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ABSTRACT High-precision light curves from space-based telescopes and precise astrometry from the Gaia satellite have revolutionized our ability to characterize exoplanet host stars. Asteroseismology has allowed for stellar parameters to be determined to remarkable precision, achieving age uncertainties as low as 10−20  per cent for Sun-like stars. We present an asteroseismic analysis of the naked-eye ($V = 5.78$), G4V star $$\nu ^2$$ Lupi (HD 136352), which hosts three small transiting planets with orbital periods of 11, 27, and 107 d. We used the latest 20-s cadence photometry data from the Transiting Exoplanet Survey Satellite (TESS) to extract stellar oscillations. Comparing these to stellar models, we find that the star has a mass of $$0.83^{+0.04}_{-0.03}$$ (ran) $$\pm 0.07$$ (sys) $$M_\odot$$, a radius of $$1.00^{+0.01}_{-0.02}$$ (ran) $$\pm 0.04$$ (sys) $$R_\odot$$, and an age of $$11.9^{+2.6}_{-1.6}$$ (ran) $$\pm 1.7$$ (sys) Gyr. We also confirm that the star is likely a member of the Galactic thick disc based on its Galactic velocities, consistent with the asteroseismic age. Based on the newly determined stellar parameters, we recalculate the planet parameters. The inner planet has a mass of $$4.55 \pm 0.40$$  $$M_{\oplus }$$ and a radius of $$1.57 \pm 0.04$$  $$R_{\oplus }$$, suggesting the planet is rocky and consisting primarily of silicates without an iron-rich core, consistent with its old age and significant alpha-element enhancement. The two outer planets have masses and radii of $$10.87 \pm 0.62$$  $$M_{\oplus }$$ and $$2.75 \pm 0.06$$  $$R_{\oplus }$$, and $$8.52 \pm 0.90$$  $$M_{\oplus }$$ and $$2.42 \pm 0.08$$  $$R_{\oplus }$$, respectively, suggesting both are sub-Neptune planets with a significant H–He atmosphere. 

Abstract On the main sequence, the asteroseismic small frequency separationδν₀₂between radial and quadrupolep-modes is customarily interpreted to be a direct diagnostic of internal structure. Such an interpretation is based on a well-known integral estimator relatingδν₀₂to a radially averaged sound-speed gradient. However, this estimator fails, catastrophically, when evaluated on structural models of red giants: their small separations must therefore be interpreted differently. We derive a single expression that both reduces to the classical estimator when applied to main-sequence stellar models and reproduces the qualitative features of the small separation for stellar models of very evolved red giants. This expression indicates that the small separations of red giants scale primarily with their global seismic properties as $\delta {\nu }_{02}\propto \Delta {\nu }^2/{\nu }_{\max }$, rather than being in any way sensitive to their internal structure. Departures from this asymptotic behavior, during the transition from the main-sequence to red giant regimes, have been recently reported in open-cluster Christensen–Dalsgaard (C-D) diagrams from K2 mission data. Investigating them in detail, we demonstrate that they occur when the convective envelope boundary passes a specific acoustic distance—roughly one-third of a wavelength at ${\nu }_{\max }$—from the center of the star, at which point radial modes become maximally sensitive to the position of the boundary. The shape of the corresponding features onϵ_pand C-D (orr₀₂) diagrams may be useful in constraining the nature of convective boundary mixing in the context of undershooting beneath a convective envelope. 

Abstract Asteroseismology has been used extensively in recent years to study the interior structure and physical processes of main-sequence stars. We consider prospects for using pressure modes (p-modes) near the frequency of maximum oscillation power to probe the structure of the near-core layers of main-sequence stars with convective cores by constructing stellar model tracks. Within our mass range of interest, the inner turning point of p-modes as determined by the Jeffreys–Wentzel–Kramers–Brillouin (JWKB) approximation evolves in two distinct phases during the main sequence, implying a sudden loss of near-core sensitivity during the discontinuous transition between the two phases. However, we also employ non-JWKB asymptotic analysis to derive a contrasting set of expressions for the effects that these structural properties will have on the mode frequencies, which do not encode any such transition. We show analytically that a sufficiently near-core perturbation to the stellar structure results in nonoscillatory, degree-dependent perturbations to the star’s oscillation mode frequencies, contrasting with the case of an outer glitch. We also demonstrate numerically that these near-core acoustic glitches exhibit strong angular degree dependence, even at low degree, agreeing with the non-JWKB analysis, rather than the degree-independent oscillations that emerge from JWKB analyses. These properties have important implications for using p-modes to study near-core mixing processes for intermediate-mass stars on the main sequence, as well as for the interpretation of near-center acoustic glitches in other astrophysical configurations, such as red giants.