2%).
Figure 10: Comparison between the calibrations of the C/Fe line depth-ratio calculated without rotation and with = 8 km s^{-1}. | |
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In Fig. 11 we plot the straight lines relative to all of the four possible combinations of the extreme
values for
Dra and
Cet, that are labelled as follows:
1)
(
Cet) = 0 km s^{-1} and
(
Dra) = 8 km s^{-1};
2)
(
Cet) = 0 km s^{-1} and
(
Dra) = 0.6 km s^{-1};
3)
(
Cet) = 8 km s^{-1} and
(
Dra) = 0.6 km s^{-1};
4)
(
Cet) = 8 km s^{-1} and
(
Dra) = 8 km s^{-1}.
Figure 11: Comparison between the calibrations of the C/Fe line depth-ratio calculated by associating to the four hotter stars a mean rotation value and considering the two extreme values for the two cooler stars. The four possible combinations are labelled as follows: 1) ( Cet)=0 and ( Dra)=8km s^{-1}; 2) ( Cet)=0 and ( Dra)=0.6 km s^{-1}; 3) ( Cet)=8km s^{-1} and ( Dra)=0.6km s^{-1}; 4) ( Cet)=8km s^{-1} and ( Dra)=8km s^{-1}. The lines corresponding to the two intermediate case (1 and 2) cannot be distinguished in this figure. | |
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The difference on the C_{0} determination with respect to the nonrotational case is maximum (18%) in case (2) (when both Cet and Dra are very slow rotators) while it's practically negligible (less 2%) in case (3) (when both Cet and Dra are rather fast rotator); in intermediate cases the error is 10%. As in the previous calculation, where a "constant'' rotation was included, the situation is better for C/Ti, for which the error is very small in case (1), 13% in case (2) and 7% otherwise.
In the light of these results, we believe that our theoretical calibrations of , for the C_{0} determination, with or without rotation, remain essentially the same.
It seems that one-dimensional photospheric models, when the individual values of
,
g and [Fe/H] for the five stars are used, can satisfactorily reproduce the observed line ratios. However, the problem of temperature calibration is not one-dimensional and we cannot manage every problem with a single biunique relation. The surface gravity appears to be the principal additional parameter that has to be taken into account; we have shown, for example, that the value of C_{0} is not very sensible to possible differences of the stellar rotation, at least in the
range of our interest. In reality, other parameters, such as the micro or macroturbulence, could play a role (Stift & Strassmeier 1995), but we don't believe they can substantially invalidate our theoretical calibration. If we want to use the line-depth ratios as indicators of
variations along the solar cycle, we must then consider, on the surface
,
the curve at
Calculating, in
K and
the partial derivative
Acknowledgements
This work has been supported also by MURST (through Cofin 2000). The authors are indebted with D. F. Gray for valuable comments and for kindly providing additional data; they thank also S. Sofia, I. Mazzitelli, F. Kupka, I. Ermolli and F. Castelli for useful discussions and comments.
The referee, K. G. Strassmeier, suggested discussing the problem of stellar rotation.