A&A 423, 1109-1117 (2004)
DOI: 10.1051/0004-6361:20047050
C. Allende Prieto1,2 - M. Asplund3 - P. Fabiani Bendicho2,4
1 - McDonald Observatory and Department of Astronomy,
University of Texas, Austin, TX 78712-1083, USA
2 -
Instituto de Astrofísica de Canarias, 38200, La Laguna, Spain
3 -
Research School of Astronomy and Astrophysics,
Mt. Stromlo Observatory, Cotter Rd., Weston,
ACT 2611, Australia
4 - Departamento de Astrofísica, Universidad de La Laguna, 38206, La Laguna, Spain
Received 11 January 2004 / Accepted 30 April 2004
Abstract
We present new observations of the center-to-limb variation of spectral
lines in the quiet Sun.
Our long-slit spectra are corrected for scattered
light, which amounts to 4-8% of the continuum intensity, by comparison
with a Fourier transform spectrum of the disk center.
Different spectral lines exhibit different behaviors,
depending on their sensitivity to the physical conditions in the photosphere
and the range of depths they probe as a function of the observing angle,
providing a rich database to test
models of the solar photosphere and line formation.
We examine the
effect of inelastic collisions with neutral hydrogen in NLTE line
formation calculations of the oxygen infrared triplet, and the
Na I
line.
Adopting a classical one-dimensional theoretical model atmosphere,
we find that the sodium transition, formed in higher layers, is more
effectively thermalized by hydrogen collisions than the high-excitation
oxygen lines. This result appears as a simple consequence of the decrease of
the ratio
with depth in the solar photosphere.
The center-to-limb variation of the selected lines
is studied both under LTE and NLTE conditions.
In the NLTE analysis, inelastic collisions with hydrogen atoms are
considered with a simple approximation
or neglected, in an attempt to
test the validity of such approximation.
For the sodium line studied, the best agreement between
theory and observation happens when NLTE is considered and inelastic
collisions with hydrogen are neglected in the rate equations.
The analysis of the oxygen triplet benefits from a very detailed
calculation using an LTE three-dimensional model atmosphere and NLTE
line formation. The
statistics favors
including hydrogen collisions with
the approximation adopted, but the oxygen abundance derived in that
case is significantly higher than the value derived from OH infrared
transitions.
Key words: Sun: photosphere - line: formation - line: profiles
Abundance analyses of late-type stars commonly rely on the assumption of Local Thermodynamical Equilibrium (LTE) and a plane-parallel homogeneous structure. Even though in many cases the derived abundances may be actually similar to the real photospheric abundances, there are well-known instances when departures from LTE and inhomogeneities can introduce significant systematic errors. Consequently, efforts have been directed toward improving the modeling techniques. One of the most significant obstacles in the way of performing reliable Non-LTE (NLTE) calculations is the incompleteness of the necessary atomic data. Because the calculations tend to be involved, and rely on approximations to account for rates that have never been measured in a laboratory or cannot be calculated reliably, it is often the case that results cannot be directly accepted without extensive testing and thorough comparison with observations.
Common tests that have been put into practice
are spectroscopic observations of
profiles of lines with well-known damping constants,
abundance determinations for stars in a cluster (expecting all members
to show the same chemical composition)
or a solar abundance analysis for non-volatile elements
(anticipating that the photospheric
abundances and those found in CI-type chondrites are identical).
The spatially resolved solar disk offers the possibility to survey a
range of formation depths for the continuum, and also for any
given spectral line.
The center-to-limb variation of spectral lines
constitutes a particularly interesting test for elements like oxygen, which
is depleted in meteorites (see e.g. Sedlmayr 1974; Kiselman 1991).
Furthermore, this type of observations may turn very useful to
distinguish between the thermalizing effect of inelastic
collisions with electrons and
hydrogen atoms, as the ratio
increases by more than one order of magnitude between
and 0.
Inspection of the available literature reveals a flagrant scarcity of high quality observations of the center-to-limb variation of line profiles. Early work did not benefit from digital detectors (e.g. Müller & Mutschlecner 1964; Müller et al. 1968). The extended study by Balthasar (1988) includes more than one hundred lines, but only reports equivalent widths and a few parameters related to the amount and shape of the line asymmetry. Equivalent widths, however, neglect much of the available information in a line profile. Equivalent widths ignore, for example, whether changes in the damping wings or the core of a line dominate the line strength variation, or even if they cancel each other to produce an equivalent width nearly independent of the position on the disk. More recent work exists, but it is limited to a small number of spectral regions (e.g. Ambruoso et al. 1992; Brandt & Steinegger 1998; Grigoryeva & Turova 1998; Langhans & Schmidt 2002; Stenflo et al. 1997).
We have obtained
observations of a number of key lines for which reliable damping
constants are available. In Sect. 2 we report our measurements and
the data reduction, in particular a procedure to subtract the scattered
light. In Sect. 3 we study the cases of
the infrared oxygen triplet and the Na I
lines in the
context of a classical model atmosphere. In Sect. 4, the case
of the oxygen triplet is reanalyzed with a time-dependent three-dimensional
model of the solar photosphere. The paper concludes with some reflections
about the results and suggestions for future work.
Solar observations of the center-to-limb variation of several spectral lines
were carried out in October 22-23, 1997, with the Gregory Coudé
Telescope (GCT) and its Czerny-Turner echelle
spectrograph (Kneer et al. 1987; Kneer & Wiehr 1989) at the
Observatorio del Teide (Tenerife, Spain).
This telescope was moved to Tenerife and refurbished
in 1985, after more than 20 years of operations
in Locarno (Switzerland).
It was dismantled in 2002 to leave room for new instrumentation.
The spectrograph was operated in orders 6-9
with a slit width between 50 and 150 m (0.4-1.3 arcsec)
achieving an estimated FWHM (
)
resolving power
in the range 57 000-240 000.
Table 1: Observations.
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Figure 1:
Correspondence between the location and extent of
the slit for each position in
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We secured spectra for 8 spectral setups in 6 different
positions across the solar disk, as summarized in Table 1. A ninth setup
was centered at about 7610 Å,
to obtain an estimate of the amount of scattered light from
saturated telluric O2 lines.
The exact slit location was always chosen to avoid active regions.
Ten consecutive equal-length exposures were obtained
at each position for each setup. The exposure time varied,
depending on the setup, between 0.5 and 2.0 s.
Positions #1 to #5 were always at heliocentric angles
=
0, 15, 30, 45, and 60 degrees
(
= 1.00, 0.97, 0.87, 0.71, and 0.50)
along a straight line crossing the center of the solar disk.
Position #6 was also selected along
the same direction, sometimes at
degrees and others at 80 degrees
(
= 0.26 or 0.17).
The slit length covered approximately 205 arcsec,
but the field was truncated by the CCD size to
160 arcsec.
In the last position, part of the slit was outside the solar disk,
and therefore when the center of the slit was at
and 80 deg,
the center of the illuminated slit was at 70 and 72 deg, respectively, or
approximately
.
Figure 1 shows the slit
extent both in angular size on the sky and in
for each position,
assuming a solar angular radius of
arcsec
(Allende Prieto et al. 2003a).
In what follows we neglect limb-darkening within the slit
length, assigning any observed intensity to the location of
the center of the illuminated part of the slit.
No calibration lamps were available. The detector was a 10242 CCD with negligible dark current (1.4 ADU/s/pixel). The CCD frames were bias subtracted, and rotated. The rotation corrects small tilts between the direction of the spectral dispersion and the CCD that ranged between 0.5 and 0.7 degrees, as derived from the drift of the centers of several spectral lines in the spatial direction. Pixel-to-pixel variations and weak fringes were automatically smeared out in the spatially-averaged spectra at each position. Wavelength calibration of the disk-center spectra was carried out by fitting a 1st to 3rd order polynomial (depending on the number of available spectral lines) to the wavelengths of the line centers, as measured in the Brault & Neckel atlas (1987; see Neckel 1994). The wavelengths were adopted from Allende Prieto & García López (1998), or measured afresh when missing from their list. The same dispersion solution was applied to the observations at other positions. Finally, a velocity shift relative to the center-of-the-disk spectra (accounting mainly for solar rotation, Earth's motion, and the limb effect) were determined by cross-correlation and applied to the spectra at other positions. Although all the observations for a given spectral setup were acquired consecutively, the time interval between the observations at the disk center and at the limb was generally significant, and therefore we deem our wavelength accuracy insufficient to study the limb effect.
The spectrum of O2 lines at about 7600 Å (A band)
confirmed the suspicion that
scattered light was substantial. The presence
of scattered light is also apparent when comparing our spectra
with the Fourier transform (FT) spectrum at the center of the disk in the
Brault & Neckel atlas.
Fourier transform spectrographs (FTS's)
are not susceptible to scattered light in the same sense
as grating spectrographs, but non-linearities in the detector can cause systematic
errors in the zero-point of the intensity scale (see the discussion in
Kurucz et al. 1984).
Scattered light leaves a characteristic distortion on the spectrum. On visual
inspection, for a given spectral resolution, the cores of the lines appear filled.
This effect is noticeably different from a reduction in spectral resolution.
As the spectrum of the quiet Sun averaged over a large
area is believed to be extremely stable, two instrumental factors
are mainly responsible for the differences between
the FT spectrum of Brault & Neckel
and our center-of-the-disk spectra: resolving power, and the amount
of scattered light. Therefore, based on the well-supported
expectation that the amount of scattered light in the FTS atlas is much less
that in our GCT spectra, we used the FTS data as a template to correct the
scattered light.
For each of the setups, we compared our center-of-the-disk spectra
with the FTS atlas to derive the amount of scattered light and
the resolving power (R' relative to the FTS
atlas) that would lead to the
best agreement between the two. Our analysis was based on
modeling the instrumental profile of the GCT spectra as a Gaussian, and
the scattered light as a constant fraction of the continuum flux for each
setup.
The comparison required the GCT spectra to be first continuum corrected.
This was accomplished by using a 6th-order polynomial and a series
of clipping iterations. Then a search was performed to determine the
best-fitting values for the FWHM of the Gaussian representing the
instrumental profile, and S, the amount of scattered light expressed as
a fraction of the pseudo-continuum intensity.
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Figure 2:
Contour plot showing the reduced ![]() ![]() ![]() |
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Figure 3:
Comparison of the spectra at the center of the disk in the region
around Ca I
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Figure 4:
Center (filled circles) and limb (solid) spectra in the region
around Ca I
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To avoid a bias due to the
original normalization, the procedure was iterated until
.
Only 3-4 iterations
were required to comply with the convergence condition. Table 1 provides
the results.
Figure 2 shows the contours of the
reduced
(normalized to 1 at the minimum) for
the setup centered around the Ca I
line.
The signal in the GCT
spectra has the potential to exceed a S/N of at least 1000.
Systematic uncertainties are more difficult to assess.
Systematic errors are expected due to shortcomings
of modeling the GCT instrumental profile as a Gaussian, and the scattered
light as independent of wavelength (for a given setup), perhaps with
an additional contribution from real variations of the photospheric
solar spectrum with time.
We estimate that our derived values of the resolving
power are good to about a few percent, and that the scattered light is
constrained to within 1%.
Both the relative scattered light, and the polynomial
representing the pseudo-continuum were applied to correct the spectra
at the center of the disk (from which the values were determined)
and also to the spectra at other positions. Figure 3 shows
the FTS spectrum smoothed to the resolution of the
GCT data, and the GCT spectrum
after correcting the scattered light. We empirically find
that the observed profiles are accurate to
1%.
This figure is also supported by an rms difference of
0.7% between
two corrected
spectra of the 6122 Å region at the center of the disk obtained
on the two consecutive observing dates.
Figure 4 shows the change from the center to the limb of the
spectral region discussed in the previous figures (upper panel),
and for the setup centered at 5300 Å (lower panel).
Many differences are
noticeable. Some lines only undergo subtle changes, often
becoming slightly broader towards the limb, but
other features' wings turn less pronounced.
Some lines become both wider and
stronger towards the limb (e.g. Co I
5301.0). The
variations in behavior among different lines
reflect their different sensitivities to the physical parameters
controlling the line formation and, as earlier recognized in the 60's,
contain precious information on the atmospheric structure.
The GCT spectra have been made publicly available through the CDS.
In the next section we will
explore the possibility of quantifying the thermalizing effect of
inelastic collisions with
electrons and hydrogen atoms in the populations of neutral
oxygen and sodium through the center-to-limb behavior
of spectral lines.
Line profiles sample a range of atmospheric depths that changes depending on the position on the disk. An analysis of the resulting changes in the line shapes is, to a first approximation, independent of the f-value of the transition, the chemical abundances, and the damping constants. A zeroth-order analysis can be based on simplified LTE calculations. Such modeling already reproduces the basic behavior for many spectral lines; their strengthening or weakening from center to limb due to the changes in the equilibrium populations with atmospheric depth. A closer look at the observed line profiles will reveal the intricacies of the line formation, such as departures from LTE, the presence of inhomogeneities, or the lack of complete frequency redistribution.
We explore first the possibility of discerning the thermalizing effect of collisions with electrons from collisions with hydrogen atoms by means of an analysis based on one-dimensional model atmospheres and NLTE line formation. When dealing with high S/N and resolving power spectroscopic observations, an analysis based on static model atmospheres yields a very poor match between calculated line profiles and observations. This is mainly the result of neglecting surface inhomogeneities - granulation - which cause the observed lines to appear asymmetric. Admittedly a handicapped approach, such a detour is convenient and oftentimes didactic.
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Figure 5:
Response functions to the temperature at ![]() ![]() |
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We examine two interesting cases which are part of
our observations: the oxygen infrared triplet, and a Na I line at 6160.8 Å.
Figure 5 shows the LTE
response to temperature perturbations of the core of these lines.
These response functions (see Ruiz Cobo & del Toro Iniesta 1992, 1994)
are calculated for the continuum-corrected line profiles, and therefore
include the sensitivity of the continuum, which strengthens the
lines when the temperature increases in the deepest layers.
In this plot, a positive value represents
an increase in the relative flux, and therefore a weakening of the line.
When changing from the center to the limb, the O I lines map fairly
well the layers in the range
.
The temperature sensitivity of the core of the Na I line shows
opposite reactions to changes in the temperature at different
heights but this line is sensitive to higher layers than the triplet.
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Figure 6:
Center (at rest with the continuum normalized to 1)
and limb (shifted to the red with the continuum normalized to 1
and subtracted 0.05) spectra for the O I infrared triplet lines.
Observations (filled circles) and calculations are compared for three
different models: LTE, NLTE/
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It has been recognized that departures from LTE are significant for these lines. The main effect is due to an infrared radiation field weaker than Planckian in the line formation region (Eriksson & Toft 1979). Fortunately, the level populations of O I are relatively insensitive to the UV radiation field in solar type stars, which simplifies the modeling. Due to cosmological and galactic implications, oxygen abundances in metal-poor stars have been extensively discussed. When modeling the oxygen triplet lines in NLTE some authors opt to neglect collisions with neutral hydrogen (e.g. Nissen et al. 2002), while others prefer to adopt a recipe first suggested by Steenbock & Holweger (1984; see, e.g., Takeda 2003), based on Drawin's formula (Drawin 1968), scaled by an empirical factor(s).
We have evaluated the NLTE populations for O I with a solar model
from Kurucz's non-overshooting grid (Kurucz 1993). The model
atom and the calculations follow those in Allende Prieto et al. (2003a,b).
We have introduced the effect of
collisional excitation and ionization
due to neutral hydrogen with the prescription of
Steenbock & Holweger, multiplying the rates by a correction
factor .
To make the test independent of the oxygen abundance,
we have adjusted it for each of the following three cases: LTE,
,
and
.
A micro-turbulence of 0.9 km s-1 was adopted - a typical value
for the disk-center (e.g. Blackwell et al. 1995).
A Gaussian macro-turbulence of 2.0 km s-1 was used, as well a Gaussian instrumental
profile consistent with the analysis in Sect. 2 (see Table 1),
for a total FWHM broadening of 0.151 Å.
Collisional damping
induced by neutral hydrogen was accounted for using the line width
at 10 000 K by Barklem et al. (2000),
assuming a temperature dependence T2/5.
Figure 6 compares observed and calculated profiles for the
different positions and the three cases considered:
LTE,
,
and
.
The first three panels illustrate the agreement between observed and
computed profiles at the extreme positions:
(center) and 0.32 (near the limb). The fourth panel quantifies
the relative agreement through
,
which has been
normalized to 1 at the minimum. To avoid obvious blending
features, only the fluxes which are more than
3% lower than the continuum level are considered in the evaluation
of
.
As collisions drive the level populations toward LTE, the NLTE line profiles
with
are an intermediate case between LTE and the
profiles.
As we anticipated, the match of the observed profiles
with a static one dimensional model is poor.
The comparison suggests that
the NLTE calculations perform much better than those assuming LTE,
in agreement with previous investigations. We can also conclude that
the effect of hydrogen collisions is quite limited in these lines (see also
Nissen et al. 2002), but the best agreement is found when they are
included, albeit the difference is marginal.
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Figure 7:
Center (at rest with the continuum normalized to 1)
and limb (shifted to the red with the continuum normalized to 1
and subtracted 0.05) spectra for Na I
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Departures from LTE in Na I lines have also been extensively studied
(e.g. Athay & Canfield 1969; Baumüller et al. 1998).
The level populations are
again highly independent of the UV radiation field.
We have used the new model
atoms and calculations we previously referred to for the oxygen case.
A micro-turbulence of 0.9 km s-1 was adopted, as in Sect. 3.1,
but a lower value of the macro-turbulence than for the oxygen triplet
was necessary to fit the profiles at the center of the
disk, 1.3 km s-1, and therefore the total FWHM for the Gaussian
broadening was 0.074 Å (note the difference in resolving power,
as summarized in Table 1). The
effective principal quantum number (n*) of the 2S upper state
is larger than 3 and therefore exceeds the range of the
tables published by Anstee & O'Mara (1995).
Barklem et al. (2000) derived damping cross-sections
for other Na I transitions with similar energies sharing the same
lower state 2P
.
For those transitions
the FWHM of the Lorentzian (collisional)
component of the line absorption profile
(per unit perturber number density at 10 000 K)
was found to be
in the range -6.86 to -7.23. Comparison with the observed
profiles at the disk center showed that, independently of the
chosen abundance, the best agreement is found for a value
close to
,
which was therefore adopted. Accepting n* = 3 for the
upper s state gives
.
Figure 7 shows the profiles at the extreme positions
(
and 1) and
the comparison with the synthetic spectra for the three cases
considered. Again, we adjusted the Na abundance for each case in order
to maximize the agreement between calculated and observed profiles
at the center of the disk. It is evident that the sensitivity of the level
populations involved in this Na I transition to collisions with H is
higher than for the states connected by the O I infrared triplet. This
is most likely the result of the line formation and the
departures from LTE taking place in this case in higher layers.
The LTE
response function to the temperature for the core of the O I triplet
lines peaks at
,
while that for
the core of the Na I
6160.8 line does so at
.
Using the Van Regemorter (1962) and Drawin approximations,
for a typical transition of a few electron volts
the ratio of the collisional rate coefficients
,
where
U=Eij/(kT), and E1 is the first-order exponential integral. Thus,
is a fairly flat function of the temperature or
the optical depth in the solar photosphere.
Between
and -1,
increases by an order of magnitude,
making the effect of collisions with hydrogen more noticeable
in higher layers.
Similarly to the case of the oxygen triplet, we find the
best agreement with observations when departures from LTE are considered,
but now the calculations that neglect
collisions with hydrogen grade best, and the differences between
the
and
1 cases are now more significant. Not even the NLTE
model
can match the wider profile of the sodium line
that is observed near the solar limb.
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Figure 8: Similar to Fig. 6 for the 3D time-dependent simulation of the solar surface. |
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Modern radiation hydrodynamics simulations offer a much more realistic representation of the lower solar atmosphere than classical static model atmospheres (Stein & Nordlund 1998; Asplund et al. 2000). In the context of these, more detailed, model atmospheres, surface convection and line broadening due to small- and large-scale velocity fields are naturally included in the calculations. This makes the use of ad hoc parameters such as macro- and micro-turbulence unnecessary, and at the same time improves substantially the agreement between observed and calculated spectral line profiles.
The consideration of several more dimensions makes calculations, however, much more involved, which, with some notable exceptions, has largely limited NLTE studies to 1D modeling. However, O I is among the few ions for which the line formation in a 3D hydrodynamical model has been studied with a complex multi-level model atom. Asplund et al. (2004) have recently shown that it is necessary to consider both, departures from LTE and a inhomogeneities, to find good agreement among different spectroscopic indicators of the solar photospheric oxygen abundance, finally settling a long-standing problem. Here we will use similar calculations to revisit the center-to-limb variation of these lines, which were analyzed with a 1D model in Sect. 3.1.
The 3D solar simulation has been described in detail in previous
papers, and the NLTE line formation calculations are identical to those
reported by Asplund et al. (2004).
We proceed similarly to the 1D analysis.
Line profiles for the O I triplet were computed for different
inclinations from the vertical axis, and
averaged over azimuth, time, and horizontal position.
The computed NLTE
profiles are the product of an average LTE profile
derived from 100 snapshots covering 50 min of solar time, and
the ratio of the NLTE and LTE line profiles derived from two snapshots.
This procedure reduces the computational burden, while introducing
negligible errors.
The statistical equilibrium equations were solved for the two considered
snapshots for two different cases:
,
and
.
Three different abundances were used in the calculations
(
(O) = 8.50, 8.70, and 8.90
)
and later the profiles were linearly
interpolated to find the best match at the disk center in each of the
three cases: LTE,
,
and
.
The calculated
profiles were smoothed by convolution with a Gaussian with a FWHM of
0.088 Å to account for the finite spectral resolution of the observations.
Figure 8 is the counterpart of Fig. 6 for the 3D case.
Admittedly, the computed line profiles are not perfect. In particular,
significant discrepancies are noticeable around the cores of the
lines.
The reader should keep in mind that the lower-right panels of Figs. 6 and 8, cannot be directly compared, as the
is normalized to unity at the minimum independently for each case:
1D and 3D.
The adopted abundances
were chosen to optimize the overall fit of the
observations,
and the agreement is considerably
better in 3D than in 1D near the line wings - even though
macro-turbulence is included in the 1D case and not in 3D.
The refined 3D analysis strengthens what the 1D study
suggested. The calculated
profiles are now close enough to the observations to justify an attempt
to quantify statistically the agreement.
Making use of the accuracy estimate empirically derived in Sect. 2, we adopt
,
and evaluate
for
in the three cases. Using only
the fluxes below 3% of the continuum level to avoid the interference
of obvious blending features between the lines,
n=135, and we find
,
195, and 92
and therefore significance levels of 0 (
< 10-81),
10-3, and 1, for LTE,
,
and
,
respectively.
In the 1D analyses we have avoided comparing absolute abundances, given
that the effect of atmospheric inhomogeneities was neglected.
Now that the triplet lines are modeled in more detail, it is tempting
to examine the absolute abundances as
an additional piece of evidence. The abundances found in each case to
match the profiles at the
center of the disk are
(O) = 8.88, 8.66 and 8.72 for
LTE,
,
and
,
respectively. The
case provides the best agreement
with the average value
derived by Asplund et al. (2004)
:
dex from forbidden and
permitted lines of O I and from infrared OH lines.
Thus, absolute abundance and
statistics
lead to apparently opposite conclusions in the case of the oxygen triplet.
We should note that the average value derived by Asplund et al. (2004)
includes permitted lines analyzed in a similar manner as in this
paper with NLTE line formation and
.
A higher oxygen
abundance from allowed lines is expected for
,
and in that
case, forbidden and permitted lines
would suggest a higher abundance (8.67-8.70) than infrared
OH lines (8.61-8.65) dex.
We should stress that the necessary atomic data for the NLTE calculations were independently compiled for the 1D and 3D cases, yet the qualitative agreement is excellent. Quantitatively, the abundance corrections necessary to make the LTE and NLTE equivalent widths agree are also remarkably similar in 1D and 3D. This was also found by Asplund et al. (2004) comparing the 3D calculations with 1D abundances based on MARCS model atmospheres. The impact of departures from LTE and that of surface inhomogeneities on the abundances cannot be generally decoupled, but the situation for the OI triplet, where this is actually a good approximation, makes the conclusions from our 1D analysis to be similar as those from a 3D study.
The type of observations presented here, as exemplified in Sects. 3
and 4,
provide a stiff test of theoretical calculations
of atmospheric structure and line formation in a solar-type photosphere.
The potential of center-to-limb observations to guide
modeling was recognized early, but
consistent
observations covering extensive spectral regions are
still missing.
We have explored how these observations can constrain
the effect of collisions with hydrogen atoms on the rate
equations for two particular cases.
Our results for Na I 6160.8
fall in line with
the conclusion drawn from theoretical calculations
by Barklem et al. (2003) for excitation of another alkali,
lithium, by inelastic collisions with H atoms (see also the discussion by
Lambert 1993).
Limited by the accuracy of the observations, and approximations
involved in computing line profiles,
we have only tested here the extreme cases of
(no collisions
with H), or
(Drawin-like formula).
Comparison with detailed NLTE calculations based on
multidimensional time-dependent model atmospheres and improved solar
observations free from scattered light may be able to answer whether
or not the use of a simplified formula scaled by a constant factor
(
1 for Na I or Li I) is a useful approach.
Our results and a quick inspection of the literature
lead to the naive conclusion that such a factor would need to
vary for different species.
For example,
the role of hydrogen collisions compared to those with electrons
increases for atmospheres more metal-poor (or cooler) than solar, and
Korn et al. (2003) have recently found that
is needed to satisfy the iron ionization balance in
several metal-poor stars.
In the case of the oxygen infrared triplet,
the solar observations are best reproduced considering inelastic
hydrogen collisions in the rate equations.
However, the oxygen abundance
obtained when hydrogen collisions are neglected (8.66 dex)
is in better agreement
with the abundance inferred from OH infrared lines (8.61-8.65 dex;
Asplund et al. 2004) than
the abundance derived from the NLTE
analysis (8.72 dex).
The higher
abundance is nevertheless still consistent with the
values from forbidden lines (8.67-8.69 dex).
The weighted average of the abundances from permitted lines derived
by Asplund et al. (2004) changes from
dex
to
dex when inelastic collisions with hydrogen are
considered with the Drawin-like formula. Therefore, this
apparent inconsistency could signal that the abundance derived
from infrared OH lines is more uncertain than the values
obtained from a detailed analysis of forbidden and permitted O I lines.
We should bear in mind that the effect of collisions with hydrogen on the statistical equilibrium is not the only important uncertainty for NLTE calculations in late-type stellar atmospheres. Electron collisions, for example, are only approximately included in our models, mainly due to the lack of reliable data for particular states. Our results should be confirmed by taking into account refined data as they become available (see, e.g., Zatsarinny & Tayal 2003). In addition, the electron density in high atmospheric layers is highly uncertain even in the 3D hydrodynamical simulations, due to the assumption of LTE. Our NLTE calculations are always `restricted', in the sense that the temperature and electron density are adopted from the LTE structure calculations.
The examples discussed in this paper show how the different ingredients involved in modeling spectral line formation can begin to be unraveled when high-quality observations of spatially resolved solar line profiles are compared with detailed calculations. High quality solar observations with low spatial resolution are still missing at most wavelengths and can play a crucial role guiding theory in the quest for accuracy in the interpretation of stellar spectra.
Acknowledgements
It is a pleasure to thank Ramón García López for encouraging us to perform these observations, and Valentín Martínez Pillet for allocating the telescope time. We are grateful to Dan Kiselman for pointing out an important mistake in an earlier version of the manuscript, and to the referee, Han Uitenbroek, and the scientific editor, Wolfgang Schmidt, for useful suggestions. The Gregory Coudé Telescope was operated by the Universitäts-Sternwarte Göttingen at the Spanish Observatorio del Teide of the Instituto de Astrofísica de Canarias. We made use of NASA's ADS, and gratefully acknowledge support from NSF (grant AST-0086321) and NASA (LTSA 02-0017-0093).