3 Non-LTE calculations for the Sun

Our non-LTE calculations for the Sun are based on the model atom described above and employ the empirical model atmosphere of Holweger & Müller (1974).
In Figs. 2 and 3 the resulting departure coefficients $b_i = n_{i,{\rm NLTE}}/n_{i,{\rm LTE}}$ are shown for a calculation with the model atom for Si I and Si II, respectively. The numbers on the left of the diagrams correspond to energy level numbers specified in Table 1. In the solar photosphere at $\tau \approx 0.1$ about $\approx$$30\%$ of the silicon atoms are neutral while $\approx$$70\%$ are singly ionized. Our calculations show that most of Si II is present in the ground state. Therefore it is not surprising that the corresponding departure coefficient indicates almost perfect LTE conditions ( $b_i \approx 1$) in the Sun. The same is true for the ground state of Si I. Furthermore, throughout the photosphere ( $\log \tau_{5000} \geq -2$), deviations from LTE are almost negligible for excited levels of Si I. In contrast, most excited levels of Si II are overpopulated with respect to LTE. In both ionization stages there are groups of energy levels whose departure coefficients closely coincide. This is due to very small energy differences within these groups and consequently a strong collisional coupling.

Figure 2: Departure coefficients of Si I in the Sun.

Figure 3: Departure coefficients of Si II in the Sun.

3.1 Abundance analysis

To enable a direct comparision between former LTE abundance determinations and the present work, the line list (Table 3) is essentially that adopted by Holweger (1973),

 

 

Table 3: Line list used for the abundance analysis in the Sun: wavelength, excitation potential $\chi _i$ of lower level, equivalent width $W_{\lambda }$ (center of disk), silicon abundances and non-LTE corrections $\Delta \log \epsilon = \log \epsilon_{\rm NLTE} - \log \epsilon_{\rm LTE}$. Oscillator strengths $\log gf$ of Si I: Garz (1973), corrected by Becker et al. (1980); Si II: Froese-Fischer (1968).

$\lambda$	mult.	$\chi _i$	$\log gf$	$W_{\lambda }$	$\log \epsilon$	$\Delta \log \epsilon$
(Å)	nr.	(eV)		(mÅ)
Si I:
5645.611	10	4.9296	-2.040	34	7.535	-0.004
5665.554	10	4.9201	-1.940	40	7.532	-0.004
5684.485	11	4.9538	-1.550	60	7.485	-0.007
5690.427	10	4.9296	-1.770	52	7.563	-0.006
5701.105	10	4.9296	-1.950	38	7.514	-0.005
5708.397	10	4.9538	-1.370	78	7.550	-0.011
5772.145	17	5.0823	-1.650	54	7.599	-0.008
5780.384	9	4.9201	-2.250	26	7.577	-0.004
5793.071	9	4.9296	-1.960	44	7.623	-0.005
5797.860	9	4.9538	-1.950	40	7.567	-0.005
5948.540	16	5.0823	-1.130	86	7.508	-0.015
6976.520	60	5.9537	-1.070	43	7.532	$0.000\;^a$
7034.901	42.10	5.8708	-0.780	67	7.493	$-0.001\;^{a,b}$
7226.208	21.05	5.6135	-1.410	36	7.498	-0.005
7680.265	36	5.8625	-0.590	98	7.626	$-0.003\;^b$
7918.382	57	5.9537	-0.510	95	7.565	$0.000\;^b$
7932.348	57	5.9639	-0.370	97	7.451	$0.000\;^b$
7970.305	57	5.9639	-1.370	32	7.663	$0.000\;^a$
Si II:
6347.110	2	8.1210	0.260	56	7.639	$-0.097\;^c$
6371.370	2	8.1210	-0.040	36	7.521	$-0.064\;^c$

a	Larger error in oscillator strength.
b	Strongly sensitive to collisional line broadening.
c	Susceptable to uncertainties in model atom.

except for the lines for which no departure coefficients were available from the non-LTE calculations. The present sample consists of 18 Si I lines from 10 different multiplets and two Si II lines from the same multiplet. The wavelengths and energy values $\chi _i$ of the lower levels of a line transition are taken from Fuhr & Wiese (1998) as found on the NIST server. The equivalent widths of Holweger (1973) refer to the center of the solar disk. Therefore the present abundance determination was carried out for the center of the disk ($\mu=1$) and a constant microturbulence of $\xi = 1.0\,{\rm km}\,{\rm s}^{-1}$ was used.
Line broadening by collisions with hydrogen atoms is treated as pure van der Waals broadening. The broadening parameter C₆ for the individual lines was calculated from the mean square atomic radii of the corresponding energy levels, based on the approximation given by Unsöld (1955). In many applications, this approximation has turned out to underestimate the real damping constants, resulting in a systematic increase of abundance with equivalent width. This was corrected by applying a correction $\Delta \log C_6$, either derived empirically or from quantum mechanical calculations (Steffen 1985; O'Mara 1976). However, no increase with equivalent width is present for the silicon abundances, as can be seen from Fig. 4. For this reason, no correction is necessary and a value of $\Delta \log C_6 = 0$ was adopted for this abundance determination. This agrees with Holweger (1973) and, moreover, confirms the upper limit of $\Delta \log C_6 \approx 0.3$ given in that work.

Figure 4: Solar silicon abundance for the lines in Table 3 over equivalent width $W_{\lambda }$. Filled symbols represent non-LTE values, unfilled LTE abundances (circles for Si I, squares for Si II). The horizontal lines illustrate the weighted mean and the standard deviation (solid).

For line broadening by electron collisions, the approximations according to Griem (1968) (ions) and Cowley (1971) (neutrals) were applied. For the Si I and Si II lines used here, radiation damping is small compared to collisional broadening, and the classical approximation for $\gamma_{\rm rad}$ is used. Oscillator strengths for all lines except for $\lambda 7034$ (9-56) and $\lambda 7226$ (7-37) are tabulated in the recent compilation by Fuhr & Wiese (1998). The derived abundances however show a large scatter with a standard deviation of $\pm 0.26$ dex. The mean silicon abundance is $\log \epsilon_{\rm LTE}= 7.468$ (LTE) and $\log \epsilon_{\rm NLTE}=7.456$ (non-LTE), respectively (Fig. 5b). Taking the oscillator strengths of Wiese et al. (1969) leads to a somewhat lower non-LTE abundance of $\log \epsilon_{\rm NLTE}= 7.405$ and a slightly smaller abundance scatter of 0.24 dex. From former investigations it is known that a much higher accuracy is possible for abundance determinations in the Sun. Obviously the internal accuracy of the Si I f-values in the Wiese et al./Fuhr & Wiese compilations is rather low. Consequently, these values were not taken into account any further. Nevertheless this set of $\log gf$-values can still be applied for the model atom. More satisfactory results were achieved with the older experimental oscillator strenghts of Garz (1973) which apparently have been adopted in the compilation by Kurucz (1993) and in the database VALD. Becker et al. (1980) have revised the absolute scale for Garz's $\log gf$-values which resulted in a general correction of +0.1 dex. Because of the small intrinsic scatter of abundances, the Garz/Becker et al. f-values are used for all Si I lines in this abundance analysis.
For the two Si II lines, different sources for the $\log gf$-values were found to agree within 0.07 dex. The oscillator strengths of Wiese et al. (1969)/Fuhr & Wiese (1998) were not used, instead those derived by Froese-Fischer (1968) were choosen.
The silicon abundances determined with LINFOR from the equivalent widths of the individual lines are listed in Table 3 and illustrated in Fig. 5a.

Figure 5: Solar silicon abundance for the lines in Table 3: a) Corrected oscillator strengths by Garz (1973) and Becker et al. (1980). b) Oscillator strengths by Fuhr & Wiese (1998). Filled symbols represent non-LTE values, unfilled LTE abundances (circles for Si I, squares for Si II). The horizontal lines illustrate the weighted mean and the standard deviation (solid) and the former (and confirmed) abundance of $\log \epsilon_{\rm Si} = 7.55$ (dashed).

To account for larger uncertainties in oscillator strengths quoted for some Si I lines with wavelengths $\lambda > 6000$ Å, these lines have been entered with a half weight in the final abundance. Furthermore, the strongest Si I lines with equivalent widths $W_{\lambda} > 90$ mÅ were found to be extremely sensitive to collisional line broadening (van der Waals, Stark), with comparatively large uncertainties in abundance. Consequently the affected lines $\lambda\lambda$ 7680, 7918 and 7932 were half weighted, too. The two Si II lines are much more susceptible to uncertainties in the atomic data for the non-LTE calculations. Hence they were given half weight as well.
With the mentioned weights, an LTE abundance of $\log \epsilon_{\rm LTE} = 7.560 \pm 0.066$ and a non-LTE abundance of $\log \epsilon_{\rm NLTE} = 7.550 \pm 0.056$ was derived, implying a mean non-LTE correction of $\Delta \log \epsilon = -0.010$. Holweger (1973) used almost the same line list but with the original oscillator strenghts measured by Garz (1973) leading to a LTE abundance of $\log \epsilon=7.65$. With the systematic correction of the $\log gf$-values, the abundance was redetermined to $\log \epsilon=7.55$ (Becker et al. 1980). Their line sample is very similar to that used in the present work and the f-values were taken from the same source. There are only small differences in the atmosphere applied by Holweger (1973), mainly slightly different abundances of the main electron donors Fe, Mg and Si, and a depth-dependent microturbulence (see Holweger 1971). Since the present study showed that for Si II and especially Si I lines in the Sun, non-LTE effects are generally small, it is not surprising that the former silicon abundance could be reproduced almost exactly.
Several parameters have been varied to investigate their influence on the silicon abundance. Table 4 shows the deviations from the abundance derived for the above described standard model.

 

 

Table 4: Abundance differences due to different model assumptions for the Sun. Silicon non-LTE mean abundances, mean non-LTE corrections and deviations $(\Delta \log \epsilon)_{\rm d}$ from the result for the standard model.

model assumption	$\log \epsilon$	$\Delta \log \epsilon$	$(\Delta \log \epsilon)_{\rm d}$
standard model	$7.550 \pm 0.056$	-0.010
Non-LTE calculations:
no Stark broadening	$7.550 \pm 0.056$	-0.010	0.000
$\Delta\log C_6=-1.0$	$7.550 \pm 0.056$	-0.010	0.000
$\Delta\log C_6=+0.5$	$7.549 \pm 0.056$	-0.011	-0.001
no line transitions	$7.565 \pm 0.065$	+0.005	+0.015
all $\gamma_{\rm rad} \times 5$	$7.549 \pm 0.056$	-0.011	-0.001
all $\sigma_{\rm PI} \times 0.75$	$7.549 \pm 0.056$	-0.011	-0.001
all $\sigma_{\rm PI} \times 1.25$	$7.551 \pm 0.056$	-0.009	+0.001
all $\sigma_{\rm e} \times 0.1$	$7.543 \pm 0.055$	-0.017	-0.007
all $\sigma_{\rm e} \times 10$	$7.557 \pm 0.061$	-0.003	+0.007
all $\sigma_{\rm H}: S_{\rm H} = 0.01$	$7.545 \pm 0.056$	-0.015	-0.005
all $\sigma_{\rm H}: S_{\rm H} = 10$	$7.556 \pm 0.060$	-0.004	+0.006
Abundance analysis:
$\Delta\log \gamma_{\rm rad} = -1.0$	$7.550 \pm 0.056$	-0.010	0.000
$\Delta\log \gamma_{\rm rad} = +1.0$	$7.543 \pm 0.056$	-0.009	-0.007
$\Delta\log C_4=-0.5$	$7.565 \pm 0.057$	-0.010	+0.015
$\Delta\log C_4=+0.5$	$7.525 \pm 0.063$	-0.009	-0.025
no Stark broadening	$7.582 \pm 0.061$	-0.011	+0.032
$\Delta\log C_6=0.6$	$7.519 \pm 0.061$	-0.009	-0.031
$\Delta\log C_6=1.0$	$7.493 \pm 0.071$	-0.008	-0.056
$\xi= 0.8\,{\rm km}\,{\rm }s^{-1}$	$7.562 \pm 0.057$	-0.010	+0.012
ATLAS	$7.448 \pm 0.058$	-0.008	-0.102

In general, the uncertainties of atomic data for the non-LTE calculations like line broadening parameters, photoionization and collision cross-sections have only a minor effect on the resulting departure coefficients. However, uncertainties in the parameters for the abundance determination produce larger, altough still small, abundance deviations. Radiative damping has only little effect on the result. As already noticed by Holweger (1973) some Si I lines are quite sensitive to Stark broadening. If Stark broadening is completely neglected, the abundances for $\lambda\lambda$ 7680, 7918 and 7932 increase by 0.11 dex and by 0.08 dex for $\lambda$ 7034. This effect is smaller (<0.03 dex) for most of the remaining lines. Much stronger is the influence of the van der Waals broadening parameter $\log C_6$. A value of $\Delta \log C_6 = +1.0$ would lead to a decrease in abundance by -0.056 dex. Again, the spectral lines $\lambda\lambda$ 7680, 7918 and 7932 are most sensitive. Adopting a microturbulence of $\xi= 0.8\,{\rm km}\,{\rm }s^{-1}$ instead of $\xi = 1.0\,{\rm km}\,{\rm s}^{-1}$ would lead to a small increase of the non-LTE abundance by +0.012 dex. Replacing the Holweger & Müller photospheric model by an ATLAS9 model (Kurucz 1992) ( $T_{\rm eff}= 5780~ {\rm K},\, \log g = 4.44$) results in a considerably lower mean non-LTE abundance of 7.448 dex. Note that the described variations of the model parameters do not represent real error limits but rather demonstrate their different influence on the resulting abundance.

   
3.2 Abundance correction due to granulation

For the present calculations, always static, plane-parallel models are applied which cannot account for horizontal temperature inhomogeneities associated with convection. These inhomogeneities are believed to have a small but non-negligible effect on the photospheric abundances, making further corrections necessary. In a first attempt, Steffen (2000a) determined abundance corrections for various elements from 2D hydrodynamics simulations which yield temperature and pressure fluctuations of the solar surface. Through comparison of 1D and 2D models, both with a depth-independent microturbulence of $1.0\,{\rm km}\,{\rm s}^{-1}$, the effect of temperature inhomogenities on the abundance can be determined. A first application is reported in Aellig et al. (1999). A more detailed description will be given in Steffen & Holweger (2001).
For the case of silicon, M. Steffen (2000b) kindly provided granulation abundance corrections for 10 representative line transistions which permitted an interpolation for the remaining lines in Table 3. The resulting abundance corrections depend on the excitation potential $\chi _i$ and equivalent width. The mean granulation abundance correction for Si I line transitions with $\chi_i \approx 5$ eV is +0.023 and increases to 0.029 dex for $\chi_i \approx 6$ eV. The abundance corrections for the two Si II lines are -0.014. Considering the same weights as in the abundance determination above, a total correction of +0.021 due to granulation results. Finally, the photospheric silicon abundance becomes $\log \epsilon_{\rm GC} = 7.571$, including granulation effects. In 3D simulations, temperature fluctuations are in general smaller than in 2D. Therefore the granulation abundance correction described here should be considered as an upper limit.