Using seismic inversions to obtain an internal mixing processes indicator for main-sequence solar-like stars

Using seismic inversions to obtain an internal mixing processes indicator for main-sequence solar-like stars

G. Buldgen Institut d’Astrophysique et Géophysique de l’Université de Liège, Allée du 6 août 17, 4000 Liège, Belgium    D. R. Reese School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK.    M. A. Dupret Institut d’Astrophysique et Géophysique de l’Université de Liège, Allée du 6 août 17, 4000 Liège, Belgium
April, 2015
Key Words.:
Stars: interiors – Stars: oscillations – Stars: fundamental parameters – Asteroseismology

Context:Determining accurate and precise stellar ages is a major problem in astrophysics. These determinations are either obtained through empirical relations or model-dependent approaches. Currently, seismic modelling is one of the best ways of building accurate stellar models, and therefore providing accurate ages. However, current methods are affected by simplifying assumptions concerning stellar mixing processes. In this context, providing new structural indicators which are less model-dependent and more sensitive to mixing processes is crucial.

Aims:We wish to build a new indicator for core conditions (i.e. mixing processes and evolutionary stage) on the main sequence. This indicator should be more sensitive to structural differences and applicable to older stars than the indicator presented in a previous paper. We also wish to analyse the importance of the number and type of modes for the inversion, as well as the impact of various constraints and levels of accuracy in the forward modelling process that is used to obtain reference models for the inversion.

Methods:First, we present a method to obtain new structural kernels in the context of asteroseismology. We then use these new kernels to build a new indicator of central conditions in stars, denoted , and test it for various effects including atomic diffusion, various initial helium abundances and various metallicities, following the seismic inversion method presented in our previous paper. We then study its accuracy for 7 different pulsation spectra including those of CygA and CygB and analyse how it depends on the reference model by using different constraints and levels of accuracy for its selection

Results:We observe that the inversion of the new indicator using the SOLA method provides a good diagnostic for additional mixing processes in central regions of stars. Its sensitivity allows us to test for diffusive processes and chemical composition mismatch. We also observe that using modes of degree can improve the accuracy of the results, as well as using modes of low radial order. Moreover, we note that individual frequency combinations should be considered to optimize the accuracy of the results.


1 Introduction

Determining stellar ages accurately is crucial when studying stellar evolution, determining properties of exoplanetary systems or characterising stellar populations in the galaxy. However, the absence of a direct observational method to measure this quantity makes such determinations rather complicated. Age is usually related empirically to the evolutionary stage or determined through model-dependent techniques like forward asteroseismic modelling of stars. This model-dependence is problematic since if a physical process is not taken into account during the modelling, we will introduce a bias in the age determinations, as well as in the determination of other fundamental characteristics like the mass or the radius (see for example Eggenberger et al. (2010) for the impact of rotation on asteroseismic properties, Miglio et al. (2015) for a discussion in the context of ensemble asteroseismology and Brown et al. (1994) for a comprehensive study of the relation between seismic constraints and stellar model parameters). It is also clear that asteroseismology probes the evolutionary stage of stars and not directly the age. In other words, we are able to analyse the stellar physical conditions but relating these properties to an age will, ultimately, always be dependent on the assumptions made during the building of the evolutionary sequence of the model. A general review of the impact of the hypotheses of stellar modelling and of asteroseismic constraints on the determination of stellar ages is presented in Lebreton et al. (2014a) and Lebreton et al. (2014b).

In that sense, the question of additional mixing processes in the context of seismic modelling is central (Dupret 2008) and can only be solved by using less model-dependent seismic analysis techniques and new generations of stellar models. These new seismic methods should be able to provide relevant constraints on the physical conditions in the central regions and help with the inclusion of additional mixing in the models. In this context, seismic inversion techniques are an interesting way to relate structural differences to frequency differences and therefore offer a new insight on the physical conditions inside observed stars. From the observational point of view, the high quality of the Kepler and CoRoT data but also the selection of the Plato mission (Rauer et al. 2014) allows us to expect enough observational data to carry out inversions of global characteristics. In the context of helioseismology, structural inversion techniques have already lead to noteworthy successes. They have provided strong constraints on solar atomic diffusion (see Basu et al. 1996b), thus confirming the work of Elsworth et al. (1990). However, their application in asteroseismology is still limited. Inversions for rotation profiles have been carried out (see for example Deheuvels et al. 2014, for an application to Kepler subgiants), but as far as structural inversions are concerned, one can use either non-linear inversion techniques (see Roxburgh 2010, 2015, for an example of the differential response technique), or linear inversion techniques applied to integrated quantities as is done in Reese et al. (2012) and Buldgen et al. (2015).

In our previous paper (see Buldgen et al. 2015), we extended mean density inversions based on the SOLA technique (Reese et al. 2012) to inversions of the acoustic radius of the star and an indicator of core conditions, denoted . We also set the basis of a general approach to determine custom-made global characteristics for an observed star. We showed that applying the SOLA inversion technique (Pijpers & Thompson 1994) to a carefully selected reference model, obtained via the forward modelling technique, could lead to very accurate results. However, it was then clear that the first age indicator was limited to rather young stars and that other indicators should be developed. Moreover, the model-dependence of these techniques should be carefully studied and there is a need to define a more extended theoretical background for these methods. The influence of the number but also the type of modes used for a specific inversion should be investigated. In the end, one should be able to define whether the inversion should be carried out or not, knowing the number of observed frequencies and the quality of the reference model according to its selection criteria.

In this study, we wish to offer an answer to these questions and provide a new indicator for the mixing processes and the evolutionary stage of an observed star. We structure our study as follows: section 2 introduces a technique to obtain equations for new structural kernels in the context of asteroseismology and applies it to the and the kernels, where is the squared isothermal sound speed, the adiabatic gradient and the current helium abundance profile. Section 3 introduces a new indicator of mixing processes and evolutionary stages, which is not restricted to young stars, as was the case for the indicator presented in Buldgen et al. (2015). Having introduced this new indicator, we test its accuracy using different physical effects such as including atomic diffusion processes with high velocities (up to times the solar microscopic diffusion velocities) in the target, changing the helium abundance, changing the metallicity and changing the solar mixture of heavy elements. Section 4 analyses the impact of the type and number of modes on the inversion results whereas Sect. 5 studies how the accuracy depends on the reference model. We also tested our method on targets similar to the binary system CygA and CygB using the same modes as those observed in Verma et al. (2014) to show that our method is indeed applicable to current observational data. Section 6 summarizes our results and presents some prospects on future research for global quantities that could be obtained with the SOLA inversion technique.

2 and structural kernels

2.1 Integral equations for structural couples in the asteroseismic context

It has been demonstrated in Gough & Thompson (1991) that one could deduce from the variational principle a linear integral relation between the perturbations of frequencies and the perturbation of structural variables. This equation is obtained by assuming the adiabatic approximation and spherical symmetry, and neglecting surface integral terms. It is only valid if the stellar models are sufficiently close to each other. If one is working with the structural pair , where is the adiabatic squared sound velocity and the density, this relation takes on the following form:


where with the stellar radius, and where the classical definition of the relative differences between the target and model for any structural quantity has been used:


In what follows, we will always use the subscript or superscript “” when referring to the observed star, “” for the reference model variables in perturbation definitions and for inverted results. Other variables, such as the kernel functions, denoted without subscripts or superscripts, are of course related to the reference model and are known in practice. Finally, one should also note that the suffix is just a index to classify the modes. Moreover, since it is clear that some hypotheses are not suitable for surface regions, a supplementary function, was added to model these so-called surface effects. It is defined as a linear combination of Legendre polynomials, normalised by the factor , which is the mode inertia normalised by the inertia of a radial mode interpolated to the same frequency. We emphasize here that neither this normalisation coefficient nor the treatment of surface effects are uniquely defined and other techniques are also used (see for example Dziembowski et al. 1990; Däppen et al. 1991; Basu et al. 1996a).

The kernels of the couple were already presented in Gough & Thompson (1991) who also mentionned the use of another method, defined in Masters (1979) to modify Eq. 1 and obtain such relations for the couple and also the couple. Other approaches to obtain new structural kernels were presented (see for example Elliott 1996; Kosovichev 1999, for the application of the adjoint equations method to this problem). This latter approach has been used in helioseismology where it was assumed that the mass of the observed star is known to a sufficient level of accuracy to impose surface boundary conditions. In the context of asteroseismology, we cannot make this assumption. Nevertheless, the approach defined in Masters (1979) allows us to find ordinary differential equations for a large number of supplementary structural kernels, without assuming a fixed mass 111The method of adjoint equations previously described could also be used but would require an additional hypothesis to replace the missing boundary condition..

Another question arises in the context of asteroseismology: what about the radius? We implicitely define our integral equation in non-dimensional variables but how do we relate the structural functions, for example defined for the observed star and defined for the reference model? What are the implications of defining all functions on the same domain in varying from to ? It was shown by Basu (2003) that an implicit scaling was applied by the inversion in the asteroseismic context. The observed target is homologously rescaled to the radius of the reference model, while its mean density is preserved. This means that the oscillation frequencies are the same, but other quantities such as the adiabatic sound speed , the squared isothermal sound speed will be rescaled. Therefore, when inverted, they are not related to the real target but to a scaled one.

This can be demonstrated with the following simple test. We can take two models a few time steps from one another on the same evolutionary sequence knowing that they should not be that different (here, we consider , main-sequence models). We then test the verification of Eq. 1 by plotting the following relative difference:


with defined as follows:


The results are plotted in blue in the left panel of Fig. 1, where we can see that this equation is not satisfied. However, one could think that this inaccuracy is related to the neglected surface terms or to non-linear effects. Therefore we carry out the same test using the kernels, plotted in red in the right panel of Fig. 1. We see that for these kernels, the equation is satisfied. Moreover, when separating the contributions of each structural term, we see that the errors arise from the term related to in the first case. Using the scaled adiabatic sound speed, however, leads to the blue symbols in the right panel of Fig. 1 and we directly see that in this case, the integral equation is satisfied. This leads to the conclusion that inversion results based on integral equations are always related to the scaled target and not the target itself, as was concluded by Basu (2003). We will see in Sect. 3 that this has strong implications on the structural information given by inversion techniques.

Figure 1: Verification of Eq. 1 for a set of modes with the being the radial modes, the being the dipolar modes and the being the quadrupolar modes. The left plot illustrates this verification for the couple when no scaling is applied to whereas the right plot illustrates the same verification for the couple when scaling is applied to , as well as the couple where no scaling is needed.

2.2 Differential equation for the and the kernels

As mentioned in the previous section, the method described in Masters (1979) allows us to derive differential equations for structural kernels. In what follows, we will apply this method to the and the kernels. However, this approach can be applied to many other structural pairs such as: , , , , …, with the squared adiabatic sound speed , the local gravity, the adiabatic gradient and the local helium abundance. We will not describe these kernels since they are straightforward to obtain using the same technique as what will be used here for the kernels. One should also note that a differential equation cannot be obtained for the couple , with the Brunt-Väisälä frequency, defined as: , without neglecting a supplementary surface term.

The first step is to assume that if these kernels exist, they should satisfy an integral equation of the type given in Eq. 1, thereby leading to:


From that point, we use the definition of to express the first integral in terms of a density perturbation. This is done using the definition of the pressure, , and the cumulative mass up to a radial position, :


where we will neglect the pressure perturbation at the surface. In what follows, we will use the non-dimensional forms , where is the stellar mass, the stellar radius and the gravitational constant, and . To avoid any confusion in already rather intricated equations, we will drop the hat notation in what follows and denote these non-dimensional variables: , and . Using Eqs. 6 and 7, one can relate perturbations to and perturbations as follows:


However, using Eq. 6, one can also relate perturbations to perturbations. Doing this, one should note that the surface pressure perturbation is usually neglected and considered as a so-called surface effect. Using non-dimensional variables and combining Eq. 6 and Eq. 7 in Eq. 8, one obtains an expression relating perturbations solely to perturbations (of course, this is an integral relation due to the definition of the hydrostatic pressure, ). One can use this relation to replace in Eq. 5 and after the permutation of the integrals stemming from the definition of the hydrostatic pressure perturbation, one obtains the following integral equation relating to :


One should be careful when solving this equation since one is confronted to multiple integrals, with certain equilibrium variables associated to or . Therefore care should be taken when integrating to check the quality of the result. To obtain a differential equation, we note that it is clear that the equation is satisfied if the integrands are equal, meaning that the kernels are related as follows:


Given this integral expression, one can simply derive and simplify the expression to obtain a second order ordinary differential equation in as follows:


where , and . The central boundary condition in terms of and is obtained by taking the limit of Eq. (12) as goes to . The additional boundary conditions are obtained from Eq. 10. Namely, we impose that the solution satisfies Eq. 10 at some point of the domain. This system is then discretised using a finite difference scheme based on Reese (2013), and solved using a direct band-matrix solver.

Two quality checks can be made to validate our solution, the first one being that every kernel satisfies Eq. 10, the second one being that they satisfy a frequency-structure relation (As Eq. 5) within the same accuracy as the classical structural kernels or . We can carry out the same analysis as in section 2.1, keeping in mind that the squared isothermal sound speed will also be implicitely rescaled by the inversion since it is proportional to , as is the squared adiabatic sound speed, . The results of this test are plotted in Fig. 2 as well as an example of the verification of the integral equation for the kernel associated with the , mode.

Figure 2: The left plot illustrates the verification of Eq. 10 for , kernel noted , where is the right hand side of this equation and is the residual. The right plot illustrates the same test as in Fig. 1 for the kernels.

The equation for the kernels is identical when using the following relation:


meaning that Eq. 12 can simply be transposed using the definitions: and . One could also start from Eq. 5, use the following definition:


and neglect the contribution. This assumption is particularly justified if one has spectroscopic constraints on the metallicity. Nevertheless, the term associated with is smaller than the three other terms and if one is probing the core regions, the contribution is already very small. Consequently, all of the terms of Eq. 14 are small compared to the integral contribution. Still, this assumption is not completely innocent if one wishes to probe surface regions. When comparing the to the , we notice that their amplitude are comparable and that the is even often larger. However, we have to consider that it will be multiplied by , which is much smaller than . Moreover, the functions are somewhat alike in central regions and thus there will be an implicit partial damping of the term when damping the contribution if it is in the cross-term of the inversion. One should notice that we can control the importance of this assumption by switching from the kernels to the kernels. Indeed, if the error should be important, the inversion result would be changed by the contribution from the neglected term. In conclusion, in the case of the inversion of we will present and use in the next sections, this assumption is justified, but this is not certain for inversions of helium mass fraction in upper layers, for which only numerical tests for the chosen indicator will provide a definitive answer.

Knowing these facts, we can search for the kernels and using the previous developments and starting from the kernels, directly obtained from or , is more straightforward. One should also note that we assume, by using Eq. 14, that the equation of state is known for the target. As illustrated in Fig. 3, we again test our solutions by plotting the errors on the integral equation (Eq. 13), and by seeing how well our solution for the , mode verifies Eq. 10. The and kernels of this particular mode are illustrated in figure 4. The kernels associated with are very similar, except for the surface regions where some differences can be seen.

Figure 3: Same as Fig. 2 for the kernels.
Figure 4: Structural kernels for the , mode associated with the couple on the left hand side and with the couple on the right hand side.

3 Indicator for internal mixing processes and evolutionary stage based on the variations of

3.1 Definition of the target function and link to the evolutionary effect

Knowing that it is possible to obtain the helium abundance in the integral equation 13, one could be tempted to use it to obtain corrections on the helium abundance in the core and therefore having insights on the chemical evolution directly. However, Fig. 4 reminds us of the hard reality associated with these helium kernels. Their intensity is only non-negligible in surface regions and it is thus impossible to obtain information on the core helium abundance using such kernels.

Another approach would be to use the squared isothermal sound speed, , to reach our goal. Indeed, we know that , and that during the evolution along the main sequence, the mean molecular weight will change. Moreover the core contraction can also lead to changes in the variation of the profile. Using the same philosophy as for the definition of the first age indicator (see Buldgen et al. 2015) and ultimately for the use of the small separation as an indicator of the core conditions (see Tassoul 1980), we build our indicator using the first radial derivative of . Using instead of allows us to avoid the dependence in which is responsible for the surface dependence of . To build our indicator, we analyse the effect of the evolution on the profile of . This effect is illustrated in Fig. 5. As can be seen, two lobes tend to develop as the star ages. The first problem is that these variations are of opposite signs, meaning that if we integrate through both lobes, the sensitivity will be greatly reduced. Therefore, we choose to base our indicator on the squared first radial derivative:


where is an appropriate weight function. First, we will consider that the observed star and the reference model have the same radius. The target function for this indicator can easily be obtained. We perturb the equation for and use an integration by parts to relate the perturbations of the indicator to structural perturbations of :


The last term on the second line is not suitable for SOLA inversions, given the neglect of surface terms in the kernels. Hence we define the function so that , thereby cancelling this term. This leads to the following expression:




The weight function must be chosen according to a number of criteria: it has to be sensitive to the core regions where the profile changes; it has to be of low amplitude at the boundaries of the domain, allowing us to do the integration by parts necessary to obtain in the expression; it should be possible to fit the target function associated with this using structural kernels from a restricted number of frequencies; moreover, Eq. (18) being related to linear perturbations, it is clear that non-linear effects should not dominate the changes of this indicator222Otherwise, using the SOLA technique, which is linear, would be impossible.. We also know that the amplitude of the structural kernels is in the centre, so should also satisfy this condition.

We define the weight function as follows:


which means that we have parameters to adjust. The case of is quickly treated. Since we know that the changes will be localised in the lobes developing in the core regions, we chose to put . and should be at least , so that the integration by part is exact and the central limit for the target function and the structural kernels is the same. depends on the effects of the non-linearities. However, since we will have to perform a second derivative of a more practical concern appears: we don’t want to be influenced by the effects of the discontinuity at the boundary of the convective envelope. At the end of the day, we use the following set of parameters: , , and . One could argue that the optimal choice for would be at the maximum of the second lobe or between both lobes to obtain the maximal sensitivity in the structural variations. These values were also tested, but the results were a little less accurate than using and they involved higher inversion coefficients and therefore higher error magnification. We illustrate the weighted profile obtained for this optimal set of parameters in the right-hand side panel of Fig. 5. Furthermore, Eq. 17 is satisfied up to , so we can try to carry out inversions for this indicator. It is also important to note that for the sake of simplicity, we do not chose to change the values of these parameters with the model, which would only bring additional complexity to the problem.

Thanks to the target function , we can now carry out inversions for the integrated quantity using the linear SOLA inversion technique (Pijpers & Thompson 1994). First, let us take some time to recall the purpose of inversions and our adaptation of the SOLA technique to integrated quantities. Historically, inversions have been used to obtain seismically constrained structural profiles (Basu et al. 1996b) as well as rotation profiles (Schou et al. 1994) in helioseismology. However, all these methods are not well suited for the inversions we wish to carry here. As discussed in Buldgen et al. (2015), the SOLA inversion technique, which uses a “kernel matching” approach is well suited to our purpose. Indeed, this approach allows us to define custom-made target functions that will be used to build a cost function, here denoted . In the case of the quantity, one has the following definition:


where is the so-called averaging kernel and is the so-called cross-term kernel defined as follows for the structural pair (For , one would replace with and with ):


and are free parameters of the inversion. is related to the compromise between the amplification of the observational error bars and the fit of the kernels, whereas is allowed to vary to give more weight to the elimination of the cross-term kernel. In this expression, is the number of observed frequencies, the are the inversion coefficients, which will be used to determine the correction to be applied on the value. is a Lagrange multiplier and the last term appearing in the expression of the cost-function is a supplementary constraint applied to the inversion which is presented in Sect. 3.2.

If the observed target and the reference model have the same radius, the inversion will measure the value of for the observed target. However, if this condition is not met, the inversion will produce a scaled value of this indicator. By defining integral equations such as Eq. 15, or even Eq. 1, we have seen in Sect. 2.1 that we made the hypothesis that both target and reference model had the same radius. However, as the frequencies scale with , the inversion will preserve the mean density of the observed target. Therefore, we are implicitly carrying out the inversion for a scaled target homologous to the observed target, which has the radius of the reference model but the mean density of the observed target. Simple reasoning shows us that the mass of this scaled target is: . Thus, as scales as , there is a difference between the target value and the measured value, . We can write the following equations:


where we have used the definition of to express the mass dependencies as radius dependencies. Therefore, we will use Eq. 24 as a criterion to determine whether the inversion was successful or not.

Figure 5: Left panel: structural changes in the scaled profile with the evolution. The models are main sequence models. Right panel: the same effects as seen when is weighted according to Eq. 19.

3.2 Non-linear generalisation

We present in this section a general approach to the non-linear generalisation presented in Reese et al. (2012) and Buldgen et al. (2015) for any type of global characteristic that can scale with the mass of the star. We can say that the frequencies scale as . Of course, they scale as the mean density, namely . However, since the inversion works with a fixed radius and implicitely scales the target to the same radius as the reference model, the iterative process associated with this non-linear generalisation will never change the model radius. Therefore, we do not take this dependence into account and simply work on the mass dependence. Let be a global characteristic, related to the mass of the star. It is always possible to define a factor so that:


And we obtain directly:


However, using the definition of the inverted correction of , one has:


with the the inversion coefficients. Using the same reasoning as in Buldgen et al. (2015) and Reese et al. (2012), we define the inversion as “unbiased” (term which should not be taken in the statistical sense.) if it satisfies the condition:


Now we can define an iterative process, using the scale factor , we will scale the reference model (in other words, multiply its mass and density by , its pressure by , leaving unchanged) and carry out a second inversion, then define a new scale factor and so on until no further correction is made by the inversion process. In other words, we search for the fixed point of the equation of the scale factor. At the iteration, we obtain the following equation for the inversion value of :


This can be written in terms of the scale factor only, noting that at the iteration, :


The fixed point is then easily obtained:


and can be used directly to obtain the optimal value of the indicator . We can also carry out a general analysis of the error bars treating the observed frequencies and as stochastic variables:


with and being the stochastic contributions to the variables. Using the hypothesis that we easily obtain the following equation for the error bars:


We note that in the particular case of the indicator , . Indeed, whereas .

3.3 Tests using various physical effects

To test the accuracy of the SOLA technique applied to the indicator, we carried out the same test as in Buldgen et al. (2015) using stellar models which would play the role of observed targets. These models included physical phenomena not taken into account in the reference models. A total of targets were constructed, with masses of , and but to avoid redundancy, we only present of them which are representative of the mass and age ranges and of the physical effects considered in our study. We tested various effects for each mass, namely those that come from microscopic diffusion using the approach presented in Thoul et al. (1994) (multiplying the atomic diffusion coefficients by a factor given in the last line of Table 1333These values might seem excessive regarding the reliability of the implementation of diffusion. We stress here that our goal was to witness the impact of significant changes on the results. However, other processes or mismatches could alter the gradient and thus be detected by the inversion.), those caused by a helium abundance mismatch, those that result from a metallicity mismatch, as well as those which stem from using a different solar heavy elements mixture. For the last case, the target was built using the Grevesse-Noels (GN, Grevesse & Noels (1993)) abundances and the reference model was built using the Asplund-Grevesse-Sauval-Scott (AGSS , Asplund et al. (2009)) abundances. All targets and reference models were built using version of the Code Liégeois d’Evolution Stellaire (CLES) stellar evolution code (Scuflaire et al. 2008b) and their oscillations frequencies were calculated using the Liège oscillation code (LOSC, Scuflaire et al. (2008a)). Table 1 summarises the properties of the targets presented in this paper.

Target Target Target Target Target Target
Mass ()
Radius ()
Age ()
Abundances AGSS09 GN93 AGSS09 AGSS09 AGSS09 AGSS09
Diffusion factor
Table 1: Characteristics of the targets. The differences between target and reference model are given in boldface.

The selection of the reference model was based on the fit of the large and small separation for modes with and using a Levenberg-Marquardt minimization code. The use of supplementary constraints will be discussed in Sect. 5 whereas the effects of the selection of the modes will be discussed in Sect. 4. The choice of frequencies is motivated by the number of observed frequencies for the system Cyg - Cyg by Kepler, which is between and (Verma et al. 2014). The inversions were carried out using the and the structural kernels.

If the inversion of shows that there are differences between the target and the reference model, then we know that the core regions are not properly represented. Whether these differences arise from atomic diffusion or helium abundance mismatch, the indicator alone could not answer this question444Constraining properly the changes due to multiple additional mixing processes with only one structural indicator is of course impossible.. Therefore, the philosophy we adopt in this paper is simply the following: ”Is the inversion able to correct mistakes in the reference models? If yes, within what accuracy range?”. The capacity of disentangling between different effects is partially illustrated in Sect. 5, but additional indicators are still required to provide the best diagnostic possible given a set of frequencies.

The results are given in Table 2 for the kernels and Table 3 for the kernels along with the respective error contributions given according to the developments of Reese et al. (2012) and Buldgen et al. (2015). We denote these error contributions: , , . These errors contributions being defined as follows:


if the couple is used. If one prefers the , becomes:


Finally, is associated to the residual contribution, in the sense that it is what remains after one has taken into account both and . The target function and their fits are illustrated in Fig. 6 for .

Table 2: Inversion results for the targets using the kernels.
Table 3: Inversion results for the targets using the kernels.
Figure 6: The left hand-side figure illustrates the averaging kernels for both structural pairs and their respective targets. The red and blue curves are nearly identical, so that only the red curve is visible. The right hand-side figure illustrates the cross-term kernels for both structural pairs, the target being .

As we can see from Tables 2 and 3, we obtain rather accurate results for all cases. This means that the inversion is succesfull and that the regularization process is sufficient for the values and .

We see in the fifth column of Tables 2 and 3 that the averaging kernel fit is usually the dominant error contribution. We will see in the next sections how this result changes with the modes used or with the quality of the reference model. If we analyse the cross-term error contribution, we see that it is generally way less important than the averaging kernel mismatch error. We also see that despite the high amplitude of the cross-term kernels associated with shown in Fig. 6, the real error is rather small and often smaller than the error associated with the helium cross-term kernel. This is due not only to the small variations in between target and reference model but also to the oscillatory behaviour of the cross-term kernel. In contrast, the cross-term kernel of the kernel has a smaller amplitude, but nearly no oscillatory behaviour and is more important in the surface regions, where the inversion is naturally less robust. Nonetheless, the results of the kernels also show some compensation. We also note that they tend to have more important residual errors. However, there is no clear difference in accuracy between the and the kernels. In the case of , we see that although the results are slightly improved, the reference value is within the errors bars of the inversion results. In an observed case, this would mean that the reference model is already very close to the target as far as the indicator is concerned. However, we wish to point out that it seems rather improbable that the only difference between a static model and a real observed star would be in its heavy elements mixture.

Analysing the residual contribution is slightly more difficult, since it includes any supplementary effect: surface terms, non-linear contributions, errors in the equation of state (when using kernels related to ), … We can see in this study that the residual error is rather constricted. This will not be the case for example if the parameter is chosen to be very small, or if the scaling effect has not been taken into account555In which case one would be searching for a result impossible to obtain.. In fact, the parameter is a regularizing parameter in the sense that it does not allow the inversion coefficients to take on extremely large values. If that would be the case, the inversion would be completely unstable because a slight error on the fit would be amplified and would lead to wrong results. This is quickly understood knowing that the inversion coefficients are used to recombine the frequencies as follows:


with the total number of observed frequencies. Where this equation is subject to the uncertainties in Eq. 17 and Eq. 5 (or, respectively Eq. 13), any error will be dramatically amplified by the inversion process. Therefore there is no gain in reducing since at some point, the uncertainties behind the basic equation of the inversion process will dominate and lead the method to failure. In this case, the inversion problem is not sufficiently regularized. Such an example is presented and analysed in the following section.

4 Impact of the type and number of modes on the inversion results

When carrying out inversions on observed data, we are limited to the observed modes. Therefore the question on how the inversion results depend on the type of modes is of uttermost importance. The reason behind this dependence is that different frequencies are associated with different eigenfunctions, in other words different structural kernels, sensitive to different regions of the star. Therefore, the inverse problem will vary for each set of modes because the physical information contained in the observational data changes. Hence, we studied targets using sets of modes. As in Sect. 3.3, we wish to avoid redundancy and will present our results for one target, namely , and different frequency sets. As a supplementary test case, we defined two target models with the properties of CygA and CygB found in the litterature (Metcalfe et al. 2012; White et al. 2013; Verma et al. 2014). Using these properties, we added strong atomic diffusion to the CygA model and used as well as the GN mixture for CygB. In constrast, the reference models used and the AGSS mixture. The characteristics of these models are also summarized in Table 4. We used the set of observed modes given by Verma et al. (2014) and ignored the isolated modes for which there was no possibility to define a large separation.

All the sets used for the test cases of this section are summarized in Table 5. The reference model was chosen as in Sect. 3.3, using the arithmetic average of the large and small frequency separations as constraints for its mass and age. Using these test cases, we first analyse whether the inversion results depend on the values of the radial order of the modes, with the help of the frequency sets , and (see Table 5). Then we analyse the importance of the modes for the inversion using set and .

The inversion results for all these targets and sets of mode are presented in Tables 6 and 7 for the kernels. A first conclusion can be drawn from the results using sets , and : low are important to ensure an accurate results. Indeed, set provides much better results than set , and even using set (which is only set extended up to for each ) does not improve the results any further. This means that modes with are barely used to fit the target.

This first result can be understood in different ways. First, using mathematical reasoning, we can say that the kernels associated with higher have high amplitudes in the surface regions and are therefore not well suited to probing central regions. Another way of understanding this problem is that when we use modes with high , we come closer to the asymptotic regime and the eigenfunctions are described by the JWKB approximation, all having a similar form and therefore not providing useful additional information. In that sense, there is a clear difference between inverted structural quantities and the information deduced from asymptotic relations, which requires high values to be valid, thereby highlighting the usefulness of inversions.

CygA-clone CygB-clone
Mass ()
Radius ()
Age ()
Abundances AGSS09 GN93
Diffusion coefficient
Table 4: Characteristics of the CygA and CygB clones.

The question of the importance of the modes is also quickly answered from the results obtained with set and . For those test cases, we reach a very good accuracy even without , unlike the previous test case using set . Moreover, we use even less frequencies than for the first sets. This is in fact crucial to determine whether one can apply an inversion in an observed case, since a few modes can change the results and make the inversion successful.

To further illustrate the importance of the octupole modes, we use the CygA and CygB clones and carry out inversions for their respective observed frequency sets. First, we use all frequencies and reach a reasonable accuracy for both targets. In the second test case, we do not use the octupole modes and one can observe a drastic change in accuracy. These results are illustrated in Table 7, where the notation ”Small” (for small frequency set) has been added to the lines associated with the results obtained without using the modes.

Looking again at the results for CygA, we again see that although the inversion improves the value of , the reference value lies within the observational error bars of the inverted result. The case of the truncated set of frequencies is even worse, since the inverted result is less accurate than the reference value. We therefore analysed the problem for the full frequency set. To do so we carried out different inversions using higher values of , the results for are illustrated in Table 7. In this case we have smaller error bars, but what is reassuring is that the result did not change drastically when we changed . This means that the problem is properly regularized around and and that we can trust the inversion results. Our advice is therefore to always look at the behaviour of the solution with the inversion parameters to see if there is any sign of compensation or other undesirable behaviour. In fact there is no law to select the value of and applying fixed values blindly for all asteroseismic observations is probably the best way to obtain unreliable results.

The case of the small frequency set is even more intriguing since the result improves greatly with . The question that arises is whether the problem is not properly regularized with or whether we are facing some fortuitous compensation effect that leads to very accurate results. If we are facing fortuitous compensation, taking slightly larger than or increasing will drastically reduce the accuracy since any change in the linear combination will affect the compensation. However, if we are facing a regularization problem, the accuracy should decrease regularly with the change of parameters (since we are slightly reducing the quality of the fit with those changes). We emphasize again that one should not choose values of the inversion parameters for which any small augmentation of the regularization would drastically change the result. In this particular case, we were confronted with insufficient regularisation and choosing corrected the problem.

Set Set Set Set Set CygA CygB
Table 5: Sets of modes used to analyse the impact of the number and type of frequencies on the inversion results