5 Balmer profiles from the Holweger-Müller solar model

5.1 The solar HM Model

For the calculation of the solar Balmer profiles we adopted the Holweger-Müller model (1974, henceforth, HM) to avoid additional complications from various solar models, already discussed, for example, by CGK. We started from the HM T- $\tau_{5000}$ relation given for 29 layers, and extrapolated-interpolated to suit the depth ranges used by our respective codes.

There are differences in the optical depth coverage of the Michigan and Trieste codes. In the first case, the T- $\tau_{5000}$ relation was interpolated-extrapolated to 135 layers, while in the second case it was interpolated for 50 layers before using it in the Kurucz codes. While the Michigan code performs integrations directly in terms of $\log(\tau_{5000})$, the use of the Kurucz codes requires a conversion from the $\tau_{5000}$ depth scale to a RHOX (or $\int\rho {\rm d}x$) depth scale, where $\rho$ is the density of the stellar gas and x is the geometrical height in the atmosphere. The conversion was obtained by computing the continuous opacity $\kappa_{5000}$ at $\lambda=$ 5000 Å by means of the ATM code from Holweger, Steffen & Steenbock (1992, private communication) and by deriving RHOX from the relation ${\rm d}\tau_{5000}$ =  $\kappa_{5000}\rho~{\rm d}x$. The original HM model was made more than a quarter of a century ago. Since that time, abundances and the continuous opacity routines have been modified, presumably for the better. This means that the current relation between $\tau_{5000}$ and $\tau_{\rm Rosseland}$ is no longer the same as in the HM paper. The latter is inconsistent with the RHOX scale of the modern Kurucz codes.

We adopted as solar abundances the meteoritic values from Grevesse & Sauval (1998) and a constant microturbulent velocity $\xi=$ 1 km s^-1.

The HM model used in the Kurucz codes is given in the Appendix A.

5.2 Predictions from the HM model

Figure 4: Comparison of the solar intensity from the center of the sun predicted by the HM model (full line) with the observations from Neckel & Labs (1984) (dashed line). The line opacity in this low-resolution calculation is entirely from the ODFs.

For clarity, we first list several categories of opacity relevant to the current problems:

1.

Standard continuous opacity: bound-free and free-free transitions in various atoms and ions, Rayleigh and Thomson scattering. These are implemented in most currently-used model atmosphere and spectrum synthesis codes;

2.

TOPBASE opacities (Seaton et al. 1992). These opacities have not yet been widely implemented in current atmosphere codes, so the impact of this important work remains to be seen;

3.

Line opacity due to transitions between tabulated atomic energy levels. Some of these lines are predicted, in the sense that they have not been observed on the laboratory, but all relevant levels have been located, typically to a fraction of a wavenumber from observed lines. We shall call these classified lines. We distinguish two categories:

(a)

Stronger lines, which contribute 1% or more to the continuous opacity at the central wavelength for point in a model atmosphere.

(b)

Weaker lines, for which the above criterion is not met;

4.

Line opacity due to transitions involving one and sometimes two levels whose locations are predicted by an atomic structure code. Wavelengths for these lines may be uncertain by 10 or more angstroms. A sizable fraction of these lines involve levels above the first ionization limit, and the levels are therefore subject to autoionization. We shall refer to these as unclassified lines. Many of these lines may have been observed in laboratory experiments. Again, we list two categories:

(a)

Stronger lines. In certain chemically peculiar stars, we know there must be many such lines because we are unable to identify a large fraction of the measurable stellar lines. There are also many unidentified lines in the solar spectrum, though they are usually weaker than a few tens of milliangstroms, and typically increase in number to the violet.

(b)

Weaker lines connecting predicted levels;

5.

"Missing'' opacity. Calculations of the solar continuum using only standard continuous opacity (No. 1 above) predict values significantly higher than the "observed'' continuum. The disparity increases toward the violet (see discussion below).

Figure 4 compares the solar intensity $I_{\lambda }$(0) from the center of the Sun measured by Neckel & Labs (1984) with $I_{\lambda }$(0) predicted using the continuous and line opacities from Kurucz (1993c) and the HM model given in Appendix A. The line opacity is treated with the opacity distribution functions (ODF), which include both classified and unclassified lines. Because the ODFs involve averages over wavelength intervals of the order of 20 Å in the 3300-6400 Å region and larger for $\lambda$ > 6400 Å, we refer to the calculation of Fig. 4 as a low-resolution synthesis.

When the opacity of both classified and unclassified lines is considered in the calculations, the agreement of the low resolution observations with the low resolution predictions seems to be rather good at the first glance. However, a closer inspection shows that the observed and computed pseudo-continuum levels agree well in the regions 4200-4500 Å and 5700-6600 Å, but that elsewhere the computed intensity is systematically larger than the observed one, with differences of the order of 5-10%. This disagreement may indicate that either the observed low-resolution central intensity is affected by uncertainties larger than the estimated limit of $\pm$1% (Neckel & Labs 1984), or that the HM model should be refined, or that the problem of the missing opacity has not been completely solved.

As far as observations are concerned we would like to remark that the absolute integrals of the solar disk-center intensity measured by Burlov-Vasiljev et al. (1995) are higher by about 6% than that of Neckel & Labs (1984) at H$_{\delta}$, 4% at H$_{\gamma}$, 2% at H$_{\beta}$, while it is about 2% lower at H$_{\alpha }$. Burlov-Vasiljev et al. (1995) estimated errors from 2.5% at 3100 Å to 2.2% at 6800 Å. This implies that the different levels of the observations at the position of H$_{\gamma}$ and H$_{\delta}$ are outside the error limits.

In Sect. 5.3 we will show that the HM model produces almost the same discrepancy as the theoretical solar Kurucz model does when high-resolution observed and computed Balmer profiles, unnormalized to the continuum level are compared.

Section 5.4 deals with the effects of the missing opacity on the Balmer profiles. Its nature is somewhat controversial, and will not be argued here. A recent reference, with citations to earlier discussion, is Peterson et al. (2001).

Limb darkening predictions from the HM model are compared in Fig. 5 with those from Neckel & Labs (1994). In this case, opacity from lines is not included in the computations in accordance with the assumption of Neckel & Labs (1994) of observations made at wavelengths free from lines contaminating the continuum. The departure of the computations from the observations in the violet can be explained with the poor chance to have regions free from lines in this part of the solar spectrum. Except for the violet wavelengths, the agreement is satisfactory.

Figure 5: Comparison between observed (points) and computed (full line) solar limb-darkening curves $I_{\lambda }$( $\cos\theta$)/ $I_{\lambda }$(0). Observations are from Neckel & Labs (1994) and computed curves are based on the HM model.

5.3 The Balmer profiles in absolute intensity

Figure 6 shows the observed and computed Balmer profiles for the disk center in absolute intensity. We have adopted the Kitt Peak observations available at the Hamburg site (ftp.hs.uni-hamburg.de; pub/outgoing/FTS-Atlas) and described by Neckel (1999, henceforth, KPN). The files include absolute intensities, as well as continuum estimates at each wavelength. The resolution of the observations is about 350 000.

The synthetic Balmer profiles were computed with the SYNTHE code and the HM model. Two different spectra were computed, the first only with the relevant Balmer line, the second one with all classified and unclassified lines. For both spectra standard continuous opacity sources were used. The second synthetic spectrum is computed with the same line opacity adopted for computing ODFs, so that it can be directly compared with the intensity from the center of the sun predicted by the ATLAS9 code and the HM model. Each synthetic spectrum was degraded at the observed resolution and it was broadened by assuming a macroturbulent velocity $\xi_{{\rm macro}}=$ 1.5 km s^-1, although Balmer profiles are independent of instrumental and macroturbulence broadenings of the order of those here adopted.

Figure 6 shows that, in agreement with Fig. 4, the observations fall below the calculated profiles, especially for H$_{\beta}$ and H$_{\delta}$. The differences are very small for H$_{\alpha }$, i.e. less than 1%, but they are of the order of 5% for H$_{\beta}$, 4% for H$_{\gamma}$, and 8% for H$_{\delta}$. This result is very similar to that obtained by CGK from the theoretical solar Kurucz model (Fig. 7 in Castelli et al. 1997), indicating that the discrepancy is rather independent of the specific solar model adopted for the computations.

The two synthetic spectra plotted in Fig. 6 indicate that the high points of the calculation including all lines generally reach the profile where only the Balmer line is included. Therefore the difference between the observed and computed intensity levels is not resolved by the inclusion of all classified and unclassified lines in the calculation. A reasonable interpretation is that the majority of the opacity from the unclassified lines is seen as relatively strong features that appear as absorption lines rather than a smooth pseudo-continuum or veil of weak features. We conclude that a direct comparison of theory and observation in absolute units cannot be made unless this discrepancy is taken into account. We do this in a crude way in the following section, where we used Balmer profiles normalized to the continuum levels in order to avoid all the uncertainties related with absolute calibration of the observed solar intensity from the disk center.

Figure 6: Two unnormalized to the continuum level $I_{\lambda }$(0) computed spectra (thin lines) are shown in each panel, with (1) only the relevant Balmer line, and (2) all classified and unclassified lines. The unnormalized observed $I_{\lambda }$(0) spectrum (KPN, thick line) generally falls below the computed spectra. The y scale gives $I_{\lambda }$(0) intensities in units of 106 erg cm-2 s-1 stear-1 Å-1, which have to be multiplied by 3.2 for H$_{\alpha }$, 5.0 for for H$\beta$ and H$\gamma$, and 5.5 for H$\delta$.

5.4 The normalized Balmer profiles

 

 

Table 2: Solar continuum specific intensity in units of 10¹⁵ cgs.

Wavelength (Å)	This work	KPN
3298.973	0.3235	0.3231
3355.431	0.3269	0.3272
3782.919	0.4083	0.4093
4020.705	0.4589	0.4591
4279.262	0.4652	0.4666
4419.404	0.4598	0.4609
4504.079	0.4540	0.4545
4861.000	0.4230	0.4179
5102.095	0.3999	0.3990
5203.252	0.3906	0.3902
5801.460	0.3435	0.3424
6109.561	0.3200	0.3189
6202.178	0.3146	0.3144
6409.847	0.2990	0.2972
6500.584	0.2907	0.2899
6802.324	0.2660	0.2663
6850.076	0.2619	0.2627
6950.356	0.2546	0.2553
6972.875	0.2536	0.2540
7000.000	0.2524	0.2524

In the current work, one of us (CRC) attempted new estimates of the continuum for the observed spectrum - less as an attempt to improve on the KPN values, as to gain some insight into the uncertainties in this endeavor. We began with spectral high points within 10 Å intervals plotted vs. wavelength, and smoothed the "envelope'' by selectively deleting points, in an obviously subjective way, to achieve an overall smooth plot. The adopted points are shown in Table 2, along with those from KPN. We make no claim that the current continuum is superior in any way to that chosen in KPN. It was simply used in the Michigan work for normalization purposes. We employed a four-point Lagrange interpolation scheme to normalize observations between the chosen points.

Figure 7: H$\alpha$ profile for the center of the solar disk normalized to the continuum level. The thin curve is the observed KPN spectrum, and the solid the CSII calculation with an assumed microturbulence $\xi _{\rm t} = 1$ km s-1. In this calculation no allowance for missing opacity has been made, and the continuum has been adopted as described.

Our independent evaluation of the continuum based on the points shown in Table 2 is in excellent agreement with KPN, with the exception of the region near H$\beta$. The value shown in Col. 2 for $\lambda$4861 interpolated with the four-point Lagrange formula, from the surrounding points, is 1.2% higher than the KPN continuum. This region appears depressed for reasons that are unclear and deserve investigation.

The continuous specific intensity using the HM model and Michigan codes matches the interpolated continuum from Table 2 at H$\alpha$ to within 1%. For H$\beta$ through H$\delta$, the calculated continua fall above the measured (as interpolated in Table 2) continua by 2.4, 3.9, and 7.8% respectively. These results agree well with those discussed in the previous section of the comparison of the observed and computed absolute intensities.

If we assume the "missing opacity'' as cause for these disagreements as well as for those shown in Fig. 6, there is at present no obviously correct way to account for it. For these calculations, we assumed this opacity has the same depth dependence as standard continuous opacity sources. We have simply scaled them by constant factors until the calculated specific continuous intensities agree with the observed chosen continuum.

When spectra normalized to the continuum levels are compared, we find an excellent agreement for H$_{\alpha }$ (Fig. 7). The results are the same both from the CSII and the SYNTHE code, and are to be compared with BPO's Fig. 8 (upper), done for the solar flux. We see good agreement in all cases. The agreement of the CSII profiles with BPO profiles is expected, since the only basic difference is the use in BPO of the PK approximation while CSII uses a full numerical convolution, a distinction we have found thus far to be unimportant.

As far as the three higher, normalized Balmer lines are concerned, the best fits to the wings are obtained when the "observed'' continua are adjusted downward from values obtained by interpolation in Table 2 - the sense is that the continuum there is too high. For H$\gamma$ and H$\delta$, the downward adjustment is 2%. The observed continuum at H$\beta$ needed a downward adjustment of 3%; problems with the continuum in this region were mentioned earlier in this section. Figure 8 shows the fit for H$\delta$. The other two Balmer line fits may be seen at the url: http://www.astro.lsa.umich.edu/users/cowley/balmers.html/

Figure 8: KPN spectrum and CSII calculation for H$\delta$.

In principle, the adjustment of the continuum requires an iteration with a new continuous opacity to the new continuum. Fortunately, the normalized Balmer profiles are not very sensitive to small adjustments for the missing opacity.