Open Access
Issue
A&A
Volume 668, December 2022
Article Number A137
Number of page(s) 16
Section Catalogs and data
DOI https://doi.org/10.1051/0004-6361/202244365
Published online 16 December 2022

© S. Garcia et al. 2022

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

After the Kepler satellite allowed us to probe the interiors of thousands of stars (Koch et al. 2010), it became clear that the theory of stellar evolution (e.g. Kippenhahn et al. 2012) still needs extra tuning to explain the observed internal stellar structures. Stellar astrophysics, since then, has experienced a fast proliferation of competing internal stellar processes that aim to explain the differences between theory and observation, notably in terms of angular momentum transport and chemical mixing (e.g. Rogers 2015; Moravveji et al. 2015, 2016; Fuller 2017; Townsend et al. 2018; Eggenberger et al. 2019; Ouazzani et al. 2019; Mombarg et al. 2022; Pedersen et al. 2021). Those processes are currently being evaluated to identify the dominant ones in different stellar classes and at different stages of stellar evolution. A critical factor in this research is the size of the samples of asteroseismically modelled pulsators in various evolutionary phases (e.g. Aerts et al. 2019; García & Ballot 2019; Córsico et al. 2019, for extensive reviews) and the amount of useful information per star. The available samples of dwarfs are currently too small to evaluate their complex and interacting processes of internal rotation and mixing, particularly for intermediate- and high-mass stars, with M > 1.4 M (Bowman 2020; Aerts 2021).

The ongoing Transiting Exoplanet Survey Satellite (TESS) mission has been providing the community with full-frame images, giving rise to hundreds of thousands of light curves of bright stars (Ricker et al. 2015). These TESS data may help to tackle the issue that the sample sizes of pulsating dwarfs are too small, with good asteroseismic modelling, which requires spectroscopy to get good fundamental parameters, while this is hard to establish for Kepler stars. As TESS continues to observe stars, the precision of measured pulsation mode frequencies increases. This will allow us to asteroseismically test future improvements of stellar structure and evolution theory from stellar pulsation computations. New methodological frameworks coupled to high-precision uninterrupted space photometric data of intermediate-mass gravity-mode pulsators have been developed, including those by Kurtz et al. (2014), Saio et al. (2015, 2021), Moravveji et al. (2015, Moravveji et al. 2016), Van Reeth et al. (2016, 2018), Ouazzani et al. (2017, 2019), Aerts et al. (2018), Szewczuk & Daszynska-Daszkiewicz (2018), Li et al. (2020), Pedersen et al. (2021), Michielsen et al. (2021), Bowman & Michielsen (2021), and Szewczuk et al. (2022). The powerful tools in these studies have been used to identify the stars with maximum asteroseismic scientific potential among the ever-increasing data provided by space missions.

The traditional approximation of rotation (TAR; Eckart 1960; Townsend 2003a; Mathis 2009) is an essential tool for performing gravity-mode asteroseismology of intermediate-mass stars, because they tend to be fast rotators, demanding the inclusion of the Coriolis acceleration into the treatment of the pulsation predictions. Ever more complex versions of the TAR, including the centrifugal and Lorentz forces, have been developed to allow for accurate predictions of gravito-inertial pulsation frequencies comparable with observations (Prat et al. 2019, 2020; Van Beeck et al. 2020; Henneco et al. 2021; Dhouib et al. 2021a,b, 2022). These various versions of the TAR inform us how gravity-mode pulsation frequencies of dwarfs are affected by rotation and magnetism, and allow us to search for such signatures in space photometry (Van Reeth et al. 2015a; Christophe et al. 2018; Szewczuk et al. 2021; Pedersen et al. 2021; Garcia et al. 2022). In terms of TESS data, Garcia et al. (2022) relied on theoretical predictions to create a catalogue of 106 gravito-inertial pulsators with detected period-pacing patterns affected by rotation. This follow-up paper extracts the asteroseismic information from that catalogue with the aim of constraining the interior rotation and buoyancy travel time in those stars, based on ~1 yr of uninterrupted space photometry. Our methodology is validated by analysing a well-studied sample of Kepler Gamma Doradus (hereafter γ Dor) stars, and comparing our results with those in the literature.

Period-spacing patterns are series of periods of (near-) consecutive g modes with identical surface geometry. The spacings between periods of modes with consecutive radial order are important observational diagnostics. They allow us to assess information contained in the pulsation mode periods and to couple this information to internal physical processes. The slope of the pattern is a reflection of the internal rotation (Bouabid et al. 2013; Van Reeth et al. 2016; Ouazzani et al. 2017), with the pulsators that have steeper patterns being the faster rotators. In addition, the structure in the patterns (by means of dips or wiggles) indicates the presence of trapped modes produced by strong internal gradients that are influenced by microscopic and/or macroscopic internal mixing (Miglio et al. 2008; Pedersen et al. 2018, 2021; Michielsen et al. 2019, 2021; Mombarg et al. 2022). These processes are poorly understood (Salaris & Cassisi 2017), but theoretical approximations of them based on the TAR and coupled to observed, identified gravito-inertial modes allow us to deduce the mixing and abundance profiles in the stellar interior, for a variety of global and local stellar parameters. Therefore, we concentrate our efforts on finding gravity- and gravito-inertial modes (hereafter, collectively referred to as g modes) in pulsators with period-spacing patterns from TESS data. In this way, we compose a TESS sample with the potential to derive the profiles of internal rotation and mixing inside γ Dor stars along the entire core-hydrogen burning phase.

Once the data are assembled, asteroseismic modelling is only as good as the identification of the pulsation modes involved in the period-spacing patterns (see Bowman & Michielsen 2021). Proper mode identification in fast rotators is a challenging problem. In this paper, we use a comprehensive approach to test the likelihood of possible low-degree mode identifications in the detected period-spacing patterns found by Garcia et al. (2022). The plan of the paper is as follows. Section 2 describes our methodology to determine the internal rotation and buoyancy travel time from period-spacing patterns. Section3 presents our asteroseismic catalogue of γ Dor stars, discusses some of their basic properties, and identifies the stars with maximum asteroseismic scientific potential. Section 4 characterises our catalogue with spectroscopic quantities. Section 5 validates our methodology by a re-analysis of a sample of Kepler. Finally, Sect. 6 presents our main conclusions.

2 Interior rotation and buoyancy travel time from g-mode period-spacing patterns

For a non-rotating, non-magnetic star, the surface geometry of a stellar pulsation mode is characterised by a spherical harmonic function with two quantum numbers: the angular degree and the azimuthal order m. Aside from and m, the radial order n correlates with the number of radial nodes of the displacement vector of the mode (e.g. Aerts et al. 2010). For such a simplified approximation of a star, pulsation periods of g modes in the asymptotic regime, that is, when the radial order n ≫ , depend on the buoyancy travel time across the mode cavity, as follows:

(1)

with N being the Brunt–Väisälä frequency defined in a cavity with inner and outer boundaries r1 and r2, respectively. It should be noted that Eq. (1) implicitly involves many details of the internal structure of the star, notably the local density p, gravity g, pressure p, temperature gradients ∇ ≡ (d ln T/d ln p), and its adiabatic version ∇ad, as well as the gradient of the mean molecular weight ∇µ (e.g. Aerts et al. 2010):

(2)

In this regime of stellar oscillations in a non-rotating, nonmagnetic star, its radial-order series of g-mode periods are equally spaced as

(3)

where αg is a phase term.

If the star is rotating, the pulsation frequencies are affected by the Coriolis and centrifugal forces. In the current work, we neglect the centrifugal force and assume rigid rotation with frequency frot. The effect of the Coriolis force on the g modes is described well by the TAR (Eckart 1960; Townsend 2003a; Mathis 2009). The geometry of stellar pulsations in rotating stars is characterised by the Hough functions, which are solutions to the Laplace tidal equations. Thus, the oscillation modes can no longer be described by spherical harmonics with and m. Rather, the eigenvalues associated with the Hough functions are λℓms, which depend on the spin parameter, defined as

(4)

with being the mode frequency in the co-rotating reference frame with rotation frequency frot. The TAR then predicts a radial-order series of g-mode periods as

(5)

In the inertial reference frame of an observer, this series is measured as (e.g. Van Reeth et al. 2016)

(6)

where modes with m > 0 and m < 0 are prograde and retrograde modes, respectively.

Corrections to theoretical predictions of g-mode periods due to magnetic fields, the centrifugal force, and differential rotation have been explored recently in Van Beeck et al. (2020), Dhouib et al. (2021b, 2022), and Henneco et al. (2021), relying on the theoretical frameworks developed by Prat et al. (2019, Prat et al. 2020) and Dhouib et al. (2021a). As with the Kepler data exploitation of g-mode pulsators so far, this paper will neglect these improvements of the TAR for a first exploration. Indeed, our aim is to assess the capability of TESS to exploit g-mode patterns. Moreover, while the additional pulsation frequency shifts caused by these neglected effects are formally detectable within the available space-based photometric observations, they are two orders of magnitude smaller than the pulsation frequency shifts caused by the Coriolis acceleration (Henneco et al. 2021; Dhouib et al. 2021b). Thus, we can safely limit this work to interpretations based on the simplified version of the TAR via Eq. (5). The neglected processes are not expected to significantly affect the obtained mode identifications or rotation rate measurements. This also allows for faster calculations for the many period-spacing patterns in our catalogue compared to the more complex dispersion relations in the above-mentioned theory papers.

2.1 Methodological framework for the estimation of the interior rotation and buoyancy travel time

The input catalogue of this study contains 106 γ Dor stars observed by the TESS space telescope in its Cycle 1 in the southern ecliptic hemisphere. These 106 g-mode pulsators exhibit 140 period-spacing patterns as reported by Garcia et al. (2022). The patterns were detected in the 1-yr-long light curves of the southern continuous viewing zone (hereafter S-CVZ). Unlike the 4-yr-long light curves assembled by the Kepler space telescope, where missing modes are infrequent thanks to the long time base (e.g. Van Reeth et al. 2015b), the 1-yr-long light curves used in this work are more likely to result in sparse observed pulsation patterns, that is, some g-mode periods in the radial-order series given by Eq. (6) may not be detected.

In this work, we fitted the detected g-mode periods, , directly. We calculated theoretical predictions for the g-mode periods, (hereafter simplified as Pnℓm), in Eq. (6). The effects of rotation were evaluated using the TAR module of GYRE1 (v6, Townsend & Teitler 2013; Townsend et al. 2018; Goldstein & Townsend 2020). It provides tabulated numerical solutions to the Laplace tidal equation, which were obtained with the spectral matrix method, as described by Townsend (2003b). In Sect. 5, we validate our method by evaluating γ Dor stars that have been studied in the literature before, and comparing the results.

The comparison between g-mode periods from asymptotic theory and the observations was done by adapting the cost function from Garcia et al. (2022) to the parameters, to be estimated as

(7)

where Pn is the theoretical mode period and ∆Pn its local period spacing, is the observed mode period, its uncertainty, its observed amplitude, and is the maximum detected amplitude in the observed Lomb-Scargle periodogram as computed by Garcia et al. (2022). The angular wavenumbers, and m, are omitted in Eq. (7) for readability. Identification of the radial orders of the observed mode periods, , with respect to the theoretically predicted period sequence, Pn, was done by finding the closest match between the dominant frequency in the observed sequence and the predicted one. In doing so we included the information about missing radial orders in the observed sequences as reported in Garcia et al. (2022) by setting their corresponding amplitude to zero. This step was paramount. Because the asymptotic model for the period-spacing pattern modelling has three free parameters, an observed pattern has to contain at least four individual pulsation modes for a successful analysis. If there are too many missing radial orders or the number of missing radial orders is uncertain, the minimum number of observed modes for a successful analysis increases.

Table 1

3D space considered for the mode identification and parameter estimation.

2.2 Identification of the angular wavenumbers

The identification of the angular wavenumbers deduced from the observed g-mode patterns was done following the labelling system introduced by Lee & Saio (1997), where gravito-inertial and Rossby modes of consecutive radial order n are labelled by the set of indices (k, m), with k ≡ ℓ − |m| 0 for gravito-inertial modes and k < 0 for Rossby modes. We computed the mode periods from the TAR for all dipole and quadrupole modes, as well as the Rossby modes (−2, −1) and (−1, −1). This resulted in ten different g-mode patterns (k, m), each of them returning a minimum value of the cost function denoted by .

To deduce the most likely identification for each detected g-mode pattern, we considered the coarse grid with parameter ranges for ∏0, frot, and αg as indicated in Table 1. These ranges for ∏0 and frot are wider compared to the ones reported in the literature for the large sample of Kepler γ Dor pulsators (e.g. Van Reeth et al. 2015b; Li et al. 2020). We thus expect all best-fit values to show well-defined minima in the cost function. The phase term αg in Eqs. (3) and (5) captures the effect of boundary layers inside the star and may attain any value (Aerts et al. 2010). However, we limit its values in Eq. (7) to the effective range 0 ≤ αg ≤ 1. In other words, we would interpret a value of αg > 1 and αg < 0 as belonging to a mode with the overtone n + 1 and n − 1, respectively. When the cost function changes monotonically within the parametric range, leading to a solution at the edges of the range, we reject that solution for the mode identification.

We adopted the global minimum among all the combinations, , as the most likely mode identification. This identification process is exemplified in Fig. 1 by using the main period-spacing pattern present in TIC 381950897, that is, the one with the dominant frequency. This period-spacing pattern has eight detected g-mode periods and eight missing mode periods, which is a common case in the catalogue. The figure shows the ratio for each mode combination. The relative heights of the bars represent how likely the identification of (k, m) is for the detected mode pattern of this star among the considered options. Whenever the identification was ambiguous, that is, if multiple bars in the identification diagnostic tool, such as the one shown in Fig. 1, had a height close to unity, we manually selected the most likely identification based on the sample distribution of the different mode geometries that were observed and reported by Li et al. (2020). These authors detected and identified 960 period-spacing patterns in a sample of more than 600 γ Dor pulsators from 4-yr-long Kepler light curves, and summarised the relative sample fractions of the different mode identifications in their Fig. 6 (represented in Fig. 2 in this work). Where necessary, we used this distribution to assess the probability of observing different mode geometries. For example, as already shown by Li et al. (2020), we expect sectoral modes (k = 0) to be more likely to be detected because rapid rotation tends to reduce the photometric visibility of the oscillations by confining the pulsational variability to the equatorial regions (Townsend 2000; Dhouib et al. 2022). Because this effect is less strong for prograde sectoral modes, they have a higher visibility in most stars.

Figure 3 shows the TAR best-fit solution for the period-spacing patterns generated with the two most likely pulsation mode identifications in Fig. 1, namely (k, m) = (0,1) and (0,2). The two solutions converged to reproduce the mode with the highest amplitude, occurring with a period of about 0.49 d, optimally. This is a consequence of using the detected mode amplitudes as weights in the cost function defined in Eq. (7).

thumbnail Fig. 1

Mode identification deduced from the main period-spacing pattern detected for TIC 381950897.

2.3 Estimation of frot and ∏0

Once the pulsation pattern was identified, we refined the estimates of ∏0 and frot by searching for a minimum of the cost function S in a denser parameter space, also indicated in Table 1. For computational reasons, this denser space was only considered for the final mode identification. The impact of the resolution of frot on the estimation of ∏0 is shown in Fig. 4 for the main pattern in TIC 381950897. The many local minima in black are due to the finite resolution of the grid and the strong correlation between ∏0 and frot. This is illustrated in Fig. 5.

The left panel represents a prograde mode, while the middle panel represents a retrograde mode. These panels indicate that a change in ∏0 can be compensated by a small change in frot. Proper estimation of both parameters therefore requires a sufficiently dense parameter grid, particularly for frot, and proper treatment of degeneracies in the computation of the uncertainties. The black curve in Fig. 4 results from the rough sampling for frot, while the red curve stems a grid step that is five times denser. As a result of the denser 3D grid search, the estimation of ∏0 results from a smoother curve without numerous local minima.

To avoid ending up with local minima for values of ∏0 differing by ~ 250 s, as illustrated for the case in Fig. 4, we replaced the actual cost function by its convex approximation and selected the five data points around its minimum. These points were then used to fit a parabola, from which we took the vertex as the final solution for ∏0. The convex envelope was computed using the Qhull library (Barber et al. 1996). The process to obtain the final value of ∏0 is shown in Fig. 6, where the red and blue vertical lines represent ∏0 obtained from the discrete dense parameter grid and the parabolic fit, respectively.

Figure 7 shows the final result of the TAR model that best fits the main period-spacing pattern of TIC 381950897. The lower panel of the figure shows the period spacings in the data where we made the representation such as to exclude the parts with missing modes in the data, although these were identified from the theoretical prediction. Therefore, the lower panel shows a lower or equal number of symbols labelled as measurements when compared to the upper panel, but we stress that the TAR fit was performed using all the detected mode periods indicated in the upper panel.

thumbnail Fig. 2

Probability distribution for the detection of different g-mode pulsation geometries (k, m), based on the observed sample distribution from the γ Dor pulsator study by Li et al. (2020).

thumbnail Fig. 3

Traditional approximation of rotation solution for the first and second most plausible pulsation mode identifications, that is, (0, 1) and (0, 2), resulting from the main detected g-mode pattern of TIC 381950897 from the distribution in Fig. 1. The grey curve is the Lomb-Scargle periodogram.

thumbnail Fig. 4

Effect of the resolution of frot in the parameter space on the determination of ∏0 for TIC 381950897. The black and red curves represent the cost function evaluated on the sparse and the dense parameter space indicated in Table 1, respectively.

thumbnail Fig. 5

Cost function evaluated for the parameter ranges for the internal rotation and buoyancy travel time. Each panel shows a typical correlation between frot and ∏0. Left: Main pattern in TIC 381950897 with mode identification (0,1). Middle: Secondary pattern in TIC 381950897 with mode identification (−2, −1). Right: Pattern of TIC 293221812 with mode identification (0, −1). In each panel, the blue contour lines indicates S = 100 Smin while the cyan lines indicate S = 10 Smin.

thumbnail Fig. 6

Further refinement of the solution shown in Fig. 4 for TIC 381950897. The red curve is the same as in Fig. 4.

2.4 Correlation structure and error estimation

Correlations between frot and ∏0 found in the parameter space were already presented in Fig. 5. The shape of the correlation in the left panel is the result of the internal rotation rate affecting the pulsation periods at two separate instances in the methodological chain: 1) from the conversion of the asymptotic spacing for non-rotating values into those for rotating values via the spin parameter; 2) from transformation of the mode periods computed in the co-rotating frame towards the line of sight (seeEqs. (3) and (6)). When frot is such that the two instances affect the pulsation periods similarly, we get a correlation structure as in the right panel of Fig. 5, as already discussed in the theory paper by Bouabid et al. (2013).

In order to deal appropriately with the variety of correlation structures, we used a residual bootstrap technique to estimate the uncertainties of frot and ∏0. The errors reported in Table A.1 are the 1σ errors that correspond to the 68% confidence interval obtained from the bootstrap distributions generated by a non-parametric residual resampling of the TAR fit. The parameter space used during the bootstrap application was centred around the best fit TAR solution and had a full range of 0.12 day−1 for frot, and a range of 1500 s for ∏0. These ranges were chosen so as to cover typical uncertainties achieved from 4-yr-long Kepler light curves based on the median of the 1σ uncertainties reported in Van Reeth et al. (2016). The sampling for αg for the bootstrap was taken to be the same as the dense parameter space indicated in Table 1.

For each detected pattern, we generated 200 datasets of the same size as the measured pattern and subsequently minimised Eq. (7) for each dataset to sample the error distribution and deduce its variance. As an example, Fig. 8 shows the bootstrap distribution for TIC 41483281, from which we estimated the 1σ error for frot and ∏0. We note that the distribution for the parameter αg is not uniform, yet it is well defined, despite the modest resolution we used for this parameter, given that it does not offer any astrophysical information but is only a nuisance parameter with the adopted approach (see Table 1).

thumbnail Fig. 7

Best TAR model for the mode with k = 0 and m = 1 in TIC 381950897. Top: Lomb-Scargle periodogram (grey curve), detected periods (vertical black lines), and the TAR fit (vertical red lines). Bottom: Period spacings of both the TAR fit (red curve) and the detected periods (black circles). Period spacings of missing modes are omitted in the bottom panel. The dotted line connects periods spacings that stretch across missing modes. The units of the vertical axis AP are kiloseconds (ks) or 103 s.

thumbnail Fig. 8

Residual bootstrap distribution for the three fitted parameters of TIC 41483281. The vertical dashed lines indicate the TAR best fit solutions.

thumbnail Fig. 9

Distribution of the identified modes for 60 TESS γ Dor pulsators in the catalogue of Garcia et al. (2022).

thumbnail Fig. 10

Distribution of internal rotation and buoyancy travel time in 611 γ Dor stars in the Kepler sample, and the 60 pulsators in our sample from the TESS S-CVZ. The probability density function (PDF) of each histogram has been normalised to unity for comparison.

3 Results for the interior rotation and buoyancy travel time of TESS γ Dor stars

We obtained frot and ∏0 for 60 γ Dor stars in the TESS S-CVZ from the catalogue of Garcia et al. (2022) from the application of our methodology. For the remaining 46 γ Dor stars, we did not obtain a converged fit. The patterns were either too short or could not be unambiguously identified because of too many missing modes. Although these 46 stars revealed period-spacing patterns in their TESS 1-yr-long light curve, they require a longer light curve to achieve mode identification. For the 60 successfully analysed stars, the results are listed in Table A.1, along with information about the mean period, mean period spacing, and the presence of acoustic modes from Garcia et al. (2022). All the fits to the patterns we obtained are displayed in the supplementary material available online, in the same format as in Fig. 7. A subset of the fits are presented in Sect. 3.2. We note that missing modes were common in the catalogue of Garcia et al. (2022) due to the limited duration of the Cycle 1 S-CVZ TESS data, unlike in the 4-yr-long Kepler light curves.

The distribution of the identified pulsation modes is presented in Fig. 9. It reveals a dominance of dipole (k = 0, m = 1) and quadruple (k = 0, m = 2) prograde sectoral modes. This distribution is consistent with the results for ~ 600 Kepler γ Dor pulsators from Li et al. (2020). The distributions of our values for frot and ∏0, and those for the Kepler sample treated by Li et al. (2020), are compared in Fig. 10. Both samples have similar averages, despite the difference in size by a factor of ten.

thumbnail Fig. 11

Improvement of the determination of frot and ∏0 for TIC 381950897 when fitting its two detected period-spacing patterns simultaneously. It concerns a prograde mode (0,1) and a retrograde mode (−2, −1) pattern. The left panel shows an overplot of the two cost functions in which the valley of S with an overall d∏0/dfrot > 0 corresponds to the prograde mode, while the other valley refers to the retrograde mode. Contour lines are omitted for better visibility but were shown in Fig. 5, where both cost functions were shown separately. The middle panel shows the cost function when both patterns are fitted simultaneously. The solution lies at the intersection of the two valleys from the left panel. The right panel shows a close-up of the middle panel. The blue contour line indicates S = 100 Smin, while the cyan one indicates S = 10 Smin.

3.1 Stars with multiple g-mode period-spacing patterns

Stars with two or more period-spacing patterns provide us with the opportunity to further constrain the stellar interior by fitting the patterns simultaneously (e.g. Van Reeth et al. 2016). Garcia et al. (2022) found 26 of the 106 g-mode pulsators to have multiple patterns. However, some of the secondary patterns had too few modes to obtain a converged fit. We managed to get converged simultaneous fits for five of the 26 stars, as listed in Table A.2. We relied on the pattern with the clearest mode identification to resolve any uncertainty in the identification of the secondary pattern, following some criteria: (1) modes with were rejected, see Fig. 1 for an example; (2) the two independent solutions for ∏0 needed to be consistent within 2σ; (3) solutions based on equality of the two independent frot were preferred, given that almost all Kepler γ Dor stars reveal near-rigid rotation (Li et al. 2020); (4) prograde sectoral modes were preferred following the mode identification distribution obtained by Li et al. (2020).

An example of a simultaneous two-pattern fit is shown in Fig. 11 for TIC 381950897. The left panel shows the cost function for frot and 0 when the two patterns are fitted independently. The middle and right panels show the better constrained solution when the two patterns are fitted simultaneously. The minimum of the cost function is indeed more localised, as shown by the comparison between the contours in Figs. 5 and 11. This translates into more precise values for frot and ∏0, as indicated in Table A.2.

3.2 Most promising TESS γ Dor stars for future asteroseismic modelling

Six stars in our sample offer particularly long patterns with few missing modes, and hence have good potential for future aster-oseismic modelling. These are TIC 374944608, TIC 350144657, TIC 381950897, TIC 349832567, TIC 141479660, and TIC 38515566. These six stars show period-spacing patterns with 12 to 15 frequencies with high amplitudes and high signal-noise ratios in the Lomb-Scargle periodograms, as presented in Fig. 12. Most of these stars are also hybrid pulsators, that is, they reveal both g and p (acoustic) modes. They also have available spectroscopic data, as discussed in the next section (and listed in Table A.1). This makes these stars prime candidates for further asteroseismic modelling, as in Mombarg et al. (2020, 2021).

Adding the TESS Cycle 3 and Cycle 5 data to the light curves of these stars (as all others in our sample) will translate into effectively 3- and 5-yr-long light curves, respectively. Although those light curves will have one or two 1-yr gaps, they will undoubtedly decrease the noise levels in the periodograms, while still delivering manageable aliasing. This will allow us to detect some of the currently missing modes and prolong the period-spacing patterns found here, in addition to increasing the frequency resolution appreciably.

thumbnail Fig. 12

Most promising TESS γ Dor stars for asteroseismic modelling, revealing the best TAR fits. (A) TIC 374944608. (B) TIC 350144657. (C) TIC 381950897. (D) TIC 349832567. (E) TIC 141479660. (F) TIC 38515566. The symbols are the same as in Fig. 7.

4 Astrophysical properties of the TESS γ Dor stars

We searched the literature for homogeneous spectroscopic parameters of our sample stars and found 34 of them to have spectra available in the Third Data Release (DR3) of the GALAH Survey (Buder et al. 2021). For 22 of them, reliable stellar parameters were determined2.

We took the spectroscopic parameters effective temperature, gravity, metallicity, and the velocity connected with the spectral line broadening, vbroad, which is an upper limit of the projected rotation velocity v sin i in the analysis method adopted by Buder et al. (2021). Indeed, the authors fitted the broadened metal lines in the medium-resolution GALAH spectra, assuming the absence of any macroturbulence, often connected with the occurrence of pulsations, notably g modes (Aerts et al. 2009). We took the luminosity for these 22 γ Dor stars from the TESS Input Catalog, Version 8 (Stassun et al. 2019, based on Gaia DR2 parallaxes). The distributions of the stellar parameters for these 22 γ Dor stars are shown in Fig. 13. These distributions for the effective temperature and surface gravity are consistent with those of Kepler γ Dor stars from high-resolution spectroscopy (Niemczura et al. 2015; Van Reeth et al. 2015b; Gebruers et al. 2021). However, the metallicities reported in the GALAH study by Buder et al. (2021) are lower than those found by Gebruers et al. (2021) based on much higher-quality spectroscopic data, which may point to major systematic uncertainties in this quantity deduced for F-type dwarfs from medium-resolution survey spectroscopy. The distribution for vbroad, when compared with the rotational broadening for Kepler γ Dor stars derived by Gebruers et al. (2021, their Fig. 5), shows that these 22 TESS γ Dor stars with identified patterns are mainly slow rotators.

Despite the small sample of only 22 γ Dor stars, we searched for correlations among frot, ∏0, and the stellar parameters shown in Fig. 13. Following Van Reeth et al. (2015b), we used multi-variate linear regression with backwards elimination, that is, each of the rotation frequencies and buoyancy travel times were modelled as a linear combination of the five stellar parameters shown in Fig. 13. The coefficient with the largest p-value above 0.05, obtained from a t-test, was then removed from the regression model before repeating it, until only explanatory variables with significant p-values remained among the stellar parameters. We found a weak correlation (R2 = 0.20) between frot and Teff, as shown in Fig. 14, and whose free parameters are listed in Table 2. Despite the small R2 value, this correlation may indicate that stars on the main sequence slow down as they age, as expected from angular momentum transport mechanisms active during stellar evolution (Aerts et al. 2019). In contrast, no significant correlation was found between frot and vbroad, which is additional evidence of the presence of macroturbulence affecting the line broadening. We also found a weak bivariate relationship (R2 = 0.23) between ∏0 on the one hand, and log g and log L on the other hand. The positive parameters of the bivariate fit listed in Table 3 and shown in Fig. 15 indicate that the buoyancy travel time increases as the luminosity or surface gravity increases. Van Reeth et al. (2016) found similar weak correlations expected from stellar evolution theory for the rotation, yet none for the buoyancy travel time.

Values for the projected rotational velocity, v sin i, were available in SIMBAD for 15 of the 22 γ Dor stars in the GALAH survey. These values come from Sharma et al. (2018). We noticed that the distribution of v sin i from SIMBAD differs from the one from the GALAH vbroad values, as shown in Fig. 16. This indicates that time-dependent pulsational line broadening is active in these stars, as expected for g modes. Considerable line-profile broadening, in addition to time-independent rotational broadening, is indeed a well-known phenomenon detected and analysed for bright γ Dor pulsators studied with high-resolution spectroscopy (De Cat et al. 2006). Time-dependent pulsational line broadening is often approximated by time-independent macroturbulence and its occurrence may prevent the proper derivation of v sin i (Aerts et al. 2014), which may explain the difference in the distributions in Fig. 16. In order to investigate this further, we searched for evidence of line asymmetries in narrow metal lines of the γ Dor stars in our sample. We looked for optimal isolated iron spectral lines following the analyses by Bruntt et al. (2008) and computed their bisector as a good diagnostic for line asymmetries. Only eight of the 15 stars have spectra of high-resolution, namely R ≳ 4800, with a signal-noise ratio S/N ≳ 100. We found indications of asymmetric lines in the following eight stars: TIC 38515566, TIC 350343297, TIC 349092320, TIC 55849446, TIC 382519218, TIC 149540525, TIC 167124706, and TIC 150392753. Some of those stars, such as TIC 350343297 in Fig. 17, show clear asymmetric lines by visual inspection, while others’ asymmetric lines are less pronounced but these are detected in their bisectors and in the change over different epochs, such as TIC 167124706 in Fig. 18.

thumbnail Fig. 13

Distribution of spectroscopic parameters and luminosity for 22 stars in our catalogue. Typical uncertainties are shown at the top of each panel. See Sect. 4 for a further description.

thumbnail Fig. 15

Multi-variate linear regression for the buoyancy travel time. Each panel shows the bivariate model fits listed in Table 3. The symbols are the same as in Fig. 14. Typical uncertainties are shown in blue in the bottom right corner.

thumbnail Fig. 14

Multi-variate linear regression for the internal rotation. The squares are the observations and the circles are the modelled values. The corresponding values are connected with a black line. The red line is the model listed in Table 2. The typical uncertainty of log Teff is shown in blue in the bottom right corner. Uncertainties of frot are smaller than the symbol sizes.

Table 2

Multi-variate linear regression result for the rotation frequency.

Table 3

Multi-variate linear regression for the buoyancy travel time.

5 Quality comparison by revisiting Kepler γ Dor stars

To validate our methodology described in Sect. 2, we applied it to Kepler 4-yr-long light curves of γ Dor stars whose frot and ∏0 were derived by Van Reeth et al. (2016). Their sample includes 37 stars that were monitored with high-resolution spectroscopy by Tkachenko et al. (2013) and were confirmed to be single γ Dor stars. The mass, age, and level of near-core mixing of these stars was deduced from forward asteroseismic modelling by Mombarg et al. (2019, 2021), so this sample constitutes the best characterised γ Dor stars to date. We revisited their Kepler light curves and deduced the g-mode frequencies following the same analysis as in Garcia et al. (2022).

Van Reeth et al. (2016) identified multiple period-spacing patterns in the 37 Kepler stars, but only the patterns for consecutive radial-order modes with (k, m) = (0, 1) were used to constrain frot and ∏0. We recovered and identified all patterns as (k = 0, m = 1). Figure 19 shows the comparison between our results and those by Van Reeth et al. (2016). The left panel shows excellent agreement for frot. Previously, Ouazzani et al. (2019, their Fig. 5) had already made a comparison of their 37 frot values deduced by Christophe et al. (2018) independently of Van Reeth et al. (2016). They found the same frot values as Van Reeth et al. (2016) for 36 of the 37 stars. The right panel of Fig.19 compares our ∏0 values with those deduced by Van Reeth et al. (2016). Overall, this figure also shows a good agreement for ∏0 because the larger deviations from the bisector are accompanied by larger uncertainties than for frot. Also, this quantity for these 37 stars was deduced independently by Christophe et al. (2018). Figure 20 compares the values from the three observational studies starting from the same light curves of the 37 pulsators. While there is overall excellent agreement, particularly for frot, Ouazzani et al. (2019) and this work both find a larger range of values for ∏0. Finally, and as an example, Fig. 21 compares the TAR obtained by Van Reeth et al. (2016) and the one from this work for a particular period-spacing pattern in KIC 7939065. The two results are in agreement and the slight offsets of some peaks are generated from the coarse parameter space for αg used in this work and defined in Table 1.

Overall, we conclude from Figs. 19-21 that our method for obtaining the interior rotation frequency and buoyancy travel time works well. Furthermore, the uncertainties we deduced from our bootstrap method are comparable to those in the literature deduced from higher-quality Kepler data. By comparing the median uncertainties obtained from the 1-yr-long TESS and the 4-yr-long Kepler light curves, we conclude that period-spacing patterns from the 1-yr-long TESS light curves can constrain the internal rotation frequency and buoyancy travel time to a precision of 0.03 day−1 and 400 s, respectively, which is about half as good as the values deduced in the literature and by our re-analyses from the 4-yr-long Kepler light curves. Therefore, the values and uncertainties for frot and ∏0 deduced from the 1-yr-long TESS light curves are robust.

thumbnail Fig. 16

Broadening velocities and projected rotational velocities of our 15 stars in common with GALAH DR3, with bona fide spectra and with reported values from Sharma et al. (2018). Histograms have been normalised to unity for comparison.

thumbnail Fig. 17

Normalised spectrum of TIC 350343297 showing asymmetric absorption lines. The FeIIλ4520.2 and FeIIλ4541.5 lines are indicated by vertical red lines as a reference. Spectrum obtained by the FEROS spectrograph (Kaufer et al. 1997, 1999) at the MPG/ESO 2.2-m telescope located at the La Silla Observatory in Chile, on 24 March 2012 with a spectral resolution of 48 000 and a signal-to-noise ratio of 104. The spectrum is not corrected by redshift.

thumbnail Fig. 18

FeIIλ4576.3 absorption line of TIC 167124706 at three different epochs. The bisector of the line is shown in red and is calculated with respect to the smoothed spectrum shown in blue. The smoothed spectrum is generated by a moving average using the window length of three data points. The top axis shows the Doppler velocity with respect to the central wavelength of the absorption line indicated with the vertical dashed line. The spectrum is not corrected by redshift. Observation dates, spectral resolutions, and signal-to-noise ratios are listed in the insets.

thumbnail Fig. 19

Comparison of internal rotation and buoyancy travel time obtained by Van Reeth et al. (2016) and this work. Different colours and symbols are different pulsation modes (k, m) specified in the legend. The symbols labelled with KIC numbers in the left panel are the same patterns labelled in the right panel.

thumbnail Fig. 20

Comparison of internal rotation and buoyancy travel time obtained by Van Reeth et al. (2016), Ouazzani et al. (2019), and this work. Histograms have been normalised to unity for comparison. The columns of different studies are shifted for visibility instead of stacked.

thumbnail Fig. 21

Comparison of the best TAR model obtained by Van Reeth et al. (2016; dashed blue lines) and this work (solid red lines) for the period-spacing pattern in KIC 7939065 reported by the aforementioned study.

6 Conclusions

We have provided rotation frequencies and buoyancy travel times for a sample of 60 γ Dor stars observed in the TESS S-CVZ during 352 days covered in Cycle 1 of the mission. These two important observables of stellar interiors can be readily decoded from period-spacing patterns of identified g-mode pulsators. While we have shown that working with 1-yr-long light curves comes with its drawbacks, mainly due to missing mode periods in the patterns hampering secure mode identification, the proper information could be deduced for 60 of the 106 γ Dor stars in our sample.

We validated our methodology by reproducing Kepler results from Van Reeth et al. (2016) and concluded that period-spacing patterns from 1-yr-long TESS light curves can constrain the internal rotation frequency and buoyancy travel time to a precision of 0.03 day−1 and 400 s, respectively.

The 60 g-mode pulsators in the TESS S-CVZ, made available in this work, offer excellent asteroseismic potential to constrain their internal mixing profiles and angular momentum transport, once additional TESS data from the ongoing cycles are added to the light curves. Indeed, the scientific value of the stars in our sample will continue to increase as the TESS mission continues beyond its nominal mission. Besides the asteroseismic dimension, this sample has spectroscopic values for ~ 60% of its stars from spectra available with one or more epochs indicated in Table A.1. With the third data release of Gaia newly available, this catalogue is a prime candidate for combined asteroseismic, spectroscopic, and astrometric modelling to test theoretical and computational tools from generalised TAR and radiative levitation.

Acknowledgements

The research leading to these results has received funding from the KULeuven Research Council (grant C16/18/005: PARADISE) and from the BELgian federal Science Policy Office (BELSPO) through PRODEX grants for the Gaia and PLATO space missions. T.V.R. gratefully acknowledges support from the Research Foundation Flanders (FWO) under grant agreement N° 12ZB620N. We thank the anonymous referee for useful comments which have allowed us to improve the text.

Appendix A Extended tables

Table A.1

TIC stars in our catalogue and their period-spacing pattern parameters, derived for each pattern that was successfully modelled.

Table A.2

TIC stars in our catalogue and their period-spacing pattern parameters. The fit used two patterns simultaneously.

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2

Flagged with flag_sp==0 in the GALAH database.

All Tables

Table 1

3D space considered for the mode identification and parameter estimation.

Table 2

Multi-variate linear regression result for the rotation frequency.

Table 3

Multi-variate linear regression for the buoyancy travel time.

Table A.1

TIC stars in our catalogue and their period-spacing pattern parameters, derived for each pattern that was successfully modelled.

Table A.2

TIC stars in our catalogue and their period-spacing pattern parameters. The fit used two patterns simultaneously.

All Figures

thumbnail Fig. 1

Mode identification deduced from the main period-spacing pattern detected for TIC 381950897.

In the text
thumbnail Fig. 2

Probability distribution for the detection of different g-mode pulsation geometries (k, m), based on the observed sample distribution from the γ Dor pulsator study by Li et al. (2020).

In the text
thumbnail Fig. 3

Traditional approximation of rotation solution for the first and second most plausible pulsation mode identifications, that is, (0, 1) and (0, 2), resulting from the main detected g-mode pattern of TIC 381950897 from the distribution in Fig. 1. The grey curve is the Lomb-Scargle periodogram.

In the text
thumbnail Fig. 4

Effect of the resolution of frot in the parameter space on the determination of ∏0 for TIC 381950897. The black and red curves represent the cost function evaluated on the sparse and the dense parameter space indicated in Table 1, respectively.

In the text
thumbnail Fig. 5

Cost function evaluated for the parameter ranges for the internal rotation and buoyancy travel time. Each panel shows a typical correlation between frot and ∏0. Left: Main pattern in TIC 381950897 with mode identification (0,1). Middle: Secondary pattern in TIC 381950897 with mode identification (−2, −1). Right: Pattern of TIC 293221812 with mode identification (0, −1). In each panel, the blue contour lines indicates S = 100 Smin while the cyan lines indicate S = 10 Smin.

In the text
thumbnail Fig. 6

Further refinement of the solution shown in Fig. 4 for TIC 381950897. The red curve is the same as in Fig. 4.

In the text
thumbnail Fig. 7

Best TAR model for the mode with k = 0 and m = 1 in TIC 381950897. Top: Lomb-Scargle periodogram (grey curve), detected periods (vertical black lines), and the TAR fit (vertical red lines). Bottom: Period spacings of both the TAR fit (red curve) and the detected periods (black circles). Period spacings of missing modes are omitted in the bottom panel. The dotted line connects periods spacings that stretch across missing modes. The units of the vertical axis AP are kiloseconds (ks) or 103 s.

In the text
thumbnail Fig. 8

Residual bootstrap distribution for the three fitted parameters of TIC 41483281. The vertical dashed lines indicate the TAR best fit solutions.

In the text
thumbnail Fig. 9

Distribution of the identified modes for 60 TESS γ Dor pulsators in the catalogue of Garcia et al. (2022).

In the text
thumbnail Fig. 10

Distribution of internal rotation and buoyancy travel time in 611 γ Dor stars in the Kepler sample, and the 60 pulsators in our sample from the TESS S-CVZ. The probability density function (PDF) of each histogram has been normalised to unity for comparison.

In the text
thumbnail Fig. 11

Improvement of the determination of frot and ∏0 for TIC 381950897 when fitting its two detected period-spacing patterns simultaneously. It concerns a prograde mode (0,1) and a retrograde mode (−2, −1) pattern. The left panel shows an overplot of the two cost functions in which the valley of S with an overall d∏0/dfrot > 0 corresponds to the prograde mode, while the other valley refers to the retrograde mode. Contour lines are omitted for better visibility but were shown in Fig. 5, where both cost functions were shown separately. The middle panel shows the cost function when both patterns are fitted simultaneously. The solution lies at the intersection of the two valleys from the left panel. The right panel shows a close-up of the middle panel. The blue contour line indicates S = 100 Smin, while the cyan one indicates S = 10 Smin.

In the text
thumbnail Fig. 12

Most promising TESS γ Dor stars for asteroseismic modelling, revealing the best TAR fits. (A) TIC 374944608. (B) TIC 350144657. (C) TIC 381950897. (D) TIC 349832567. (E) TIC 141479660. (F) TIC 38515566. The symbols are the same as in Fig. 7.

In the text
thumbnail Fig. 13

Distribution of spectroscopic parameters and luminosity for 22 stars in our catalogue. Typical uncertainties are shown at the top of each panel. See Sect. 4 for a further description.

In the text
thumbnail Fig. 15

Multi-variate linear regression for the buoyancy travel time. Each panel shows the bivariate model fits listed in Table 3. The symbols are the same as in Fig. 14. Typical uncertainties are shown in blue in the bottom right corner.

In the text
thumbnail Fig. 14

Multi-variate linear regression for the internal rotation. The squares are the observations and the circles are the modelled values. The corresponding values are connected with a black line. The red line is the model listed in Table 2. The typical uncertainty of log Teff is shown in blue in the bottom right corner. Uncertainties of frot are smaller than the symbol sizes.

In the text
thumbnail Fig. 16

Broadening velocities and projected rotational velocities of our 15 stars in common with GALAH DR3, with bona fide spectra and with reported values from Sharma et al. (2018). Histograms have been normalised to unity for comparison.

In the text
thumbnail Fig. 17

Normalised spectrum of TIC 350343297 showing asymmetric absorption lines. The FeIIλ4520.2 and FeIIλ4541.5 lines are indicated by vertical red lines as a reference. Spectrum obtained by the FEROS spectrograph (Kaufer et al. 1997, 1999) at the MPG/ESO 2.2-m telescope located at the La Silla Observatory in Chile, on 24 March 2012 with a spectral resolution of 48 000 and a signal-to-noise ratio of 104. The spectrum is not corrected by redshift.

In the text
thumbnail Fig. 18

FeIIλ4576.3 absorption line of TIC 167124706 at three different epochs. The bisector of the line is shown in red and is calculated with respect to the smoothed spectrum shown in blue. The smoothed spectrum is generated by a moving average using the window length of three data points. The top axis shows the Doppler velocity with respect to the central wavelength of the absorption line indicated with the vertical dashed line. The spectrum is not corrected by redshift. Observation dates, spectral resolutions, and signal-to-noise ratios are listed in the insets.

In the text
thumbnail Fig. 19

Comparison of internal rotation and buoyancy travel time obtained by Van Reeth et al. (2016) and this work. Different colours and symbols are different pulsation modes (k, m) specified in the legend. The symbols labelled with KIC numbers in the left panel are the same patterns labelled in the right panel.

In the text
thumbnail Fig. 20

Comparison of internal rotation and buoyancy travel time obtained by Van Reeth et al. (2016), Ouazzani et al. (2019), and this work. Histograms have been normalised to unity for comparison. The columns of different studies are shifted for visibility instead of stacked.

In the text
thumbnail Fig. 21

Comparison of the best TAR model obtained by Van Reeth et al. (2016; dashed blue lines) and this work (solid red lines) for the period-spacing pattern in KIC 7939065 reported by the aforementioned study.

In the text

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