A 20 Second Cadence View of Solar-type Stars and Their Planets with TESS: Asteroseismology of Solar Analogs and a Recharacterization of π Men c

TESS currently observes 1000 stars per sector at 20 second cadence during the extended mission, 600 of which are selected through the TESS Guest Investigator and Directors Discretionary programs. Figure 1 shows an H-R diagram and stellar radius versus TESS magnitude for the ≈4900 unique stars with T_eff < 8000 K observed during the first 10 sectors of the extended mission (2020 July 4 to 2021 April 2), using effective temperatures and radii from the TESS Input Catalog (Stassun et al. 2018, 2019). The TESS magnitude distribution in Figure 1(b) shows a pile-up at T ≈ 8 mag, which predominantly corresponds to the 400 stars per sector observed for calibration purposes for the TESS Science Processing Operations Center (SPOC, Jenkins et al. 2016). The calibration stars are requested to be bright but unsaturated, resulting in a tendency toward more evolved red giant stars. The main-sequence sample consists of a large number of optically faint M dwarfs, which are monitored to study stellar flares (Günther et al. 2020; Feinstein et al. 2020), and a brighter sample of solar-type stars. Note that Figure 1 does not show compact stars (such as hot subdwarfs and white dwarfs), which are observed at 20 second cadence for asteroseismology (e.g., Bell et al. 2019; Charpinet et al. 2019) and to search for transiting planets (e.g., Vanderburg et al. 2020).

Zoom In Zoom Out Reset image size

To test the photometric precision, we downloaded all 20 second light curves obtained in Sectors 27–36 from the Mikulski Archive for Space Telescopes (MAST). We used the PDC-MAP light curves provided by the SPOC, which have been optimized to remove instrumental variability (Smith et al. 2012; Stumpe et al. 2012, 2014). We performed standard data processing steps, retaining only data with quality flags set to zero. We then binned each light curve to 2 minute cadence and 60 minute cadence, high-pass filtered the data with a first-order 0.5 day Savitzky–Golay filter (Savitzky & Golay 1964), and calculated the standard deviation of the binned light curves (hereafter referred to as time-domain scatter) to provide a measure of the photometric precision on those timescales. We performed the same procedure using the original 2 minute light curves for the same stars, which are a standard SPOC data product and provide a benchmark for comparison to the new 20 second light curves. We calculated the photometric precision for each sector and each star to test the dependence of the noise properties on varying conditions between different sectors.

Figures 2(a) and (b) show the measured time-domain scatter for each star and each sector as a function of TESS magnitude over timescales of 2 minutes (left panels) and one hour (right panels). As expected, the noise increases toward fainter magnitudes due to photon, sky, and read noise. For each data set, we identified stars dominated by stellar variability by scaling the TESS noise model from Sullivan et al. (2015) upward by 40%, and marking all stars with a time-domain scatter above that level (gray points).

Zoom In Zoom Out Reset image size

Figure 2 demonstrates that the 20 second light curves binned to 2 minute cadence show a strong magnitude-dependent improvement in precision compared to the original 2 minute cadence light curves. To illustrate this more clearly, Figure 2(c) shows the ratio of the two measurements, again as a function of TESS magnitude. The average scatter for 20 second light curves is ≈25% lower at T = 6 mag, ≈10% lower at T = 8 mag, and reaches parity with the 2 minute light curves around T = 13 mag. The same effect is seen for light curves binned to 1 hr cadence (Figure 2(d)), but with larger scatter. Table 1 lists the median ratios in bins of one magnitude (orange circles in the bottom panels of Figure 2) for each data set, which may be used to approximate the precision of 20 second data relative to that of 2 minute data in the magnitude range T = 6–16 mag.

Table 1. Noise Ratios between 20 second and 2 minute Data

TESS Magnitude	${[{\sigma }_{20{\rm{s}}}/{\sigma }_{2{\rm{m}}}]}_{2{\rm{m}}}$	${[{\sigma }_{20{\rm{s}}}/{\sigma }_{2{\rm{m}}}]}_{1{\rm{h}}}$
6.0	0.749 ± 0.007	0.766 ± 0.011
7.0	0.810 ± 0.005	0.771 ± 0.008
8.0	0.894 ± 0.001	0.877 ± 0.003
9.0	0.923 ± 0.002	0.925 ± 0.005
10.0	0.942 ± 0.002	0.955 ± 0.003
11.0	0.974 ± 0.001	0.985 ± 0.004
12.0	0.984 ± 0.001	0.996 ± 0.004
13.0	0.998 ± 0.001	1.006 ± 0.004
14.0	1.009 ± 0.001	1.012 ± 0.004
15.0	1.013 ± 0.001	1.027 ± 0.003
16.0	1.018 ± 0.002	1.032 ± 0.003

Note. Precision ratios apply for light curves retaining only quality flags set to zero. Keeping quality flags related to cosmic rays significantly degrades the 20 second data precision for bright stars (see text).

Download table as:  ASCIITypeset image

The improvement shown in Figure 2 can partially be explained by the difference in cosmic-ray rejection algorithms applied to 20 second and 2 minute data. Specifically, 20 second data do not undergo onboard cosmic-ray mitigation. Instead, cosmic-ray mitigation is performed through postprocessing by the SPOC, which identifies cadences affected by cosmic rays and attempts to correct their flux values. The onboard processing removes exposures with the highest and lowest flux for each stack of ten 2 second exposures, which leads to a 20% reduction in effective exposure time for 2 minute data (Vanderspek et al. 2018). This shorter effective exposure time would correspond to a precision penalty of ≈10% for 2 minute data if exposures were randomly rejected. Pre-flight simulations predicted a penalty closer to ≈3% after taking into account that only exposures with the lowest and highest flux values are rejected (Z. Berta-Thompson 2021, private communication). Since the improvement in Figure 2 is significantly larger than 3%, this implies that sources in addition to photon noise must contribute to the distribution.

One likely reason is pointing jitter, which for brighter stars should lead to larger changes in pixel values, which are then preferentially removed during the onboard cosmic-ray rejection for 2 minute data. To test this, we calculated time-domain scatter for the two halves of Sector 34, which had significantly different pointing performance. Figure 3 confirms that the second orbit of Sector 34, which has larger pointing jitter, shows a stronger improvement of 20 second compared to 2 minute data, especially for the brightest stars. The improvement becomes negligible for stars fainter than T ≈ 13 mag.

Zoom In Zoom Out Reset image size

We also repeated the calculations using three different quality-flag masks as defined in the Lightkurve package (v2.0.10): "Hardest" (rejecting all data with nonzero quality flags, as done above), "Hard" (rejecting data with severe and cosmic-ray flags only), and "Default" (rejecting data with severe flags only). ⁴⁴ Figure 4 compares the ratio of the time-domain scatter for the three mask combinations. For 2 minute data the mask choice has only a small impact (≲1% on average) and yields identical results when comparing the "Hard" and "Default" masks due to the onboard cosmic-ray rejection. Removing data with cosmic-ray-related quality flags yields significantly lower noise for 20 second data, especially for bright stars (T < 13 mag). The shape of the distribution is similar to the bottom panels of Figure 2, demonstrating that removing cadences identified as cosmic rays during postprocessing is important for the improved precision of 20 second data for bright stars. For faint stars (T > 13 mag) the choice of quality-flag mask has little influence on the precision for 20 second cadence data, which implies that the corrections applied to cadences affected by cosmic rays during postprocessing are more efficient for faint stars. We conclude that on average the best photometric precision for 20 second data is achieved when keeping only quality flags set to zero for bright stars (T < 13 mag).

Zoom In Zoom Out Reset image size

The results presented here were predicted in pre-flight simulations, which showed that spacecraft jitter would lead to excess noise when applying the onboard cosmic-ray mitigation for the brightest stars but would provide significant noise improvement for the larger number of faint stars (Z. Berta-Thompson, 2021, private communication). Additional effects that may impact the relative precision of 2 minute and 20 second light curves include the size of photometric apertures, which are calculated separately for each cadence. While a detailed investigation of these and other effects is left for future work, the confirmation of the pre-flight expectations presented here has significant ramifications for the allocation of 20 second cadence target slots, which are a scarce resource. Specifically, for stars brighter than T < 13 mag (and especially for T < 8 mag) 20 second data provide improved photometric precision irrespective of the timescale of astrophysical variability. Conversely, stars with T > 13 mag gain little from being observed in 20 second cadence unless the detection of astrophysical variability requires fast sampling (such as stellar flares or the detection of pulsations and transits for compact objects such as white dwarfs or subdwarfs).

To search for solar-like oscillations in the 20 second cadence sample, we analyzed all 84 solar-type stars observed as part of Cycle 3 Guest Investigator Program 3251 ⁴⁵ (PI Huber). We performed the same data processing steps described in the previous section and manually inspected the power spectra of each star. We only included data from Sectors 27 and 28 in this study.

We detected clear oscillations in three bright solar-like stars: γ Pav (F9V, V = 4.2 mag), ζ Tuc (F9.5V, V = 4.2 mag), and π Men (G0V, V = 5.7 mag). For π Men, the SPOC light curve for Sector 27 showed scatter that is about a factor of two larger than expected. We therefore constructed a custom light curve from the target pixel files using the Lightkurve software package (Lightkurve Collaboration et al. 2018). We selected a larger aperture than had been used to construct the SPOC light curve, thereby capturing more of the flux from π Men. The larger aperture was the single biggest factor in improving the quality of the light curve, and we confirmed that it captured all flux from saturated pixels. The light curve was extracted using simple aperture photometry. Then, to further correct for instrumental trends in the raw light curve, background pixels that were not within the target aperture were used to identify the four most significant trends via principal component analysis. The raw light curve was then detrended against these principal components, resulting in our corrected light curve. We created light curves for π Men and ζ Tuc using this method for both 20 second and 2 minute cadence data. For γ Pav we used regular SPOC PDC-MAP light curves.

Figure 5 shows the power spectrum of each star centered on the power excess due to solar-like oscillations. Note that we removed the transits of π Men c from the light curve prior to our analysis. The location of the power excess from oscillations predominantly depends on stellar surface gravity (Brown et al. 1991), and the observed excess at ≈2500 μHz for each star is consistent with predicted values from the TESS Asteroseismic Target List (ATL, Schofield et al. 2019). For comparison, the top panels of Figure 5 show power spectra calculated using the 2 minute cadence light curves of the same stars. We observe a strong improvement in S/N in all three stars, highlighting the benefit of the TESS 20 second light curve products for the study of solar-like oscillations in Sun-like stars. Indeed, this demonstrates that a few sectors of 2 minute data for π Men are insufficient for a detection of oscillations, which only become significant with 20 second cadence light curves.

Zoom In Zoom Out Reset image size

In addition to the lower time-domain noise, the improvement in S/N in Figure 5 can be attributed to the reduced amplitude attenuation enabled by the shorter integration times of 20 second cadence observations. The fractional amplitude attenuation caused by time-averaging of a signal with frequency f is given by

$$\begin{eqnarray}&&A=\mathrm{sinc}(\pi {{ft}}_{\exp }),\end{eqnarray} \tag{ 1 }$$

where $\mathrm{sinc}(x)=(\sin x)/x$ and ${t}_{\exp }$ is the exposure time. For observations with no dead time between exposures, such as those obtained by TESS and Kepler, the exposure time is equal to the sampling time, and thus the fractional attenuation in power can be written as

$$\begin{eqnarray}&&P={\mathrm{sinc}}^{2}\left(\displaystyle \frac{\pi }{2}\displaystyle \frac{f}{{f}_{\mathrm{Nyq}}}\right),\end{eqnarray} \tag{ 2 }$$

where f_Nyq = 1/(2Δt) is the Nyquist frequency for a time series with a constant sampling rate Δt.

Figure 6 shows the power spectra for the three stars on a log–log scale with lines showing the fractional power attenuation given by Equation (2). Figure 6 demonstrates that the longer sampling time of 2 minutes causes power attenuation of up to 30% at frequencies corresponding to the power excess of Sun-like stars. In contrast, the rapid sampling for 20 second cadence alleviates power attenuation, demonstrating the importance of 20 second cadence data for the study of solar analogs using asteroseismology with TESS.

Zoom In Zoom Out Reset image size

Several groups of coauthors used various methods of analysis to extract global oscillation parameters (e.g., Huber et al. 2009; Mosser & Appourchaux 2009; Mathur et al. 2010; Mosser et al. 2012a; Benomar et al. 2012; Corsaro & De Ridder 2014; Lundkvist 2015; Stello et al. 2017; Campante 2018; Nielsen et al. 2021; Chontos et al. 2021a), many of which have been extensively tested on Kepler, K2, and TESS data (e.g., Hekker et al. 2011; Verner et al. 2011; Zinn et al. 2020; Stello et al. 2021). In most of these analyses, the contributions due to granulation noise and stellar activity were modeled by a combination of Harvey-like functions (Harvey 1988) and a flat contribution due to photon noise. The frequency of maximum power (${\nu }_{\max }$) was measured either by heavily smoothing the power spectrum or by fitting a Gaussian function to the power excess. We calculated final ${\nu }_{\max }$ values given in Table 5 as the median over a total of 11 different methods, with uncertainties calculated by adding in quadrature the standard deviations over all methods and the median formal uncertainty. The ${\nu }_{\max }$ measurement uncertainties range from ≈2% to 4%.

To extract individual frequencies, different groups of coauthors applied either traditional iterative sine-wave fitting, i.e., pre-whitening (e.g., Lenz & Breger 2005; Kjeldsen et al. 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. 2012b; Corsaro & De Ridder 2014; Corsaro et al. 2015, S. Breton 2021 in preparation). For each star, we compared results and required at least two independent methods to return the same frequency within uncertainties. For the final list of frequencies we adopted values from one fitter who applied pre-whitening, 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 frequency lists are given in Tables 2, 3, and 4.

To measure the large frequency separation, Δν, we performed a weighted linear fit to all identified radial modes. Uncertainties were calculated by adding in quadrature the median formal uncertainty and the standard deviation for all estimates, yielding an average Δν uncertainty of 0.8% (Table 5). Figure 7 shows the power spectra in échelle format (Grec et al. 1983) using these Δν values, with extracted frequencies overlaid. As expected from Figure 5, the frequency extraction was most successful for γ Pav and ζ Tuc, yielding 6–7 dipole modes and strong constraints on the small frequency separation between modes with l = 0 and 2, which is sensitive to the gradient of sound speed near the core and thus stellar age (Christensen-Dalsgaard 1988). The S/N for π Men is lower due its fainter magnitude, but still allowed the extraction of several radial and nonradial modes. The offset of the l = 0 ridge in each échelle diagram, which is sensitive to the properties of the near-surface layers of the star (e.g., Christensen-Dalsgaard et al. 2014), is consistent with expectations from Kepler measurements for stars with similar Δν and T_eff (White et al. 2011).

Zoom In Zoom Out Reset image size

Table 5. Stellar Parameters

	γ Pav	ζ Tuc	π Men
Hipparcos ID	1599	105858	26394
HD Number	203608	1581	39091
TIC ID	425935521	441462736	261136679
VT magnitude	4.21	4.23	5.65
TESS magnitude	3.67	3.72	5.11
π (mas)	108.01 ± 0.11	116.18 ± 0.13	54.683 ± 0.035
fbol (10−7 erg s−1 cm−2)	5.47 ± 0.18	5.34 ± 0.17	1.409 ± 0.058
L (L⊙)	1.461 ± 0.049	1.232 ± 0.039	1.469 ± 0.061
${\nu }_{\max }$(μHz)	2693 ± 95	2660 ± 99	2599 ± 69
Δν (μHz)	119.9 ± 1.0	125.9 ± 0.8	116.7 ± 1.1
Teff (K)	6168 ± 130	5924 ± 130	5980 ± 130
[Fe/H] (dex)	−0.66 ± 0.09	−0.21 ± 0.09	0.07 ± 0.07
M⋆ (M⊙)	0.934 ± 0.021(ran) ± 0.026(sys)	0.985 ± 0.020(ran) ± 0.018(sys)	1.091 ± 0.026(ran) ± 0.016(sys)
R⋆ (R⊙)	1.057 ± 0.008(ran) ± 0.009(sys)	1.044 ± 0.008(ran) ± 0.006(sys)	1.136 ± 0.009(ran) ± 0.006(sys)
ρ⋆ (g cm–3)	1.115 ± 0.004(ran) ± 0.014(sys)	1.222 ± 0.004(ran) ± 0.017(sys)	1.050 ± 0.003(ran) ± 0.013(sys)
$\mathrm{log}g$ (cgs)	4.359 ± 0.003(ran) ± 0.007(sys)	4.394 ± 0.003(ran) ± 0.006(sys)	4.365 ± 0.004(ran) ± 0.006(sys)
Age (Gyr)	5.9 ± 0.6(ran) ± 1.0(sys)	5.3 ± 0.5(ran) ± 0.2(sys)	3.8 ± 0.7(ran) ± 0.4(sys)

Note. V and TESS magnitudes are from the Tycho-2 catalog (Høg et al. 2000) and TESS Input Catalog (Stassun et al. 2018), and parallaxes are from Gaia EDR3 (Lindegren et al. 2021). Effective temperatures and metallicities are from Aguilera-Gómez et al. (2018), with uncertainties calculated as described in the text. All other quantities are determined in this work. For modeling γ Pav we adopted [M/H] = −0.56 ± 0.09 dex based on α-element abundances in Bensby et al. (2005; see text).

Download table as:  ASCIITypeset image

Due to the brightness of our stars, their atmospheric parameters such as effective temperature and metallicity have been extensively studied in the literature. We adopted T_eff and [Fe/H] from Aguilera-Gómez et al. (2018), which were homogeneously derived from high-resolution spectroscopy. These values fall within 1σ of the median of T_eff and [Fe/H] from 15 to 30 independent studies based on both photometry and spectroscopy. We adopted a 2% systematic error in T_eff, which accounts for uncertainties in the fundamental T_eff scale based on the accuracy of angular diameters measured using optical long-baseline interferometry (White et al. 2018; Tayar et al. 2020). This estimated uncertainty was added in quadrature to the formal 50 K spectroscopic errors. ⁴⁶ We adopted a systematic uncertainty of 0.062 dex in [Fe/H] to account for method-specific offsets (Torres et al. 2012). Note that γ Pav is a metal-poor star with significant α-element enhancement of [α/Fe] = 0.13 ± 0.06 dex, calculated using individual abundances from Bensby et al. (2005). Using the conversion by Salaris et al. (1993) yields [M/H] =−0.56 ± 0.09 dex, which we adopted for model grids that do not specifically account for α-element enhancement.

To calculate bolometric fluxes (f_bol), we fitted the spectral energy distribution of each target using broadband photometry following Stassun & Torres (2016). Independent estimates were calculated from Tycho V_T and B_T photometry (Høg et al. 2000), combined with bolometric corrections from MIST isochrones (Choi et al. 2016) as implemented in isoclassify (Huber et al. 2017). Interstellar extinction was found to be negligible in both methods, consistent with the short distances of all three targets. We also extracted f_bol estimates from the infrared flux method, as described in Casagrande et al. (2011). Our final f_bol estimates were calculated as the median over all methods, with uncertainties calculated by adding the mean uncertainty and scatter over all methods in quadrature. The final f_bol uncertainties are 3% to 4%, consistent with the expected systematic offsets (Zinn et al. 2019; Tayar et al. 2020). Finally, we combined f_bol values with Gaia EDR3 parallaxes (Lindegren et al. 2021) to calculate luminosities, which provide an independent constraint for asteroseismic modeling. The results are summarized in Table 5.

Different groups of coauthors used a number of approaches to model the observed oscillation frequencies, including different stellar evolution codes (ASTEC, GARSTEC, MESA, and YREC, Demarque et al. 2008; Christensen-Dalsgaard 2008; Weiss et al. 2008; Paxton et al. 2011, 2013, 2015; Choi et al. 2016), oscillation codes (including ADIPLS and GYRE, Antia & Basu 1994; Christensen-Dalsgaard 2008; Townsend & Teitler 2013), and modeling methods (including AIMS, AMP, ASTFIT, BeSSP, BASTA, PARAM, and YB, Metcalfe et al. 2009; Stello et al. 2009; Basu et al. 2010; Gai et al. 2011; Silva Aguirre et al. 2015; Creevey et al. 2017; Serenelli et al. 2017; Rodrigues et al. 2014, 2017; Ong et al. 2021; Ball & Gizon 2017; Mosumgaard et al. 2018; Rendle et al. 2019). The adopted methods applied corrections for the surface effect (Kjeldsen et al. 2008; Ball & Gizon 2014). Model inputs included the spectroscopic temperature and metallicity, individual frequencies, Δν, and luminosity. To investigate the effects of different input parameters, modelers were asked to provide solutions with and without taking into account the luminosity constraint from Gaia.

Overall, the modeling efforts yielded consistent results and we were able to provide adequate fits to the observed oscillation frequencies, as expected for stars with properties close to those of the Sun. The modeling results excluding and including the luminosity were consistent, demonstrating that there is no strong disagreement between the luminosity implied from asteroseismic constraints and that from Gaia. To make use of the most observational constraints, we used the set of nine modeling solutions that used T_eff, [Fe/H], frequencies, and the luminosity as input parameters. From this set of solutions, we adopted the self-consistent set of stellar parameters derived using MESA following Ball & Gizon (2017), which showed the smallest difference to the median derived mass when averaged over all three stars.

Table 5 lists our final stellar parameters for each star. For properties derived from asteroseismology (radius, mass, density, surface gravity, and age) we quote random errors using the formal uncertainty of the adopted method following Ball & Gizon (2017) and systematic errors as the standard deviation of the parameter over all methods. The results show that random and systematic errors make approximately equal contributions to the error budget, highlighting the importance of taking into account effects from different model grids. This is particularly pronounced for the mean stellar density, which formally can be measured with very high precision through the relation of the large separation to the integral of the sound speed (Ulrich 1986). The average uncertainties (calculated by adding random and systematic errors in quadrature) are ≈1% in radius, ≈3% in mass, ≈1% in density, and ≈20% in age, comparable to uncertainties from asteroseismology of Kepler stars (Silva Aguirre et al. 2017; Celik Orhan et al. 2021). Systematic age uncertainties are largest for γ Pav, consistent with larger differences in model predictions for metal-poor stars.

Mosser et al. (2008) presented an asteroseismic analysis of γ Pav based on five nights of radial velocity observations with HARPS. Our results show that the identification of even and odd degree modes was reversed in Mosser et al. (2008) due to the difficulty of ambiguously extracting frequencies from single-site ground-based data. Despite the different mode identification the derived mass is broadly compatible, but we measure a significantly younger age (5.9 Gyr compared to 7.3 Gyr). The younger age for γ Pav derived here is consistent with asteroseismic red giant populations that show a flat age–metallicity relation (and thus mostly constant star formation history) for stars in the Galactic disk (Casagrande et al. 2016; Silva Aguirre et al. 2018).

Magnetic activity cycles are one of the most poorly understood aspects of stellar evolution, but they play an important role in establishing empirical age indicators such as chromospheric activity (Mamajek & Hillenbrand 2008) and the spin-down of stars (gyrochronology, Barnes 2003). Stars with asteroseismic ages and characterized chromospheric activity cycles are critical ingredients for understanding and calibrating the interplay between rotation, age, and activity. The measurement of activity cycles requires decades-long observations, which are typically only available for bright stars such as those included in the Mt. Wilson survey (Baliunas et al. 1995). TESS has already demonstrated this powerful synergy for the solar analog α Men (Chontos et al. 2021b) and the binary 94 Aqr (Metcalfe et al. 2020), thereby providing benchmarks for calibrating empirical age indicators.

The asteroseismic detections in bright solar analogs presented here provide additional benchmarks for calibrating activity–age relations. Figure 8 shows the activity–age relation for a sample of spectroscopic solar twins from Lorenzo-Oliveira et al. (2018) with measured chromospheric activity from Ca ii H&K lines ($R{{\prime} }_{\mathrm{HK}}$) and ages derived through isochrone fitting. We overplot several bright stars with asteroseismic ages (Creevey et al. 2017; Chontos et al. 2021b; Metcalfe et al. 2020, 2021), including the solar analogs in this paper, with $R{{\prime} }_{\mathrm{HK}}$ placed on the same scale as Lorenzo-Oliveira et al. (2018) using the mean S-index values compiled by Boro Saikia et al. (2018) and the T_eff values in Table 5. The resulting $R{{\prime} }_{\mathrm{HK}}$ values were corrected for metallicity effects (0.5 × [M/H] following Saar & Testa 2012, private communication). We omitted γ Pav from the plot because its low metallicity falls outside the calibration range. Error bars for the Lorenzo-Oliveira et al. (2018) sample were omitted because they do not take into account systematic errors from different model grids, as was done for the asteroseismic sample.

Zoom In Zoom Out Reset image size

Figure 8 shows that the asteroseismic sample covers the critical regime at old ages (≳3 Gyr) where the activity–age relation flattens. Interestingly, we observe that stars with similar ages and masses (such as the Sun and ζ Tuc) have similar $R{{\prime} }_{\mathrm{HK}}$ values, while stars with similar ages but significantly different masses (such as α Men A and 94 Aqr Aa, with 0.94 M_⊙ and 1.22 M_⊙) show a significant spread in $R{{\prime} }_{\mathrm{HK}}$. This implies that the spread in the activity–age relation is probably linked to a spread in stellar mass and thus the depth of the convection zone, analogous to the mass (or zero-age main-sequence temperature) dependence of gyrochronology relations (e.g., van Saders et al. 2016). Additional asteroseismic results from TESS 20 second data will be required to quantify such differential effects in activity–age relations. Additional extended mission observations in 20 s cadence will also help to decrease the error bars on asteroseismic ages by enabling the detection of a larger number of oscillation frequencies and will complement the existing database of active solar analogs already measured by Kepler (Salabert et al. 2016).

π Men joins the population of ≈110 exoplanet host stars that have been characterized using asteroseismology (Figure 9). The majority of the sample comes from Kepler (Huber et al. 2013a; Lundkvist et al. 2016), which has led to important insights into demographics of small-planet radii and eccentricities (Van Eylen & Albrecht 2015; Van Eylen et al. 2018a, 2019) and their dynamical formation histories through the measurement of asteroseismic spin-axis inclinations (Huber et al. 2013b; Chaplin et al. 2013; Benomar et al. 2014; Lund et al. 2014; Campante et al. 2016; Kamiaka et al. 2019; Zhang et al. 2021). While the reanalyses of Kepler data and new data from the K2 mission have added some detections (Van Eylen et al. 2018b; Chontos et al. 2019; Lund et al. 2019), the number of asteroseismic host stars has stagnated over the past few years. Furthermore, similar to the general asteroseismic and host star sample, most Kepler stars are faint and distant and thus difficult to characterize using ground-based observations.

Zoom In Zoom Out Reset image size

First results from TESS have already started to expand the population with detected oscillations in nearby exoplanet host stars, including newly discovered transiting exoplanets such as TOI-197 (Huber et al. 2019) and TOI-257 (Addison et al. 2021) and known exoplanet hosts (Campante et al. 2019; Nielsen et al. 2020). Ground-based radial velocity campaigns have yielded asteroseismic detections in some bright nearby exoplanet hosts including π Men (Kunovac Hodžić et al. 2021), but are generally limited to single-site observations, causing ambiguities in the mode identification. The detection of oscillations presented here makes π Men the closest and brightest star with a known transiting planet for which detailed asteroseismic modeling is possible, and highlights the strong potential of 20 second data to increase the asteroseismic host star sample.

The 20 second cadence data provide an opportunity to recharacterize π Men c, the first transiting planet discovered by TESS (Huang et al. 2018). In particular, the asteroseismic constraints on stellar radius and mean density, both measured with an accuracy of ≈1%, allow the opportunity to resolve degeneracies between eccentricity, impact parameter, and transit duration. This degeneracy often limits the accuracy of derived planet radii (Petigura 2020) and can provide a constraint on the orbital eccentricities of small planets (Van Eylen & Albrecht 2015; Van Eylen et al. 2019). In addition to the sub-Neptune sized π Men c, the system includes a massive, nontransiting substellar companion on an eccentric orbit with a period of ≈5.7 yr discovered using radial velocities (Jones et al.2002).

We used exoplanet (v0.4.0; Foreman-Mackey et al. 2021) to perform a joint fit of the TESS 20 second light curve and archival radial velocities spanning over ≈20 yr from UCLES/AAT (Jones et al. 2002), HARPS (Gandolfi et al. 2018; Huang et al. 2018), CORALIE, and ESPRESSO (Damasso et al. 2020). We follow Damasso et al. (2020) in splitting the HARPS, CORALIE, and ESPRESSO data sets based on expected zero-point offsets, calculating nightly bins, and parameterizing separate offsets and jitter terms (σ) for each of the eight radial velocity (RV) data sets. We also added a linear RV trend to the model to account for unknown outer companions. The transit model was parameterized with a photometric zero-point offset, an extra photometric jitter term (σ_TESS), conjunction times (T₀), orbital periods (P), impact parameters (b), quadratic limb darkening parameters (u₁, u₂), eccentricity parameters ($\sqrt{e}\,\sin \omega$, $\sqrt{e}\,\cos \omega$), mean stellar density (ρ_⋆), and radius ratio (R_p/R_⋆). We included a Gaussian process (GP) using a single simple harmonic oscillator kernel, consisting of a timescale (ρ_GP), amplitude (σ_GP), and a fixed quality factor $Q\,=\,1/\sqrt{2}$, to account for instrumental and stellar variability in the TESS light curve. For computational efficiency we only used 1 day chunks of the light curve centered on each transit. We used informative priors for the stellar mean density and radius based on the derived asteroseismic parameters (Table 5) and wide Gaussian priors for the quadratic limb darkening parameters to account for uncertainties in model atmosphere predictions. The final model has 36 parameters, which were sampled using four chains with 1500 draws each and tested for convergence using the standard Gelman–Rubin statistic. The priors and summary statistics for our joint transit and RV model are listed in Table 6. Our results agree well with previous analyses of the π Men system (e.g., Damasso et al. 2020; Günther & Daylan 2021), but provide significantly improved eccentricity constraints for π Men c (see Section 4.3).

Table 6.  π Men Planet Parameters

Parameter	Prior	Value
T0,b (BTJD)	${ \mathcal N }(-466.8,1.0)$	$-{463.99}_{-0.76}^{+0.78}$
T0,c (BTJD)	${ \mathcal N }(1519.8,0.1)$	${1519.8016}_{-0.0085}^{+0.0086}$
Pb (days)	${ \mathcal N }(2090,10)$	${2089.17}_{-0.40}^{+0.40}$
Pc (days)	${ \mathcal N }(6.27,0.01)$	${6.26789}_{-0.0001}^{+0.0001}$
ρ⋆	${ \mathcal N }(1.050,0.013)$	${1.052}_{-0.013}^{+0.013}$
$\sqrt{{e}_{{\rm{b}}}}\,\cos {\omega }_{{\rm{b}}}$	${ \mathcal U }(-1,1)$	${0.7039}_{-0.0014}^{+0.0015}$
$\sqrt{{e}_{{\rm{b}}}}\,\sin {\omega }_{{\rm{b}}}$	${ \mathcal U }(-1,1)$	$-{0.3863}_{-0.0033}^{+0.0036}$
$\sqrt{{e}_{{\rm{c}}}}\,\cos {\omega }_{{\rm{c}}}$	${ \mathcal U }(-1,1)$	${0.014}_{-0.172}^{+0.170}$
$\sqrt{{e}_{{\rm{c}}}}\,\sin {\omega }_{{\rm{c}}}$	${ \mathcal U }(-1,1)$	${0.06}_{-0.251}^{+0.251}$
bc	${ \mathcal U }(0,1)$	${0.591}_{-0.144}^{+0.056}$
u1	${ \mathcal N }(0.28,0.2)$	${0.30}_{-0.12}^{+0.12}$
u2	${ \mathcal N }(0.28,0.2)$	${0.22}_{-0.15}^{+0.15}$
Rp/R⋆	${ \mathcal U }(0,1)$	${0.01716}_{-0.00030}^{+0.00024}$
Kb (m s−1)	${ \mathcal N }(200,20)$	${194.55}_{-2.73}^{+2.83}$
Kc (m s−1)	${ \mathcal N }(1.5,1.5)$	${1.48}_{-0.20}^{+0.21}$
σGP (ppm)	IG(3, 500)	${51.1}_{-8.0}^{+10.9}$
ρGP (days)	$\mathrm{log}{ \mathcal N }(5,10)$	${2.61}_{-0.64}^{+0.89}$
σTESS (ppm)	$\mathrm{log}{ \mathcal N }(0,10)$	${111.6}_{-1.8}^{+1.8}$
σAAT (m s−1)	${ \mathcal U }(0.1,100)$	${4.46}_{-0.98}^{+1.16}$
${\sigma }_{{\mathrm{HARPS}}_{\mathrm{pre}}}$ (m s−1)	${ \mathcal U }(0.1,100)$	${2.72}_{-0.30}^{+0.35}$
${\sigma }_{{\mathrm{HARPS}}_{\mathrm{post}}}$ (m s−1)	${ \mathcal U }(0.1,100)$	${2.30}_{-0.17}^{+0.20}$
${\sigma }_{{\mathrm{CORALIE}}_{98}}$ (m s−1)	${ \mathcal U }(0.1,100)$	${13.13}_{-3.03}^{+4.53}$
${\sigma }_{{\mathrm{CORALIE}}_{07}}$ (m s−1)	${ \mathcal U }(0.1,100)$	${11.54}_{-2.42}^{+3.45}$
${\sigma }_{{\mathrm{CORALIE}}_{14}}$ (m s−1)	${ \mathcal U }(0.1,100)$	${4.29}_{-0.75}^{+0.84}$
${\sigma }_{{\mathrm{ESPRESSO}}_{\mathrm{pre}}}$ (m s−1)	${ \mathcal U }(0.1,100)$	${1.04}_{-0.27}^{+0.40}$
${\sigma }_{{\mathrm{ESPRESSO}}_{\mathrm{post}}}$ (m s−1)	${ \mathcal U }(0.1,100)$	${1.34}_{-0.20}^{+0.24}$
Derived parameters for π Men b
eb	⋯	${0.6447}_{-0.0012}^{+0.0011}$
ωb	⋯	$-{28.75}_{-0.25}^{+0.27}$
ab/R⋆	⋯	${625.6}_{-2.5}^{+2.5}$
ab (au)	⋯	${3.315}_{-0.031}^{+0.031}$
$M\sin {i}_{{\rm{b}}}\,({M}_{{\rm{J}}})$	⋯	${9.99}_{-0.19}^{+0.19}$
Mb (MJ)	⋯	${13.07}_{-0.87}^{+1.16}$
Derived parameters for π Men c
ec	⋯	${0.066}_{-0.047}^{+0.086}$
ωc	⋯	${38.7}_{-135.2}^{+78.0}$
ac/R⋆	⋯	${12.977}_{-0.052}^{+0.052}$
ac (au)	⋯	${0.06876}_{-0.00065}^{+0.00065}$
ic (deg)	⋯	${87.37}_{-0.11}^{+0.40}$
Rc (R⊕)	⋯	${2.131}_{-0.042}^{+0.037}$
Mc (M⊕)	⋯	${4.50}_{-0.63}^{+0.66}$

Note. See text for a description of all parameters. Nuisance parameters (photometric and RV instrument offsets, linear RV trend) are omitted from the table. ${ \mathcal N }$, ${ \mathcal U }$, and IG denote normal, uniform, and inverse gamma distributions. The mass of π Men b was calculated using i = 499 ± 50 (De Rosa et al. 2020).

Download table as:  ASCIITypeset image

Figure 10 shows the phase-folded transit light curve for π Men c, and Figure 11 shows the radial velocity data for π Men b and c with best-fitting models. Figure 10 illustrates that the transits are well sampled in 20 second data, while 30 minute cadence would significantly smear out the durations of ingress and egress. A detailed comparison of 20 second compared to 2 minute cadence data for deriving transit parameters will be presented in a future study (C. S. K. Ho et al. 2021, in preparation). As demonstrated by Damasso et al. (2020), the combination of long baseline and high precision (in particular from ESPRESSO) for the available radial velocity data set provides exquisite constraints on both planets (Figure 11). We measure the radius and mass of π Men c to ≈2% and ≈13%, which include systematic errors on stellar parameters.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Orbital eccentricities, inclinations, and obliquities provide valuable information to constrain formation scenarios for close-in exoplanets. In particular, they help to distinguish dynamically "hot" formation pathways such as high-eccentricity migration triggered by planet–planet scattering (Chatterjee et al. 2008; Nagasawa et al. 2008) or Kozai–Lidov cycles (Kozai 1962; Lidov 1962; Fabrycky & Tremaine 2007) from in situ formation or migration in the protoplanetary disk (Cossou et al. 2014). While dynamical architectures have been extensively studied for hot Jupiters (e.g., Winn et al. 2010; Albrecht et al. 2012), constraints for sub-Neptune sized planets are still relatively scarce, in particular for systems with known outer companions (Rubenzahl et al. 2021).

π Men provides an excellent opportunity to study the dynamical formation pathway for a close-in sub-Neptune sized planet. The combination of Hipparcos and Gaia astrometry recently revealed that the orbit of π Men b is misaligned with that of π Men c (De Rosa et al. 2020; Xuan & Wyatt 2020), while Rossiter–McLaughlin observations show a 24° ± 4° projected obliquity between the host star and π Men c (Kunovac Hodžić et al. 2021). Taken together these observations provide evidence for a dynamically hot formation pathway for π Men c. However, key dynamical properties such as the orbital eccentricity of π Men c have so far been poorly constrained.

Figure 12 shows the marginalized posterior distribution for the eccentricity of π Men c derived from our fit to the 20 second photometry alone, and also using the joint transit and RV fit. Both are consistent with a low eccentricity for π Men c, and the joint fit places an upper limit of <0.1 (68%), a factor of two tighter than previous constraints (Damasso et al. 2020). The posterior mode is consistent with a circular orbit, which implies that any initially high eccentricity caused by a dynamically hot formation has been damped over the 3.8 ± 0.8 Gyr lifetime of the system. Tidal dissipation rates are highly uncertain, mainly owing to the unknown planetary tidal quality factor (Q) and tidal Love number (k_2,1), which quantify the strength of the energy dissipation of tides in the planet and the perturbation of the gravitational potential at its surface due to star–planet tidal interactions (Ogilvie 2014; Mathis 2018). Assuming that π Men c has tidally circularized, we can place an upper limit on Q/k_2,1 assuming equilibrium tides (Goldreich & Soter 1966; Hut 1981; Xuan & Wyatt 2020):

$$\begin{eqnarray}&&\displaystyle \frac{Q}{{k}_{\mathrm{2,1}}}\lt \displaystyle \frac{21\pi }{2}\displaystyle \frac{{M}_{\star }}{{M}_{c}}{\left(\displaystyle \frac{{R}_{c}}{{a}_{c}}\right)}^{5}\displaystyle \frac{\tau }{{P}_{c}},\end{eqnarray} \tag{ 3 }$$

where τ is the age of the star and P_c is the orbital period of π Men c. Substituting values from Tables 5 and 6, we arrive at Q/k_2,1 ≲ 2400. Assuming that π Men c has a rocky core composed of iron and silicates in terrestrial proportions with a radius of ≈1.5 R_⊕ and a mass of ≈4.5 M_⊕ as inferred from planetary interior modeling (García Muñoz et al. 2021), the ab initio computations by Tobie et al. (2019) imply k_2,1 ≈ 0.4 for π Men c (compared to 0.3 for the Earth, Wahr 1981). Using Q/k_2,1 ≲ 2400 leads to Q ≲ 970, consistent with predictions that a ≈4.5 M_⊕ planet with a terrestrial iron proportion should have 200 < Q < 1000 depending on the considered viscosity (Tobie et al. 2019).

Zoom In Zoom Out Reset image size

If π Men c arrived at its present-day configuration through high-eccentricity migration, as suggested by the orbital misalignments, our data imply that it has completed tidal circularization. Assuming that the eccentricity excitation occurred through Kozai–Lidov cycles with π Men b, this result would imply a present-day mutual inclination of π Men c to π Men b of ∼40° or ∼140° (De Rosa et al. 2020). Furthermore, the circular orbit would suggest that π Men c is not undergoing low-eccentricity migration, which has been suggested as a possible formation pathway for producing ultrashort-period planets (Pu & Lai 2019). Finally, the results imply that circular orbits for close-in sub-Neptune sized planet cannot be used to rule out dynamically hot formation scenarios. However, we note that dynamically "cold" formation pathways, such as disk migration or in situ formation, can still explain the observed properties of π Men c and thus cannot be completely excluded.

A key dynamical constraint for the π Men system is the inclination of the stellar spin axis, which would yield the full 3D architecture of the system. While the current S/N is insufficient to reliably measure this inclination using asteroseismology (Gizon & Solanki 2003; Ballot et al. 2006; Kamiaka et al. 2018), additional TESS 20 second observations (especially beyond the current extended mission, enabling >1 yr coverage) may enable some constraints on this important parameter.

The dearth of planets with radii around 1.8 R_⊕ in the Kepler sample (Fulton et al. 2017) has sparked several efforts to investigate the origin and evolution of close-in sub-Neptune sized planets, including studies of small planets in the K2 sample (Hardegree-Ullman et al. 2020) and the dependence of the radius valley on both stellar mass (Fulton & Petigura 2018; Cloutier & Menou 2020; Van Eylen et al. 2021) and age (Berger et al. 2020; David et al. 2021; Sandoval et al. 2021). A remarkable feature of the radius valley is that it is devoid of planets for a sample of well characterized stars and planets using asteroseismology (Van Eylen et al. 2018a), suggesting that the dominant formation mechanism may be a relatively rapid process such as photoevaporation (Owen & Wu 2017). While there is evidence that many "gap planets" in the general Kepler sample are linked to underestimated uncertainties in transit fits from long-cadence photometry (Petigura 2020), recent studies have shown evidence for a transition of sub-Neptune to super-Earth sized planets on gigayear timescales (Berger et al. 2020) and a shift in the radius gap with stellar age (David et al. 2021), consistent with slower processes such as core-powered mass loss (Ginzburg et al. 2018; Gupta & Schlichting 2020). If the primary mechanism for sculpting the radius gap operates on gigayear timescales, we should find examples of old planets with ages similar to π Men c that are currently located in the gap.

Figure 13 compares our measured radius for π Men c with the sample of small planets orbiting asteroseismic Kepler stars from Van Eylen et al. (2018a). Unlike some other studies, our analysis firmly places π Men c at the upper edge of the radius gap at its orbital period. This confirms previous results that π Men c has probably held on to a significant volatile envelope even after 3.8 Gyr and is consistent with recent Hubble Space Telescope transmission spectroscopy as well as planetary interior and long-term evolution modeling, which imply that π Men c has an atmosphere of moderate to high molecular weight (García Muñoz et al. 2020, 2021).

Zoom In Zoom Out Reset image size

The position of π Men c at the upper edge of the radius gap also confirms the lack of genuine "gap planets" in the asteroseismic host star sample. Since the asteroseismic sample is biased toward older solar-type stars with ages ranging from ≈2–12 Gyr and contains planets with well characterized radii, this suggests that the evolution of the radius valley may be restricted to ≲2 Gyr. However, we note that, depending on the actual composition of its core and envelope, it is also possible that π Men c will eventually evolve through the radius gap.