TESS asteroseismology of $\beta$ Hydri: a subgiant with a born-again dynamo

TESS observed $\beta$ Hyi at 20 s cadence during Sectors 67 and 68, corresponding to 2023 July 1 – 2023 August 25, with the usual gaps mid-sector and between sectors for data downlink. We used 20 s instead of 2 min data due to the significant improvement in photometric precision for bright stars (Huber et al., 2022). Given the brightness of $\beta$ Hyi (${\rm Tmag}=2.216$), we anticipated count rates of approximately $2\times 10^{7}~{}\rm s^{-1}$, which is consistent with the rates seen in the Science Processing Operations Center (SPOC; Jenkins et al., 2016) simple aperture photometry (SAP) light curves for both sectors. However, the corrected SPOC Pre-search Data Conditioning SAP (PDCSAP) light curve for both sectors shows a mean value of approximately $1.14\times 10^{7}~{}\rm s^{-1}$, substantially lower than both expectations and the uncorrected version. Both SPOC light curves also show noise levels that are significantly higher than anticipated for such a bright star.

From experience, we know that the normal TESS data processing can struggle with bright sources such as $\beta$ Hyi. Accordingly, we followed the procedures described in Nielsen et al. (2020) and Metcalfe et al. (2023b) to extract custom light curves for each sector. In essence, our aperture masks are created by starting from the pixel with the largest flux, and adding additional pixels one at a time until the signal-to-noise ratio (S/N) no longer improves. The resulting light curve is then detrended against centroid pixel coordinates and high-pass filtered with a cutoff frequency of $100~{}\mu\rm Hz$ to minimize residual contributions from spacecraft positional jitter. Figure 2 compares our unfiltered light curve (red) to the PDCSAP data product (black), showing the approximately factor of two reduction in the root-mean-square (rms) noise level of the light curve.

To look for any signature of rotation in the unfiltered light curve described in Section 2.1, we started by interpolating the gaps with inpainting techniques based in a sparsity prior (Elad et al., 2005) using a multiscale discrete cosine transform (Starck & Murtagh, 2006; García et al., 2014a; Pires et al., 2015). This procedure has been successfully applied to determine the surface rotation periods of main-sequence stars, subgiants, and red giants (e.g. Santos et al., 2019, 2021; Lam et al., 2021; García et al., 2023). The resulting light curve was re-binned by 100 points to a final cadence of $\sim$33 minutes.

The rotation period estimate was obtained from three different but complementary methods, which are illustrated in Figure 3. The initial method involved a period-time analysis (top left panel), using a Morlet wavelet-based decomposition (Torrence & Compo, 1998). We projected the results onto the period axis, creating the global wavelet power spectrum (GWPS; Mathur et al., 2010; García et al., 2014b). Starting from the highest peak, we iteratively fit Gaussian functions (green line, top right panel) and subtracted them from the GWPS until no further power remained. The central value of the tallest Gaussian function serves as our initial estimate of $P_{\rm{rot}}$. We adopt the full width at half maximum as the measurement uncertainty—a conservative approach that considers the possibility of surface latitudinal differential rotation. A second estimate of $P_{\rm{rot}}$ was obtained from the auto-correlation function (ACF; McQuillan et al., 2013) of the light curve (middle panel). Finally, we multiplied the GWPS and the ACF to obtain the composite spectrum (CS; Ceillier et al., 2016, 2017), which emphasizes signals common to both analyses (bottom panel).

After bench-marking several different surface rotation pipelines, Aigrain et al. (2015) showed that combining different analysis techniques is a powerful method to determine a reliable value of $P_{\rm{rot}}$. Starting with the GWPS, the main periodicity corresponds to $\sim$23 days. Unfortunately, this value falls within the exclusion cone of the wavelet analysis, indicated by the hashed region in the top left panel of Figure 3. Reliable results typically require a light curve longer than three times the rotation period. The ACF shows a signal around 24.7 days, which is further enhanced in the CS (24.4 days). Given the star’s brightness, we believe that the modulation is likely of stellar origin. Based on the current TESS data analysis, we conclude that the rotation period is $P_{\rm rot}=23.0\pm 2.8$ days. However, additional observations may be required to confirm this estimate. Note that previous single-sector data do not provide useful constraints.

To obtain a constraint on the stellar luminosity, we performed an analysis of the broadband spectral energy distribution (SED) of $\beta$ Hyi together with the Gaia DR3 parallax, following the procedures described in Stassun & Torres (2016) and Stassun et al. (2017, 2018). No systematic offset was applied (see, e.g., Stassun & Torres, 2021). We adopted the $JHK_{S}$ magnitudes from 2MASS, the W3–W4 magnitudes from WISE, the $UBV$ magnitudes from Mermilliod (2006), and the Strömgren $ubvy$ magnitudes from Paunzen (2015). Together, the available photometry spans the full stellar SED over the wavelength range 0.4–20 $\mu$m.

We performed a fit using Kurucz stellar atmosphere models, with the effective temperature ($T_{\rm eff}$) from North et al. (2007), and the surface gravity ($\log g$) and metallicity ([Fe/H]) from Bruntt et al. (2010). The extinction $A_{V}$ was fixed at zero due to the very close proximity of the star ($d=7.5$ pc). The resulting fit has a reduced $\chi^{2}$ of 0.9. Integrating the model SED gives the bolometric flux at Earth, $F_{\rm bol}=1.979\pm 0.059\times 10^{-6}$ erg s^-1 cm^-2. Taking $F_{\rm bol}$ together with the Gaia parallax (Gaia Collaboration et al., 2021) yields the bolometric luminosity $L_{\rm bol}=3.45\pm 0.10$ L_⊙.

The power spectrum of the filtered TESS light curve for $\beta$ Hyi is shown in the top panel of Figure 4. We observe a clear series of regularly spaced peaks centered at a frequency near $\nu_{\rm max}$=1038 $\mu$Hz, consistent with previous detections of solar-like oscillations. To extract individual frequencies, three teams applied Lorentzian mode-profile fitting (e.g. García et al., 2009; Handberg & Campante, 2011; Appourchaux et al., 2012; Mosser et al., 2012; Corsaro & De Ridder, 2014; Corsaro et al., 2020; Nielsen et al., 2021; Breton et al., 2022). For each mode, we required two independent methods to return the same frequency within uncertainties. The consensus list of frequencies was further refined from visual inspection of the power spectrum. For the final list we adopted values from a single method, with uncertainties derived by adding in quadrature the median formal uncertainty and the standard deviation of the extracted frequencies from all methods that identified a given mode. Table 1 lists the identified frequencies, which agree with the corresponding modes from Bedding et al. (2007) within the uncertainties.

The bottom panel of Figure 4 shows an échelle diagram with a large separation of $\Delta\nu=57.9$ $\mu$Hz and the extracted frequencies. We identified seven radial ($l=0$), nine dipole ($l=1$), and seven quadrupole ($l=2$) modes. The dipole modes show a clear avoided crossing near 950 $\mu$Hz, a departure from the regular frequency spacing due to the interaction of gravity modes in the core and pressure modes in the envelope (Aizenman et al., 1977). Avoided crossings are often found in the oscillations of subgiants, and are a powerful diagnostic of the stellar age (e.g. Benomar et al., 2012).

To determine the fundamental properties of $\beta$ Hyi, several modeling teams attempted to match the oscillation frequencies identified in Section 3.1, along with the $T_{\rm eff}$ from North et al. (2007) and the [Fe/H] from Bruntt et al. (2010) with errors inflated to account for the systematic noise floor suggested by Torres et al. (2012), as well as the luminosity constraint from Section 2.3. The five independent analyses used stellar evolution models from ASTEC (Christensen-Dalsgaard, 2008a), GARSTEC (Aguirre Børsen-Koch et al., 2022), MESA (Paxton et al., 2011), and YREC (Demarque et al., 2008). The resulting determinations of the stellar properties were in very good agreement, with a relative dispersion of 2% in radius (1.796–1.840 $R_{\odot}$), 6% in mass (1.059–1.127 $M_{\odot}$), and 22% in age (6.1–7.5 Gyr). For consistency with the rotational evolution modeling in Section 4.1, below we provide additional details about the results from YREC and MESA (AMP 2.0; Metcalfe et al., 2023a).

The asteroseismic modeling with YREC generally followed the same procedures described in Metcalfe et al. (2020) for the subgiant 94 Aqr Aa. However, the characteristics of the model grids were different for $\beta$ Hyi, and the treatment of spectroscopic constraints was modified slightly. The initial model grid for $\beta$ Hyi was constructed with masses between 1.02 and 1.20 $M_{\odot}$ in increments of 0.01 $M_{\odot}$, initial helium abundances between 0.25 and 0.29, initial [Fe/H] between $-$0.15 and +0.15, and a mixing length parameter between 1.4 and 2.1. After evaluating the most likely mass, a finer grid was constructed with masses between 1.09 and 1.12 $M_{\odot}$ and increments of 0.005 $M_{\odot}$, with all other parameters spanning the same ranges as in the initial grid. This finer grid was then used for a Monte Carlo analysis of the spectroscopic parameters. For each realization of the spectroscopic constraints, the most likely asteroseismic model was identified. The final stellar properties were obtained from the distribution of most likely models resulting from this procedure.

The MESA results were obtained from version 2.0 of the Asteroseismic Modeling Portal (AMP)¹¹1https://github.com/travismetcalfe/AMP2, which originally used models from the Aarhus stellar evolution and pulsation codes (Christensen-Dalsgaard, 2008a, b). The AMP code was released in 2009 (Metcalfe et al., 2009; Woitaszek et al., 2009), and several minor revisions followed as the quality of asteroseismic data from the Kepler mission gradually improved (Mathur et al., 2012; Metcalfe et al., 2014; Creevey et al., 2017). The first major revision coupled the same optimization method to the MESA (Paxton et al., 2011) and GYRE (Townsend & Teitler, 2013) codes. Although most of the input physics in the new version of AMP were chosen to be the same as in the previous version, there were two major updates that addressed the dominant sources of systematic error in the analysis of Kepler data sets. First, although the Aarhus models included diffusion and settling of helium (Michaud & Proffitt, 1993), the treatment of heavier elements was numerically unstable. The MESA models include diffusion and settling of both helium and heavier elements (Thoul et al., 1994). Second, the original version of AMP included an empirical correction for inadequate modeling of near-surface layers (Kjeldsen et al., 2008), while the updated version uses a physically-motivated correction that has now become the standard in the field (Ball & Gizon, 2014).

The asteroseismic radius, mass, and age of $\beta$ Hyi from the YREC and MESA modeling procedures are listed in Table 3.2. The consistency of the asteroseismic properties from these two independent model grids and fitting methodologies suggests that they are robust at the indicated level of precision.

We performed rotational evolution modeling following a procedure similar to that described in Metcalfe et al. (2020). We used a tracer code that computes the torque on the star given the braking law of van Saders et al. (2016) and the stellar structure as a function of age. We utilized two different evolutionary codes to predict the stellar structure. The first was a YREC model grid that assumed a chemical enrichment law for helium as a function of metallicity but a free mixing length parameter for convection, with input physics described in Metcalfe et al. (2020). The second grid was constructed with MESA, tuned to match the input physics of AMP 2.0 (Metcalfe et al., 2023a) and described in detail by Saunders et al. (2024). This grid allowed both the mixing length and helium to be free parameters.

For the braking law, we adopted the form of van Saders et al. (2016) and the calibration of Saunders et al. (2024), which determined best-fit braking law parameters using a Bayesian hierarchical framework to model both individual asteroseismic and open cluster system parameters, as well as the parameters of the braking law itself, which were assumed to be the same for all stars in the sample. We adopted their critical Rossby number $\mathrm{Ro_{crit}=0.94\pm 0.04\ Ro_{\odot}}$ ($\mathrm{Ro_{\odot}=2.33}$) and braking normalization $f_{K}=7.64$ for the YREC model grid, and $\mathrm{Ro_{crit}=0.91\pm 0.03\ Ro_{\odot}}$ ($\mathrm{Ro_{\odot}=2.05}$) and $f_{K}=5.46\pm 0.51$ for the MESA model grid. We also adopted their fits of the standard spin-down model to the data (without WMB), where $f_{K,YREC}=8.55$ and $f_{K,MESA}=6.11$.

For both model grids, we used $T_{\rm eff}$, [Fe/H], $L_{\rm bol}$, and their 1$\sigma$ uncertainties as surface constraints. We adopted a wide Gaussian prior on mass ($\sigma=0.5\ M_{\odot}$) from asteroseismology and bulk starting metallicity ($\sigma=0.3$ dex) but a tight asteroseismic prior on age ($6.46\pm 0.13$ Gyr for the YREC grid, and $6.26\pm 0.57$ Gyr for the MESA grid). This was done to help mitigate the “overcounting” of constraints: the asteroseismic modeling adopted the measured surface properties as constraints, so knowledge of them is already reflected in the asteroseismic mass. Age is weakly constrained by measurements of $T_{\rm eff}$, [Fe/H], and $L_{\rm bol}$, but very tightly constrained by the small frequency separation and the avoided crossing. In this sense the asteroseismic age is more independent of the surface constraints than the mass. For this reason we adopted the tight prior on the asteroseismic age, but only very broad priors on other asteroseismic properties. We assumed a mixing length prior of $1.7\pm 0.2$ bounded [1.4, 2.0] for both grids. In the YREC grid, the initial helium abundance was treated with a chemical enrichment law; in the MESA grid it was free to vary, and assigned a prior of $0.26\pm 0.10$, bounded [0.22, 0.28]. We considered values below the primordial helium abundance to allow for the possibility of systematic errors in the stellar models, and to avoid truncating the posterior probability distributions. Note that the measured rotation period from Section 2.2 was not used as an input constraint, but only as a consistency check with the predictions from two different sets of braking models.

For the YREC standard spin-down model we adopt the best-fit Saunders et al. (2024) braking law parameters without uncertainties, and predict a rotation period of $39^{+6}_{-7}$ days. For the WMB model, we predict a period of $19.4\pm 3.5$ days. In the MESA grid, we use the neural network framework of Saunders et al. (2024) to predict periods using No-U-Turn Sampling (Hoffman et al., 2014), which is a modified Hamiltonian Markov Chain Monte Carlo algorithm. The adopted $\mathrm{Ro_{crit}}$ and $f_{K}$ were treated as parameters with tight Gaussian priors given by their quoted uncertainties in Saunders et al. (2024). The standard spin-down model predicts a rotation period of $35^{+11}_{-8}$ days, while the WMB model predicts $19^{+5}_{-4}$ days. These values are substantially unchanged if we adopt the asteroseismic properties of the lower mass model from MESA, with the same wide mass prior now centered at $1.059\ M_{\odot}$ and a tight age prior of $6.10\pm 0.19$ Gyr.

In both grids, despite differing model physics and assumptions about the helium, the WMB model predicts a period consistent with that recovered from the current TESS photometry, albeit slightly shorter. As found by Saunders et al. (2024), this is a common pattern in more evolved stars, and may arise from our simplified braking prescription. Beyond $\mathrm{Ro_{crit}}$ magnetic braking is assumed to cease entirely, but in reality there may still be some minimal spin-down that subtly slows the rotation (along with the increasing moment of inertia) during the latter half of the main-sequence lifetime. Regardless, standard spin-down models predict much longer rotation periods that are inconsistent with the observations.

Figure 5 illustrates the evolution of the Rossby number in the YREC model grid as a function of stellar age, with the current observed position of $\beta$ Hyi marked and curves for both the standard spin-down (purple) and WMB (orange) cases. Standard spin-down models predict a relatively large Rossby number by the age of $\beta$ Hyi, while WMB models predict a Rossby number much closer to the solar value, where stellar activity cycles are most prominent (Egeland, 2017).

The properties of $\beta$ Hyi provide a second example of the “born-again” dynamo phenomenon that was previously suggested to explain the cycle in 94 Aqr Aa (Metcalfe et al., 2020). In that scenario, $\beta$ Hyi started as an F-type star on the main-sequence, where it probably had a short-period activity cycle like $\iota$ Hor (Metcalfe et al., 2010). The cycle would gradually grow longer as the rotation rate slowed due to magnetic braking over the next $\sim$2 Gyr, until it reached the critical Rossby number for the onset of WMB. During the second half of its main-sequence lifetime, the rotation rate would remain almost constant while the cycle would grow longer and weaker before disappearing entirely (Metcalfe & van Saders, 2017). However, when hydrogen shell-burning began and the star expanded and cooled onto the subgiant branch, the longer turnover time in the deeper convection zone would overwhelm the longer rotation period from conservation of angular momentum. For stars slightly more massive than the Sun, these evolutionary effects can push the Rossby number back below $\mathrm{Ro_{crit}}$, reinvigorating the large-scale dynamo and briefly sustaining an activity cycle once again before ascending the red giant branch. The current position of $\beta$ Hyi in Figure 5 generally supports this evolutionary scenario, which only returns below $\mathrm{Ro_{crit}}$ in the WMB scenario.

The activity cycle of 94 Aqr Aa is much more prominent in Ca HK than the cycle in $\beta$ Hyi. The IUE data for $\beta$ Hyi show a magnetic maximum in 1986.9, followed by a rise toward the subsequent maximum in late 1998 (top panel of Figure 1). The Ca HK observations from mid-2007 to 2013 span the next predicted maximum in late 2010, and show a slightly higher mean activity level during the 2010 season followed by a gradual decline from 2011–2012 (bottom panel of Figure 1). Earlier Ca HK measurements during the cycle minimum in 1993 showed a similar range ($S=0.151$–0.171; Henry et al., 1996), so the cycle amplitude in the optical appears to be substantially smaller than in the ultraviolet. This is a consequence of the larger dynamic range for magnetic variability at higher energies, which also explains the clear X-ray cycle in $\alpha$ Cen A (Ayres, 2023) that has not been detected in the optical. These stars might have been classified as “flat activity” targets from their optical data alone, but the availability of higher-energy observations revealed their cycles. By contrast, Mount Wilson observations of 94 Aqr Aa show a robust cycle in Ca HK (Metcalfe et al., 2020).

The difference between the Ca HK variability of $\beta$ Hyi and 94 Aqr Aa may be a consequence of their relative Rossby numbers. Because rotation and activity are strongly coupled prior to the onset of WMB, $\mathrm{Ro_{crit}}$ can be converted into a critical activity level ($\log R^{\prime}_{HK}\!\!\approx\!-4.95$; Brandenburg et al., 2017). For any given star, this critical activity level corresponds to a specific Mount Wilson S-index ($\mathrm{S_{crit}}$), which depends on the B$-$V color (Noyes et al., 1984). For $\beta$ Hyi this value is $\mathrm{S_{crit}}=0.165$, which falls above most of the observations shown in the bottom panel of Figure 1. By contrast, the critical activity level for 94 Aqr Aa is $\mathrm{S_{crit}}=0.18$, which falls within the range of variability in the Mount Wilson measurements ($S=0.136$–0.196; Metcalfe et al., 2020). Thus, the cycle is weaker in $\beta$ Hyi because it is closer to $\mathrm{Ro_{crit}}$, while 94 Aqr Aa is well below $\mathrm{Ro_{crit}}$ and has a stronger cycle.