Stars ring like bells, and those faint vibrations let scientists look deep inside them. The surprise is that even our best pictures of stellar interiors can still be wrong at the core. Even the best models of sun-like stars have oscillation frequencies that differ significantly from the frequencies observed in real stars.

Those differences contain clues about what is wrong inside the model. The method uses those frequency differences to infer differences in the star's internal structure. It extends a technique used extensively for the Sun to a wider group of stars.

The study examines twelve sun-like stars and can infer differences in sound speed within the innermost thirty percent of each star by radius. The aim is to understand structural differences between stellar models and observations using a method called asteroseismic structure inversion, which uses the differences between a star's frequencies and its model to infer localized information about internal structure.

Think of it like listening to a bell: the pattern of its ringing can reveal whether the metal is thicker or thinner in particular places, even when the inside cannot be seen. With a suitable reference model, the differences between the frequencies of an observed star and the model can be used to infer the underlying structural differences.

The calculation uses sensitivity maps that express how strongly each oscillation frequency responds to a small change in the star's structure. After finding a reference model, the frequency differences between the star and the model are corrected for effects near the surface.

The calculation then combines those corrected frequency differences with sensitivity maps to obtain localized differences in sound speed between the observed star and the reference model. The results are expressed across different distances from the center of each star.

A positive result means that the sound speed is higher in the star than in the model. These measurements test whether the inner structure of each star matches its best-fitting model. Across the twelve stars, some clear mismatches appear, while others agree with the model within their uncertainty, revealing where stellar models still need refinement.

The differences between stars and models are linked with the central abundances of carbon and nitrogen measured in the model. When central hydrogen is lower, reactions beyond the proton-proton chain become easier, primarily increasing the energy generated by the CNO cycle.

When the central hydrogen amount is lower, reactions beyond the star's usual hydrogen-burning chain can happen more easily, especially the reactions that produce more energy in the core. The two stars with the largest differences have points where the inferred internal sound speed differs from the models by more than ten percent, and those differences are larger than the measurement uncertainties.

A test using two models with a difference of about fifteen percent recovered the correct shape of that difference. One set of sensitivity maps recovered the correct value within the uncertainties at every target location. The study used the stars' vibrations to infer detailed core structure in sun-like stars with radiative cores, expanding the group studied with this approach from two stars to twelve.

The stars fall into three broad groups: models that agree with the stars, models whose sound speed is higher than the star's, and models whose sound speed is lower than the star's. In half of the target stars, the best-fit model fully agrees with the observations.

In the others, the inferred sound-speed profile differs significantly from the model. Five stars have cores where the model's sound speed is too low, while one star shows the opposite behavior. For the two stars with the largest differences, changing the assumed nuclear reaction rates or the way the core blocks energy can reduce the mismatch.

But those changes do not fully resolve the differences, so the stars are pointing to gaps that remain in the models of how stellar cores work. The stars reveal a pattern: models of some older stars seem to underestimate the speed of sound in their cores, and changing the assumed nuclear physics helps but does not fully fix the mismatch.