Open Access
Issue
A&A
Volume 677, September 2023
Article Number A68
Number of page(s) 9
Section Stellar structure and evolution
DOI https://doi.org/10.1051/0004-6361/202347196
Published online 06 September 2023

© The Authors 2023

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

Rotation is an important ingredient of stellar evolution models (Maeder 2009). However, our understanding of the physical processes inside stars induced by their internal rotation is marked by some lingering deficiencies. Thanks to asteroseismology, we have access to a tool for measuring the internal profile from so-called rotational splitting of a star’s non-radial mode frequencies (Ledoux 1951). This was first turned into practice for the pressure modes of the Sun (Deubner et al. 1979) and subsequently for gravity modes in a white dwarf (Winget et al. 1991), as well as for low-order pressure and gravity modes in a young massive B-type dwarf (Aerts et al. 2003; Dupret et al. 2004).

The study of internal stellar rotation became an established field of research once the photometric light curves assembled with the NASA Kepler telescope reached a sufficiently long duration to resolve rotationally split frequencies. This led to estimates for the core rotation frequency from split dipole mixed modes in red giants (e.g. Beck et al. 2012, 2014; Mosser et al. 2012; Deheuvels et al. 2012; Gehan et al. 2018) and in subgiants (e.g. Deheuvels et al. 2014, 2020). Rotationally split multiplets also allowed to deduce the near-core rotation frequency of main sequence stars (e.g. Kurtz et al. 2014; Saio et al. 2015; Moravveji et al. 2016; Schmid & Aerts 2016; Van Reeth et al. 2016; Li et al. 2019, 2020). A few more detections of internal rotation for multiple intermediate- and high-mass stars have been done with the BRITE constellation as well (e.g. Sowicka et al. 2017; Kallinger et al. 2017). The refurbished version of the Kepler project (K2) subsequently delivered internal rotation measurements of white dwarfs (Hermes et al. 2017), while those of subdwarfs are summarised in Charpinet et al. (2018). Applications of rotation inversions delivered the entire rotation profile inside various types of slowly rotating stars, instead of just the (near-) core and envelope values (Deheuvels et al. 2014, 2015; Triana et al. 2015, 2017; Di Mauro et al. 2016; Bazot et al. 2019). Aerts et al. (2019) provided an overarching summary of all internal rotation measurements and discussed them in terms of stellar evolution and angular momentum transport mechanisms.

A particularly challenging case to measure internal stellar rotation from the splitting of nonradial mode frequencies occurs when the modes have frequency values comparable to the frequency of rotation. In such a case, the Coriolis acceleration cannot be treated as a small perturbative effect with respect to the other acting forces in the computations of the eigenmodes of the star. A proper treatment of the nonradial oscillation modes in such a situation can be done adopting the traditional approximation of rotation (TAR; Lee & Saio 1987, 1997; Townsend 2003; Mathis 2009; Bouabid et al. 2013). The TAR is an excellent approximation for modes with spin parameters s  ≡  2Ω/ω ≥ 1 with Ω the rotation frequency and ω the frequency of the mode. It is valid in the regime where 2Ω < N and ω ≪ N with N the Brunt-Väisälä (BV) frequency and in stars not flattened too much by the centrifugal acceleration (Ouazzani et al. 2017; Dhouib et al. 2021a,b; Henneco et al. 2021), provided that the horizontal displacement caused by the mode is dominant over the vertical one. This latter condition is fulfilled for gravity modes of stars in the core hydrogen burning phase of their evolution, as proven by long-term high-resolution time-series spectroscopy of slowly pulsating B (SPB) stars (Aerts et al. 1999; Mathias et al. 2001; De Cat & Aerts 2002; Briquet et al. 2003; De Cat et al. 2005) and γ Doradus (γ Dor) stars (Mathias et al. 2004; Aerts et al. 2004; De Cat et al. 2006). Space asteroseismology meanwhile showed that these conditions are also well met for most of the gravito-inertial and Rossby modes active in the rapidly rotating radiative envelope of intermediate- and high-mass pulsating dwarfs, revealing spin parameters roughly in the range s ∈ [10, 30] (Neiner et al. 2012a,b; Aerts et al. 2017; Saio et al. 2018).

Methods for measuring the internal rotation profile, Ω(r), of gravito-inertial mode pulsators have been developed since the four-year light curves assembled with the NASA Kepler space telescope allowed for the identification of series of such modes with consecutive radial order (Van Reeth et al. 2015, 2016, 2018; Ouazzani et al. 2017; Christophe et al. 2018; Li et al. 2019, 2020; Takata et al. 2020). Applications of these methods have led to rigorous and consistent estimates of the internal rotation frequency for hundreds of intermediate-mass stars (see Aerts 2021, for a summary). Most of the asteroseismic measurements deliver the internal rotation frequency at the bottom of the radiative envelope, where a boundary layer occurs between the convective core and the radiative envelope. The gravito-inertial mode kernels have their strongest probing power in that boundary layer (Van Reeth et al. 2016; Ouazzani et al. 2017; Pedersen et al. 2018; Michielsen et al. 2019, 2021; Mombarg et al. 2021; Vanlaer et al. 2023). For a fraction of the pulsators with gravito-inertial modes, an envelope or surface estimate of the rotation is also available from pressure modes and rotational modulation, respectively (Van Reeth et al. 2018; Li et al. 2020; Sekaran et al. 2021).

Gravity and super-inertial gravito-inertial modes with ω > 2Ω do not propagate in the convective core of dwarfs (e.g. Prat et al. 2016, 2018) and can hence not deliver their core rotation (e.g. Kurtz et al. 2014; Triana et al. 2015, for early attempts from rotation inversion for a γ Dor and SPB star, respectively). A major breakthrough on this front was achieved by Ouazzani et al. (2020), who came up with a way to determine the rotation in the convective core from inertial modes restored purely by the Coriolis acceleration. Their theoretical study showed convincingly that coupling between a pure inertial mode trapped in the convective core and a sub-inertial gravito-inertial mode (with ω < 2Ω) propagating in the surrounding radiative envelope may occur for models of rapidly rotating γ Dor stars. Ouazzani et al. (2020) found such mode coupling to result in a clear dip signature at particular frequencies in period spacing patterns of envelope gravito-inertial modes. Their theoretical study was inspired by the predictions of Saio et al. (2018) to explain observed period spacing patterns in the young rapidly rotating γ Dor pulsator KIC 5608334. To interpret the asteroseismic data of this star, Saio et al. (2018) compared predictions based on the TAR with computations taking full account of the Coriolis acceleration from the method developed by Lee & Baraffe (1995) and found the latter to reveal a dip structure in the period spacing diagram, while the TAR predictions do not.

The findings by Ouazzani et al. (2020) then led Saio et al. (2021) to revisit the Kepler data of γ Dor stars and study those with a single clean dip signal in their period spacing pattern from the perspective of coupling between inertial core modes and gravito-inertial envelope modes. They found 16 γ Dor stars with such mode coupling and deduced the rotation frequency in their convective core. This gives slightly faster rotation in the core than in the envelope at a level of 10% differentiality or less. They also found an anticorrelation between the level of differentiality and the evolutionary stage of the pulsators expressed as the hydrogen mass fraction of the core, Xc, which ranges from 0.7 to 0.2 for the 16 pulsators with mode coupling.

Following onto Ouazzani et al. (2020) and Saio et al. (2021), a deeper theoretical understanding of the dip structure in period spacing patterns of γ Dor stars and of their 10% level of differentiality between the core and envelope rotation was offered by Tokuno & Takata (2022). Their study also provides theoretical expressions for a dimensionless parameter, ε, which they assume to attain values between 0 and 1 and expressing a level of optimal circumstances for inertial modes in the core to couple to sub-inertial gravito-inertial modes in the envelope.

While a dip structure in mode period spacings due to coupling between inertial core modes and gravito-inertial envelope modes has been well established in several γ Dor stars, no such signature has been reported yet for SPB stars. However, the inner structure of γ Dor and SPB stars is similar in the sense that they have a well-developed convective core surrounded by an often rapidly rotating boundary layer that connects the core to a radiative envelope. The members of both classes of gravito-inertial pulsators cover the entire range from slow rotation to almost critical rotation. It is not yet understood why the members of these two classes of pulsators reveal period spacing patterns with somewhat different morphological properties. Indeed, while many of the γ Dor stars have long period spacing patterns involving tens of modes of consecutive radial order, often with just one dip or none at all (Van Reeth et al. 2015; Keen et al. 2015; Bedding et al. 2015; Li et al. 2019, 2020; Sekaran et al. 2021), most of the SPB stars have shorter patterns with oscillatory behaviour and/or multiple dips (Degroote et al. 2010; Pápics et al. 2012, 2014, 2017; Szewczuk & Daszyńska-Daszkiewicz 2015, 2018; Szewczuk et al. 2021; Pedersen et al. 2021). The detected periodicity in the dip pattern of slow to moderate rotators among the γ Dor and SPB stars is relatively well understood in terms of a strong μ-gradient (∇μ) due to a receding convective core as the star evolves, causing strong mode trapping (Kurtz et al. 2014; Saio et al. 2015; Schmid & Aerts 2016; Murphy et al. 2016; Pedersen et al. 2018; Michielsen et al. 2019, 2021; Li et al. 2019; Wu et al. 2020). This phenomenon caused by ∇μ, or by buoyancy glitches in general, gives a specific morphology to period spacing patterns that also occurs in the absence of fast rotation, as studied analytically by Miglio et al. (2008) and Cunha et al. (2019). However, some γ Dor and SPB stars reveal periodic and rather shallow dips intertwined with one or a few sharp dips, which may point to the simultaneous occurrence of mode trapping due to a strong μ-gradient and coupling between inertial and gravito-inertial modes.

Moreover, the presence of a core magnetic field may also give rise to a dip structure, even when the TAR is used (Prat et al. 2019, 2020; Van Beeck et al. 2020; Dhouib et al. 2022). Being able to unravel the physical causes of all the observed dips thus offers the future potential to detect and measure internal magnetic field profiles in addition to the rotation profiles and check if they are consistent with the predictions for the internal magnetic field strengths by Aerts et al. (2021). This requires a way to unravel the signature of mode coupling from the one due to ∇μ, in the presence of rapid rotation and possible magnetic fields when modeling observed dips. With this goal in mind, we aim to provide a numerical value for the coupling coefficient by relying on the best asteroseismic models of a sample of gravito-inertial pulsators consisting of both γ Dor and SPB stars. We wish to investigate if numerical values of the parameters ε and introduced by Tokuno & Takata (2022) from asteroseismic models provide a useful prediction about the occurrence or absence of core-to-envelope mode coupling. In particular, our goal is to investigate whether γ Dor stars and SPB stars have different or similar values for ε and . We will also test if the values of ε and indeed occur in the interval [0, 1] as assumed by Tokuno & Takata (2022) and whether they are correlated with typical properties of the convective core of γ Dor and SPB stars, which have quite different size and mass. Finally, we will also search for relationships between the coupling coefficients and the hydrogen mass fraction (as a good proxy for the evolutionary stage) or the properties of μ and ∇μ in the boundary layer between the convective core and the radiative envelope for both types of pulsators.

2. Asteroseismic inference of the coupling coefficient, ε

The theoretical work by Tokuno & Takata (2022) defined a new parameter, ε, capturing the level of opportunity for an interaction between a pure inertial mode in the rotating convective core and a sub-inertial gravito-inertial mode propagating in the rotating radiative envelope. We therefore call ε the “coupling coefficient” (Tokuno & Takata 2022 did not provide any terminology for this parameter). In their analytical expressions derived for ε, Tokuno & Takata (2022) relied on some assumptions, one being weak interaction between the oscillation modes in the core and envelope, such that 0 < ε ≪ 1. They found ε to decrease from 0.343 at stellar birth to 0.018 near the terminal age main sequence of a γ Dor star model of 1.5 M with solar metallicity and rotating at 2.2 d−1 (25.44 μHz) computed by Saio et al. (2021).

The theoretical work by Tokuno & Takata (2022) provided a physical understanding of observed mode couplings in γ Dor stars. It triggered our current study with the aim to derive values of ε for concrete gravito-inertial pulsators. Following up on Aerts et al. (2021), we compute such numerical values of ε for the 63 gravito-inertial pulsators in that study, relying on their best forward asteroseismic models. These were computed by Mombarg et al. (2021) for the 37 γ Dor stars and by Pedersen et al. (2021) for the 26 SPB stars using statistical methodology inspired by Aerts et al. (2018). While some additional g-mode pulsators have been modeled in the literature (see Aerts 2021, for a review), the adopted methods are too diverse to add them to the sample. Moreover, their numerical seismically calibrated models are not available to us. Thus, we restricted this study to the homogeneously treated sample of 63 pulsators already considered in Aerts et al. (2021). Two γ Dor stars in this sample reveal the envisioned coupling between an inertial core and a gravito-inertial envelope mode.

2.1. Internal profiles at the convective core boundary

We first recall the ingredients and definition of ε for the two extreme cases considered by Tokuno & Takata (2022), namely a continuous profile of the hydrogen mass fraction at the convective core boundary versus a discontinuity at that boundary resulting in a sharp spike of the BV frequency N(r). Figure 1 shows the profiles of the hydrogen and helium mass fraction, mean molecular weight (μ) and its gradient (∇μ) for the 63 pulsators. Aerts et al. (2021) already included the profiles for N(r) in their Fig. 2. Here, we show ∇μ, which provides by far the largest contribution to N(r) in the boundary layer between the convective core and the radiative envelope. Indeed, many of the γ Dor and all of the SPB stars experience a receding convective core throughout their main sequence evolution, leaving behind a gradient in μ. This gradient is completely dominant over the contribution of the entropy gradient shown as dotted lines in the bottom panels of Fig. 1. This figure reveals that both smooth and abrupt profiles occur. Hence, we used both formulations for the coupling coefficient deduced by Tokuno & Takata (2022) and applied them to all the stars. The upper two panels of Fig. 1 show that the 26 SPB stars cover the entire main sequence, while the sample of 37 γ Dor stars does not contain stars so close to hydrogen exhaustion in the core.

thumbnail Fig. 1.

Profiles of hydrogen (full lines) and helium (dotted lines) mass fraction (top), of the mean molecular weight (middle), and of a zoom-in on its gradient in the area of the convective core (bottom) for the 37 γ Dor (left) and 26 SPB (right) stars. These profiles were retrieved from the best forward asteroseismic models computed by Mombarg et al. (2021) and by Pedersen et al. (2021), respectively.

Continuous hydrogen and helium mass fraction profiles at the core boundary correspond to continuous density profiles. Assuming a constant density in the convective core, which is a good approximation for the case of continuous profiles, Tokuno & Takata (2022) derived the following expression for the coupling coefficient ε for a mode with frequency, ω :

(1)

where ra is defined as the inner boundary of the envelope mode propagation cavity where ω equals the BV frequency N. This equation shows that the opportunity of mode coupling increases with increasing rotation frequency and decreases with the steepness of the wall caused by the stratification. In practice ra is close to the radius of the convective core (denoted here as Rcc), which is the position where N2 becomes negative towards the stellar centre. In the entire inner region where N2 < 0, all the material is homogeneously mixed, such that ∇μ = 0 (cf. bottom panels of Fig. 1). However, ra is determined from the outer side towards the core, hence it depends on the properties of the boundary layer between the convective core and the envelope. The structure of this layer is still uncertain and asteroseismology has been used to try and unravel its chemical (Pedersen et al. 2018) and temperature (Michielsen et al. 2019) properties. Forward asteroseismic modeling of 26 SPB stars by Pedersen et al. (2021), Pedersen (2022) pointed out that convective penetration with an adiabatic temperature gradient and full chemical mixing in the boundary layer is preferred above a radiative temperature gradient accompanied by a smooth diffusive mixing profile for somewhat over half of the stars. Hence, this sample study of 26 pulsators showed that there is no unified best way to describe the temperature profile for all SPB pulsators. For this reason, Michielsen et al. (2021) dissected the chemical mixing profile and the temperature gradient in the boundary layer of one of these 26 SPB pulsators in more detail. They compared models with a radiative temperature gradient with those having a gradual transition between an adiabatic and radiative gradient, where the transition is based on the Péclet number. The latter type of gradient is inspired by numerical simulations (Viallet et al. 2013) and compares the ratio of advective versus diffusive transport in the boundarylayer between a convection and radiative zone. Michielsen et al. (2021) selected the SPB with the longest period spacing pattern to test their more detaild physical description of chemical mixing in the boundary layer, stitching a penetrative and diffusive mixing profile and assessing the nature of the temperature gradient. This turned out to be a challenging task, given the dominance of ∇μ over the temperature gradient in that layer (cf. Fig. 1). They found KIC 7760680, which rotates at 25% of its critical rate, to reveal a fully radiative rather than Péclet-based temperature gradient in their models with convective boundary mixing. Unraveling the temperature and mixing profiles for γ Dor pulsators is even harder than in the case of SPB stars (Mombarg et al. 2019) because the effects of microscopic atomic diffusion cannot be ignored for these pulsators. Moreover, the role of radiative levitation for element mixing is unclear when treated without incorporating rotational mixing in the modeling (Mombarg et al. 2020). Although they are computationally intense, predictions of oscillation mode properties that take into account the joint effects of radiative levitation and rotation offer a promising route to bring the asteroseismic probing power of the boundary layer of γ Dor pulsators to the next level (Mombarg et al. 2022).

For the case of discontinuous profiles of the hydrogen mass fraction and of N2 at the convective core boundary, Tokuno & Takata (2022) introduced an alternative expression for the coupling coefficient, while also omitting the assumption of a constant density, namely:

(2)

where the limit has to be taken from the radiative envelope towards the convective core, that is, going from the envelope outside of the core towards the centre of the star.

In the mathematical limit of dN2/dr → ∞, one gets ε → 0 preventing mode coupling even for the case of continuous profiles. On the other hand, for strictly discontinuous profiles resulting in a steep BV frequency spike, one would get N0 → ∞ and hence , again excluding mode coupling. We evaluate the difference between the results from both expressions in Eqs. (1) and (2) in the next section.

2.2. Inferred values of ε and

The numerical values for ε and are plotted as circles and squares, respectively, in Fig. 2 as a function of the near-core rotation frequency Ω taken from Aerts et al. (2021). First, we obtained values for ε and between 0 and 1, as assumed by Tokuno & Takata (2022). The numerical stellar evolution models computed by Mombarg et al. (2021) and Pedersen et al. (2021) have a dense mesh for their core boundary layers. This is necessary for proper computation of the eigenmodes as they have high radial order and thus many nodes need to get resolved for proper numerical approximation of the displacement vector. These dense meshes give us good evaluation capacity for the derivative in Eq. (1) from its linearised algebraic-differential version. Since we here consider the general case of inertial modes in the core with a range of frequencies ω, we compute the gradient from the value of N2 and r in the three cells closest to the core boundary defined by N2(r) < 0 and retain the largest value of ε among the three, as a good representation of maximal mode coupling in the case of a continuous BV profile.

thumbnail Fig. 2.

Values for ε in the case of a continuous (circles) and for in the limit of a discontinuous (squares) BV profile plotted as a function of the near-core rotation frequency. The γ Dor and SPB stars are indicated in red and blue, respectively. The two γ Dor stars for which Saio et al. (2021) detected a signature of coupling between a core and envelope mode are overplotted with black symbols, namely, KIC 11907454 (cross) and KIC 12066947 (plus). Linear correlation coefficients, rc and rd, between ω and ε, respectively , are listed for the two samples.

The dotted lines in Fig. 2 indicate the range covered by ε and per star, connecting the values obtained for the two approximations expressed by Eqs. (1) and (2), which can be considered as representing the two extreme cases of reality. For some stars the two estimates are very similar but for others the difference between ε and is large. This difference depends entirely on the shape of the ∇μ profile near the convective core boundary.

The strongest potential coupling between the modes in the core and envelope systematically occurs for ε computed from Eq. (1), as shown by comparing the circles and squares per star in Fig. 2. The values we get for ε are according to the theoretical expectations in Tokuno & Takata (2022). We find that is close to zero for many stars. Low values for both approximations of the coupling coefficient occur for a considerable fraction of the SPB pulsators, whose ∇μ value is often some ten times larger than those for γ Dor stars, with a sharp drop towards the convective core. A major conclusion thus is that mode coupling is harder to establish for SPB than for γ Dor stars. As discussed in the previous section, this is in agreement with Kepler space photometry, where mode coupling between core and envelope is not yet reported for any of the SPB stars, while it was found in various of the γ Dor stars and interpreted as such by Saio et al. (2021). We refer to the latter paper for an extensive discussion and illustrations of the delicate balance between the local rotation and two pulsation frequencies of the two involved modes and their eigenmode properties for the mode coupling to become effective.

Saio et al. (2021) found two of the γ Dor stars in our sample to have a dip in the period spacing pattern that is characteristic of the coupling between a core inertial mode and an envelope gravito-inertial mode, namely, KIC 11907454 and KIC 12066947. These have a relatively high ε value and are among the most rapid rotators, as indicated in Fig. 2.

The lack of any mode coupling detection for SPB stars may simply be due to an observational bias, as there are a factor 20 less such stars with period spacing patterns from space photometry compared to γ Dor stars. The SPB star KIC 8255796 may be the best candidate, as it has a typical Lorentz-shape dip in its period spacing pattern (Pedersen et al. 2021, see Fig. 15 in the supplementary material). However, its pattern is among the shortest ones, constituting of only nine identified dipole modes. This SPB has a mass of 5.7 M and a relatively slowly rotating near-core boundary layer (some 19% of the star’s critical rate). Moreover, it has a low ε = 0.0118 and is the most evolved of all the 26 SPB stars in the sample while, as we will show below, we expect the coupling capacity to be weakest in that stage of the main sequence. Therefore, we consider it unlikely for its observed dip structure to be due to mode coupling.

Since we wish to understand how mode coupling between core and envelope modes comes about, we now investigate relationships between ε and with respect to other parameters characterizing the convective core. Following their definitions in Eqs. (1) and (2), a positive correlation occurs with Ω; namely, higher rotation rates lead to higher coupling coefficients. This is indeed found for both ε and (shown in Fig. 2), where the linear correlation coefficients for the continuous and discontinuous cases are denoted as rc and rd, respectively. They are almost equal (rc = 0.64 and rd = 0.66) for the SPB pulsators, while the relation is more pronounced for the continuous case of the γ Dor star models (rc = 0.89 versus rd = 0.51).

2.3. Relation with stellar core parameters

Figure 3 shows a fairly strong positive correlation between ε or and the relative size of the convective core for SPB stars. The trend also occurs for ε of γ Dor stars but with a lower rc value – yet it is absent for . This difference can be understood from the properties of the fractional radius among the two samples. Indeed, the γ Dor stars cover a narrower range in Rcc/R (with R the radius of the star) than the SPB stars, implying a weaker correlation between ε and convective core size than for the SPB stars. Moreover, γ Dor stars cover a range in stellar mass (M) such that some stars in the sample exhibit a growing convective core with a steep BV profile during the core hydrogen burning, while others have a receding core and a smoother varying BV frequency. This mixed behaviour in terms of convective core size evolution makes it intrinsically less obvious to have a tight correlation between ε and Rcc and explains the absence of any relation with . The two γ Dor stars with detected mode coupling do not stand out in terms of their Rcc/R value.

thumbnail Fig. 3.

Same as Fig. 2, but plotted as a function of the extent of the convective core, expressed as a fraction of the total radius.

For both samples, the convective core mass (expressed as relative fraction of the total stellar mass) has a lower linear correlation with respect to ε than the fractional radius of the convective core and the correlation essentially disappears for the γ Dor stars (see Fig. 4). Again, this is to be expected as the γ Dor sample covers stars with a growing and decreasing convective core mass. For both samples rc is lower for Mcc than for Rcc because the amount of hydrogen that gets injected into the core at the expense of CNO products is more dependent on the cause and character of the overshoot and envelope mixing than the size of the core. We also checked for correlations between ε and the values of Rcc or Mcc themselves instead of their fractional dimensionless values, but this leads to even lower rc and rd values than those listed in Figs. 3 and 4.

thumbnail Fig. 4.

Same as Fig. 2, but plotted as a function of the mass in the convective core, expressed as a fraction of the total stellar mass.

Both Rcc/R and Mcc/M evolve as the core hydrogen burning progresses. The manner in which they do depends greatly on the level and character of the internal mixing in the core boundary layer and in the envelope (Moravveji et al. 2015, 2016; Mombarg et al. 2019, 2021; Pedersen et al. 2021; Pedersen 2022). Thus, a comparison between ε or and the evolutionary stage is also of interest. Figure 5 reveals the relationships between either ε or and the central core hydrogen mass fraction, Xc, with respect to the initial value at birth. This fraction represents the stage of the main sequence evolution. We find a relatively mild trend that mode coupling in the continuous BV case is more likely in the early stages of stellar evolution. This trend is similar to the one found for Rcc/R, that is somewhat tighter for the SPB stars than for the γ Dor stars. This is in line with the fact that the convective core of SPB stars decreases monotonically from birth to central hydrogen exhaustion for the entire sample, while this is not the case for the γ Dor sample. No connection was found between and the evolutionary stage for γ Dor stars, while ε and behave similarly with respect to the evolutionary stage for the SPB class.

thumbnail Fig. 5.

Same as Fig. 2, but plotted as a function of the main sequence stage, defined as the current core hydrogen mass fraction divided by the initial hydrogen mass fraction at birth.

The chemical evolution of the star is roughly captured by the value of the mean molecular weight in the envelope versus that in the core, μe/μc. This ratio is 1 at the star’s birth and decreases as the chemical evolution of the star continues. However, in contrast to the evolutionary stage shown in Fig. 5, μe/μc depends directly on the effect of internal mixing in the radiative envelope. We show the relationship between ε or and μe/μc in Fig. 6. This gives the same picture as the one seen for the main sequence stage.

thumbnail Fig. 6.

Same as Fig. 2, but plotted as a function of μe/μc.

2.4. On the Schönberg-Chandrasekhar limiting mass

A similar yet slightly different way of looking at the evolutionary aspect of the mode coupling coefficients is via the so-called Schönberg-Chandrasekhar limiting mass, MSC (Schönberg & Chandrasekhar 1942). This quantity is formally defined as the maximum mass a helium core can have after hydrogen exhaustion in the core in order to remain inert, that is to withstand the pressure of the encompassing stellar envelope without contracting. When the value of MSC is surpassed, the helium core will start to shrink and the star will evolve on a fast contraction time scale rather than on a slow nuclear time scale.

An analytical expression for MSC has been deduced from the virial theorem in the case of non-rotating polytropic stellar models with an isothermal helium core (Stein 1966; Cox & Giuli 1968). Despite the fact that real stars do not adhere to a polytropic equation of state, the approximation for the Schönberg-Chandrasekhar limit deduced from numerical stellar models was found to be essentially the same as for the polytropic approximation (Kippenhahn et al. 2013):

(3)

Maeder (1971) investigated the effect of uniform rotation on MSC, concluding that its value does not change appreciably. It decreases due to rotation by at most 3%, even for fast rotation close to the critical value. It is thus meaningful to compute the current value of MSC/M from Eq. (3) for our sample of rapidly rotating gravito-inertial mode pulsators, some of which are close to hydrogen exhaustion in their core (Fig. 5). Figure 7 shows the relation between ε or and the quadratic dependence on μe/μc via the computed value of MSC from the asteroseismically calibrated μ profiles shown in Fig. 1.

thumbnail Fig. 7.

Same as Fig. 2, but plotted as a function of the Schönberg-Chandrasekhar core mass limit.

The correlation between MSC/M and ε plotted in Fig. 7 is obviously consistent with the one displayed for the linear dependence, μe/μc, in Fig. 6. The obtained numerical values for MSC/M in the current evolutionary stage of the stars computed from Eq. (3) resemble those of the fractional convective core mass used in Fig. 4. The expression in Eq. (3) was deduced for standard stellar models at core hydrogen exhaustion, irrespective of the kind and level of mixing that the star underwent during the main sequence. While Maeder (1971) found that rotation hardly affects it, the chemical evolution of the star due to near-core boundary mixing (Michielsen et al. 2021) and envelope mixing (Pedersen et al. 2021) does play a role in the behaviour of μ. We therefore compare the asteroseismically inferred values for Mcc/M at the current evolutionary stage of the stars with the current value of MSC/M computed via Eq. (3).

Our sample contains stars with masses between 1.38 M and 9.52 M. This lower limit is close to the value where stars transition from being below to above their MSC limit at the hydrogen exhaustion. As long as the helium core mass remains below MSC, the core can stay inert at hydrogen exhaustion and the subsequent hydrogen shell burning happens on a nuclear timescale. If, in contrast, the helium core mass exceeds MSC, it will start to contract while hydrogen burns in a shell and so this stage of evolution happens on a much shorter contraction time scale. For this reason, the value of MSC is an important quantity for the star’s evolution. We compare the current values of Mcc/M and MSC/M according to Eq. (3) for our 63 sample stars in Fig. 8, where the symbols are linearly scaled with the total stellar mass. During their evolution, stars evolve from the right to the left and those with a shrinking convective core also from the top to the bottom in such a diagram. It can be seen that all of the γ Dor stars have a convective core mass unrelated to and below their Schönberg-Chandrasekhar limit. They will steadily evolve almost horizontally to the left in the figure as they approach hydrogen depletion, because they hardly experience envelope mixing and thus keep their μe unchanged. Since none of them are close to hydrogen exhaustion and given their mass range, it is not expected that they would have already surpassed their MSC value.

thumbnail Fig. 8.

Convective core mass versus Schönberg-Chandrasekhar mass limit expressed as a fraction of the total mass. The γ Dor and SPB stars are indicated in red and blue symbols, respectively. The symbol size scales linearly with the total stellar mass, within the two extreme masses occurring in the samples, as indicated in the legend.

The more massive SPB stars, on the other hand, have convective cores tightly correlated with the evolution of their MSC/M limit, already surpassing that limit on the main sequence for the more massive sample stars as expected. For completeness, we also used the new fourth-degree polynomial fit for MSC proposed by Chowdhury & Sarkar (2023, Eq. (15) in their manuscript) instead of the second-degree formula in Eq. (3). This gives the same results as those in Fig. 8 for the linear correlation coefficients of rc and rd and for the plot to remain within the symbol sizes.

Pedersen (2022) predicted the values of the helium core masses at core hydrogen exhaustion of the 26 SPB stars based on their current asteroseismic modes and Mcc values. She concluded that due to their levels of envelope mixing, most of these SPB stars will have higher helium core masses than those resulting from standard stellar evolution without extra mixing. Her results are in line with convective core mass estimations from massive eclipsing binaries (Tkachenko et al. 2020; Johnston 2021). These binary and asteroseismic results for higher-than-standard convective core masses have not yet been taken into account in chemical yield computations guiding the overall chemical evolution models of galaxies (Karakas & Lattanzio 2014; Kobayashi et al. 2020). However, the higher levels of core masses that have been measured will have a major impact on yield prediction models, given that the uncertainties of such models for intermediate-mass stars mainly come from unknown internal mixing.

3. Discussion and future prospects

In this paper, we provide numerical values for the coupling coefficients, ε and , which are unit-less measures below value 1 indicative of the opportunity to have inertial modes in rapidly rotating convective cores couple to gravito-inertial envelope modes. We deduced numerical values for ε and from forward asteroseismic models of 63 gravity-mode pulsators, covering a mass range from 1.38 M to 9.52 M. Our findings are in agreement with the theoretical expectations and interpretations recently proposed by Tokuno & Takata (2022). From the perspective of the ε or values with respect to the near-core rotation frequency, the opportunity for mode coupling is similar for SPB and γ Dor star models. In practice, however, the sample of the few tens of SPB stars with identified modes from period spacing patterns available in the literature contains slower rotators than many of the γ Dor stars in the sample of published pulsators with proper diagnostic gravity-mode patterns, which is 20 times larger (see Aerts 2021, for an overview).

We find that the inferred values of ε or offer a useful diagnostic to hunt for core-to-envelope mode coupling. Yet the values of ε and alone do not allow us to distinguish among the few gravito-inertial pulsators with a core mode coupled to a gravito-inertial mode from the majority of them that do not reveal this coupling phenomenon. The values of ε and for the two γ Dor stars included in our sample of 37 do not stand out from the other rapid rotators in terms of their core properties deduced from forward modeling of their identified gravito-inertial modes. A visual inspection of characteristic dips in the period spacing patterns as done by Saio et al. (2021) thus remains the best (and currently only) way to find pulsators with coupled inertial core modes. These selected targets then allow for the derivation of the rotation frequency in their convective core via the matching of the frequencies of the inertial and gravito-inertial modes, given the near-core rotation frequency measured from the tilt of their period spacing pattern(s). Such type of core rotation measurement was first proposed by Ouazzani et al. (2020) and further elaborated upon by Saio et al. (2021). This is currently the best way to constrain the internal rotation profile from gravity-mode pulsators, given the challenges encountered for frequency inversions for this type of pulsators caused by nonlinear effects, notably the dense occurrence of avoided crossings (Vanlaer et al. 2023).

The observational challenge remains to distinguish dips in the period spacing patterns stemming from inertial mode coupling rather than from periodic deviations induced by mode trapping at the bottom of the envelope due to a strong μ-gradient in that position, because both phenomena occur simultaneously. This also explains the current absence of any mode coupling detection in SPB stars so far, given their much larger ∇μ value and more extended μ-gradient zone in the near-core boundary layer compared to the one of γ Dor stars (cf. Fig. 1). On the other hand, this observational challenge is in agreement with the population statistics. Indeed, we currently have about 28 SPB stars with period spacing patterns and none of them have a convincing dip structure that would be expected for inertial mode coupling. Saio et al. (2021) found 16 of the current sample of 670 γ Dor stars to have the signature (2.4%). While it concerns a low number of stars, these are crucial targets to map the internal rotation (and possibly the magnetic field, see e.g. Van Beeck et al. 2020; Dhouib et al. 2022) of intermediate- and high-mass stars. In this respect, the potential of the TESS mission has yet to be explored. Indeed, the 61 TESS γ Dor and 2 SPB stars with detected period spacing patterns found by Garcia et al. (2022a,b) are promising in this respect, but their period-spacing patterns from 352 d light curves are too short to offer proper dip structures caused by mode coupling or mode trapping. However, progress can be ensured by analyzing the thousands of light curves for the SPB and γ Dor candidates from TESS data assembled throughout the nominal and extended mission, covering more than five years. The catalogues from Pedersen et al. (2019), Antoci et al. (2019), and Skarka et al. (2022) are only the tip of the iceberg in discovery space for TESS gravito-inertial asteroseismology.

Acknowledgments

The research leading to these results has received funding from the KU Leuven Research Council (grant C16/18/005: PARADISE). C.A. acknowledges financial support from the Research Foundation Flanders (FWO) under grant K802922N (sabbatical leave); she is grateful for the kind hospitality offered by CEA/Saclay during her sabbatical work visits in the spring of 2023. The authors are grateful to May Gade Pedersen and Joey Mombarg for providing their forward asteroseismic models in electronic format, to Alex Kemp for valuable comments on the manuscript prior to its submission, and to the referee for constructive comments and encouragements to expand the manuscript with more details.

References

  1. Aerts, C. 2021, Rev. Mod. Phys., 93, 015001 [Google Scholar]
  2. Aerts, C., De Cat, P., Peeters, E., et al. 1999, A&A, 343, 872 [NASA ADS] [Google Scholar]
  3. Aerts, C., Thoul, A., Daszyńska, J., et al. 2003, Science, 300, 1926 [Google Scholar]
  4. Aerts, C., Cuypers, J., De Cat, P., et al. 2004, A&A, 415, 1079 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  5. Aerts, C., Van Reeth, T., & Tkachenko, A. 2017, ApJ, 847, L7 [Google Scholar]
  6. Aerts, C., Molenberghs, G., Michielsen, M., et al. 2018, ApJS, 237, 15 [Google Scholar]
  7. Aerts, C., Mathis, S., & Rogers, T. M. 2019, ARA&A, 57, 35 [Google Scholar]
  8. Aerts, C., Augustson, K., Mathis, S., et al. 2021, A&A, 656, A121 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  9. Antoci, V., Cunha, M. S., Bowman, D. M., et al. 2019, MNRAS, 490, 4040 [Google Scholar]
  10. Bazot, M., Benomar, O., Christensen-Dalsgaard, J., et al. 2019, A&A, 623, A125 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  11. Beck, P. G., Montalban, J., Kallinger, T., et al. 2012, Nature, 481, 55 [Google Scholar]
  12. Beck, P. G., Hambleton, K., Vos, J., et al. 2014, A&A, 564, A36 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  13. Bedding, T. R., Murphy, S. J., Colman, I. L., & Kurtz, D. W. 2015, Eur. Phys. J. Web Conf., 101, 01005 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  14. Bouabid, M. P., Dupret, M. A., Salmon, S., et al. 2013, MNRAS, 429, 2500 [Google Scholar]
  15. Briquet, M., Aerts, C., Mathias, P., Scuflaire, R., & Noels, A. 2003, A&A, 401, 281 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  16. Charpinet, S., Giammichele, N., Zong, W., et al. 2018, Open Astron., 27, 112 [NASA ADS] [CrossRef] [Google Scholar]
  17. Chowdhury, S., & Sarkar, T. 2023, MNRAS, 523, 518 [NASA ADS] [CrossRef] [Google Scholar]
  18. Christophe, S., Ballot, J., Ouazzani, R. M., Antoci, V., & Salmon, S. J. A. J. 2018, A&A, 618, A47 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  19. Cox, J. P., & Giuli, R. T. 1968, Principles of Stellar Structure (New York: Gordon and Breach) [Google Scholar]
  20. Cunha, M. S., Avelino, P. P., Christensen-Dalsgaard, J., et al. 2019, MNRAS, 490, 909 [Google Scholar]
  21. De Cat, P., & Aerts, C. 2002, A&A, 393, 965 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  22. De Cat, P., Briquet, M., Daszyńska-Daszkiewicz, J., et al. 2005, A&A, 432, 1013 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  23. De Cat, P., Eyer, L., Cuypers, J., et al. 2006, A&A, 449, 281 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  24. Degroote, P., Aerts, C., Baglin, A., et al. 2010, Nature, 464, 259 [Google Scholar]
  25. Deheuvels, S., García, R. A., Chaplin, W. J., et al. 2012, ApJ, 756, 19 [Google Scholar]
  26. Deheuvels, S., Doğan, G., Goupil, M. J., et al. 2014, A&A, 564, A27 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  27. Deheuvels, S., Ballot, J., Beck, P. G., et al. 2015, A&A, 580, A96 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  28. Deheuvels, S., Ballot, J., Eggenberger, P., et al. 2020, A&A, 641, A117 [EDP Sciences] [Google Scholar]
  29. Deubner, F. L., Ulrich, R. K., Rhodes, E. J. J., 1979, A&A, 72, 177 [NASA ADS] [Google Scholar]
  30. Dhouib, H., Prat, V., Van Reeth, T., & Mathis, S. 2021a, A&A, 652, A154 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  31. Dhouib, H., Prat, V., Van Reeth, T., & Mathis, S. 2021b, A&A, 656, A122 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  32. Dhouib, H., Mathis, S., Bugnet, L., Van Reeth, T., & Aerts, C. 2022, A&A, 661, A133 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  33. Di Mauro, M. P., Ventura, R., Cardini, D., et al. 2016, ApJ, 817, 65 [NASA ADS] [CrossRef] [Google Scholar]
  34. Dupret, M. A., Thoul, A., Scuflaire, R., et al. 2004, A&A, 415, 251 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  35. Garcia, S., Van Reeth, T., De Ridder, J., & Aerts, C. 2022a, A&A, 668, A137 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  36. Garcia, S., Van Reeth, T., De Ridder, J., et al. 2022b, A&A, 662, A82 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  37. Gehan, C., Mosser, B., Michel, E., Samadi, R., & Kallinger, T. 2018, A&A, 616, A24 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  38. Henneco, J., Van Reeth, T., Prat, V., et al. 2021, A&A, 648, A97 [EDP Sciences] [Google Scholar]
  39. Hermes, J. J., Gänsicke, B. T., Kawaler, S. D., et al. 2017, ApJS, 232, 23 [Google Scholar]
  40. Johnston, C. 2021, A&A, 655, A29 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  41. Kallinger, T., Weiss, W. W., Beck, P. G., et al. 2017, A&A, 603, A13 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  42. Karakas, A. I., & Lattanzio, J. C. 2014, PASA, 31, e030 [NASA ADS] [CrossRef] [Google Scholar]
  43. Keen, M. A., Bedding, T. R., Murphy, S. J., et al. 2015, MNRAS, 454, 1792 [Google Scholar]
  44. Kippenhahn, R., Weigert, A., & Weiss, A. 2013, Stellar Structure and Evolution (Heidelberg: Springer Verlag) [Google Scholar]
  45. Kobayashi, C., Karakas, A. I., & Lugaro, M. 2020, ApJ, 900, 179 [Google Scholar]
  46. Kurtz, D. W., Saio, H., Takata, M., et al. 2014, MNRAS, 444, 102 [Google Scholar]
  47. Ledoux, P. 1951, ApJ, 114, 373 [Google Scholar]
  48. Lee, U., & Baraffe, I. 1995, A&A, 301, 419 [NASA ADS] [Google Scholar]
  49. Lee, U., & Saio, H. 1987, MNRAS, 224, 513 [Google Scholar]
  50. Lee, U., & Saio, H. 1997, ApJ, 491, 839 [Google Scholar]
  51. Li, G., Bedding, T. R., Murphy, S. J., et al. 2019, MNRAS, 482, 1757 [Google Scholar]
  52. Li, G., Van Reeth, T., Bedding, T. R., et al. 2020, MNRAS, 491, 3586 [Google Scholar]
  53. Maeder, A. 1971, A&A, 14, 351 [NASA ADS] [Google Scholar]
  54. Maeder, A. 2009, Physics, Formation and Evolution of Rotating Stars (Heidelberg: Springer Verlag) [Google Scholar]
  55. Mathias, P., Aerts, C., Briquet, M., et al. 2001, A&A, 379, 905 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  56. Mathias, P., Le Contel, J. M., Chapellier, E., et al. 2004, A&A, 417, 189 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  57. Mathis, S. 2009, A&A, 506, 811 [CrossRef] [EDP Sciences] [Google Scholar]
  58. Michielsen, M., Pedersen, M. G., Augustson, K. C., Mathis, S., & Aerts, C. 2019, A&A, 628, A76 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  59. Michielsen, M., Aerts, C., & Bowman, D. M. 2021, A&A, 650, A175 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  60. Miglio, A., Montalbán, J., Noels, A., & Eggenberger, P. 2008, MNRAS, 386, 1487 [Google Scholar]
  61. Mombarg, J. S. G., Van Reeth, T., Pedersen, M. G., et al. 2019, MNRAS, 485, 3248 [NASA ADS] [CrossRef] [Google Scholar]
  62. Mombarg, J. S. G., Dotter, A., Van Reeth, T., et al. 2020, ApJ, 895, 51 [Google Scholar]
  63. Mombarg, J. S. G., Van Reeth, T., & Aerts, C. 2021, A&A, 650, A58 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  64. Mombarg, J. S. G., Dotter, A., Rieutord, M., et al. 2022, ApJ, 925, 154 [NASA ADS] [CrossRef] [Google Scholar]
  65. Moravveji, E., Aerts, C., Pápics, P. I., Triana, S. A., & Vandoren, B. 2015, A&A, 580, A27 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  66. Moravveji, E., Townsend, R. H. D., Aerts, C., & Mathis, S. 2016, ApJ, 823, 130 [NASA ADS] [CrossRef] [Google Scholar]
  67. Mosser, B., Goupil, M. J., Belkacem, K., et al. 2012, A&A, 548, A10 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  68. Murphy, S. J., Fossati, L., Bedding, T. R., et al. 2016, MNRAS, 459, 1201 [Google Scholar]
  69. Neiner, C., Floquet, M., Samadi, R., et al. 2012a, A&A, 546, A47 [CrossRef] [EDP Sciences] [Google Scholar]
  70. Neiner, C., Mathis, S., Saio, H., et al. 2012b, A&A, 539, A90 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  71. Ouazzani, R.-M., Salmon, S. J. A. J., Antoci, V., et al. 2017, MNRAS, 465, 2294 [Google Scholar]
  72. Ouazzani, R. M., Lignières, F., Dupret, M. A., et al. 2020, A&A, 640, A49 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  73. Pápics, P. I., Briquet, M., Baglin, A., et al. 2012, A&A, 542, A55 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  74. Pápics, P. I., Moravveji, E., Aerts, C., et al. 2014, A&A, 570, A8 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  75. Pápics, P. I., Tkachenko, A., Van Reeth, T., et al. 2017, A&A, 598, A74 [Google Scholar]
  76. Pedersen, M. G. 2022, ApJ, 930, 94 [NASA ADS] [CrossRef] [Google Scholar]
  77. Pedersen, M. G., Aerts, C., Pápics, P. I., & Rogers, T. M. 2018, A&A, 614, A128 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  78. Pedersen, M. G., Chowdhury, S., Johnston, C., et al. 2019, ApJ, 872, L9 [Google Scholar]
  79. Pedersen, M. G., Aerts, C., Pápics, P. I., et al. 2021, Nat. Astron., 5, 715 [NASA ADS] [CrossRef] [Google Scholar]
  80. Prat, V., Lignières, F., & Ballot, J. 2016, A&A, 587, A110 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  81. Prat, V., Mathis, S., Augustson, K., et al. 2018, A&A, 615, A106 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  82. Prat, V., Mathis, S., Buysschaert, B., et al. 2019, A&A, 627, A64 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  83. Prat, V., Mathis, S., Neiner, C., et al. 2020, A&A, 636, A100 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  84. Saio, H., Kurtz, D. W., Takata, M., et al. 2015, MNRAS, 447, 3264 [Google Scholar]
  85. Saio, H., Kurtz, D. W., Murphy, S. J., Antoci, V. L., & Lee, U. 2018, MNRAS, 474, 2774 [Google Scholar]
  86. Saio, H., Takata, M., Lee, U., Li, G., & Van Reeth, T. 2021, MNRAS, 502, 5856 [Google Scholar]
  87. Schmid, V. S., & Aerts, C. 2016, A&A, 592, A116 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  88. Schönberg, M., & Chandrasekhar, S. 1942, ApJ, 96, 161 [CrossRef] [Google Scholar]
  89. Sekaran, S., Tkachenko, A., Johnston, C., & Aerts, C. 2021, A&A, 648, A91 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  90. Skarka, M., Žák, J., Fedurco, M., et al. 2022, A&A, 666, A142 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  91. Sowicka, P., Handler, G., Dębski, B., et al. 2017, MNRAS, 467, 4663 [Google Scholar]
  92. Stein, R. 1966, Stellar Evolution, Proceedings (New York: Plenum Press) [Google Scholar]
  93. Szewczuk, W., & Daszyńska-Daszkiewicz, J. 2015, MNRAS, 453, 277 [CrossRef] [Google Scholar]
  94. Szewczuk, W., & Daszyńska-Daszkiewicz, J. 2018, MNRAS, 478, 2243 [Google Scholar]
  95. Szewczuk, W., Walczak, P., & Daszyńska-Daszkiewicz, J. 2021, MNRAS, 503, 5894 [NASA ADS] [CrossRef] [Google Scholar]
  96. Takata, M., Ouazzani, R. M., Saio, H., et al. 2020, A&A, 635, A106 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  97. Tkachenko, A., Pavlovski, K., Johnston, C., et al. 2020, A&A, 637, A60 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  98. Tokuno, T., & Takata, M. 2022, MNRAS, 514, 4140 [NASA ADS] [CrossRef] [Google Scholar]
  99. Townsend, R. H. D. 2003, MNRAS, 340, 1020 [Google Scholar]
  100. Triana, S. A., Moravveji, E., Pápics, P. I., et al. 2015, ApJ, 810, 16 [Google Scholar]
  101. Triana, S. A., Corsaro, E., De Ridder, J., et al. 2017, A&A, 602, A62 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  102. Van Beeck, J., Prat, V., Van Reeth, T., et al. 2020, A&A, 638, A149 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  103. Van Reeth, T., Tkachenko, A., Aerts, C., et al. 2015, ApJS, 218, 27 [Google Scholar]
  104. Van Reeth, T., Tkachenko, A., & Aerts, C. 2016, A&A, 593, A120 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  105. Van Reeth, T., Mombarg, J. S. G., Mathis, S., et al. 2018, A&A, 618, A24 [NASA ADS] [EDP Sciences] [Google Scholar]
  106. Vanlaer, V., Aerts, C., Bellinger, E. P., & Christensen-Dalsgaard, J. 2023, A&A, 675, A17 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
  107. Viallet, M., Meakin, C., Arnett, D., & Mocák, M. 2013, ApJ, 769, 1 [NASA ADS] [CrossRef] [Google Scholar]
  108. Winget, D. E., Nather, R. E., Clemens, J. C., et al. 1991, ApJ, 378, 326 [NASA ADS] [CrossRef] [Google Scholar]
  109. Wu, T., Li, Y., Deng, Z.-M., et al. 2020, ApJ, 899, 38 [NASA ADS] [CrossRef] [Google Scholar]

All Figures

thumbnail Fig. 1.

Profiles of hydrogen (full lines) and helium (dotted lines) mass fraction (top), of the mean molecular weight (middle), and of a zoom-in on its gradient in the area of the convective core (bottom) for the 37 γ Dor (left) and 26 SPB (right) stars. These profiles were retrieved from the best forward asteroseismic models computed by Mombarg et al. (2021) and by Pedersen et al. (2021), respectively.

In the text
thumbnail Fig. 2.

Values for ε in the case of a continuous (circles) and for in the limit of a discontinuous (squares) BV profile plotted as a function of the near-core rotation frequency. The γ Dor and SPB stars are indicated in red and blue, respectively. The two γ Dor stars for which Saio et al. (2021) detected a signature of coupling between a core and envelope mode are overplotted with black symbols, namely, KIC 11907454 (cross) and KIC 12066947 (plus). Linear correlation coefficients, rc and rd, between ω and ε, respectively , are listed for the two samples.

In the text
thumbnail Fig. 3.

Same as Fig. 2, but plotted as a function of the extent of the convective core, expressed as a fraction of the total radius.

In the text
thumbnail Fig. 4.

Same as Fig. 2, but plotted as a function of the mass in the convective core, expressed as a fraction of the total stellar mass.

In the text
thumbnail Fig. 5.

Same as Fig. 2, but plotted as a function of the main sequence stage, defined as the current core hydrogen mass fraction divided by the initial hydrogen mass fraction at birth.

In the text
thumbnail Fig. 6.

Same as Fig. 2, but plotted as a function of μe/μc.

In the text
thumbnail Fig. 7.

Same as Fig. 2, but plotted as a function of the Schönberg-Chandrasekhar core mass limit.

In the text
thumbnail Fig. 8.

Convective core mass versus Schönberg-Chandrasekhar mass limit expressed as a fraction of the total mass. The γ Dor and SPB stars are indicated in red and blue symbols, respectively. The symbol size scales linearly with the total stellar mass, within the two extreme masses occurring in the samples, as indicated in the legend.

In the text

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