Uncertainties in the solar photospheric oxygen abundance

TL;DR

This paper uses Bayesian inference with 3D models and solar atlases to estimate the solar oxygen abundance, accounting for various uncertainties and correlations, resulting in a robust confidence interval.

Contribution

It introduces a Bayesian framework to rigorously quantify confidence limits on solar oxygen abundance derived from 3D models and observations.

Findings

Most probable oxygen abundance: 8.86±0.04

Correlations identified between model parameters and abundance estimates

Robust confidence limits established considering multiple error sources

Abstract

The purpose of this work is to better understand the confidence limits of the photospheric solar oxygen abundance derived from three-dimensional models using the forbidden [OI] line at 6300 \AA , including correlations with other parameters involved. We worked with a three-dimensional empirical model and two solar intensity atlases. We employed Bayesian inference as a tool to determine the most probable value for the solar oxygen abundance given the model chosen. We considered a number of error sources, such as uncertainties in the continuum derivation, in the wavelength calibration and in the abundance/strength of Ni. Our results shows correlations between the effects of several parameters employed in the derivation. The Bayesian analysis provides robust confidence limits taking into account all of these factors in a rigorous manner. We obtain that, given the empirical three-dimensional model and the atlas observations employed here, the most probable value for the solar oxygen abundance is $\log(\epsilon_O) = 8.86\pm0.04$. However, we note that this uncertainty does not consider possible sources of systematic errors due to the model choice.

Figures

Tables

Table 1. Table 1: Adopted atomic parameters. γ r ​ a ​ d subscript 𝛾 𝑟 𝑎 𝑑 \gamma_{rad} , γ S ​ t ​ a ​ r ​ k subscript 𝛾 𝑆 𝑡 𝑎 𝑟 𝑘 \gamma_{Stark} and γ W ​ a ​ a ​ l ​ s subscript 𝛾 𝑊 𝑎 𝑎 𝑙 𝑠 \gamma_{Waals} are the radiative, the Stark and the van der Waals damping parameters (units 10 8 superscript 10 8 10^{8} r ​ a ​ d 𝑟 𝑎 𝑑 rad s − 1 superscript 𝑠 1 s^{-1} ). The log( g ​ f 𝑔 𝑓 gf ) are taken from: a Storey & Zeippen ( 2000 ) , b Johansson et al. ( 2003 ) , c VALD database.

Ion	Wavelength [nm]	Ex. Pot. [eV]	log(gf)	Conf. (lower)	Conf. (Upper)	${\gamma }_{rad}$	${\gamma }_{Stark}$	${\gamma }_{Waals}$	$\sigma$	$\alpha$
[O i]	630.0304	0.000	-9.717a	${}_{}{}^{3}P_2^{}$	${}_{}{}^{1}D_2^{}$	0.0	0.05	1.00	…	…
Ni i	630.0335	4.266	-2.253b	${}_{}{}^{3}D_1^{}$	${}_{}{}^{3}P_0^{}$	2.63	0.054	1.82	…	…
Ni i	630.0355	4.266	-2.663b	${}_{}{}^{3}D_1^{}$	${}_{}{}^{3}P_0^{}$	2.63	0.054	1.82	…	…
Sc ii	630.0678	1.507	-1.898c	${}_{}{}^{3}P_2^{}$	${}_{}{}^{3}D_2^{}$	2.30	0.05	1.30	…	…
Fe i	630.15012	3.654	-0.718c	${}_{}{}^{5}P_2^{}$	${}_{}{}^{5}D_2^{}$	…	…	…	834.4	0.243
Fe i	630.24940	3.686	-1.13	${}_{}{}^{5}P_1^{}$	${}_{}{}^{5}D_0^{}$	…	…	…	850.2	0.239

Table 2. Table 2: Priors selected for each parameter of our analysis.

Parameter	Range	Type
log(${ϵ}_o$)	(8.55, 9.20)	Flat
log(${ϵ}_{Ni}$)	(5.84, 6.36)	Flat
v [km/s]	(-1, 1)	Flat
Enhancement factor	(0.5, 2.3)	Flat
$\sigma$	(0.0001, 0.02)	Jeffreys
Cont. slope [$I/I_c$ $\dot{}$ $\AA {}^{-1}$]	(-0.1, 0.1)	Flat
Cont. y intercept [$I/I_c$]	(0.9, 1.1)	Flat

Table 3. Table 3: Parameters median values and deviations for both atlases.

Atlas	log(${ϵ}_o$)	log(${ϵ}_{Ni}$)	v [km/s]	Enh. factor	$\sigma$
Neckel	$8.861\pm 0.014$	$6.22\pm 0.03$	$0.05\pm 0.01$	$1.92\pm 0.03$	$0.0015\pm 0.0001$
Delbouille	$8.856\pm 0.006$	$6.18\pm 0.01$	$0.014\pm 0.005$	$2.05\pm 0.01$	$0.00120\pm 0.00005$

Table 4. Table 4: Comparison of the results for the different experiments.

			log(${ϵ}_o$)	log(${ϵ}_{Ni}$)	v [km/s]	Enh. factor	$\sigma$
Both lines	Neckel	prior Ni	$8.864\pm 0.013$	$6.21\pm 0.03$	$0.05\pm 0.01$	$1.92\pm 0.03$	$0.0015\pm 0.0001$
		no prior	$8.861\pm 0.014$	$6.22\pm 0.03$	$0.05\pm 0.01$	$1.92\pm 0.03$	$0.0015\pm 0.0001$
	Delbouille	prior Ni	$8.858\pm 0.006$	$6.18\pm 0.01$	$0.014\pm 0.005$	$2.05\pm 0.01$	$0.00120\pm 0.00005$
		no prior	$8.856\pm 0.006$	$6.18\pm 0.01$	$0.014\pm 0.005$	$2.05\pm 0.01$	$0.00120\pm 0.00005$
Only [O i]	Neckel	prior Ni	$8.910\begin{array}{c}+0.015\\ -0.019\end{array}$	$6.02\pm 0.05$	$0.17\begin{array}{c}+0.03\\ -0.04\end{array}$	$1.70\pm 0.04$	$0.00052\pm 0.00005$
		no prior	$8.940\begin{array}{c}+0.007\\ -0.012\end{array}$	$5.90\begin{array}{c}+0.06\\ -0.04\end{array}$	$0.23\begin{array}{c}+0.01\\ -0.02\end{array}$	$1.74\pm 0.04$	$0.00050\pm 0.00005$
	Delbouille	prior Ni	$8.937\begin{array}{c}+0.004\\ -0.007\end{array}$	$5.88\begin{array}{c}+0.04\\ -0.03\end{array}$	$0.171\begin{array}{c}+0.007\\ -0.013\end{array}$	$1.99\pm 0.02$	$0.00036\pm 0.00005$
		no prior	$8.940\pm 0.002$	$5.85\begin{array}{c}+0.02\\ -0.01\end{array}$	$0.178\begin{array}{c}+0.003\\ -0.004\end{array}$	$2.00\pm 0.01$	$0.00036\pm 0.00002$

Equations

$$p(\mathbf{\theta }|D,I)=\frac{p(\mathbf{\theta }|I)p(D|\mathbf{\theta },I)}{p(D|I)}.$$

$$p({\theta }_1|D,I)=\int \mathrm{d}{\theta }_2\dots \mathrm{d}{\theta }_{n}p(\mathbf{\theta }|D,I)$$

$$O({\lambda }_i)=I(\mathbf{\theta },{\lambda }_i)+n({\lambda }_i),$$

$$p(D|\mathbf{\theta },I)=\prod _{i=1}^n\frac{1}{\sqrt{2\pi }\sigma }\exp \left[-\frac{-{\left[O({\lambda }_i)-I(\mathbf{\theta },{\lambda }_i)\right]}^2}{2{\sigma }^2}\right],$$

Full text

*Methods. *We worked with a three-dimensional empirical model and two solar intensity atlases. We employed Bayesian inference as a tool to determine the most probable value for the solar oxygen abundance given the model chosen. We considered a number of error sources, such as uncertainties in the continuum derivation, in the wavelength calibration and in the abundance/strength of Ni.

*Results. *Our results shows correlations between the effects of several parameters employed in the derivation. The Bayesian analysis provides robust confidence limits taking into account all of these factors in a rigorous manner. We obtain that, given the empirical three-dimensional model and the atlas observations employed here, the most probable value for the solar oxygen abundance is $\log(\epsilon_{O})=8.86\pm 0.04$. However, we note that this uncertainty does not consider possible sources of systematic errors due to the model choice.

1 Introduction

The solar chemical composition is still under debate, being particularly important the case of the oxygen (the third most abundant element in the Sun after hydrogen and helium). One of the abundance set traditionally accepted considers an oxygen abundances of log($\epsilon_{O}$) = 8.93$\pm$0.04 (Grevesse et al., 1984), in the astronomical usual scale referred to hydrogen (where $\log(\epsilon_{O})=log(N_{X}/N_{H})+12$). This abundance was later downward revised to log($\epsilon_{O}$) = 8.83$\pm$0.06 (Grevesse & Sauval, 1998). With this abundance set, there is excellent agreement between solar interior models predictions and the helioseismology. However, using more recent three-dimensional theoretical models, lower solar metallicities have been obtained. For example, Asplund et al. (2004) reached log($\epsilon_{o}$) = 8.66$\pm$0.05, which may spoil the agreement with the helioseismology (see Basu & Antia, 2008). Note however that a recent study suggests that we might be underestimating interior opacities which could restore the agreement with helioseismology (Bailey et al., 2015). Given that oxygen is a very important element for stellar interior models, adopting such a dramatic revisions would have implications over a broad range of topics in astrophysics.

Further works have not yet converged to a satisfactory resolution of the issue. On one hand, some studies found a relatively low oxygen abundance (e.g., Socas-Navarro & Norton, 2007; Pereira et al., 2009; Grevesse et al., 2010); but, on the other hand, high oxygen abundance has also been obtained (e.g., Centeno & Socas-Navarro, 2008; Ayres, 2008; Socas-Navarro, 2015b). Perhaps an intermediate value, such as log($\epsilon_{O}$) $\approx$ 8.75 (e.g., Caffau et al., 2008, 2015) might be able to satisfy all of the constraints if the uncertainties had been underestimated thus far (i.e., the model atmosphere is typically assumed to be perfect except in Socas-Navarro 2011). In conclusion, the oxygen abundance issue remains open and largely controversial.

A key factor in the abundance derivation is the solar atmosphere model used. It seems natural that a three-dimensional model should be preferred over a one-dimensional one. However, when the three-dimensional model is a numerical simulation and the one-dimensional model has been obtained empirically by fitting observations, it is not clear that the former is better than the latter to fit observations (Ayres, 2008).

Deriving element abundances is far from trivial. The results are not determined solely by the solar atmosphere model used. They are also dependent of the solar atlas adopted to fit the model and even some details on the calibration (choice of the continuum level, spectrum rectification, wavelength calibration, etc). Recently, some studies had noted the differences between the solar atlases (e.g., Caffau et al., 2008, 2009; Doerr et al., 2016) and the need to use more than one in the analysis. Another obvious factor is the value of the line strength employed (parameterized in terms of $\log(gf)$), which have been discussed in some works (e.g., Storey & Zeippen, 2000; Johansson et al., 2003).

In this paper, we present a study of the oxygen abundance in the solar photosphere with the novelty that we applied Bayesian inference to properly disentangle the effects of the relevant parameters involved. In order to do that, we used a three-dimensional empirical model of the solar atmosphere (Socas-Navarro, 2011, 2015b) to fit the forbidden [O i] line at 630.03 nm observed in two different solar atlases. In Section 2, we describe the solar model used, the syntheses made and the solar atlas observations. A brief summary of the Bayesian inference and a table with our priors are given in Section 3. Then, we present the results in Section 4 and, finally, some conclusions in Section 5.

2 Model atmosphere and observational data

2.1 Solar model used and synthesis

The solar atmosphere model that is used in this work was derived by Socas-Navarro (2011, 2015a) based on observations of the spectro-polarimeter (SP; Lites et al., 2001) of the solar optical telescope onboard the Hinode satellite (Kosugi et al., 2007). The observations have a field of view very close to the disk center and the wavelength range covers from 6300.89 to 6303.27 Å. The spectrum at each pixel was inverted using the code NICOLE (Socas-Navarro et al., 2015) to determine a column with the height stratification of temperature, line-of-sight velocity and magnetic field vector. The reader is referred to the papers cited above for more details on the model.

Our aim is to compare the synthetic line profiles with those observed in different atlases of the solar spectrum. The spatial resolution of these observations is extremely poor (of the order of tens of arcsec). Thus, we mimick this resolution by averaging all profiles synthesized in all pixels of the snapshot. As a consequence, the forward model in this three-dimensional snapshot turns out to be very time consuming. Given that the Bayesian analysis that we explain in the following section requires the evaluation of several tens of thousand of forward models, it is convenient to pre-compute a grid of models and carry out the Bayesian inference using a simulator that just interpolates on the grid of models (O’Hagan, 2006). This greatly accelerates the inference by several orders of magnitude.

The precomputed database is built using a cartesian grid in all parameters of the model. The model parameters can be separated in two different classes: slow parameters, that require the recomputation of the synthesis in the whole snapshot because they affect the radiative transfer, and fast parameters, that can be applied directly to the profiles with simple manipulations. The first two slow parameters are the oxygen and nickel abundances (note that the [O i] feature analyzed here has a Ni i blend). The third slow parameter is an enhancement factor for the velocities in the lower layers of the snapshot. This parameter is introduced because the model of Socas-Navarro (2011) is constructed fitting the strong Fe i lines at 630.1 and 630.2 nm, and it misses some of the dynamics in the lower layers of the atmosphere, precisely where the oxygen line forms. Fortunately, the nearby Sc ii line, very similar in strength and formation height to the [O i] line, is an excellent calibration tool for this missing turbulence, which we parameterize as a multiplicative enhancement factor applied on the velocities of the lower layers. For more details, see the discussion in Socas-Navarro (2011). The fast parameters are: two for a linear correction to the continuum, a global velocity shift that takes into account imprecision in the wavelength calibration (this correction is important because a degeneration exists between the ratio of Ni and O abundances,this is explained in more detailed in sect 4.2 below) and the uncertainty of the fit.

To end up with a well sampled database, we took ten possible values for each one of the slow parameters, resulting a grid of 1000 models. The oxygen and nickel abundances were varied between typical values reported in the literature (see e.g., Grevesse et al., 1984; Anders & Grevesse, 1989; Grevesse & Sauval, 1998; Asplund et al., 2009). Their minimum and maximum values are shown in Table 2 together with those for the enhancement factor.

It is important to stress that we considered the nickel abundance in the analysis because the spectral feature at 630.0 nm is actually the result of a blend between the [O i] line at 630.03 nm and a Ni i line at 630.03 nm. We considered the two major isotopes 58Ni i and 60Ni i, with a log(gf) value of -2.11 and a fraction of 72% and 28% respectively for each isotope (Johansson et al. (2003); see Table 1).

For the synthesis with NICOLE , we chose a wavelength range from 630.0 to 630.3 nm, including also the Fe i lines at 630.15 and 630.25 nm. We introduced the Fe i lines because the [O i] line is formed in the far wing of the 630.15 nm Fe i line and we need to include this effect in our synthesis, since it affects the continuum estimate.

We introduce an additional simplification in the forward modeling, given that the model of Socas-Navarro (2011) has $200\times 200$ spatial pixels in the field of view. Synthesizing the wavelength range in each one of the 1000 models in our grid would be very demanding computationally. For this reason, we carried out a Montecarlo selection of a subset of 1000 pixels, as a good compromise between accuracy and computational time, that produce an average profile that is “equivalent” to the average profile of the whole snapshot. By “equivalent” we mean that the differences between the average intensity profile of the full map and that of the subset is smaller than $0.10\%$ in the oxygen line.

In summary, we constructed a grid of 1000 models where each model has 1000 pixels. For each one of them, we synthesized the spectral lines of interest using NICOLE, taking into account the atomic information compiled in Table  1. For the sake of reproducibility, the table also includes the references for the log($gf$) values. The log(gf) value for the Fe i at 630.25 nm was taken from Socas-Navarro (2011). The radiative, Stark and van der Waals damping parameters are also provided for each line except the Fe lines. For those, we give instead the $\alpha$ and $\sigma$ damping parameters using the method of Anstee & O’Mara (1995) obtained with the code of Barklem et al. (1998).

2.2 Observational data

In this study we also intend to assess the influence of the choice of solar observation on the abundance determination. We used two different intensity atlases of the solar photosphere which have a good signal-to-noise (S/N) ratio and have been widely used in previous works (e.g. Caffau et al., 2008, 2009; Ayres, 2008; Maiorca et al., 2009).

The first one is the disk center intensity atlas of the solar spectrum from 3000 Å to 10000 Å by Delbouille et al. (1973). The spectrum was obtained at the International Scientific Station of the Jungfraujoch and may be downloaded from the BASS2000 web server111http://bass2000.obspm.fr/solar_spect.php. The second atlas that we considered was the Kitt Peak intensity FTS (Fourier Transform Spectrometer) atlas by Neckel & Labs (1984). This atlas was produced with the FTS instrument at the McMath telescope and spans the wavelength range from 3290 to 12510 Å.

3 Bayesian analysis

We employ the formalism of Bayesian inference (e.g., Gregory, 2005) to quantify the uncertainty in our conclusions derived from the use of a prescribed model, a bunch of free parameters and a set of priors (information known a priori). The aim of the Bayesian analysis is to compute the posterior probability distribution associated with all model parameters by taking into account the information provided by the observations and all our a-priori information. It relies on two fundamental tools. The first one is the Bayes theorem, that describes very simply how the prior information is updated with the acquisition of new observations, to the posterior information. If we denote the model parameters of interest with the vector $\boldsymbol{\theta}$, and the observations with $D$, the Bayes theorem states that

[TABLE]

In the Bayes theorem, $p(\boldsymbol{\theta}|I)$ is the prior probability distribution of $\theta$, that encodes all the a-priori information we know about the parameters (i.e., if a quantity is positive, that a certain region of the space of parameters is more probable than others, etc.). $p(D|\boldsymbol{\theta},I)$ is the likelihood, which encodes the information about the model parameters that can be extracted from the observations. $p(\boldsymbol{\theta}|D,I)$ is the posterior distribution, that describes everything that we know about the model parameters. $p(D|I)$ is the model evidence, which is an unimportant constant in our case, because it does not depend on $\boldsymbol{\theta}$. Finally, $I$ refers to any important context information that is necessary for the inference. In our case, for instance, the specific radiative transfer model that we use. The second fundamental tool is the marginalization, that is used to obtain the posterior distribution for any parameter taking into account the uncertainties and correlations with other parameters:

[TABLE]

In this work we considered seven free parameters, that we consider necessary for explaining all the expected variability of the line profiles: the oxygen abundance (which is the main parameter of interest), the abundance of nickel (necessary to reproduce the line shape because the oxygen line of interest is blended with a Ni i line), a global wavelength shift to account for possible inaccuracies in the wavelength scale, an enhancement factor for the dynamics that is described above, two free parameters that are used to correct the continuum level and its slope, and finally a parameter $\sigma$ that describes the (unknown) uncertainty of the observations and also quantifies the quality of our modeling.

In any Bayesian inference, it is crucial to define the priors. In our case, we define the priors as flat (meaning there is no preference) for almost all parameters in the ranges shown in Table 2. The ranges have been chosen taking into consideration the results obtained in previous studies (see e.g., Grevesse et al., 1984; Anders & Grevesse, 1989; Grevesse & Sauval, 1998; Asplund et al., 2009) but trying not to discard parts of the space of parameters that might be compatible with the observations. The only non-flat prior is that of $\sigma$, for which we adopted a Jeffreys prior, $p(\sigma)\propto\sigma^{-1}$, because it is a scale parameter that can potentially have values spanning several orders of magnitude (e.g., Gregory, 2005).

For computing the likelihood, we assume the following generative model:

[TABLE]

which states that the observations for the $i$-th wavelength point, $O(\lambda_{i})$, can be modeled with $I(\boldsymbol{\theta},\lambda_{i})$ plus some uncertainty that has zero mean and variance $\sigma^{2}$. Following the standard approach (e.g., Gregory, 2005), the likelihood has the form of an uncorrelated multivariate normal distribution, which reduces to:

[TABLE]

where $n$ is the number of observed wavelengths.

We sample the posterior distribution using the nested sampling tecnique by Skilling (2004) 222We use the implementation of the Python package nestle, which can be obtained from http://kbarbary.github.io/nestle/..

4 Results

4.1 Line fit

Figure 1 shows our fits of the [O i] and Sc ii lines in our model (black lines) with respect to the atlases (blue). The left panel uses the Neckel atlas and the right one uses the Delbouille atlas. Instead of a single fit, we provide a sampling of models that are compatible with the observations. They are obtained by synthesizing models with parameters extracted from the posterior distribution. Figure 1 clearly shows that our model fits very well both atlases, even in very weak lines like these. However, the models present some discrepancies with the observations in the continuum region in between the two lines, where the models set the continuum to a higher level. Given that we have this effect in both atlases, this might be produced by some unknown absorption in this region. The lines of interest are very weak, thus, this region could affect our abundance determination indirectly by setting the continuum level on a wrong level. We tried to minimize this effect by introducing the Sc ii line in the analysis, which then sets a much better continuum level estimation. We have plans to refine this approach in the future by using a modeling that is able to absorb these deficiencies (similar to the nonparametric approach based on Gaussian Processes of Czekala et al., 2015).

4.2 Marginal posterior distributions

The joint and marginal posterior distributions for the five main parameters (oxygen and nickel abundances, velocity, enhancement factor and $\sigma$) are shown in Fig. 2. The blue distributions correspond to the Neckel atlas while the green distributions correspond to the Delbouille atlas. We did not show the distribution for the two parameters of the continuum because they are nuisance parameters that are only needed to have a good fit of the line, but they do not provide relevant information.

For both atlases, the joint and marginal posterior distributions show a Gaussian-like shape, where those corresponding to the Neckel atlas always present a larger width. This width indicates that the uncertainties in the abundances inferred from the Neckel atlas are larger than those obtained from the Delbouille atlas. Moreover, the marginal distribution for $\sigma$ also shows that this parameter is smaller for the case of the Delbouille atlas. Thus, our model seems to provide more constrained model parameters and better fits for the Delbouille atlas.

It is encouraging that the marginal posterior for the oxygen abundance is consistent in both atlases, with that inferred from the Delbouille atlas being contained inside the uncertainty associated with the Neckel atlas. It is interesting that the maximum marginal a posteriori (the peak of the marginal posterior) is located at roughly the same value for the two cases.

This is not the same with the rest of parameters. The marginal posteriors corresponding to the Delbouille atlas tend to be shifted towards smaller values, except for the enhancement factor, where the opposite happens. In this last parameter, the marginal distributions even indicate that the inferred enhancement factors are not compatible (at least to 3 standard deviations).

If we look at the panels with the joint posteriors (showing the correlation between pairs of parameters), we see a clear correlation between the abundances of oxygen and nickel. This correlation is a direct consequence of the fact that the [O i] line is a blend: we can obtain a fit equally good if we take a model with a higher nickel abundance and a lower oxygen abundance and viceversa. Furthermore, these two parameters (oxygen and nickel abundances) are correlated with the velocity shift. This correlation makes sense for small velocities, since a wavelength shift displaces the line and one can still obtain a good fit by appropriately modifying the abundances. For example, a blue shift can be compensated with a lower nickel abundance and a higher oxygen abundance.

All marginal posterior distributions have a Gaussian-like shape. Therefore, we summarize them in Table 3 by providing the median and the uncertainty defined by the percentiles 16 and 84 (equivalent to the standard $1\sigma$ uncertainty in the Gaussian case). It is clear in the table that the most probable value for the oxygen abundance is the same (down to the second decimal) in both atlases. The value obtained from this study is, hence, $\log(\epsilon_{O})=8.86\pm 0.01$. This value is within the category of high solar oxygen abundances: lower than the $8.93\pm 0.02$ value in Grevesse et al. (1984), but compatible with the $8.83\pm 0.06$ revision in Grevesse & Sauval (1998).

4.3 More experiments

In order to quantify how much the impact of the continuum level and the correlation between the oxygen and nickel abundances are, we carried out two more studies. They are intended also to check the robustness of our results.

First, we carried out the same analysis as before but reducing the wavelength range to just take into account the [O i] line. Although we do not display the fits or the marginal posteriors for simplicity, the summary of the results is displayed in Table 4 (see rows corresponding labeled as ”Only [O i]” and ”no prior”). For completeness, we repeat in this table the results of the previous section for an easier comparison (labeled as ”Both line” and ”no prior”). Thus, modifying the range of wavelengths, the most probable oxygen abundance changes from log($\epsilon_{O}$) = 8.86, a value compatible with Grevesse & Sauval (1998), to log($\epsilon_{O}$) = 8.94, a value compatible with Grevesse et al. (1984). Because we are neglecting the wing of the Sc ii line and the inferred abundance is different, we conclude that a good estimate of the continuum is crucial for the estimation of abundances. However, this large oxygen abundance seems improbable at the light of the really low nickel abundance inferred which is incompatible with all previous reports in the literature (that usually give $\log(\epsilon_{\mathrm{Ni}})>6$).

The second test is motivated by the low nickel abundance of the previous results. In this case, we redo the analysis but including a different prior for the nickel abundance. This prior is set to a Gaussian distribution with mean $\mu=6.17$ and standard deviation of $\sigma=0.07$, as reported in Scott et al. (2009). We intentionally increase the value of $\sigma$ to decrease the informativeness of the prior and let the data drive the results. The results of this new analysis are also shown in Table 4 (with labels ”prior Ni”). The results show that the inference only with the [O i] line are not very reliable because, when including the Gaussian prior for the nickel abundance, the modification in the oxygen and nickel abundances are greater than the previously quoted uncertainties. This fundamentally means that the result is strongly dependent on the prior. However, this is not the case when using the [O i] and Sc ii lines together, where we obtain results that are essentially insensitive to the prior.

4.4 Abundance and log($gf$) factor

We are aware that an important source of uncertainty in the determination of abundances are the atomic parameters of the spectral lines, specifically the value of log($gf$). We have tried to use the most recent determinations, but it is true that a slightly different value of log($gf$) inside its uncertainty would produce different oxygen and nickel abundances. Therefore, one can consider our determination of the abundance to be indeed an inference over the product $gf\epsilon_{O}$, which remains valid for weak lines.

The $gf$ value used in this work (Johansson et al., 2003) is accepted as the most accurate, but its uncertainty can go up to 10%. This uncertainty translates into $log(gf)=-9.717\pm 0.043$. If this is taken into account, it would induce an uncertainty in the estimated oxygen abundance of $log(\epsilon_{O})=8.86\pm 0.04$.

5 Conclusions

There are many parameters involved in the determination of abundances. To end up with an accurate determination of abundances, it is crucial to put emphasis on appropriately considering all of them. This is the motivation of this work and it represents a first step towards a reliable determination of the solar abundances. Several conclusions can be extracted from our work:

• •

We have used a very flexible generative model that contains seven parameters. Some of these parameters are nuisance parameters that are of no diagnostic interest but which are necessary for a good modeling of the spectral line. Including these nuisance parameters and marginalizing over them is crucial for a reliable determination of abundances.

• •

Including nearby spectral lines turns out to be very important, because they modify the continuum level. We have found that a good characterization of the continuum level in these weak lines is crucial for the inference of abundances.

• •

We have found some differences when applying exactly the same modeling to different atlases. Therefore, we find it may be convenient to infer solar abundances using the largest panoply of observations. However, we find a reliable oxygen abundance with just 0.003 dex difference between the studies with the two solar atlases.

• •

A reliable independent determination of the spectral line atomic parameters is very important. Otherwise, this uncertainty propagates accordingly onto the inferred abundances.

• •

Given the interdependencies among all model parameters and the uncertainty in the observations, one should pursue a fully Bayesian approach taking into account all effects simultaneously. Ideally, such an approach should also be considered for aggregating all sources of uncertainty and checking, through a meta-analysis, for the compatibility of all the inferred abundances that exist in the literature.

Despite the many factors that we take into account, we found a good agreement in our results for the oxygen abundance. Thus, we can conclude that, based on the three-dimensional model that we used, the most probable value for the oxygen solar abundance is $log(\epsilon_{0})=8.86\pm 0.04$. This value is classified as a high solar oxygen abundance and it is compatible with the results of Grevesse & Sauval (1998).

Finally, a caveat is in order. Note that all results given in this paper are conditioned on the empirical model of Socas-Navarro (2011, 2015a) and the radiative transfer of NICOLE being correct. If this is not the case, the conclusions might change. The advantage of the Bayesian framework is that it is transparent (every probability distribution in Eq. (1) is conditioned on the a-priori information $I$).

Acknowledgements.

Financial support by the Spanish Ministry of Economy and Competitiveness through projects AYA2014-60476-P and Consolider-Ingenio 2010 CSD2009-00038 are gratefully acknowledged. AAR acknowledges financial support through the Ramón y Cajal fellowship. MCA acknowledges Fundación la Caixa for the financial support received in the form of a PhD contract. This research has made use of NASA’s Astrophysics Data System Bibliographic Services. This work has made use of the VALD database, operated at Uppsala University, the Institute of Astronomy RAS in Moscow, and the University of Vienna. We acknowledge the community effort devoted to the development of the following open-source Python packages that were used in this work: numpy, matplotlib, nestle, and seaborn.