A star’s tiny frequency mismatches can reveal that its model has the wrong core physics. This study turns those mismatches into localized sound-speed measurements inside twelve distant, Sun-like stars. The theoretical oscillation frequencies of even the best asteroseismic models of solar-like oscillators show significant differences from observed oscillation frequencies.

Structure inversions use these frequency differences to infer underlying differences in stellar structure. Structure inversion results have been used extensively to study the Sun, but results for other stars have so far been limited. Applying sound-speed inversions to more stars can probe stellar theory over a larger range of conditions.

The study presents results for twelve main-sequence solar-type stars with masses between one and one point one five solar masses, probing the isothermal sound speed in the innermost thirty percent by radius. Half the target stars agree fully with their best-fit models. In the others, the inversions reveal significant differences: five model cores have sound speeds that are too low, while one star shows the opposite behavior.

For the two stars with the most significant differences, changing nuclear reaction rates or core opacities reduces, but does not fully resolve, the differences. The combination of high-precision photometric time-series data from Kepler, astrometric parallax data from Gaia, and high-resolution spectroscopic measurements of effective temperatures and metallicities provides an opportunity to test stellar evolution theory at unprecedented precision.

Asteroseismology uses oscillation frequencies obtained from stellar light curves and provides a direct way to test the physics of stellar interiors. The frequencies are sensitive to the internal structure of the star. By constructing stellar evolution models that reproduce observed oscillation frequencies and surface properties such as luminosity, effective temperature, and metallicity, asteroseismology can study atomic diffusion, rotation, magnetic fields, and convection.

Even for stars with the highest-quality asteroseismic data, discrepancies remain between models and observations. The tension suggests that stellar models need improvement, but it does not directly identify what those improvements should be. Asteroseismic structure inversions use the differences between the frequencies of an observed star and its model to infer potential underlying structural differences.

The goal of a structure inversion is to infer differences between the actual stellar structure and a reference model. Because the inversion equation uses a linear perturbation approach, the reference model must be suitably close to the actual star. The reference model is typically the best-fit model obtained through forward modeling.

The study created a grid of sixteen thousand three hundred eighty-four tracks using version r22.05.01 of MESA. The grid varies initial mass, initial helium mass fraction, metallicity, and mixing-length parameter using a Sobol sequence, with the covered parameter ranges given in Table 1.

The models use GS98-scaled metal abundances, OPAL and Ferguson opacity tables, and a MESA blend of OPAL, SCVH, FreeEOS, and Skye equation-of-state data. The models use the pp cno_extras_o18_ne22.net reaction network, reaction rates from JINA REACLIB and NACRE, electron screening, thermal neutrino loss rates, time-dependent local convection, gravitational settling, an Eddington-gray atmosphere, and GYRE for adiabatic frequencies.

Table one lists the parameter ranges used to construct the stellar model grid. The authors vary mass from zero point seven to one point two solar masses, initial helium from zero point two four to zero point two nine, initial metallicity from zero point zero zero zero five to zero point zero seven, and the mixing-length parameter from one point three to two point four.

These grid limits define the forward-modeling search used to obtain reference models for later structure inversions. For each target star, the reference model is found by fitting the observed frequencies, effective temperature, and metallicity of that star.

The frequency-fit statistic uses the number of frequencies and distinguishes observations from model frequencies. Before that statistic is calculated, the model frequencies receive the two-term surface correction from Ball and Gizon. The grid is scanned for the parameters mass, initial helium abundance, initial metallicity, mixing-length parameter, and central hydrogen abundance that minimize the frequency-fit statistic.

The analysis interpolates in central hydrogen abundance along each track, but not between tracks, and keeps models within six standard deviations of effective temperature and metallicity and within ten standard deviations of the Gaia FLAME luminosity. With a suitable reference model, the differences between observed-star frequencies and reference-model frequencies can be used to infer underlying structure differences.

Stellar structure kernels express how sensitive an oscillation-mode frequency is to a small perturbation in the structure. The relative frequency difference compares an observed mode with the corresponding reference-model mode, while the kernel functions describe the structure variables being considered.

The mode kernel functions are obtained through a linear perturbation of the oscillation equation. The study uses the isothermal sound speed and helium mass fraction as a structure-variable pair. A surface-effect term can be included in the kernel equation, but this work instead corrects for the surface effect when calculating the frequency differences.

Each oscillation mode is sensitive to many points within the star, so the study uses optimally localized averages, or OLA, to obtain localized information from a linear combination of frequency differences. When the averaging kernel is normalized and the cross term is small, the linear combination of frequency differences acts like a weighted mean over the structure difference.

If the coefficients give the averaging kernel most of its amplitude around one target radius, the same frequency combination provides a localized average difference in the structure variable at that radius. The localized averaging kernels are constructed with multiplicative optimally localized averages, or MOLA.

Its weight function penalizes averaging-kernel amplitude away from the target radius. The inversion requires choosing an error-suppression parameter and a cross-term-suppression parameter, alongside the target radius. Although the normalization of the weight function can vary, changing it only changes the relative weight of one term and can be counteracted by changing the trade-off parameters, so the inversion results do not change.

The study uses the isothermal sound speed and helium mass fraction pair because the helium kernels have low amplitude except in helium ionization zones, naturally suppressing the cross-term kernel at the target radii. Using helium mass fraction as a structure variable requires an equation-of-state assumption.

For stars other than the Sun, the resulting error is not significant compared with the larger asteroseismic frequency uncertainties. To remain consistent with the MESA equation-of-state blend, the needed partial derivatives of the first adiabatic exponent are obtained directly from MESA’s equation-of-state module.

For stars other than the Sun, imprecise stellar mass and radius are an additional complication. Because stellar frequencies scale with mean density, a mean-density mismatch can offset the inversion results. To minimize this problem, the inversion uses the relative difference in dimensionless squared isothermal sound speed, with dimensionless mode kernels and dimensionless frequency differences.

For each star, the study attempts inversions at six target radii: zero point zero five, zero point one zero, zero point one five, zero point two zero, zero point two five, and zero point three zero of the stellar radius, although some stars provide suitable kernels at only five radii.

The disagreement between each target star and its model is summarized across the target radii with an inversion chi-squared statistic. The inversion results have associated uncertainties, and errors between target radii are correlated because they use the same underlying data.

After finding a reference model, the analysis calculates surface-term-corrected dimensionless frequency differences between the target star and the model. Calibration models are used to choose the error-suppression parameter at each target radius and obtain the averaging kernels.

The inversion coefficients and frequency differences then yield localized differences in dimensionless squared isothermal sound speed. The Kepler LEGACY and KAGES samples provide oscillation data for ninety-five stars. The analysis keeps reference models with radiative cores throughout main-sequence evolution and a frequency-fit chi-squared below twenty.

Those criteria produce suitable reference models for thirty-four stars, and twelve have approximately thirty-five observed frequencies—enough to form well-localized averaging kernels. Figure 1 compares asteroseismic inversions with best-fitting stellar models for twelve stars, plotting relative differences in dimensionless squared isothermal sound speed across fractional radii from near the core outward.

Points above the dashed line indicate higher sound speed in the star, while vertical and horizontal bars show inversion uncertainty and averaging-kernel width; colors encode statistical significance. This matters because the figure shows where, and with what confidence, the observed stellar interiors differ from their reference models.

The twelve stars divide into three groups: models that agree with observations, models whose dimensionless squared isothermal sound speed is too high, and models whose value is too low. After accounting for uncertainties, six stars are in the agreement group, five are in the group where the model value is too high, and one is in the group where it is too low.

Half the sample therefore shows significant differences, suggesting limitations in the reference-model physics that most often produce internal values of dimensionless squared isothermal sound speed that are too low. Figure 2 shows Spearman rank correlations between each star’s maximum-significance inversion result and properties of its reference model, along with rotation and activity indicators.

The strongest displayed association is with frequency separation ratios, at zero point eight one, while correlations with mass and rotation are also shown, with bootstrap uncertainties marked by black error bars. This matters because it helps the authors investigate why some models agree with the inferred sound-speed structure while others show significant disagreement.

The strongest correlation is with the chi-squared statistic for frequency-separation ratios, with a Spearman correlation coefficient of zero point eight one. That statistic measures how well the reference model matches the observed frequency-separation ratios r ten and r zero two.

These ratios are insensitive to the surface effect, so they provide another measure of how well the model reproduces internal structure. The strong correlation supports the interpretation that the inversion differences are internal-structure differences rather than problems in near-surface layers.

The discrepancies correlate positively with the model’s central abundances of carbon twelve and nitrogen fourteen, with correlation coefficients of zero point seven two and zero point six six, and with energy generated by the CNO cycle, with a coefficient of zero point six one.

A similar-strength correlation runs in the opposite direction with central hydrogen abundance, with a coefficient of negative zero point six two. These properties are mutually correlated. When central hydrogen is lower, reactions beyond the pp-chain can occur more easily, primarily increasing energy generated by the CNO cycle.

Overall, the study finds a moderate correlation between the significance of the inferred sound-speed differences and more evolved main-sequence stars. Figure three compares reference-model frequencies with observations for the solar analogs sixteen Cyg A and sixteen Cyg B.

The left and center panels show frequency échelle diagrams before and after the two-term Ball and Gizon surface correction, while the right panels show the frequency-separation ratios r ten and r zero two. Open symbols mark the models and filled symbols the observations, with colors and shapes identifying spherical degree, making the agreement in both frequencies and interior-sensitive ratios visible.

Figure four shows inversion results for the solar analogs sixteen Cygni A and sixteen Cygni B, plotted against fractional radius. Blue points represent the authors’ results, while orange points reproduce the results from Bellinger and colleagues in twenty seventeen, with their reported mass and radius used to calculate the plotted quantity.

The paired measurements and error bars show how the two analyses compare throughout the stellar interiors, making this figure a direct check on the inferred structure of these well-studied Kepler stars. The analysis used asteroseismology to infer detailed core structure in the best solar-type stars observed by Kepler, focusing on main-sequence stars with radiative cores and expanding the number studied with structure inversions from two to twelve.

MESA provided the reference-model grid, calibration models provided the inversion parameters, and the resulting inversions measured relative differences in dimensionless squared isothermal sound speed between each reference model and target star. The sample contains six stars whose reference models agree with observations, one whose model value is higher than the star’s, and five whose model value is lower than the star’s.

The results show significant correlations suggesting that older main-sequence models with more energy generated by the CNO cycle have larger differences between model and star. The inversions show that half the models disagree significantly with their stars, most often underestimating the internal sound speed.

Nuclear reaction rates and core opacities help, but do not fully solve the mismatch.