MNRAS 430, 2979–2985 (2013)
doi:10.1093/mnras/stt103
The abundance of silicon in the solar atmosphere
A. M. K. Shaltout,1‹ M. M. Beheary,1 A. Bakry1 and K. Ichimoto2
1 Department
2 Kwasan
of Astronomy and Meteorology, Faculty of Science Al-Azhar University, Cairo, Egypt
and Hida Observatories, Kyoto University, Kurabashira Kamitakara-cho, Takayama-city, 506-1314 Gifu, Japan
Accepted 2013 January 14. Received 2012 December 23; in original form 2012 August 11
ABSTRACT
High-resolution solar spectra were used to determine the silicon abundance (εSi ) content by
comparison with Si line synthesis relying on realistic hydrodynamical simulations of the solar
surface convection, as 3D inhomogeneous model of the solar photosphere. Based on a set of
19 Si I and 2 Si II lines, with accurate transition probabilities as well as accurate observational
data available, the solar photospheric Si abundance has been determined to be log εSi (3D) =
7.53 ± 0.07. Here we derive the photospheric silicon abundance taking into account non-LTE
effects based on 1D solar model, the non-LTE abundance value we find is log εSi (1D) =
7.52 ± 0.08. The photospheric Si abundance agrees well with the results of Asplund and more
recently published by Asplund et al. relative to previous 3D-based abundances, the consistency
given that the quoted errors here are (±0.07 dex).
Key words: line: formation – line: profiles – Sun: abundances – Sun: atmosphere – Sun: photosphere.
1 I N T RO D U C T I O N
The solar chemical composition is a fundamental yardstick in astronomy but has been heavily debated in recent times. The solar Si
abundance is of particular importance since it is the anchor of the
meteoritic scale due to the depletion of volatile elements such as H
in meteorites: the photospheric and meteoritic abundances of Si are
forced to agree. Also, silicon is one of the main electron contributors
in the atmospheres of late-type stars (Holweger 1973; Wedemeyer
2001). Thus the solar photospheric Si abundance requires detailed
and careful study.
Important discrepancies still exist among the values of the solar abundance of silicon published during the past 10 years. The
solar photospheric silicon abundance for departures from LTE has
been obtained by Wedemeyer (2001) and Shi et al. (2008). An extensive investigation of silicon lines taking into account the threedimensional (3D) structure of the solar atmosphere has been performed by Asplund (2000) and Asplund, Grevesse & Sauval (2005)
and Asplund et al. (2009). 3D hydrodynamic models have allowed
us to get rid of many uncertainties in abundance determination, especially those coming from the use of micro- and macroturbulence
velocity parameters (Asplund et al. 2000).
In this paper, we performed a new analysis of a sample of unblended Si I and Si II lines for which accurate transition probabilities
and accurate damping constants have been computed by Kurucz
(2007) and other sources which were taken from the improved
quantum mechanical broadening treatment of Anstee & O’Mara
(1995) and Barklem & O’Mara (1997). The detailed explanations
E-mail: [email protected]
on the non-LTE calculations for the adoption of the Si I and Si II
model atom included in the present work and finally the results of
the solar Si abundance based on 1D LTE and 1D non-LTE are presented in Section 2. Finally, the results concerning the solar Si abundance using 3D inhomogeneties model atmosphere are discussed in
Section 3.
2 N O N - LT E C A L C U L AT I O N S
2.1 Model atom
The model atom used for this study is essentially the same as that
used by Wedemeyer (2001). It comprises 115 energy levels and 84
line transitions of neutral and singly ionized silicon (see table 1 in
Wedemeyer 2001). Generally, for Si I (ionization limit at 8.15 eV)
75 energy levels up to 7.77 eV and 53 line transitions were taken
into account. For Si II (ionization limit at 16.35 eV) 40 energy levels
up to 15.65 eV and 31 line transitions were included. For clarity, the
energy levels are listed in table 1 in Wedemeyer (2001), together
with statistical weights, and the labels of the levels in the next
higher ionization stage. A summary of all oscillator strengths and
line transitions used for non-LTE calculations is given in table 2
(also given in Wedemeyer 2001).
We took the photoionization cross-sections with complex structure from the calculations of the TOPbase compilation at the
website: (http://vizier.u-strasbg.fr/topbase/topbase.html) (see also
Seaton et al. 1992; Nahar & Pradhan 1993; Seaton, Mihalas &
Pradhan 1994) for almost all the energy levels used, except for a
few highly excited levels [these levels are marked with an asterisk (*) in table 1 in Wedemeyer 2001] for which the photoionization cross-sections were not available from the TOPbase server.
C 2013 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society
2980
A. M. K. Shaltout et al.
For transitions missing in the TOPbase tables, the Kramers Gaunt
approximation for hydrogen-like atoms, as given by Allen (1973),
was used. For our non-LTE calculations, we take into account the inelastic collisions with electrons and neutral hydrogen atoms for both
excitation and ionization. The electron collisional cross-sections for
optically forbidden transitions are calculated from Allen (1973) formula. For the electron collisional cross-sections of optically allowed
transitions, the traditional formula of van Regemorter (1962) was
applied using all available f-values and additional values from the
Opacity Project or, if unknown the cross-sections are treated in the
same way as the optically forbidden transitions but setting collision strength = 10 (Inga Rentzsch-Holm, 1996). The ionization
cross-sections for electron collisions are calculated from the Seaton
(1962) formula. Inelastic collisions with neutral hydrogen atoms
are calculated as outlined by Steenbock & Holweger (1984) on the
basis of the approximations given by Drawin (1968, 1969). Based
on our non-LTE calculations, an empirical scaling factor of SH =
1.0 and 0.1 was applied for both collisional excitation and ionization by neutral hydrogen with the formula given by Steenbock &
Holweger (1984).
2.2 Non-LTE code, model atmosphere and silicon departure
coefficients
The code used in this study, which simultaneously solves the equations of statistical equilibrium and radiative transfer, is the 1D nonLTE MULTI code, written by Carlsson (1986, 2011). The radiativestatistical equilibrium equations have been solved for a 115-level
silicon model, considering 84 bound–bound transitions in detail. For
the calculation of the solar silicon abundance we adopted the semiempirical solar model atmosphere of Holweger & Müller (1974,
hereafter HM). We took the HM model as input for the MULTI and
KURUCZ codes from Cowley & Castelli (2002). In addition, the HM
model is available from a link on F. Castelli web site at the url:
http://wwwuser.oat.ts.astro.it/castelli/sun/ hm50meteorat9.dat. The
HM model taken directly as is for T(τ 5000 ), Pg (τ 5000 ), ρ(τ 5000 ) and
Ne (τ 5000 ). Here, T, Pg , ρ and Ne are the temperature (K), gas pressure (dyn cm−2 ), density of the stellar gas (g cm−3 ) and electron
density (cm−3 ) as a function of the standard optical depth at 5000
Å, respectively. At first, we calculate the departure coefficients bi =
Ni,non-LTE /Ni, LTE [here Ni, LTE and Ni,non-LTE denote the level occupation numbers in LTE and non-LTE, respectively] given as a function
of optical depth. Figs 1 and 2 show the departure coefficients for all
the most important terms of Si I and Si II levels with the hydrogen
collision parameter (SH = 1.0, 0.1) as a function of the standard
optical depth at 5000 Å. The resulting departure coefficients for our
final selection of the hydrogen collision parameter (SH = 0.1) are
presented in Fig. 2, which is the same as that assumed in Wedemeyer
(2001) and Shi et al. (2008).
As is evident from Figs 1(b) and 2(b) most of the Si II is present
in the ground state. Since the levels 79 and 80 have slightly negative
departure coefficients in the line forming layers, these levels are in
LTE. It is quite clear from the resulting departure coefficients of Si II
that it is almost perfectly in LTE conditions, hence bi ≈ 1, in the
Sun. The same is obvious for the ground state of Si I (Figs 1a and
2a). This is most pronounced in level 6 (log bi ≈ 0.2). The differences between the departure coefficients of the higher levels of Si I
(i > 8) are very small. They follow LTE out to the upper photosphere as seen in the levels 14, 26 and 28. Through the photosphere
at the level (log τ 5000 ≥ −2), it is clear that departures from LTE
are negligible for the excited levels of Si I. On the other hand, most
excited levels of Si II are almost overpopulated with respect to LTE.
Figure 1. Si I (a) and Si II (b) departure coefficients for the final selection
of the hydrogen collision parameter (SH = 1.0) as a function of the standard
optical depth calculated for the HM model.
This is shown in levels 100, 103 and 115. There are groups of excitation energies whose departure coefficients closely coincide. It
is the result of a small excitation energy differences within these
groups and therefore a strong collisional coupling. Furthermore,
the resulting departure coefficients are consistent with Wedemeyer
(2001), showing similar findings to that found in his analysis.
2.3 Atomic data
The Si lines used for the abundance analysis are the same as in
Holweger (1973). Our adopted Si lines and atomic data are given in
Table 1. It should be noted that the abundance determinations were
calculated using 3D solar model atmosphere based on a sample
line list consisting of 19 Si I and two Si II lines (described in detail
in Section 3). Based on non-LTE abundance calculations, our line
list consists of 18 Si I and two Si II lines. The necessary oscillator
strengths for Si I lines were taken from Becker et al. (1980), which
are measured by Garz (1973) but corrected using more accurate lifetime measurements. For the two Si II lines, the oscillator strengths
of Bautista et al. (2009) were used. The wavelengths and excitation
potentials are from Wedemeyer (2001). The damping constants and
Stark broadening are taken from the Vienna Atomic Line Database
(VALD; Kurucz 2007). For most lines, we took collisional broadening parameters calculated for individual lines from VALD (Kurucz
The abundance of silicon
Figure 2. Si I (a) and Si II (b) departure coefficients for the final selection
of the hydrogen collision parameter (SH = 0.1) as a function of the standard
optical depth calculated for the HM model.
2007). For the remainder set of Si lines are interpolated in the
Anstee & O’Mara (1995) and Barklem & O’Mara (1997) tables
where possible, as explained in Table 1.
2.4 Abundance analysis
The lines used in the abundance analysis are listed in Table 1, together with the excitation potentials, f-values, collisional damping
constants [radiative damping, equivalent widths and finally the logarithms scale of abundances (3D non-LTE, 1D LTE) together with
1D non-LTE corrections, respectively]. The final Si abundances are
illustrated in Fig. 3. To account for the larger uncertainties in the
f-values for some Si I lines, these lines enter the final abundance
estimate with half weight, as marked in Table 1. In addition, the
strongest Si I lines λ 768.02, λ 791.83 and λ 793.23 nm (equivalent
width Wλ > 9 pm) were found to be extremely susceptible to the
adopted collisional line broadening (van der Waals, Stark). The influence of van der Waals and Stark broadening for the three strong
Si I lines (λ 768.0, λ 791.8 and λ 793.2 nm) is discussed. If we
neglect entirely the van der Waals collisional broadening, the abundances show an increase by 0.209, 0.397 and 0.425 dex, respectively.
Similarly, the neglect of Stark broadening changes the abundances
by 0.111 dex for λ 768.0 nm, whereas the abundances for λ 791.8
2981
and λ 793.2 nm show an increase by (0.032 and 0.029 dex). To be
conservative, we give these lines half weight into the final abundance determination. For the two Si II lines, they are much more
sensitive to uncertainties in the model atom for non-LTE calculations. Consequently, they are entered with half weight as well. For
computing silicon lines the code is WIDTH9 (Kurucz 1993), which
produces LTE abundances for any silicon line. For most of Si lines,
the van der Waals broadening is taken from VALD data set (Kurucz 2007), while for the reminder of Si lines the Van der Waals
are interpolated in the Anstee & O’Mara (1995) and Barklem &
O’Mara (1997) tables. Only the most recent work on the silicon
lines (e.g. Kurucz 2007) has included the new radiative and Stark
broadening of VALD data. They are available from a link at the url:
http://cfaku5.cfa.harvard.edu/ATOMS/1400. To calculate the solar
abundance, we adopted the equivalent widths for Si lines from Holweger (1973). Throughout this analysis a microturbulent velocity
ξ = 1.0 km s−1 was assumed. This value is the same as that adopted
by Wedemeyer (2001) in his 1D non-LTE study of photospheric
Si abundance. Based on the mentioned weights, the solar silicon
abundance for the LTE case was log εSi (LTE) = 7.54 ± 0.096 from
20 Si lines with the 1D HM solar model.
To make a comparison between LTE abundance determinations
and the present work, a line list consists of 18 Si I lines from 10
different multiplets and two Si II lines from the same multiplet, except for Si I λ 674.1 nm for which the departure coefficient was
not available from the non-LTE calculations. Successively the departure coefficients were computed; we were ready to calculate
the non-LTE abundance log εSi (non-LTE), as well as the abundance correction (relative to the LTE case) log ε = log εnon-LTE
− log εLTE , for a given equivalent width. The resulting non-LTE
silicon abundances were calculated with the WIDTH9 program written by Kurucz (1993). It was partly modified according to Takeda
(1991) to incorporate the non-LTE effect for the appropriate level
departure coefficients given as a function of the optical depth. The
non-LTE solar photospheric Si abundance has been determined to
be log εSi (non-LTE) = 7.52 ± 0.08 on the logarithmic abundance
scale defined to have log εH = 12. The mean non-LTE abundance
correction is −0.02 dex. Holweger (1973) adopted the original
f-values measured by Garz (1973) based on the 1D LTE abundance leading to log εSi = 7.65. Our non-LTE Si abundance is
only slightly less than the value (εSi = 7.55 ± 0.056) published by
Wedemeyer (2001) where the same set of transition probabilities
was used. Half of this difference (0.03 dex) is owing to the adoption
of the improved quantum mechanical broadening treatment, and
collisional line data were taken from VALD (Kurucz 2007) instead
of Unsöld (1955) pressure broadening and the remaining part to the
adopted f-values and HM model. Fig. 3 illustrates the individual
abundances for the adoption of the HM model with two sets of
f-values. The Becker et al. (1980) f-values and those of Bautista
et al. (2009) both lead to the small scatter of abundances. The estimated silicon abundance based on 1D non-LTE analysis is log εSi =
7.52 ± 0.08.
We show the effect of atomic data and atmospheric parameters
on the resulting non-LTE abundance corrections in detail. Table 2
shows a summary of the various test calculations, the mean non-LTE
silicon abundances and the mean non-LTE corrections for the Sun.
The first entry denotes the respective model assumption applied.
In order to test the effect of the scaling factor for the non-LTE
calculations, we adopt the collision hydrogen parameter (SH = 1.0,
0.1) with this test. The same is true for the collisional cross-sections
by electron (Se = 1.0, 0.1), where Se is the scaling factor applied
for the collisions with electron. Generally, adopting an empirical
2982
A. M. K. Shaltout et al.
Table 1. List of neutral and ionized silicon lines included in the abundance analysis for 3D non-LTE, 1D
LTE and 1D non-LTE calculations. Displayed are wavelength (nm); lower excitation energy in (eV); oscillator
strength; equivalent widths in (pm); natural line damping width (rad s−1 ); Stark broadening parameter (γ Ne , rad
s−1 cm3 ); hydrogen collision broadening parameter (γ /NH rad s−1 cm3 ); abundances obtained with 3D model;
1D LTE abundances obtained with HM model and 1D non-LTE abundance corrections derived from individual
photospheric lines in the Sun; log ε = log εnon-LTE - log ε LTE .
λ
(nm)
Si I:
564.5611
566.5554
568.4485
569.0427
570.1105
570.8397
577.2145
578.0384
579.3071
579.7860
594.8540
674.1628a
697.6520b
703.4901b
722.6208
768.0265b
791.8382b
793.2348b
797.0305
Si II:
634.7110c
637.1360c
El
(eV)
log gff
Wλ g
(pm)
γ rad
γs
4.9296
4.9201
4.9538
4.9296
4.9296
4.9538
5.0823
4.9201
4.9296
4.9538
5.0823
5.9800
5.9537
5.8708
5.6135
5.8625
5.9537
5.9639
5.9639
−2.04
−1.94
−1.55
−1.77
−1.95
−1.37
−1.65
−2.25
−1.96
−1.95
−1.13
−1.65
−1.07
−0.78
−1.41
−0.59
−0.51
−0.37
−1.37
3.40
4.00
6.00
5.20
3.80
7.80
5.40
2.60
4.40
4.00
8.60
1.60
4.30
6.70
3.60
9.80
9.50
9.70
3.20
8.54
8.53
8.54
8.54
8.54
8.54
8.62
8.53
8.53
8.53
8.61
7.50
7.52
7.94
7.96
7.45
7.51
7.60
7.51
−4.56
−4.57
−5.05
−4.57
−4.74
−4.56
−4.12
−4.43
−4.37
−4.32
−4.45
−3.60
−3.66
−3.63
−4.46
−3.83
−4.11
−4.18
−4.11
8.1200
8.1200
0.19
−0.11
5.60
3.60
9.09
9.08
−5.04
−5.04
γ vdw
ε Si
(3D)
ε Si
(1D)
log ε
(1D)
− 7.29e
− 7.30e
− 7.30e
− 7.30e
− 7.29e
− 7.29e
− 7.23e
− 7.32e
− 7.02d
− 6.98d
− 6.96d
− 6.92e
− 7.02e
− 7.13e
− 7.32e
− 7.17e
− 6.76d
− 6.75d
− 6.76d
7.510
7.430
7.470
7.470
7.480
7.480
7.500
7.510
7.540
7.500
7.470
7.530
7.550
7.520
7.510
7.520
7.500
7.510
7.580
7.497
7.501
7.490
7.547
7.484
7.572
7.572
7.525
7.561
7.501
7.427
7.600
7.549
7.516
7.490
7.662
7.489
7.374
7.599
−0.012
−0.015
−0.022
−0.019
−0.015
−0.032
−0.021
−0.011
−0.015
−0.013
−0.034
–
−0.006
−0.004
+0.007
−0.014
−0.015
−0.015
−0.006
− 7.61d
− 7.61d
7.720
7.740
7.820
7.708
−0.091
−0.063
a Line for which the departure coefficient was not available from the non-LTE calculations. Consequently, this line
(λ 674.1 nm) was not used for 1D LTE abundance determinations.
Lines which are entered into the final abundance estimate with half weight due to uncertainties in collisional
line broadening and oscillator strengths. Apart from these lines, all our lines are quite insensitive to the adopted
collisional broadening.
c The two Si II lines are given half weight due to much more sensitive to uncertainties in the atomic data for
non-LTE calculations.
d Lines for which the collisional line broadening data were interpolated in the tables of Anstee & O’Mara (1995)
and Barklem & O’Mara (1997).
e Collisional line broadening data for Si I were taken from VALD (Kurucz 2007).
f From Becker, Zimmermann & Holweger (1980) for Si I lines and Bautista et al. (2009) for the two Si II lines.
g From Holweger (1973).
b
scaling factor of SH = 0.1 instead of SH = 1.0 leads to a small
decrease in the non-LTE abundance by 0.01 dex. Using a scaling
factor Se = 0.1 led to a somewhat smaller non-LTE abundance with
a difference of 0.02 dex. Consequently, the uncertainty of atomic
data for the non-LTE calculations like collision cross-sections has
only a minor effect on the resulting departure coefficients.
We take into account the effect of the collisional parameters (radiation damping, Stark and van der Waals) to study their influence
on the non-LTE abundance corrections throughout this analysis.
When comparing collisional parameters it was noticed that radiation damping and Stark broadening have only a small effect on the
result of abundance by 0.01 dex and 0.02 dex, respectively. When
we neglect the Stark broadening, the abundances for λ 703.4 & λ
768.0 increase by 0.093 and 0.111 dex. Similarly, the Stark broadening for Si I line λ 697.6 nm affects the abundance by 0.04 dex. It is
clear that this effect is slightly less than 0.03 dex for the remainder
of Si I lines. By contrast, significantly stronger is the influence of
the van der Waals broadening. Neglecting the van der Waals broadening clearly shows that the mean non-LTE abundance increases by
0.12 dex, when comparing with the silicon abundance inferred
(7.52 ± 0.08) derived from the present work. We noticed that the
van der Waals broadening is much more sensitive for Si I lines λ
697.6, λ 703.4, λ 768.0, λ 791.8 and λ 793.2 nm. Lastly, the microturbulent velocity has been varied to ξ = 0.8 km s−1 instead
of 1.0 km s−1 . The non-LTE Si abundance would lead to a small
increase of 0.01 dex. Our findings concerning the results of the influence of atomic data on the abundance analysis is consistent with
Wedemeyer (2001).
3 3 D H Y D RO DY N A M I C A L M O D E L
C A L C U L AT I O N S
3.1 Line formation code, 3D solar model
and observational data
The spectrum synthesis computations are all performed with the
1D MULTI code written by Carlsson (1986, 2011). The solar
photospheric Si abundances using the 3D model atmosphere are
The abundance of silicon
2983
files fitting, which are necessary in classical 1D spectral analysis
(Asplund et al. 2000). The observational data of the observed spectrum are taken from the solar atlas of Brault & Neckel (1987) and
Neckel (1999).
3.2 Abundance analysis
Figure 3. The silicon solar abundance values against the equivalent width
for 20 silicon lines as derived with the HM model, hydrogen collision
parameter (SH = 0.1) and two sets of f-values by using ξ = 1.0 km s−1 . Filled
symbols represent non-LTE values, unfilled are LTE abundances (circles for
Si I, diamonds for Si II). The horizontal lines illustrate the weighted mean
and the standard deviation (solid).
Table 2. Silicon non-LTE mean abundances and mean non-LTE
abundance corrections were derived by using different model assumption for the Sun. The quoted non-LTE abundances are based on
the 1D HM solar model.
Model assumption
Non-LTE calculationsa :
Se × 1.0
Se × 0.1
SH × 1.0
SH × 0.1
Abundance calculationsb :
No radiation damping
No Stark broadening
ξ = 0.8 km s−1
log εnon-LTE
log ε
7.53 ± 0.080
7.51 ± 0.070
7.53 ± 0.080
7.52 ± 0.080
−0.01
−0.03
−0.01
−0.02
7.53 ± 0.090
7.54 ± 0.090
7.53 ± 0.080
−0.02
−0.03
−0.03
Note. The Se and SH are the collisional cross-sections by electron
and neutral hydrogen atoms respectively.
a For non-LTE calculations, the mean non-LTE abundance corrections were calculated for each parameter with a 1D LTE Si abundance
of log ε Si (LTE) = 7.54 ± 0.096.
b According to the abundance analysis, the mean non-LTE abundance
corrections were determined for non-LTE and LTE abundances based
on the assumption of each model.
based on the assumption of non-LTE. The Si abundances based on
the 3D solar model are determined from line profile fitting. The
adopted procedure is normally referred to as 1.5D radiative transfer, i.e. each column is treated as a 1D model atmosphere in which
the radiative transfer is computed without allowing for rays entering other atmospheric columns. In particular the model atmosphere
comes from a realistic 3D magnetohydrodynamic MHD simulation
of the solar surface convection performed with the code of Stein &
Nordlund (e.g. Stein & Nordlund 1989, 1998). The snapshot is the
same as that analysed by Carlsson et al. (2004) and the interested
reader is referred to that paper for further details. Note that the 3D
non-LTE abundance is determined with a single snapshot (i.e. one
snapshot of a realistic 3D MHD model of the solar photosphere).
Moreover, the turbulent velocity field in the stellar atmosphere is
automatically obtained in the 3D model without the need to specify
free parameters (like micro- and macroturbulence) for line pro-
Our Si lines and the atomic data used throughout this analysis are
described in Section 2.3. We derived the silicon abundance from
line profile fitting for 21 selected lines. With the 3D solar model
as input to the 1D MULTI code, changing the silicon abundance, we
computed a grid of synthetic spectra. As seen in Table 1, the solar
Si abundances derived from this study are presented in Column 8.
Examples of profile fits to Si I and Si II lines with the 3D model are
given in Fig. 4. When comparing the line profile fittings it is noticeable that there exists a good agreement between the predicted and
the observed line profiles for most of the Si lines, except for a few
lines which are less suitable for abundance determinations. This is
obvious from the comparison of the predicted 3D line profiles fitting
with observations, as already shown with the Si I lines λ 791.8 and
λ 793.2 nm. The main reason for poor agreement in the line profile
fitting is more likely due to the uncertainties in collisional broadening, f-values or suspected blends. The individual Si abundances
as a function of equivalent width are illustrated in Fig. 5. The solar
3D model shows no abundance trend with equivalent width, neither
excitation potential nor wavelength. This is the result of the fact
that the span in excitation potential is too small to delineate any
possible trend. The lack of trend leads to the suggestion that it is not
necessary to include free parameters like microturbulent velocity
in the spectral synthesis, besides in the elemental abundance, when
properly including the Doppler shifts inherent in convection simulations. This trend is shown here in Fig. 5 for using a least-squares
fit to the Si abundances (solid line). This type of diagram was used
by Asplund (2000) in his 3D study of photospheric Si abundance
and it should be noted that it is principally useful for demonstrating the spread of derived abundances and general abundance trends
with changing equivalent width. The quoted mean 3D Si abundance
based on the assumption of non-LTE shows several lines given half
weight. Based on the weights mentioned previously, the mean solar
photospheric Si abundance is log εSi = 7.53 ± 0.07.
For our 3D non-LTE calculations with 21 photospheric Si lines,
we adopt the 3D Si result as the best estimate of the solar abundance
log εSi = 7.53 ± 0.07, where the uncertainty (±0.07 dex) is the
standard deviations. In the present work, a recalculation of the solar
photospheric Si abundance using a realistic 3D inhomogeneous
model atmospheres has exhibited only a difference of 0.02 dex with
the value (7.51 ± 0.04) published by Asplund (2000) based on
the assumption of LTE. On the other hand, Asplund et al. (2009)
derived log εSi = 7.51 ± 0.03 using a 3D solar model and taking
non-LTE effects into account. Fig. 5 displays a small scatter of the
Si I and Si II abundances, confirming that the adopted f-values in our
calculations have a good internal accuracy.
4 CONCLUSIONS
The current work presents a new determination of the solar Si abundance by means of detailed statistical equilibrium calculations using
both a semi-empirical 1D (HM) model and one snapshot of a realistic 3D MHD model of the solar photosphere taken from Carlsson
et al. (2004). Here we derive the photospheric silicon abundance taking into account non-LTE effects in the same 3D model. Deviations
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A. M. K. Shaltout et al.
Figure 4. The synthesized Si I (for λ 569.0 and λ 768.0 nm) and Si II (for λ 637.1 nm) line profiles (diamonds) compared with observed Fourier Transform
Spectrograph (FTS) profiles (solid lines). The predicted profiles (diamonds) have been calculated using the 3D inhomogenous model atmosphere. However,
the photospheric Si results using 3D model atmospheres are based on the assumption of non-LTE.
The abundance of silicon
2985
paper. One of the authors (A. M. K. Shaltout) wants to express his
great thanks to the Egyptian government for supporting this study.
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Figure 5. The individual silicon abundances derived from the profiles fitting
of Si I (solid circles) and Si II (open circles) as a function of the equivalent
width (Holweger 1973). These abundances are the results of the 3D model
atmospheres based on the assumption of non-LTE. Note that the equivalent
widths of Holweger (1973) are only listed here to allow easy identification and were not used for the abundance determinations. The solid line
represents a least-squares fit to the Si abundances. This trend supports the
conclusion that no microturbulent velocity is necessary to include in the
spectral synthesis besides the elemental abundance.
from LTE in silicon are found to be small in the Sun. The abundances based on the Holweger (1973) equivalent widths are as for
the 1D calculations. A re-analysis of the departures from LTE using
1D solar model atmospheres have revealed only the mean non-LTE
silicon abundance correction with respect to standard LTE calculations of log ε = −0.02 dex compared with previously published
estimates, leading to a solar Si abundance of log εnon-LTE = 7.52 ±
0.08. Taking advantage of realistic 3D hydrodynamical model atmospheres into account, the solar photospheric Si abundance has
been derived to be log εSi (3D) = 7.53 ± 0.07. Thus the resulting
abundance is very similar to the value advocated by Asplund (2000)
and more recently by Asplund et al. (2009).
AC K N OW L E D G E M E N T S
The first author wishes to thank the National Astronomical Observatory of Japan (NAOJ) for supporting the series lectures of
the Practical Radiative Transfer at NAOJ, June 13–16, 2011. Mats
Carlsson is thanked for useful comments, help with non-LTE calculations and for providing the 3D solar model. The VALD database
has been providing the collisional parameters data, which is gratefully acknowledged by the authors. We would like to thank P.
Barklem for using his program for computing quantum mechanical broadening data for the Si lines. The authors wish to thank the
referee for a thorough and constructive report that improved our
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