Online material

Appendix A: Estimate of intrinsic errors of the method

To estimate the domain of validity of the method, it is necessary to determine how sensitive the distribution of points in Fig. 2 is to variations of the physical parameters used to construct the model grid. Note that the linear fit given by Eq. (5) was obtained considering each model as an independent measurement without uncertainties. Indeed, the whole dataset is composed of independent subsets of models: the evolutionary tracks. As a consequence, the standard procedure for calculating coefficient errors and/or the goodness of the fit are meaningless here. Instead, we sought to estimate the errors committed by studying how variations of the physical parameters with which the model dataset was constructed (Table 1) modify the shape or the thickness of the strip shown in Fig. 2.

In particular, we examined the Δν- relation by analyzing the maximum variation of a given parameter p at once (Table 1), leaving the remaining parameters free to vary. Then, we calculated the size of the model strip at a given value of Δν_i with a given uncertainty ± u_i, defined as (A.1)and the intrinsic error committed in the estimate of from Fig. 2 for a given value Δν_i with an uncertainty of u_i is given by (A.2)That is, we considered the largest possible error for a given parameter variation.

These error estimates are necessarily dependent on the uncertainty in the observed value of the large spacings. We studied this dependence by calculating ϵ_p(x) for a set of u_i values (in μHz) ranging from 0 to 10   μHz. Figure A.1 shows the evolution of the errors with u_i, which is roughly linear. Note that for u_i ≲ 2   μHz, values lower than 0.12   g   cm^-3 are predicted for . Therefore, very low uncertainties in Δν are required for a good determination of the mean density. For instance, to derive an uncertainty of ± 0.02 in , one would need to measure Δν with a precision u_i lower than 1   μHz. Note that this is the precision reached by García Hernández et al. (2013) in the study of periodicities of the δ Scuti star HD174966. Indeed, ϵ_p are intrinsic errors of our method. The total error σ (see Table 2) on the mean density also depends on all the constraints considered to model the studied star.

For an ideal perfect measurement of the large spacing, the lowest precision is given by the strongest rotation effect (due to the star deformation) on Δν (see Sect. 2.3). For very rapidly rotating objects (stars rotating faster than 40% of the Keplerian velocity) the effect of rotation on the large spacing is stronger than 2.3   μHz, which means an intrinsic error on of approximately 0.12gcm^-3. For slower stars, this intrinsic error becomes smaller. The largest errors predicted for the estimate of the mean density range from 11% to 21% of the total variation of in the main sequence.

We recall that such variations correspond to the worst case, and therefore they must be regarded as an upper limit. When other observational constraints are considered (e.g., metallicity, gravity, effective temperature), the errors in the diagnostics proposed here can drop drastically (GH09).

	Fig. A.1 Dependence of errors (Eq. (A.2)) with the uncertainty in Δν for the four quantities considered. A value of Δν = 60   μHz (about the middle of the main sequence) is assumed.
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	Fig. A.2 Overall scheme of the basic data model adopted in TOUCAN. The physical variables from equilibrium models and the asteroseismic variables are listed on the left and right panels.
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Appendix B: TOUCAN

Here we present TOUCAN, the first virtual observatory tool for asteroseismology developed by the Spanish Virtual Observatory (SVO)³, with which performed the entire workflow of our study. In this section we show the tool in the context of necessities of the asteroseismic community, describe its main objectives and characteristics, and detail its current workflow. Note that such an application is constantly evolving and some of the snapshots provided here might be different in the future. But the main purpose and ultimate objectives will remain the same.

Appendix B.1: Context

Stellar physics experiments today have significantly progress because of the rapid development of one of its main laboratories: stellar seismology, which is the only technique that allow probing the interior of stars to gain detailed knowledge of the internal structure and the physical processes occurring there. In the last decades we have witnessed a significant development of this technique, mainly because of the increase of the quantity and quality of the observations, particularly from space and ground-based multisite campaigns. From space, a significant amount of high-quality asteroseismic data is available from MOST⁴(Walker et al. 2003), CoRoT⁵(Baglin 2003), and Kepler⁶(Gilliland et al. 2010), launched in 2009. Other missions such as GAIA (Perryman 2003) and PLATO (Catala 2009), will increase the available datasets by a factor of several hundreds. From the ground, dedicated photometric and spectroscopic follow-up observations for the above-mentioned space missions (e.g. Poretti et al. 2009; Uytterhoeven et al. 2010,for CoRoT and Kepler missions, respectively) are necessary for a better characterization of the stars observed by the satellites.

A proper understanding of this huge amount of information requires a similar leap forward on the theoretical side (see Suárez 2010, for a recent review on this topic). Today, simulations of complex systems produce huge amounts of information that are difficult to manipulate, analyze, extract, and publish. Significant advance has been made in this regard, for instance, the Asteroseismic Modeling Portal (AMP, Metcalfe et al. 2009) or MESA (Paxton et al. 2011) codes. However, the main problem comes from the necessity of dealing with theoretical models developed by different groups, with different codes, numerical approximations, physical definitions, etc. This lack of homogeneity makes it difficult to design automatic tools to simultaneously work with different models and/or applications able to use the models on the fly.

On the observational side, these problems have been successfully solved thanks to Virtual Observatory (VO), which is an international initiative whose main objective is to guarantee easy access and analysis of the information residing in astronomical archives and services. Nineteen VO projects are now funded through national and international programs, all projects working together under the IVOA⁷ to share expertise and best practices and develop common standards and infrastructures for the VO.

In this context, the Spanish VO (SVO), which joined IVOA in June 2004, is deeply involved in the development of standards that guarantee a full interoperability between theory and observations and among theoretical collections themselves. In particular, SVO actively participated in the development of the VO access protocol for theoretical spectra⁸ and is currently working on a more general protocol called S3⁹. Examples of theoretical models published in the VO framework can be found at the SVO theoretical model server¹⁰.

Fig. B.1 Illustration of a Herzprung-Russel diagram of the models matching all the input criteria simultaneously. Dots represent the effective temperature and luminosity of all the valid models. Lines represent evolutionary tracks. For clarity, only some of the models used in this work are depicted. A color version of the plot is accessible in the online version of the paper. (This figure is available in color in electronic form.)

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Appendix B.2: Characteristics

TOUCAN is a tool conceived to work with VO-compliant models. In the VO, models are described according to the same data model and accessed using the same access protocol which solves all the problems of data discovery, data access, and data representation of non-VO tools.

The tool is intended to have a wider applicability in asteroseismology, and more generally in stellar physics and in any other field for which stellar models are required. To summarize, the main characteristics are:

• Efficiency. TOUCAN typically queries multiplemodel databases in seconds.

• Collections of models are handled easily and with user-friendly web interfaces.

• The only software required is a web browser.

• Tables, figures, and model collections are fully downloadable.

• Designed for the easy and fast comparison of very different and heterogenous models.

• Visualization tools are available. Some of the plots presented here were built with the TOUCAN graphic tools (see Appendix B.4).

• The tool offers new scientific potential, which is otherwise technically impossible or time consuming. The multivariable analysis performed in this work only required a systematic query to TOUCAN for the different parameters needed.

Furthermore, TOUCAN has also been designed for an easy and quick interpretation of the asteroseismic data of running space missions such as MOST, CoRoT, and Kepler, as well as future missions such as the PLAnetary Transits and Oscillations (PLATO), currently an M3 candidate in the ESA Cosmic Vision program, or the Transiting Exoplanet Survey Satellite (TESS), a new NASA space mission scheduled for launch in 2017. For this purpose, the next steps in the development of TOUCAN will be:

• The inclusion of new collections of models, namely solar-likeand giant-like asteroseismic models. This will be achieved bycalculating new model datasets with our own codes, and byadapting other model databases, built with differently codes anddifferent physics, to TOUCAN.

• Implementation of a direct link between TOUCAN and other existing VO services, allowing the search for observed physical parameters stored in VO-compliant databases (those of the space missions), and using them as inputs in TOUCAN.

Fig. B.2 Large spacing, Δν, as a function of the mean density for a large set of models described in Sect. 3. Large spacings were calculated using all the frequencies available per ℓ, up to ℓ = 3 (more details in the text). Time evolution reads from right to left, as the mean density of stars in the main sequence decreases with time. Plots were obtained using TOUCAN graphical utilities. A color version of the plot is accessible in the online version of the paper. (This figure is available in color in electronic form.)

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Appendix B.3: VO service

TOUCAN has been designed following the Virtual Observatory standards and requirements. This means that in parallel to the web interface, the system can also be accessed from other VO applications using the S3 protocol to obtain in a standard way information about:

• The available combinations of evolutionary and seismologicalmodels.

• The query parameters, their physical description, and the available range of values.

• The list of models that match the query criteria and their properties.

• The stellar shell structure and the oscillation spectrum for each model.

From a technical point of view, this feature is very important, since it allows the tool to work with multiple model databases, no matter where they are located physically. Moreover, this opens the possibility of interconnecting TOUCAN with existing astronomical archives, catalogs, etc., in particular, those being constructed using asteroseismic space data that are already in VO-compliant form.

Appendix B.4: Workflow

The TOUCAN workflow we describe here is general, and was therefore applied for the present work. It is composed of three main steps:

• Input parameter specification

• Summary of the results and check-out

• Model selection and online analysis.

One of the most critical steps when building a tool to handle different theoretical models in a compatible way is the identification of the mandatory parameters to represent the physics involved, and their mapping into a common set of variables. For this, we developed a prototype data model (Fig. A.2) for asteroseismology that contains 17 star global properties (effective temperature, surface gravity, luminosity, etc.), 44 star shell variables (density, pressure, temperature, etc.), and 35 seismic properties (frequency ranges, fundamental radial mode, large and small separation, etc.). For a maximum interoperability, we used the most common definitions in the field for these variables, with the aim of setting the basis of VO standards for asteroseismology.

After selecting the model parameters, TOUCAN queries the user-specified model database. Here we used our own model database described in Sect. 2.1. The results obtained from these queries are shown to the user in different formats, with the possibility of managing them and, more importantly, of using TOUCAN online graphic tools, which allow the researcher to easily do online asteroseismology, for instance by:

• Visually examining the resulting models, with some statisticsand sorting possibilities for an efficient handling of the results.

• Selecting individual or multiple files to be analyzed (including a shell variables analysis for equilibrium models) with the graphic tools, which allows the user to download the generated plots (an example of HR diagram built with the TOUCAN graphic tools is shown in Fig. B.1).

• Allowing the user to select individual or multiple files to be downloaded (e.g. complete evolutionary tracks) in the original codes’ output formats and in VO table formats. This provides compatibility with other VO-compliant visualization tools like TOPCAT¹¹.

• To download plots in “png” format, by placing the mouse pointer on the plot window and clicking the mouse’s right button.

comparisons of different and heterogeneous models, or even model collections.

Moreover, these tools allow the user to perform statistical works on theoretical properties of multiple variables at the same time. The present work is an example of such a work: the relation between the large spacings and the mean density for δ Scuti stars shown in the contour plot (Fig. 2) was built using the data obtained from Fig. B.2 during this research workflow.

© ESO, 2014