Expanding the frontiers of cool-dwarf asteroseismology with ESPRESSO. Detection of solar-like oscillations in the K5 dwarf $\epsilon$ Indi

Fuelled by space photometry, asteroseismology is vastly benefitting the study of cool main-sequence stars, which exhibit convection-driven solar-like oscillations. Even so, the tiny oscillation amplitudes in K dwarfs continue to pose a challenge to space-based asteroseismology. A viable alternative is offered by the lower stellar noise over the oscillation timescales in Doppler observations. In this letter we present the definite detection of solar-like oscillations in the bright K5 dwarf $\epsilon$ Indi based on time-intensive observations collected with the ESPRESSO spectrograph at the VLT, thus making it the coolest seismic dwarf ever observed. We measured the frequencies of a total of 19 modes of degree $\ell=0$--2 along with $\nu_{\rm max}=5305\pm176\:{\rm \mu Hz}$ and $\Delta\nu=201.25\pm0.16\:{\rm \mu Hz}$. The peak amplitude of radial modes is $2.6\pm0.5\:{\rm cm\,s^{-1}}$, or a mere ${\sim} 14\%$ of the solar value. Measured mode amplitudes are ${\sim} 2$ times lower than predicted from a nominal $L/M$ scaling relation and favour a scaling closer to $(L/M)^{1.5}$ below ${\sim} 5500\:{\rm K}$, carrying important implications for our understanding of the coupling efficiency between pulsations and near-surface convection in K dwarfs. This detection conclusively shows that precise asteroseismology of cool dwarfs is possible down to at least the mid-K regime using next-generation spectrographs on large-aperture telescopes, effectively opening up a new domain in observational asteroseismology.


Introduction
Asteroseismology has seen remarkable advances thanks to missions such as Convection, Rotation and planetary Transits (CoRoT; Baglin et al. 2006) and Kepler/K2 (Borucki et al. 2010;Howell et al. 2014).These missions have provided exquisite space photometry, enabling the detailed study of the interiors of solar-type and red-giant stars, which exhibit convection-driven solar-like oscillations (for a recent review, see Aerts 2021).The ongoing Transiting Exoplanet Survey Satellite (TESS; Ricker et al. 2015), along with the upcoming PLAnetary Transits and Oscillations of stars (PLATO; Rauer et al. 2014) and Nancy Grace Roman (Spergel et al. 2015) space telescopes, are set to revolutionise the field as they are expected to raise the yield of known solar-like oscillators to a few million stars, or by two orders of magnitude over previous missions combined (Gould et al. 2015;Miglio et al. 2017;Hon et al. 2021;Goupil et al. 2024).
Despite this success story, space-based asteroseismology faces a challenge regarding K dwarfs.Owing to the low luminosities of K dwarfs, their oscillation amplitudes are extremely small (below a few parts per million or, equivalently, 10 cm s −1 ; Kjeldsen et al. 2008;Verner et al. 2011;Corsaro et al. 2013) and thus hard to detect, even with multi-year Kepler photometry (e.g.Kepler-444;Campante et al. 2015).As a result, only a few dwarfs cooler than the Sun have detected solar-like oscillations to date, and none cooler than ∼5000 K (see Fig. 1).
A viable alternative to space photometry is offered by Doppler observations.Stellar noise due to non-oscillatory fluctuations associated with activity and granulation is substantially lower in Doppler than it is in photometry (Harvey 1988).Consequently, radial-velocity (RV) observations have a higher signalto-noise ratio (S /N) over the typical timescales of the oscillations (by an order of magnitude in power for the Sun; Grundahl et al. 2007).This motivated a number of pre-Kepler groundbased campaigns on cool dwarfs with the then state-of-the-art spectrographs such as the High Accuracy Radial velocity Planet Searcher (HARPS; Mayor et al. 2003) and the Ultraviolet and Visual Echelle Spectrograph (UVES; Dekker et al. 2000).Observing runs like those on τ Ceti (G8 V; Teixeira et al. 2009), 70 Ophiuchi A (K0 V; Carrier & Eggenberger 2006), and α Centauri B (K1 V; Kjeldsen et al. 2005) are the epitome of such efforts, having helped set a lower effective temperature (T eff ) bound on cool-dwarf asteroseismology.However, long readout times and/or relatively small apertures meant that these early campaigns would remain limited to the very brightest dwarfs.K dwarfs have since become a primary focus in searches for potentially habitable planets (Lillo-Box et al. 2022;Mamajek & Stapelfeldt 2023).Moreover, owing to their ubiquity and long lives, they are unique probes of local Galactic chemical evolution (Adibekyan et al. 2012;Delgado Mena et al. 2021).The time is thus ripe to systematically extend asteroseismology to these cooler dwarfs via ultra-high-precision RV observations  et al. 2017;Hatt et al. 2023), and radial-velocity campaigns (red diamonds; see e.g.Arentoft et al. 2008;Kjeldsen et al. 2008, and references therein).The stellar background sample (grey dots) is taken from the TESS Input Catalog (TIC; Stassun et al. 2019).The Sun is represented by its usual symbol.Approximate spectral type ranges (F, G, and K) are delimited by the vertical dashed lines.ϵ Indi (K5 V) is the coolest seismic dwarf observed to date (its interferometric radius and effective temperature were used to place it in the diagram; Rains et al. 2020).
that make use of next-generation spectrographs on large-aperture telescopes.The combination of a large collecting area, instrumental stability, and high spectral resolution makes the Echelle SPectrograph for Rocky Exoplanets and Stable Spectroscopic Observations (ESPRESSO; Pepe et al. 2021), mounted on the Very Large Telescope (VLT) at the European Southern Observatory (ESO), Paranal, Chile, particularly suitable for this purpose.
An exploratory campaign on the seventh magnitude K3 dwarf HD 40307 was conducted in December 2018 as part of the ESPRESSO Guaranteed Time Observations (GTO), as described in Sect.5.3.2 of Pepe et al. (2021).However, due in part to the target's relative faintness, only a tentative claim of p-mode detection (at the level of 3-4 cm s −1 ) could be made.In this letter we overcome this drawback as we report on the recent campaign conducted with ESPRESSO on the fourth magnitude K5 dwarf ϵ Indi A (HD 209100, HR 8387; hereafter ϵ Indi), a target providing a nearly ten times greater flux than HD 40307.We are able to firmly establish the presence of solar-like oscillations in the RV data of ϵ Indi, thus making it the coolest seismic dwarf observed to date.
We observed ϵ Indi for six consecutive half nights with ESPRESSO in September 2022.Observations were carried out in single Unit Telescope (single-UT) high-resolution (1 × 1 bin- ning and fast readout) mode.Weather conditions were generally favourable, with photometric and/or clear skies over the first two nights, and spells of thin cirrus clouds and relatively high winds during the remaining nights.We obtained 2084 spectra with a fixed exposure time of 25 s and a median cadence of one exposure every 60 s (which corresponds to a Nyquist frequency of 8.3 mHz).The spectra were subsequently reduced using version 3.0.0 of the ESPRESSO data reduction software (DRS), having adopted a K6 stellar binary mask to compute cross-correlation functions (CCFs), from which RVs and associated CCF parameters (see Appendix A) were derived.
The resulting (raw) RVs are shown in the top panel of Fig. 2. The slow modulation of the time series is likely a manifestation, as seen in this short-duration data set, of the 18-day signal (corresponding to half the rotation period) due to rotational activity variations identified by Feng et al. (2019).Moreover, the time series displays intranight trends, presumably due to a combination of instrumental drift and stellar convection.To remove the slow modulation and intranight trends, we high-pass filtered the time series one night at a time using a triangular smoothing function (∼2 hr cutoff).The detrended time series thus obtained is shown in the middle panel of Fig. 2. Its dispersion (rms scatter) is 30cm s −1 and greater than the photon noise, which we attribute to the presence of oscillations (the average photon-noise uncertainty per data point is < σ RV >= 22 cm s −1 ; see bottom panel of Fig. 2).

Computation of the power spectrum
We based our analysis of the power spectrum on the discrete Fourier transform (DFT) of the detrended RV time series.This involved using the measurement uncertainties, σ RV,i , as statistical weights in calculating the power spectrum (according to w i = 1/σ 2 RV,i ).In order to optimise the noise floor in the power spectrum, these weights were further adjusted to account for a small fraction of bad data points (37 data points, or ∼2% of the total, were removed) as well as night-to-night variations in the Fig. 3. Noise-optimised power spectrum of ϵ Indi.The power spectrum has been oversampled for visual purposes.A clear power excess due to solar-like oscillations can be seen centred just above 5mHz.The vertical dashed line represents the measured ν max (see text for details).The inset shows the spectral window, with prominent sidelobes (daily aliases) due to the single-site nature of the observations.noise level.We followed a well-tested procedure in adjusting the weights (for details, see e.g.Bedding et al. 2004;Arentoft et al. 2008).
The resulting noise-optimised power spectrum is shown in Fig. 3, which displays a clear power excess due to solar-like oscillations centred just above 5 mHz (typical periods of ∼3 min).This is in agreement with the predicted frequency of maximum oscillation amplitude, 2 mHz, scaled by solar values (Brown et al. 1991;Kjeldsen & Bedding 1995), where g is the surface gravity (log g = 4.61 ± 0.29 dex; Gomes da Silva et al. 2021).We proceeded to measure ν max based on a heavily smoothed version of the power spectrum (see Sect. 3.3 for details).After correcting for the background noise in the power spectrum, an estimate ν max = 5305 ± 176 µHz was obtained.
The spectral window is shown as an inset in Fig. 3, and reveals prominent sidelobes caused by the daily gaps in the RV data.The average photon-noise level in the amplitude spectrum, as measured at high frequencies (above 6.5 mHz, i.e. beyond the frequency range occupied by the p modes), is 0.94 cm s −1 .For comparison, the high-frequency noise level reported for the observations of α Centauri B is 1.39 cm s −1 (Kjeldsen et al. 2005).

Oscillation frequencies
The frequencies of acoustic (p) modes of high radial order, n, and low angular degree, ℓ, are well approximated by the asymptotic relation (Tassoul 1980): Here ∆ν is the large separation between modes of like degree and consecutive order, being a probe of the mean stellar density; δν 0ℓ is the small separation between modes of different degree and is sensitive to variations in the sound speed gradient near the core in main-sequence stars; and the dimensionless offset, ε, is determined by the reflection properties of the surface layers.Observed solar-like oscillations in main-sequence stars are expected to follow this relation closely.We thus used this prior Notes.Quoted uncertainties depend on the S /N of the corresponding mode peaks, and were calibrated using simulations (e.g.Kjeldsen et al. 2005).We opted to list mode frequencies without correcting for the lineof-sight motion (Davies et al. 2014).Given the non-negligible magnitude of this effect (∼0.7 µHz at 5000 µHz), we advise applying this correction when directly comparing the observed individual frequencies to model frequencies.
information to guide the mode identification and extraction (as described below), bearing in mind the presence of daily aliases in the power spectrum (appearing at splittings of ±11.57µHz, or ±1 cycle per day, about genuine peaks).
Extracted mode frequencies are listed in Table 1 and displayed in échelle format in the top panel of Fig. 4. Owing to the short duration of the RV time series, individual modes are only partially resolved (see also Sect. 3.4).Modes were thus extracted using a standard iterative sine-wave fitting procedure, also known as prewhitening (e.g.Bedding et al. 2010).A total of 19 modes of degree ℓ = 0-2 were extracted across seven orders down to S /N = 2.5.The full procedure for identifying and extracting oscillation frequencies consisted in the following steps: 1. We measured the strongest peak (5121.61µHz) within the frequency range occupied by the p modes and used it to compute a modified comb response (e.g.Kjeldsen et al. 1995) over a range of trial large separations.The comb response peaks at ∼201.3 µHz, which we adopted as a first estimate of ∆ν. 2. Guided by this estimate, we identified the sequence of (nearly) regularly spaced peaks below and above the dominant mode at 5121.61 µHz (and hence sharing the same ℓ).
We measured seven such modes, based on which the large separation, ∆ν = 201.25 ± 0.16 µHz, was computed.Based on the value for ∆ν and the frequency of the dominant mode, we inferred ε = 1.451 ± 0.019, consistent with the empirical results in the literature obtained for other cool dwarfs (cf.White et al. 2011aWhite et al. ,b, 2012;;Lund et al. 2017), and hence with these being radial (ℓ = 0) modes.3. We extracted these modes from the time series through iterative sine-wave fitting.By collapsing the resulting prewhitened power spectrum about the position of the ℓ = 0 ridge, we saw a power excess at lower frequencies, which we assigned to the ℓ = 2 ridge, being able to resolve the small separation (δν 02 ∼ 14.5 µHz).Guided by this, we then identified a sequence of five quadrupole (ℓ = 2) modes in the prewhitened power spectrum, based on which we obtained δν 02 = 15.28 ± 0.45 µHz.We note that δν 02 is a decreasing function of frequency (or n), as expected (e.g.Lund et al. 2017).4. Finally, we collapsed the power spectrum about the midpoint between consecutive radial modes.A clear power excess corresponding to the ℓ = 1 ridge could be seen below the midpoint frequency (δν 01 ∼ 4.4 µHz).Guided by this, we identified a sequence of seven dipole (ℓ = 1) modes, providing a direct measurement of the small separation, δν 01 = 3.46 ± 1.48 µHz.We estimated the power ratio between the ℓ = 1 and 0 ridges to be ∼1.3, in accordance with the predicted spatial response of Doppler observations (Kjeldsen et al. 2008;Schou 2018).
Figure B.1 shows the prewhitened power spectrum after extracting all 19 identified modes, where it can be seen that they account for most of the power within the p-mode frequency range.
The problem of mode identification, in the sense of assigning a pair (n, ℓ) to the extracted frequencies, may not be a trivial one to solve.There are numerous instances of seismic studies that led to uncertainty on the mode identification (e.g.Carrier & Eggenberger 2006;Appourchaux et al. 2008;Bedding et al. 2010), which is made worse in the case of single-site groundbased observations.As an additional check on the above observational procedure, we adopted a model-based approach to mode identification based upon the work of White et al. (2011a,b), in order to verify whether the observed position of the ℓ = 0 ridge in the échelle diagram is consistent with expectations from stellar models.Based on the stellar model grid of Li et al. (2023), we calculated, for each model in that grid, a likelihood function, L i ∼ exp(−χ 2 i /2), where the discrepancy function, χ 2 i , is given by the sum of the error-normalised discrepancies for T eff , [Fe/H], and ∆ν, adopted as observational constraints.We next computed a model prediction of the quantity ε ∆ν, which gives the absolute position of the ℓ = 0 ridge in an échelle diagram (cf.Eq. 1).This was simply done by constructing a likelihood-weighted histogram of ε ∆ν for the models in the grid (see bottom panel of Fig. 4).According to this, the power ridge just below 100 µHz in the top panel of Fig. 4 should correspond to ℓ = 2,0 (rather than ℓ = 1), thus providing support to our adopted mode identification.We note the presence of a small offset between the model-based and observed (shown as a vertical dotted line) ε ∆ν.This is to be expected, and is due to the fact that model frequencies were not corrected for the surface effect (see e.g.Ball & Gizon 2014).We will be investigating the magnitude of the surface effect in this T eff regime in a follow-up study.

Oscillation amplitudes
The measured amplitudes of individual modes are affected by the stochastic nature of the excitation and damping.We hence followed the procedure described in Kjeldsen et al. (2005Kjeldsen et al. ( , 2008) ) to measure the oscillation amplitude envelope in a way that is independent of these effects.In short, we heavily smoothed the power spectrum by convolving it with a Gaussian having a full width at half maximum (FWHM) of 4 ∆ν; converted to power density; fitted and subtracted the background noise; and multiplied by ∆ν/c and took the square root, thus converting to amplitude per radial mode (a value of c = 4.09 was adopted, representing the effective number of modes per order for full-disk velocity observations, normalised to the amplitudes of radial modes; Kjeldsen et al. 2008).
The envelope peak amplitude thus obtained is osc = 2.6 ± 0.5cm s −1 , or a mere ∼14% of the solar value ( osc,⊙ = 18.7cm s −1 , as measured using stellar techniques and averaged over one full solar cycle; Kjeldsen et al. 2008).The associated uncertainty was estimated as the standard deviation resulting from having applied the above procedure to the power spectra of 2000 artificial time series, generated using the asteroFLAG Artificial DataSet Generator, version 3 (AADG3; Ball et al. 2018).Each simulated time series contained as input all extracted mode frequencies (cf.Ta- Figure 5 shows the amplitude per radial mode as a function of T eff (colour-coded according to the chromospheric emission ratio, log R ′ HK ) for ϵ Indi and a number of cool dwarfs with published measurements.We note that the amount of smoothing of the power spectrum affects the exact height of the smoothed amplitude envelope, and hence the estimate of osc .Measurements for the Sun and α Centauri A and B (Kjeldsen et al. 2008), as well as for τ Ceti (Teixeira et al. 2009) were obtained following the same procedure as described above.The value plotted for 70 Ophiuchi A (Carrier & Eggenberger 2006) corresponds to the upper bound on the amplitudes of the highest mode peaks detected and has no associated uncertainty, while an estimate (no error provided) of the mode amplitudes for the solar twin 18 Scorpii is given in Bazot et al. (2011).Finally, our reanalysis of the ESPRESSO GTO radial-velocity data of HD 40307 showed no p-mode detection (see Appendix D), and so the plotted value corresponds to an upper limit.
Based on calculations by Christensen-Dalsgaard & Frandsen (1983), Kjeldsen & Bedding (1995) suggested a scaling of the oscillation amplitudes of p modes in Doppler velocity of the form osc ∝ (L/M) s , with s = 1, and where L and M are respectively the stellar luminosity and mass.The numerical value of the exponent s has since been revised theoretically, based on models of main-sequence stars, and found to lie in the range 0.7-1.5 (see e.g.Houdek et al. 1999;Samadi et al. 2005Samadi et al. , 2007)).On the other hand, observational 1 studies based on large ensembles of main-sequence and subgiant Kepler stars have constrained s to the approximate range 2 0.5-1.0,depending on T eff (Verner et al. 2011;Corsaro et al. 2013).
We show, in Fig. 5, two scalings of the mode amplitudes corresponding to the extrema of the theoretical range in s (i.e.s = 0.7 and 1.5).The displayed mode amplitude measurements hint at a transition to a scaling closer to (L/M) 1.5 below ∼5500 K (cf.Verner et al. 2011, where a negative ds/dT eff gradient was determined).This is supported by the mode amplitudes measured herein for ϵ Indi and the upper limit on the mode amplitudes obtained for HD 40307.The fraction of magnetically active stars among K dwarfs is higher than among G dwarfs (e.g.Jenkins et al. 2011;Gomes da Silva et al. 2021).At the same time, increasing levels of activity are known to suppress the amplitudes of solar-like oscillations (García et al. 2010;Chaplin et al. 2011;Bonanno et al. 2014;Campante et al. 2014), which could therefore be the underlying cause for the apparent transition between scaling relations when moving down in T eff in Fig. 5.We find no significant correlation between log R ′ HK and 1 Observational results draw mostly from studies conducted in photometry.Photometric mode amplitudes, once corrected to bolometric amplitudes, are expected to scale as A bol ∝ osc T 1−r eff (with r = 1.5 if assuming adiabatic oscillations; Kjeldsen & Bedding 1995), thus allowing for conversion between photometric and Doppler observations. 2The procedure described in Kjeldsen et al. (2005Kjeldsen et al. ( , 2008) ) has been widely implemented in automated analysis pipelines with the goal of calibrating scaling relations.Corsaro et al. (2013) make exclusive use of amplitudes derived following this procedure.Verner et al. (2011) compare different analysis pipelines, several of which adopt this procedure; for the sake of homogeneity, we consider only the latter pipelines here (i.e.Huber et al. 2009;Hekker et al. 2010;Mathur et al. 2010).The error on ∆ν is too small to be discerned.
T eff (Spearman's rank correlation coefficient, ρ = −0.17,and large p-value, p ≫ 0.05) for the displayed cool-dwarf sample.However, given the limited size of this sample, we refrain from making more general considerations regarding the role of activity in this context, and advocate for the inclusion of its effect in the calibration of the mode-amplitude scaling in this T eff regime as more targets are observed.Finally, it is worth noting that ϵ Indi

Oscillation lifetimes
Solar-like oscillations are stochastically excited and damped by near-surface convection.The power spectrum of a single mode that is observed for long enough will appear as an erratic function concealing a Lorentzian profile, the width of which indicates the mode lifetime (e.g.Anderson et al. 1990).If, as in the present case, the observations are not long enough to fully resolve the Lorentzian profile, then the effect of the finite mode lifetime is to randomly shift each mode peak from its true position by a small amount (e.g.Bedding et al. 2004;Kjeldsen et al. 2005).
Measuring this scatter provides an opportunity to infer the mode lifetimes in ϵ Indi.We inferred the mode lifetimes by measuring the scatter of the observed frequencies of radial modes (which are not impacted by rotation) about their power ridge in the échelle diagram and comparing with simulations (see Appendix E).The top panel of Fig. E.1 shows the outcome of this calibration procedure for the ϵ Indi observing window.Although an upper bound on the mode lifetimes is weakly constrained, it is safe to say that lifetimes are at least a few days long (≳ 3 d).For context, the average mode lifetime in the Sun, measured in the range 2.8-3.4 mHz, is 2.88 ± 0.07 d (Chaplin et al. 1997), being slightly longer than for τ Ceti (1.7 ± 0.5 d; Teixeira et al. 2009), and in line with that measured for α Centauri B (3.3 +1.8  −0.9 d at 3.6 mHz and 1.9 +0.7 −0.4 d at 4.6 mHz; Kjeldsen et al. 2005).

Conclusion and outlook
In this letter we have presented the definite detection of solar-like oscillations in the bright K5 dwarf ϵ Indi based on radial-velocity observations carried out with the ESPRESSO spectrograph.This campaign hence unambiguously demonstrates the potential of ESPRESSO for cool-dwarf asteroseismology, effectively opening up a new observational domain in the field.Measured mode amplitudes for ϵ Indi are approximately two times lower than predicted from a nominal L/M scaling relation, favouring a scaling closer to (L/M) 1.5 below ∼5500K.A calibration of the mode-amplitude scaling relation in this T eff regime is thus called for as more targets are observed for asteroseismology.Mode amplitudes are determined by a delicate balance between the energy supply and the mode damping, both being directly connected to the turbulent velocity field associated with convection (Houdek & Dupret 2015).The measurement of oscillation modes in K dwarfs will hence allow us to constrain the dynamical coupling between pulsations and near-surface convection in a regime yet unexplored.Moreover, measured mode amplitudes, used in combination either with 1D non-local time-dependent convection models (Chaplin et al. 2005) or with state-of-the-art 3D stellar atmosphere simulations (Zhou et al. 2021), will enable predictions of amplitudes in photometry.This information is key to accurately estimating the PLATO seismic yield (Miglio et al. 2017;Goupil et al. 2024) and can potentially influence the PLATO pipeline development strategy (Cunha et al. 2021).
Furthermore, ϵ Indi is the only known system containing Ttype brown dwarfs for which a test of substellar cooling with time and a coevality test of model isochrones are both made possible (Chen et al. 2022).These tests would greatly benefit from having a precise seismic age for the host star, as it still remains a major source of uncertainty in the evolutionary and atmospheric modelling of the system.We display, in Fig. 6, the location of ϵ Indi in a C-D diagram (Christensen-Dalsgaard 1984), showing δν 02 versus ∆ν.Inspection of this diagram implies a seismic stellar age < 4 Gyr, consistent with most literature measurements (see Chen et al. 2022, and references therein), which include activity-based estimates, as well as ages from kinematics and isochrone fitting.Detailed asteroseismic modelling of ϵ Indi will be the subject of a follow-up study.implied by the observed scatter.Based on the simulation results, the observed scatter of dipole modes is too large to be explained by a combination of mode damping and rotational splittings.If, however, a strong dipole magnetic field is present, this could in principle lead to extra damping of the ℓ = 1 modes (with respect to the ℓ = 0 modes).The unpublished, single-epoch Zeeman-Doppler imaging (ZDI) map from Vidotto et al. (2016) is indeed dominated by a strong dipole field (56% of the field energy is in the dipole component, according to their Table 2).A similar field, if concurrent with the ESPRESSO observations, could partially account for the enhanced scatter about the ℓ = 1 ridge.

Fig. 1 .
Fig. 1.Stellar radius-effective temperature diagram highlighting seismic detections from Kepler and TESS photometry (blue circles; Mathur et al. 2017; Hatt et al. 2023), and radial-velocity campaigns (red diamonds; see e.g.Arentoft et al. 2008; Kjeldsen et al. 2008, and references therein).The stellar background sample (grey dots) is taken from the TESS Input Catalog (TIC; Stassun et al. 2019).The Sun is represented by its usual symbol.Approximate spectral type ranges (F, G, and K) are delimited by the vertical dashed lines.ϵ Indi (K5 V) is the coolest seismic dwarf observed to date (its interferometric radius and effective temperature were used to place it in the diagram; Rains et al. 2020).

Fig. 2 .
Fig. 2. Time series of ESPRESSO radial-velocity measurements of ϵ Indi.Top: Raw time series (after removal of a constant RV offset).The solid red curves represent smoothing functions applied on a nightly basis (see text for details).Middle: Detrended time series (after highpass filtering).Bottom: Internal (photon-noise limited) RV precision as returned by the ESPRESSO DRS.The horizontal dotted line represents the instrumental noise level of 10 cm s −1 quoted by Pepe et al. (2021).

Fig. 4 .
Fig. 4. Outcome of the mode identification and extraction procedure.Top: Replicated échelle diagram displaying the mode frequencies extracted for ϵ Indi (cf.Table1).The smoothed power spectrum is shown in grey scale.The vertical dashed line gives the measured ∆ν value.Bottom: Likelihood-weighted histogram of ε ∆ν for the models in the grid (see text for details).The observed ε ∆ν is represented by a vertical dotted line.

Fig. 5 .
Fig. 5. Amplitude per radial mode as a function of T eff for ϵ Indi and a number of cool dwarfs with published measurements.Reported statistical uncertainties for the mode amplitudes are comparable in size to the plotted symbols.Data points are colour-coded according to the corresponding log R ′ HK ratio.Except for ϵ Indi (see Appendix C) and the Sun (Mamajek & Hillenbrand 2008), all log R ′ HK measurements are from Gomes da Silva et al. (2021).Two scalings of the mode amplitudes are shown, differing in terms of the exponent s (s = 0.7, dotted curve; s = 1.5, solid curve).The adopted L/M relation is from stellar models in a 4.57 Gyr, solar-metallicity isochrone computed with the PAdova and TRieste Stellar Evolution Code (PARSEC; Bressan et al. 2012).

Fig. 6 .
Fig. 6.C-D diagram, showing δν 02 vs ∆ν.Models from the grid of Li et al. (2023) are colour-coded according to age (left) and mass (right).The location of ϵ Indi is indicated by the black symbol in both panels.The error on ∆ν is too small to be discerned.

Fig. E. 1 .
Fig. E.1.Mode lifetime calibration (top: radial modes; bottom: dipole modes) for the ϵ Indi observing window.The solid lines are the result of simulations (see text for details) and show frequency scatter vs S /N for a range of input mode lifetimes.The measured frequency scatter is represented by a black symbol in both panels, placed at a fiducial S /N = 3.5 level (corresponding to the typical S /N of radial modes near ν max ).