Analysis of the linear approximation of seismic inversions for various structural pairs

Context. Thanks to the space-based photometry missions CoRoT and Kepler, we now benefit from a wealth of seismic data for stars other than the sun. In the future, K2, Tess, and Plato will complement this data and provide observations in addition to those already at hand. The availability of this data leads to questions on how it is feasible to extend kernel-based, linear structural inversion techniques to stars other than the sun. Linked to the inversion problem is the question of the validity of the linear assumption. In this study, we analyse the limitations of this assumption with respect to changes of structural variables. Aims. We wish to provide a more extended theoretical background to structural linear inversions by doing a study of the validity of the linear assumption for various structural variables. We thus point towards limitations in inversion techniques in the asteroseismic and helioseismic cases. Methods. First, we recall the origins of the linear assumption for structural stellar inversions and explain its importance for asteroseismic studies. We also briefly recall the impact of unknown structural quantities such as the mass and the radius of the star on structural inversion results. We then explain how kernels for new structural variables can be derived using two methods, one suited to asteroseismic targets, the other to helioseismic targets. For this second method, we present a new structural pair, namely the (A, Y) structural kernels. The kernels are then tested in various numerical experiments that enable us to evaluate the weaknesses of different structural pairs and the domains of validity of their respective linear regime. Results. The numerical tests we carry out allow us to disentangle the impact of various uncertainties in stellar models on the verification of the linear integral relations. We show the importance of metallicity, the impact of the equation of state, extra-mixing, and inaccuracies in the microphysics on the verification of these relations. We also discuss the limitations of the method of conjugated functions due to the lack of extremely precise determinations of masses or radii in the asteroseismic context.