Astron. Astrophys. 323, 909-922 (1997)

4. The stellar parameters of Procyon

From a rather conservative point of view one may state that only the Sun provides well-defined parameters in the regime of dwarf F and G stars. There is, however, recent progress in stellar diameter measurements which confirms the results of Hanbury Brown et al. (1974) who determined Procyon's angular diameter to be  mas from intensity interferometry. The new Mark III Optical Interferometer measurements of Mozurkewich et al. (1991) yield the same value  mas, but with a considerably reduced error. With this angular diameter measurements the most direct methods to determine the effective temperature result in (cf. Steffen 1985)

if we make use of the bolometric magnitudes from Code et al. (1976), or the measurement of the integrated flux. In the latter case we take the average from Code et. al. (1976): 18.08 0.76, Beeckmans (1977): 18.14 1.14, Blackwell & Shallis (1977): 18.0 0.8 and Smalley & Dworetsky (1995): 18.638 0.868 (in units of erg cm s^-1) with an estimated uncertainty of erg cm s^-1. Hence we get a mean value for the effective temperature

While this is already a relatively small error for stellar abundance analyses, Procyon - being a visual binary (  years,  AU) - supplies a very precise value for the surface gravity, too. According to we need to know the stellar mass and radius. The latter is obtained from the above mentioned angular diameter in combination with the stellar distance. Irwin et al. (1992) derive an absolute parallax of  arcsec and compare this to the recent value published by the U. S. Naval Observatory  arcsec (cited by Irwin et al.). Along with the data from Mozurkewich et al. (1991), Procyon's radius is then found to be and , respectively.

From the orbital parameters of Procyon and its white dwarf companion Irwin et al. also supply a value for the stellar mass with or , depending on whether they use their own parallax value or the one proposed by USNO, and therefore

i.e. one may adopt

although an error even twice as much could be entitled "very precise" in the context of stellar atmosphere analyses.

One has, however, to concede that from the stellar evolutionary point of view Procyon is too massive by . But this well-known discrepancy (cf. Steffen 1985, Irwin et al. 1992, and references therein) can be solved if - as Irwin et al. suggest - the separation of the binary components has been systematically overestimated by approximately 0.2 arcsec. Irwin et al. point out that observations of this kind are extremely difficult since the magnitude difference of the primary and secondary in the Procyon system is quite large (). It is also worth mentioning that their analysis with respect to the visual binary separation is based on Lick and Yerkes data from 1897 to 1913. For this reason they suggest that a modern observational study of the Procyon separation should be initiated to remove this uncertainty and further studies are obviously underway (cf. Dyson et al. 1994, Walker et al. 1994), or will profit from the availability of the H IPPARCOS catalogue in 1997.

For the time being, if one simply adopts a mass of the surface gravity is changed to , but this inevitably means that the parameter does not benefit very much from the precise knowledge of the stellar mass.

At this stage, having established Procyon's effective temperature and surface gravity by very direct methods, we turn to the iron abundance and model atmosphere analyses. Let us first briefly recall the problem Steffen demonstrated ten years ago: in his analysis of the Procyon spectrum of Griffin & Griffin (1979) an ATLAS6 model atmosphere (Kurucz 1979) with K and produced inconsistent abundances from lines of neutral and ionized stages of the same element. Abundances derived from ionized lines are systematically higher than those resulting from neutral lines in all cases where both kinds of lines are available. Steffen is able to reconcile this dichotomy by increasing the effective temperature to 6750 K, but favours 6500 K to be the most probable value, albeit the ionization equilibrium calls for an unrealistic in this case. Instead of looking for compromises by taking some kind of "mean values" as others did before and later, Steffen emphasizes this discrepancy and suggests NLTE effects and temperature inhomogeneities associated with convective motions (cf. Gray 1981b, Dravins 1987) as possible explanations. But irrespective of these problems he also points out that abundances derived from lines of nearly completely ionized elements (like Mg II, Si II, Fe II...) are quite insensitive to NLTE effects, the effective temperature and uncertainties in the temperature structure. Consequently, lines of this kind are most reliable for abundance analyses in our temperature range provided the value for the surface gravity is known.

Contrary to the analysis of Steffen, who had to work with photographic material obtained during several years, our data are CCD-based and homogeneous due to the large wavelength range achieved with one exposure. Therefore, as a first step, it was reassuring that our analysis gave the same results as Steffen's to within the error bars (cf. Fig. 8 and 9). The model atmosphere we use for Procyon is similar to Steffen's ATLAS6 model, except for that we use for the mixing-length in the convective energy transport. The wings of the Balmer lines H and H indicate the effective temperature K. With this value and taking Fe II lines and high-excitation ( eV) Fe I lines into account, the ionization equilibrium provides , a microturbulence value km s^-1 (in the usual manner that lines of different strength should give the same abundance) and a metal abundance [Fe/H] .

Fig. 8. Equivalent width measurements in the F OCES spectrum of Procyon compared to those of Steffen (1985). The mean difference in the sense (this work - Steffen) is -0.8 mÅ

Fig. 9. The ionization equilibrium of iron in the model atmosphere analysis of Procyon with K. Dashed lines indicate the abundances of Fe I (filled circles) and Fe II (open circles) as a function of surface gravity. The dotted curves are the 1 error bars (rms) valid at , the intersection of Fe I and Fe II abundances. Lines of neutral iron depend on the precise value of the effective temperature with [Fe/H]  dex for a change of K. Fe II instead is very sensitive to the surface gravity parameter, but almost independent to a change in the effective temperature ( [Fe/H]  dex for K). The discrepancy at , the astrometric surface gravity of Procyon, amounts [Fe/H]  dex. To reconcile the Fe I and Fe II abundances an unrealistic high effective temperature of K would be required

Note that the value for the effective temperature is 30 K beneath the one of our previous analysis (Fuhrmann et al. 1993) which was based on Kurucz (1979) opacities. This is however analogous to the Sun, where the wings of the Balmer lines indicate 5750 K, as has been discussed in Sect. 3. If we would instead adopt a value of 6530 K the ionization equilibrium would still be very discrepant with . To reconcile the surface gravity value with the astrometric one, we have to increase the effective temperature to K (cf. Fig. 9), which is of course beyond a reasonable limit.

As a consequence, this inevitably means that our standard model atmosphere analysis is restricted by severe limitations in the ionization equilibrium. Because neutral iron exists to merely about 1% in the line forming regions of Procyon, it is very susceptible to details in the temperature structure. Small turbulent motions capable to produce temperature inhomogeneities or deviations from the Saha-Boltzmann population numbers obviously have their impact on the line formation process and there is evidence that especially NLTE effects are at work (Watanabe & Steenbock 1985).

In this situation it is natural to ask whether we can postpone these difficulties and follow instead another, more trustworthy path to derive the surface gravity parameter. This in fact seems possible from the analysis of strong lines, as proposed in what follows.

As a rule, it is known that stars like Procyon which possess an extreme ionization balance in e.g. Fe I/Fe II or Mg I/Mg II, have the opposite advantage of being almost insensitive to pressure changes in the line formation of weak neutral lines. That is, provided one can find a scarcely populated species, which nevertheless forms weak and strong lines in the same stellar spectrum, the determination of the surface gravity value can be achieved.

In this respect we advocate Mg I to be a good tracer of the surface gravity parameter, because

• (a) the oscillator strengths, especially the ones of the Mg Ib lines are fairly well-known, consequently it is a straightforward task to derive the collisional damping constant from the line shape in the solar flux spectrum
• (b) the Mg Ib lines lie in a spectral region where an accurate (0.5 - 1%) placement of the continuum is feasible
• (c) although 5167 is heavily blended, redundancy in the measurements is achieved from 5172 and 5183
• (d) magnesium has approximately the same ionization potential and cosmic abundance as iron. Hence it is one of the most abundant elements, and - in our temperature range - the neutral stage is much less populated than the ionized one, i.e. weak neutral lines become insensitive to the surface gravity parameter
• (e) the strong Mg Ib lines show wings even in metal-poor stars of [Fe/H] , which is most important for our stellar sample (e.g. HD 19445)

In Fig. 10 we apply the information stored in the Mg I lines to the spectrum of Procyon: in the left column 5711 and the Mg Ib lines 5172 and 5183 are shown for the surface gravity value =3.58 as derived from the ionization equilibrium. 5711 is practically an unblended absorption line and - as is obvious from panel (b) - insensitive to a change in from 3.58 to 4.00 ( mÅ). The theoretical profiles (dashed lines) have been convolved with a rotational broadening component km s^-1, a radial-tangential macroturbulence km s^-1 (both adopted from Gray 1981a) and the instrumental profile (a Gaussian of km s^-1). The final profile fit to 5711 in panel (a) is now achieved by simply adjusting the magnesium abundance. Once we have fixed the Mg I abundance from this and other weak Mg I lines, it serves as the input parameter to the Mg Ib lines. They possess strong wings and respond to a change in , as shown in panels (d) and (f). It is important to realize that the magnesium abundance derived from weak Mg I lines has no meaning in an absolute sense. It merely serves as the input parameter to the strong Mg Ib lines, which makes this differential procedure a very robust one. If instead, we would have done the whole analysis with K, i.e. a 60 K higher value, the surface gravity would result in .

Fig. 10. The spectroscopic surface gravity determination of Procyon from the analysis of Mg I lines. Left column: line profiles of Mg I 5711 (top) and the Mg Ib lines 5172 and 5183 (below) for a surface gravity value , as derived from the ionization equilibrium. The profile of 5711 shows no wings and is practically independent of the surface gravity, as illustrated in panel b, where line formation is done for a value . Among other weak Mg I lines, 5711 therefore serves to fix the value of the Mg I abundance. This information is then used in the analysis of 5172 and 5183 by altering the surface gravity value until the observed line shape is reproduced as shown in panels d and f. In the case of Procyon is found by this method, which is 0.42 dex higher than derived from the ionization equilibrium and only slightly below , the very precisely known value from astrometric data

In practice the method is of course iterative. In the case of Procyon it takes two iterations to obtain the final parameters. In the first step we get from the profile fit to 5172 and 5183 and derive a new value for the microturbulence and iron abundance (e.g. [Fe/H] is increased from to -0.02). In addition, these two parameters are now derived exclusively from Fe II lines to make sure that the LTE assumption is fulfilled and to be independent of details in the temperature structure and effective temperature value. After a slight reiteration of the effective temperature from Balmer lines the next step already provides the final spectroscopic parameters: K [Fe/H] km s^-1

Note, as explained above, the effective temperature turns out to be somewhat low compared to the direct methods from the angular diameter measurements, but this has practically no influence on the other three parameters, that is, irrespective of small uncertainties in the effective temperature value it seems possible to find very reliable parameters for the surface gravity, the iron abundance and the microturbulence. Reasonable error limits in the case of Procyon are  dex, [Fe/H]  dex and km s^-1.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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