Asteroseismology and Spectropolarimetry of the Exoplanet Host Star λ Serpentis

The Transiting Exoplanet Survey Satellite (TESS) observed λ Ser during Sector 51 (2022 April 22–2022 May 18) in 20 s cadence, which has been demonstrated to have superior photometric precision to 2 minute data for bright stars (Huber et al. 2022). We were able to improve on the standard Science Processing Operations Center (SPOC) data product by using our own method to extract the light curve (Nielsen et al. 2020). We approached the postage stamp image pixel by pixel, extracting a time series for each pixel. We then took the brightest pixel, which was also the closest to the nominal position of the star, as our initial time series. The pixel time series quality figure of merit was parameterized by

where f_i is the flux at cadence i, and N is the length of the time series. Using the first differences of the light curve acts to whiten the time series and thus correct for its nonstationary nature (Nason 2006); similar approaches have been used in astronomical time series analysis by García et al. (2011), Buzasi et al. (2015), Prša et al. (2019), and Nielsen et al. (2020), among other authors. We then added the light curve from the pixel that most decreased our figure of merit and repeated the process until the light-curve quality as measured by our quality measure q ceased to improve. The resulting pixel collection was adopted as our aperture mask. Finally, we detrended the light curve produced using this mask against the centroid pixel coordinates by fitting a second-order polynomial with cross terms. Similar approaches have been used for K2 data reduction (see, e.g., Vanderburg & Johnson 2014).

The overall noise level of our light curve, measured point to point, is approximately 7% better than that of the SPOC product, and we were able to recover more points: 67,491, as compared to 55,445 in the SPOC light curve (or 58,528 if the requirements are relaxed to include points with nonzero quality flags). This improved the duty cycle from 52% or 55% to almost 64%. The top panel of Figure 1 shows the power spectrum of the resulting time series with clear evidence for solar-like oscillations centered near 1900 μHz. We use this same approach in the Appendix to quantify nondetections of oscillations in eight additional TESS targets.

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To extract individual frequencies, four different groups of coauthors applied either iterative sine-wave fitting (e.g., Kjeldsen et al. 2005; Lenz & Breger 2005; Bedding et al. 2007) or 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. 2015; Li et al. 2020; Breton et al. 2022). For each mode, we required at least two independent methods to return the same frequency within the uncertainties. 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.

The bottom panel of Figure 1 shows an echelle diagram with a large separation of Δν = 89.5 μHz and the extracted frequencies. We identified six radial (l = 0) and eight dipole (l = 1) modes, but we were unable to identify any quadrupole (l = 2) modes with confidence. Mode identification was confirmed using well-known patterns between Δν and the frequency offset (White et al. 2011) and by comparison with similar stars (e.g., KIC 7296438) from the Kepler LEGACY sample (Lund et al. 2017). The final frequency list is shown in Table 1, providing a primary input for the asteroseismic modeling described in Section 3.1.

Spectropolarimetric observations of λ Ser were obtained on 2021 May 24 using the Potsdam Echelle Polarimetric and Spectroscopic Instrument (PEPSI; Strassmeier et al. 2015) installed at the 2 × 8.4 m Large Binocular Telescope (LBT). The instrumental setup, resulting in R = 130,000 observations over the 475–540 and 623–743 nm wavelength regions, and the data reduction procedures were the same as described in Metcalfe et al. (2019). Considering the low-amplitude polarization signal, a multiline method is necessary to achieve a magnetic field detection. Here we employed the least-squares deconvolution (LSD; Kochukhov et al. 2010) technique to derive high-quality intensity and circular polarization profiles. The line mask necessary for this analysis was constructed from the output of an "extract stellar" request to the VALD database (Ryabchikova et al. 2015) with stellar atmospheric parameters from Brewer et al. (2016). This line list contained 1300 metal lines deeper than 10% of the continuum in the wavelength region covered by our PEPSI data. Combining information from these lines yielded an LSD Stokes V profile with an uncertainty of 4.6 ppm, which showed a clear magnetic signal (see Figure 2). We measured a mean longitudinal magnetic field G from these observations and estimated the strength of an axisymmetric dipole magnetic field to be B_d = 3.8 G using the line profile modeling technique described in Metcalfe et al. (2019) and adopting i = 50° (see Section 3.2). However, the resulting synthetic profile (dotted red line in Figure 2) did not provide an adequate description of the observations, suggesting the presence of nonaxisymmetric global field components. Consequently, we generalized the modeling by allowing an inclined dipole geometry. This produced a better fit to the observed Stokes V profile with B_d = 6.0 G and a magnetic obliquity of β = 515 (dashed blue line in Figure 2).

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Additional spectropolarimetric observations of λ Ser covering a broad range of rotation phases were collected with Neo-NARVAL during the summer of 2021, allowing us to model the detailed morphology of its large-scale magnetic field. The complete data set included 19 visits to the star, secured between 2021 July 7 and 2021 August 17, with a maximum of one observation per night. Neo-NARVAL is an upgrade to the NARVAL instrument at Télescope Bernard Lyot (TBL; Aurière 2003). Neo-NARVAL echelle spectra collect a broad optical wavelength region (370–1000 nm) in a single frame, with a spectral resolution close to R = 65,000. Every polarimetric sequence is obtained from the combination of four exposures taken with the two half-wave Fresnel rhombs rotated about the optical axis (Semel et al. 1993). Each polarized sequence provides simultaneous access to a Stokes I spectrum and another Stokes parameter (circular or linear polarization). The data set gathered for λ Ser was restricted to the Stokes V parameter, since the amplitude of the Zeeman signatures is expected to be largest in circular polarization (Landi Degl'Innocenti 1992). Each sequence also provides a "null" spectrum, which should contain only noise and serves as a diagnostic of possible instrumental or stellar contamination to the polarized spectrum.

The Neo-NARVAL upgrade, installed in 2019, consisted of a new detector and enhanced velocimetric capabilities (López Ariste et al. 2022). At the time of our observations, the instrument suffered from a loss of flux in the bluest orders, later identified as a fiber link issue. The new reduction pipeline (López Ariste et al. 2022) provided an unsatisfactory extraction of spectral orders affected by a very low signal-to-noise ratio (S/N), so we discarded from our reduced data all spectral bins with a wavelength below 470 nm. The LSD analysis and interpretation of these data are described in Section 3.2.

We observed λ Ser with the Chandra High Resolution Camera-Imaging detector (HRC-I; ObsID 22307) on 2020 April 25 in a single pointing with a net exposure time of 6103 s. This instrument was preferred over the Advanced CCD Imaging Spectrometer (ACIS) because a growing contamination layer on the ACIS optical blocking filter severely curtails the low-energy response below about 1 keV. The HRC-I data were reprocessed using the Chandra Interactive Analysis of Observations (CIAO; Fruscione et al. 2006) software version 4.15 and calibration database version 4.10.2. Since the HRC-I has essentially no intrinsic energy resolution, the analysis entailed examining the source photon event list for significant variability, extracting photon events attributed to λ Ser, and converting the observed count rate (CR) into a source flux.

An image of the detected events in the vicinity of λ Ser is shown in Figure 3. Overlaid is the adopted source extraction region (yellow), together with the innermost of two background regions (cyan) employed to estimate the background signal. The source region was placed at the centroid of the detected events and had a radius of 15, which corresponds to an encircled energy fraction of 95%. The background region illustrated in Figure 3 was an annulus with inner and outer radii of 35 and 7'', respectively, centered on the source. We also estimated the background rate using a much larger annulus, covering the radius interval 130''–165'', to check for the presence of spatial variations in the background. In both cases, the net source CR was 0.049 ± 0.003 counts s⁻¹ after correction for the encircled energy fraction. The extracted source counts were examined for variability using the Gregory–Loredo algorithm (Gregory & Loredo 1992); no significant variability was detected.

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In addition to the Chandra observation, λ Ser was observed in soft X-rays during the ROSAT era, initially as part of the all-sky survey (scans were acquired in 1990 August) and later during the pointed phase of the Guest Investigator program (a PSPCB exposure in 1997 February, toward the end of the mission). The CRs for the two ROSAT observations were taken from facility catalogs hosted by the High-Energy Science and Archive Research Center at the NASA Goddard Space Flight Center, as accessed through W3browse. ³¹ The rass2rxs catalog listed CR = 0.083 ± 0.016 counts s⁻¹ for a sky survey exposure of 511 s. The pointings catalog rospspctotal reported CR = 0.072 ± 0.006 counts s⁻¹ for an exposure of 2.73 ks. The energy bandpass is the ROSAT standard, 0.1–2.4 keV. Documentation for these databases can be obtained through W3browse.

The CRs from ROSAT and Chandra were converted to X-ray fluxes at Earth using a method to derive an optimum energy conversion factor for each camera system. The approach applied a sequence of coronal emission-measure distributions (EMDs) convolved with detector-dependent sensitivity curves to calculate X-ray surface fluxes for the target, finding the optimum EMD realization that achieved consistency between the calculated and model surface fluxes (each EMD level corresponds to a specific predicted X-ray surface flux). The approach was described by Ayres & Buzasi (2022) and is based on the empirical EMD models derived by Wood et al. (2018) from Chandra Low-Energy Transmission Grating spectra of nearly two dozen F–M dwarfs. Detector sensitivity curves were calculated for the reference grid of each EMD using WebPIMMS ³² for the unabsorbed 0.1–2.4 keV X-ray flux, a solar abundance APEC plasma model, and an interstellar column of 1 × 10¹⁸ cm⁻², appropriate for a nearby (d = 11.9 pc) star. There was good consistency among the three independent X-ray fluxes: f_X = 4.0 ± 0.4 × 10⁻¹³ erg cm⁻² s⁻¹. At the Gaia distance, this corresponds to L_X = 6.8 ± 0.6 × 10²⁷ erg s⁻¹, or = 27.83, about seven times larger than the sunspot cycle average Sun (Ayres & Buzasi 2022).

We can estimate the mass-loss rate of λ Ser by combining the X-ray luminosity determined above with the stellar radius inferred from asteroseismology (see Section 3.1). For stars with mass-loss rates determined directly from Lyα measurements and other techniques, there is an empirical relation between the mass-loss rate and the X-ray flux per unit surface area, (Wood et al. 2021). The resulting estimate is slightly above the solar value, .

We used synoptic observations of the S-index of chromospheric activity from the Mount Wilson Observatory (MWO) HK Project (Wilson 1978; Baliunas et al. 1996) and the Keck High Resolution Echelle Spectrometer (HIRES; Baum et al. 2022) to measure the rotation period of λ Ser and characterize its long-term magnetic variability. The S-index was defined by the MWO HK Photometer (HKP) and measured the ratio of emission from 0.1 nm cores of the chromospheric Ca ii H and K lines to the sum of two nearby 2 nm pseudocontinuum bandpasses (Vaughan et al. 1978). This long-used proxy for stellar magnetic activity reveals the presence of decadal-scale cycles in the Sun (e.g., White & Livingston 1981; Egeland et al. 2017) and Sun-like stars (e.g., Baliunas et al. 1995; Hall et al. 2007; Egeland 2017). The passage of surface active regions modulates the S-index such that when sampled at a sufficient cadence, the stellar rotation period can be obtained (Baliunas et al. 1983, 1996; Donahue et al. 1996). The MWO HK Project observed λ Ser from its inception in 1966 until its termination in 2003. Previous studies have reported results on partial records of λ Ser, but here we analyze the complete time series obtained by MWO.

We extend the MWO observations using Keck Observatory HIRES-2 data obtained for the California Planet Search and published in Baum et al. (2022). These HIRES observations (R ∼67,000) cover the Ca ii H and K region, and the S-index is obtained by reducing the spectra and integrating the bandpasses of the MWO HKP-2 spectrophotometer (Vaughan et al. 1978; Isaacson & Fischer 2010). While some data from Baum et al. (2022) were adjusted with a constant shift to be consistent with MWO, no such shift was applied to the data set for λ Ser. The composite data set is shown in the top panel of Figure 4 with the seasonal means indicated. Some low-frequency variation is apparent by visual inspection of the seasonal means. Activity falls from the beginning of the observations in 1966 (S = 0.1629) until the global minimum seasonal mean in 1979 (S = 0.1509). From there, activity rises until about 1988 (S = 0.1620) and remains relatively constant thereafter. The global maximum seasonal mean occurs in 2001 (S = 0.1679), but this appears to be an intermittent outburst reaching levels comparable to 1999 and 2018. The standard deviation of the seasonal means including and after the 1988 season is σ_S = 0.00236 (N = 29), which is about half of the standard deviation for the whole series (σ_S = 0.00460, N = 50). For reference, solar minimum has S = 0.1621, and the mean cycle maximum is S = 0.177 (Egeland et al. 2017).

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We employed the Lomb–Scargle periodogram (Lomb 1976; Scargle 1982; Horne & Baliunas 1986) to search for periodicity in the separate (MWO, Keck) and composite time series. A Monte Carlo of 100,000 trials was used, drawing from a Gaussian distribution with the same variance as the data and using the original sampling, to determine the periodogram power threshold level for a 0.1% false-alarm probability (FAP), i.e., the probability that a periodogram peak could be generated by Gaussian noise. Peaks above this level are considered significant and were ranked in order of decreasing power. From the MWO time series, with a duration of 36.3 yr, the top three significant peaks are found at 39.1, 6.7, and 10.7 yr, and the remaining four significant peaks have periods of less than 3.5 yr. While the Lomb–Scargle periodogram method is capable of detecting harmonic periodicity that extends beyond the duration of the data, such low-frequency peaks in a more complex time series can be a result of the data window and must be viewed with extreme caution. From the Keck HIRES-2 time series, with a duration of 14.3 yr, no significant low-frequency periods are detected within the data window, and a peak of 20.5 yr is found beyond it. Finally, from the composite data set, significant peaks are found at 9.65 ± 0.09, 7.00 ± 0.05, and 13.4 ± 0.2 yr, the first two corresponding well to the MWO-only time series. These results compare well to the Egeland (2017) study that combined the MWO data with observations from the Lowell Observatory Solar Stellar Spectrograph (SSS), where peaks of 43.3, 6.7, and 12.8 yr were found. In that study, the long-period peak was viewed with skepticism due to a step discontinuity in the SSS data corresponding to a CCD upgrade in the SSS. The HIRES-2 data do not suffer from such a discontinuity, and the power at low frequencies from the composite data set is diminished, indicating that the ∼40 yr "cycle" suggested by the MWO data alone is not supported by these extended observations. Variability on the scale of ∼7 to ∼13 yr is the most prominent and reliable; however, the variations in λ Ser are not as clean and Sun-like as the solar cycle, and the term "cycle" for these periodicities should be applied with caution, as discussed more generally in Egeland (2017). The periodogram in Figure 4 is expressed in units of normalized amplitude , where P_N is the usual periodogram power normalized by the variance. This normalization is useful for comparing time series of different lengths and judging cycle quality, as it (1) is independent of the number of observations, (2) has a maximum value of 1, and (3) is a relative measure of signal purity for a given peak (see discussion in Egeland 2017).

Individual seasons of the MWO and Keck time series with more than 20 observations were analyzed to search for rotational modulation, following previous efforts by Donahue (1993) and Donahue et al. (1996). As with the cycle search, a Lomb–Scargle periodogram was employed using a 100,000 trial Monte Carlo to determine the power threshold for a 5% FAP. For a search range between 10 and 40 days, peaks above the 5% FAP threshold are reported as seasonal rotation periods. When significant peaks were found, a 100,000 trial periodogram Monte Carlo was used, adjusting the observations within their errors to determine the period uncertainty. We compare our results to the previous work of Donahue (1993) in Table 2. Our analysis largely confirms the earlier results, which found rotation signals in six seasons ranging from 23.6 to 28.8 days. We also find significant periods in six seasons, though not the same six found by Donahue, with rotation ranging from 19.4 to 28.0 days. The differences may be ascribed to the following: (1) Donahue analyzed individual S-index observations, while we used a nightly average from typically three observations per night; (2) the MWO time series were recalibrated after Donahue's work; and (3) different observation rejection criteria were employed. No significant rotation periods were found in the lower-cadence MWO data beyond the 1993 season, which were not analyzed by Donahue (1993), nor in the Keck HIRES-2 data. The Keck observations tend to be clustered around a few dates within a season, making them unsuitable for a rotation period search.

To summarize, our analysis of the composite MWO and Keck S-index time series indicates a mean rotation period of 24.3 ± 2.7 days, with strong indications of differential rotation, . Significant long-term variability at 9.7, 7.0, and 13.4 yr was found, but it does not appear to be dominated by a single period that would indicate a "clean" cycle, as for the Sun. A large-amplitude long-period variation of approximately 40 yr is apparent in the MWO data, but the 54 yr composite data set does not support this periodicity being cyclic. Extended uniform data sets are required to determine whether such long-period cycles exist in λ Ser or other stars.

As an independent determination of the stellar properties, we performed an analysis of the broadband spectral energy distribution (SED) of λ Ser and the Gaia DR3 parallax (with no systematic offset applied; see, e.g., Stassun & Torres 2021) to derive an empirical measurement of the stellar radius, following the procedures described in Stassun & Torres (2016), Stassun et al. (2017), and Stassun et al. (2018). We adopted the UBV magnitudes from Mermilliod (2006), the B_T V_T magnitudes from Tycho-2, the Strömgren ubvy magnitudes from Paunzen (2015), the JHK_S magnitudes from 2MASS, the W1–W4 magnitudes from WISE, the G_BP G_RP magnitudes from Gaia, and the far-UV (FUV) magnitude from GALEX. Together, the available photometry spans the full stellar SED over the wavelength range 0.2–22 μm.

We performed a fit using Kurucz stellar atmosphere models with the effective temperature (T_eff), surface gravity (), and metallicity ([M/H]) from Brewer et al. (2016), with uncertainties inflated to account for a realistic systematic noise floor. The remaining free parameter is the extinction A_V , which we fixed at zero due to the star's proximity (d = 11.9 pc). The resulting fit (Figure 5) has a reduced χ² of 1.1, excluding the GALEX FUV flux, which indicates a moderate level of activity. Integrating the (unreddened) model SED gives the bolometric flux at Earth, F_bol = 4.481 ± 0.052 × 10⁻⁷ erg s⁻¹ cm⁻². Taking the F_bol together with the Gaia parallax gives the bolometric luminosity, L_bol = 1.984 ± 0.023 L_⊙, which together with T_eff gives the stellar radius, R = 1.349 ± 0.024 R_⊙. In addition, we can estimate the stellar mass from the empirical relations of Torres et al. (2010), giving M = 1.17 ± 0.07 M_⊙, which is consistent with that obtained directly from R and (M = 1.11 ± 0.12 M_⊙). These estimates of the radius and mass can be compared to the adopted values from asteroseismology in Section 3.1.

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As a probe of the physical environment between the photosphere and the corona, we also observed λ Ser using the Cosmic Origins Spectrograph on the Hubble Space Telescope (HST). We obtained a low-resolution G140L FUV spectrum with an integration time of 1976 s on 2020 September 2 (program 15991), which was reduced with standard pipeline processing. We computed integrated fluxes for isolated emission lines in the rest frame of the star above nearby pseudocontinua, together with rms errors. We estimated continua from linear fits to clusters of low points on either side of the line in question. When the target line was blended, two methods were employed. Some weaker blends were removed by fitting a Voigt function to better mimic the convolved instrumental profile and line shape, leaving the residual target line for flux integration as before. In some cases, we fit the entire complex of lines with multiple Voigt functions. The results are shown in Table 3. Tests on isolated lines demonstrated that straight integration and Voigt fitting yielded similar results, typically within ±5%. In several cases, multiple nearby lines of the same ion were combined. Following Ayres (2020), we also measured a 10 Å segment of relatively line-free FUV pseudocontinuum centered at 1506 Å.

We can compare the FUV fluxes of λ Ser to the Sun and α Cen A, a somewhat older (5.3 Gyr; Joyce & Chaboyer 2018), more metal-rich ([Fe/H] = +0.24; Morel 2018) solar analog (see Table 4). Lower chromospheric surface fluxes in Cl i and O i (with temperatures of peak emissivity = 3.8–3.9) are 1.3–1.5 times larger in λ Ser; the upper chromospheric C ii flux () is similarly ≈1.5× enhanced. Moving to lines formed in the stellar transition region, C iii () is ≈2× enhanced in λ Ser relative to the Sun, and Si iv ( = 4.9) is further enhanced at a factor of 2.3 and 1.9 relative to the Sun and α Cen A, respectively (the higher metallicity of α Cen A may boost its emission). In the C iv doublet ( = 5.0), the enhancement in λ Ser returns to a factor of ≈1.5 (here some optical depth effects may play a role). These results are generally in line with the more active corona of λ Ser—which shows 3.8× the solar surface F_X —and the reduced activity enhancements relative to the corona that are expected for lower T_peak emission in the chromosphere and transition region (e.g., Ayres & Buzasi 2022, among many).

Several density-sensitive line ratios are available in the HST spectra. We use the intersection of these results to estimate the electron density in the transition region. The Si iii (1892 Å)/C iii (1909 Å) ratio, with , yields using Keenan et al. (1987). The ratio C iii (1908 Å)/Si iv (1402 Å), with , gives . The ratios O iii (1666 Å)/Si iv (1402 Å) and C iii (1908 Å)/O iii (1666 Å), also with , were less certain, affected by larger errors in the O iii line; these yield and , respectively (using Keenan et al. 1988). Another diagnostic, C iii (1908 Å)/Al iii (1863 Å), implies (employing Keenan et al. 1990). A hotter diagnostic, the ratio O iv (1401 Å)/O iv (1407 Å) at (using results of Brage et al. 1996), unfortunately gives only a weak limit of due to large errors on the fluxes. Combining the cooler diagnostics, we find an average at . For comparison, using the hotter O iv ratio in the Sun (e.g., Rao et al. 2022). This suggests that λ Ser has transition region densities similar to or perhaps slightly lower than the Sun at a fixed temperature, which is consistent with its slightly lower surface gravity. We caution, however, that assumptions intrinsic to the line ratio method make the results uncertain (see discussion in Judge 2020).

In summary, λ Ser has chromospheric and transition region fluxes broadly consistent with a star that is somewhat more coronally active, with a slightly lower density than the Sun.

Using the oscillation frequencies listed in Table 1, the spectroscopic constraints on T_eff and [M/H] from Brewer et al. (2016), and the luminosity from Section 2.5, five teams attempted to infer the properties of λ Ser from asteroseismic modeling. A variety of stellar evolution codes and fitting methods were employed, including ASTEC/AMP (Christensen-Dalsgaard 2008; Metcalfe et al. 2009), GARSTEC/BASTA ( Weiss et al. 2008; Aguirre Børsen-Koch et al. 2022), MESA (Paxton et al. 2015; Li et al. 2023), and YREC (Demarque et al. 2008). We found reasonable agreement between the results for the stellar radius and mass, with individual estimates ranging from R = 1.33 to 1.38 R_⊙ and M = 1.03 to 1.13 M_⊙, but there was a significant spread in stellar age with inferences between 5.4 and 8.6 Gyr around a median value of 7.0 ± 0.8 Gyr. For consistency with the rotational evolution modeling in Section 3.3, we adopted the modeling results from YREC, which yielded the median estimates of radius and mass with an age at the young end of the distribution (see Table 5).

Table 5. Adopted Properties of the Exoplanet Host Star λ Ser

	λ Ser	Source
Teff (K)	5901 ± 78	1
[M/H] (dex)	+0.04 ± 0.07	1
(dex)	4.22 ± 0.08	1
B − V (mag)	0.60	2
(dex)	−5.004	2
Prot (days)		3
∣Bd∣ (G)	2.73, 2.12	4
∣Bq∣ (G)	1.90, 2.21	4
∣Bo∣ (G)	1.37, 2.44	4
LX (1027 erg s−1)	6.8 ± 0.6	5
Mass-loss rate ()	1.6 ± 0.2	5
Luminosity (L⊙)	1.984 ± 0.023	6
Mass (M⊙)	1.09 ± 0.04	7
Radius (R⊙)	1.363 ± 0.031	7
Age (Gyr)	5.4 ± 0.7	7
Torque (1030 erg)		8

References. (1) Brewer et al. (2016); (2) Baliunas et al. (1996); (3) Section 2.4; (4) two ZDI reconstructions in Section 3.2; (5) Section 2.3; (6) Section 2.5; (7) Section 3.1; (8) Section 3.4.

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The YREC results were obtained from a grid of models that were constructed with the Yale Stellar Evolution Code (Demarque et al. 2008). All models were constructed with the same microphysics inputs: OPAL opacities (Iglesias & Rogers 1996) supplemented with low-temperature opacities from Ferguson et al. (2005), the OPAL equation of state (Rogers & Nayfonov 2002), and nuclear reaction rates from Adelberger et al. (1998), except for the ¹⁴N (p, γ)¹⁵O reaction, for which we use the rate of Formicola et al. (2004). Additionally, the models included gravitational settling of helium and heavy elements using the formulation of Thoul et al. (1994), with the diffusion coefficient modified using the mass-dependent factor of Viani & Basu (2017).

We first determined mass and from the global asteroseismic parameters using the Yale–Birmingham (YB) pipeline (Gai et al. 2011). This step informed us of the mass range; we used the inferred mass and a ±3σ range around it, i.e., 0.99–1.25 M_⊙, to construct a grid of models. For each mass, models were created with seven values of the mixing length parameter spanning α_MLT = 1.5–2.3 and initial helium abundances from 0.20 to 0.32. The initial [M/H] of the models spanned the range 0.0 to +0.3 dex to account for the diffusion and settling of heavy elements. The models were evolved from the zero-age main sequence. Models along a track were output within ±3σ of the returned by the YB pipeline, and their frequencies were calculated with the code of Antia & Basu (1994).

The properties of λ Ser were determined as follows. We first corrected for the surface term using the two-term correction proposed by Ball & Gizon (2014). The corrected frequencies were used to define a χ²,

with N being the number of modes, which was then used to determine a likelihood function,

with C being the normalization constant. We also defined a likelihood for each of the other observables, T_eff, [M/H], and luminosity L. For instance, the likelihood for effective temperature was defined as

with

where σ_T is the uncertainty on the effective temperature, and D is the constant of normalization. We similarly defined the likelihoods for [M/H] and L. The total likelihood for each model is then

The quantity is a prior that we used to down-select models with ages greater than 13.8 Gyr; without this prior, the likelihood distribution would have a sharp cutoff. We define as

where the age τ is in units of gigayears, and σ_τ is chosen to be 0.1 Gyr.

The medians of the marginalized likelihoods of the ensemble of models were used to determine the stellar properties, after converting them to a probability density by normalizing the likelihood by the prior distribution of the property. We repeated the exercise by perturbing each of the nonseismic inputs (T_eff, [M/H], and L) by a normally distributed random amount with variance given by the observational errors. The distribution given by the ensemble of medians was used to determine the final stellar properties shown in Table 5.

Like the PEPSI data presented in Section 2.2, polarized Zeeman signatures in the individual lines of reduced Neo-NARVAL spectra are dominated by noise. We employed the LSD method to extract a cross-correlation line profile from a list of photospheric lines (Donati et al. 1997; Kochukhov et al. 2010). The line mask was chosen to be closest to the fundamental parameters of λ Ser in the grid of Marsden et al. (2014), keeping lines deeper than 40% of the continuum with no telluric contamination. The normalized Landé factor of the LSD profiles is close to 1.2, while their normalized wavelength is equal to 650 nm.

Owing to the lack of lines bluer than 470 nm, we ended up with slightly less than 1700 available spectral lines and an S/N of LSD profiles (per 1.8 km s⁻¹ velocity bin) between 11,000 and 24,000, with a mean value of 19,000. Even after the LSD processing, the Zeeman signatures remained barely visible by eye, which is consistent with the small polarized amplitude found by PEPSI (see Section 2.2). Applying the criterion of Donati et al. (1992, 1997), we obtain only four marginal detections (FAP between 10⁻⁴ and 10⁻⁵), with all other observations considered nondetections. The LSD profiles are shown in Figure 6.

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Running ZDI (Semel 1989) on polarization signatures close to the detection threshold is not ideal, but previous studies have shown that it is possible to reconstruct magnetic maps even when the signatures are dominated by noise, provided that a sufficient number of observations are combined together in the inversion process (Petit et al. 2010, 2022). Here we use the ZDI implementation of Folsom et al. (2018a, 2018b) in a procedure closely following the one presented by Petit et al. (2021). The surface magnetic field is described by the set of spherical harmonic equations in Donati et al. (2006), and we limited the expansion to because the magnetic model was not improved by including higher-order terms.

We adopted a projected rotational velocity of 2 km s⁻¹ (Brewer et al. 2016), resulting in an inclination angle of 50° when combined with our estimates of P_rot and R. The radial velocity of our LSD profiles was −65.9 km s⁻¹, which is about 0.5 km s⁻¹ larger than recent estimates (Soubiran et al. 2018). Following Petit et al. (2008), we ran a series of ZDI inversions assuming different values of the rotation period and found that the best fit was obtained for P_rot = 26.87 days. Following this estimate, the rotational phase Φ of each observation was calculated using the following ephemeris:

where the initial Heliocentric Julian date HJD₀ was from our first observation at 2459406.373. It is clear from Figure 6 that the phase coverage is very good between Φ = 0 and 0.4 (observed over two consecutive rotation cycles), while only two observations have phases above 0.5. We also performed a search for differential rotation following the method of Petit et al. (2002) but failed to measure a surface shear. The target reduced χ² of the ZDI inversion was fixed at 1.075 because adopting lower values led to clear signs of overfitting (visible as a sharp increase in the average field strength and complexity). The resulting map is shown in Figure 7, while the synthetic Stokes V LSD profiles are plotted with red lines in Figure 6.

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The reconstructed magnetic geometry has an average field strength of 2 G, with a maximum local peak strength of 5 G. A majority of the magnetic energy (87%) shows up in the poloidal field component, more specifically in the dipole component that hosts 71% of the poloidal magnetic energy. The dipole strength is equal to 2.9 G, and it is very inclined with respect to the spin axis, with a negative pole located at a latitude of ∼11°. Unsurprisingly, this very nonaxisymmetric magnetic configuration leads to only 9% of the magnetic energy in modes with m = 0.

Considering the low amplitude of the polarization signatures in the Neo-NARVAL LSD profiles, we carried out an independent ZDI reconstruction with an alternative inversion code (Kochukhov et al. 2014; Rosén et al. 2016; Lehtinen et al. 2022). This inversion adopted the same P_rot, , i, and , resulting in a qualitatively similar magnetic field distribution to the one illustrated in Figure 7 but with a somewhat stronger and more structured magnetic field map. In this case, we found an average field strength of 3.7 G and a maximum local strength of 8.9 G. The contributions of the poloidal and toroidal components are nearly equal, with the dipole component containing 48% of the total magnetic field energy and 31% of the poloidal field energy. The discrepancies of these parameters with the outcome of the first ZDI reconstruction likely reflect intrinsic limitations of ZDI based on low-S/N data. Nevertheless, the dipole field characteristics (strength 2.1 G, obliquity 98° toward the positive pole) are similar to those obtained in the first inversion.

To estimate the rate of angular momentum loss for λ Ser, we use the braking law of Finley & Matt (2018; see Section 3.4). This braking law requires the polar strengths of the dipole, quadrupole, and octupole components of the stellar magnetic field as inputs. These can be obtained from the reconstructed ZDI maps. However, the magnetohydrodynamic simulations used to construct the Finley & Matt (2018) braking law were run using only axisymmetric magnetic field modes, whereas the ZDI maps contain both axisymmetric and nonaxisymmetric components. In order to calculate the equivalent polar field strengths needed for the braking law from each ZDI map, we used the method we employed in Metcalfe et al. (2022). Briefly, this method calculates the magnetic flux in each of the dipole, quadrupole, and octupole components of the ZDI map, i.e., accounting for both the axisymmetric and nonaxisymmetric components. We then determine the polar field strengths of a purely axisymmetric dipole, quadrupole, and octupole that reproduces the respective magnetic fluxes of each component from the ZDI map. These are the field strengths reported in Table 5 (first and second reconstruction) and used in the braking law.

We fit a rotational evolution model to λ Ser following the methodology described in Metcalfe et al. (2020). We use slightly different values for two braking law parameters: a braking normalization of f_k = 8.53 for the standard law and f_k = 8.97 and Ro_crit = 2.01 for the weakened magnetic braking law, derived from calibrating the braking law to the asteroseismic rotator sample of Hall et al. (2021), open clusters, and the Sun (Saunders et al. 2023). This amounts to a braking normalization that is ∼40% higher and an Ro_crit that is 7% lower than that used in Metcalfe et al. (2020) for the weakened law and a braking normalization that is ∼25% higher for the standard law. These changes would tend to make a star of a given age and mass spin more slowly, although weakened magnetic braking occurs at slightly faster rotation rates.

We search for a best-fit model that matches our observed surface temperature, luminosity, and surface metallicity, with asteroseismic priors on the mass, age, and mixing length as described in Metcalfe et al. (2020). Our best-fit model reproduces all surface observables and priors (with the exception of rotation) within 1σ. For a standard model, we predict a rotation period of 34 ± 6 days, while weakened magnetic braking predicts a period of 18 ± 2 days. The weakened magnetic braking model is consistent with the most rapid seasonal rotation rate observed for λ Ser. If λ Ser is in fact viewed at a moderate inclination and has solar-like differential rotation, we might expect the observed rotation period to be slower than the equatorial rotation period that is predicted by the solid-body stellar models. However, its mean rotation period is in mild tension with both the weakened magnetic braking and the standard case and does not conclusively distinguish between the two scenarios.

Bringing together the magnetic field properties derived in Section 3.2, the mass-loss rate estimated in Section 2.3, the range of rotation periods measured in Section 2.4, and the asteroseismic radius and mass from Section 3.1, we can use the prescription of Finley & Matt (2018) to estimate the wind braking torque of λ Ser. We repeat the calculation using the magnetic field properties (B_d , B_q , B_o ) from two independent ZDI reconstructions that relied on the same set of Stokes V profiles (see Section 3.2). In Table 5, we report the average torque resulting from the two calculations, and we adopt half of the difference between them as the uncertainty arising from the magnetic field properties (11%). The total uncertainty on the torque includes additional contributions from the rotation period (6%), mass-loss rate (4%), radius (4%), and mass (1%), and it reflects the range of possible torques when all quantities are shifted by ±1σ.

The estimated wind braking torque for λ Ser is shown relative to several other stars in Figure 8. We calculate the Rossby number for each star from the Gaia G_BP − G_RP color using the asteroseismic calibration from Corsaro et al. (2021). We estimate the wind braking torque following the methodology outlined in Metcalfe et al. (2021, 2022), while the solar point (⊙) comes from Finley et al. (2018). On this scale, the empirical value of the Rossby number that corresponds to the onset of weakened magnetic braking is Ro_crit = 0.46 (dashed line). The horizontal error bar for λ Ser corresponds to the range of seasonal rotation periods identified in Section 2.4, while the vertical error bar is dominated by uncertainties in the strength and morphology of the large-scale magnetic field (see Section 3.2), with progressively smaller contributions from the rotation period, mass-loss rate, radius, and mass. Even considering the uncertainties, the wind braking torque for λ Ser is much higher than for other stars with a similar Ro (see τ Cet; Metcalfe et al. 2023).

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The relatively high wind braking torque for λ Ser cannot be easily attributed to an erroneous measurement. Our previous LBT observations of flat activity stars (ρ CrB, 16 Cyg A and B) resulted in null detections, while the Stokes V signature for λ Ser is strong and consistent with the lower S/N measurements from TBL. There is considerable scatter in the Wood et al. (2021) relation between X-ray flux and mass-loss rate, but the two subgiants in the calibration both have higher mass-loss rates than predicted from their X-ray flux. Despite the low mean activity level, the rotation rate inferred from MWO observations appears consistently in multiple seasons and across the complete data set. The radius and mass inferred from asteroseismology are both precise, and they agree with the independent estimates from the SED in Section 2.5. The difficulty of matching the stellar properties with rotational evolution models that assume either standard spin-down or weakened magnetic braking also suggests that λ Ser may have taken an unusual path to its present configuration. The asteroseismic age is consistent with the activity–age relation for solar analogs (Huber et al. 2022), but the remaining uncertainty prevents an unambiguous interpretation of the other stellar properties.