Asteroseismology of the Hyades with K2: first detection of main-sequence solar-like oscillations in an open cluster
Mikkel N. Lund, Sarbani Basu, Víctor Silva Aguirre, William J. Chaplin, Aldo M. Serenelli, Rafael A. García, David W. Latham, Luca Casagrande, Allyson Bieryla, Guy R. Davies, Lucas S. Viani, Lars A. Buchhave, Andrea Miglio, David R. Soderblom, Jeff A. Valenti, Robert P. Stefanik, Rasmus Handberg
MMon. Not. R. Astron. Soc. , 1–10 (2016) Printed October 16, 2018 (MN L A TEX style file v2.2)
Asteroseismology of the Hyades with K2: first detection ofmain-sequence solar-like oscillations in an open cluster
Mikkel N. Lund , (cid:63) , Sarbani Basu , Víctor Silva Aguirre , William J. Chaplin , ,Aldo M. Serenelli , Rafael A. García , David W. Latham , Luca Casagrande ,Allyson Bieryla , Guy R. Davies , , Lucas S. Viani , Lars A. Buchhave , Andrea Miglio , ,David R. Soderblom , Je ff A. Valenti , Robert P. Stefanik , and Rasmus Handberg School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark Department of Astronomy, Yale University, PO Box 208101, New Haven, CT 06520-8101, USA Institute of Space Sciences (CSIC-IEEC), Campus UAB, Carrer de Can Magrans, s / n E-08193 Cerdanyola del Vallès (Barcelona), Spain Laboratoire AIM, CEA / DRF - CNRS - Univ. Paris Diderot - IRFU / SAp, Centre de Saclay, 91191 Gifsur-Yvette Cedex, France Harvard-Smithsonian Center for Astrophysics, 60 Garden Street Cambridge, MA 02138 USA Research School of Astronomy and Astrophysics, Mount Stromlo Observatory, The Australian National University, ACT 2611, Australia Centre for Star and Planet Formation, Natural History Museum of Denmark & Niels Bohr Institute, University of Copenhagen,Øster Voldgade 5-7, DK-1350 Copenhagen K, Denmark Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
Accepted 2016 August 24. Received on 2016 August 23; in original form 2016 April 7
ABSTRACT
The Hyades open cluster was targeted during Campaign 4 (C4) of the NASA K2 mis-sion, and short-cadence data were collected on a number of cool main-sequence stars. Here,we report results on two F-type stars that show detectable oscillations of a quality that al-lows asteroseismic analyses to be performed. These are the first ever detections of solar-likeoscillations in main-sequence stars in an open cluster.
Key words:
Asteroseismology — methods: data analysis — galaxies: star clusters: individ-ual: Hyades — stars: rotation — stars: individual: EPIC 210444167 (HIP 20357, vB 37);EPIC 210499243 (HIP 19877, vB 20)
The Hyades is one of the youngest and closest open clusters to oursolar system; its close proximity of only ∼
47 pc means that it hasbeen extensively studied, and serves as an important benchmarkfor distances in our Galaxy (see Perryman et al. 1998, for a re-view). Because of its youth (with an isochrone-based age estimatedto be around 550 −
625 Myr) it contains many rapidly rotating starswhose rotation rates can be readily determined, hence it is com-monly used as an anchor in calibrating gyrochronology relationswhich link rotation rates to stellar ages.Asteroseismology — the study of stellar oscillations — of-fers independent measures of stellar properties. Results from the
Kepler mission have shown the power of asteroseismology in rela-tion to characterisation and age dating of both field and cluster stars(Gilliland et al. 2010; Basu et al. 2011; Stello et al. 2011; Miglio (cid:63)
E-mail: [email protected] et al. 2012; Chaplin et al. 2014; Silva Aguirre et al. 2015). Regret-tably, the nominal
Kepler mission did not observe nearby clusters,but K2, the repurposed
Kepler mission (Howell et al. 2014), will al-low us to study many interesting clusters. In this Letter we presentthe first asteroseismic analysis of main-sequence (MS) stars in theHyades, specifically two MS solar-like oscillators. We note in pass-ing that White et al. (in prep.) from K2 observations, and Beck et al.(2015) from a ground-based campaign, have detected oscillationsin three Hyades red-giants.The development of the paper proceeds as follows: In Sec-tion 2 we discuss the reduction of K2 data. We also describe theother known properties of the targets and present a set of new radialvelocity data designed to confirm cluster membership and identifyshort-period binaries, and introduce in Section 3 the spectroscopicanalysis of the stars. In Section 4.1 we discuss how the asteroseimicdata were used to determine stellar parameters. In Section 4.2 wepresent our analysis of the signatures of rotation; the seismic mod-elling is presented in Section 4.3, with distance estimates using the c (cid:13) a r X i v : . [ a s t r o - ph . S R ] A ug M. N. Lund et al. asteroseismic properties being compared to other distance indica-tors in Section 4.4. We end with a discussion of our findings inSection 5.
The Hyades open cluster, seen in the constellation of Taurus, wasobserved in short-cadence (SC; ∆ t ≈ using the K2P pipeline (Lund et al. 2015). Briefly, K2P defines pixel masks for targets in a given frame by using an unsu-pervised clustering algorithm on pixels above a given flux thresh-old. Subsequently, an image segmentation algorithm is run on eachpixel-cluster to adjust the pixel mask should two or more targetshappen to fall within it. The light curves were rectified using amodified version of the KASOC filter (Handberg & Lund 2014) toremove trends from the apparent motion of the targets on the CCDand other instrumental signatures. Power density spectra were cre-ated using a least-squares sine-wave fitting method, normalised bythe rms -scaled version of Parseval’s theorem (see Kjeldsen 1992;Frandsen et al. 1995).We searched the power spectra of all observed stars for indica-tions of seismic excess power — two targets were identified, EPIC210444167 and 210499243; from here on we will refer to these asE167 and E243. Based on proper motion and radial velocity studiesby, e. g., Schwan (1991), Perryman et al. (1998), and de Bruijne,Hoogerwerf & de Zeeuw (2001) both targets are members of theHyades. In Figure 1 we show the power spectra for the targets. Thestars have spectral types F5 IV-V (E167; Gray, Napier & Winkler2001) and F5 V (E243; Gebran et al. 2010).The star E243 has been studied before. E243 was specificallyhighlighted in de Bruijne, Hoogerwerf & de Zeeuw (2001) for ly-ing above the Hyades main-sequence ( ∆ V ∼ .
07 mag), and it wasspeculated if stellar variability or activity could be responsible forthis, but at that time a good estimate of the rotational velocity wasunavailable. In the Catalog of Components of Double and Multi-ple stars (CCDM; Dommanget & Nys 2002) the star (A) is listedwith two secondary components (B and C), both with magnitudesin the range V ∼ . ∆ V ∼ .
02 mag). We note, however, that the components listedin the CCDM are at very large separations of 137.5 (cid:48)(cid:48) (AB) and151.4 (cid:48)(cid:48) (AC), corresponding to ∼
35 and ∼
39 pixels on the
Kepler
CCD. The B component (Ba) has itself a faint companion (Bb),and the C component is a spectroscopic binary. From proper mo-tions, radial velocity (RV) data from the Harvard-Smithsonian Cen-ter for Astrophysics (CfA), and colours neither Ba, Bb, nor C areassociated with E243 or the Hyades cluster. Neither of these targetsfall within the assigned pixel mask of E243, and the photometryfor E243 A is thus una ff ected by B and C. E243 was furthermorefound to be a single system from an analysis of speckle imagesby Patience et al. (1998); this does not necessarily, however, ruleout a very close companion within the 0 (cid:48)(cid:48) .
05 confusion limit of thespeckle analysis.Radial velocities can provide additional constraints on the pos-sibility that close unresolved companions are contaminating the downloaded from the KASOC database; CCDM J04158 + / WDS J04158 + light of E167 and E243. Both stars have been monitored for morethan 35 years using RV instruments at CfA, and both appear to besingle-lined, with no direct evidence for light from a companion.The velocities for E167 appear to be constant, but there is sugges-tive evidence for acceleration in a long-period spectroscopic orbitfor E243 (Figure A1). There is insu ffi cient information to put astrong constraint on the possible light contamination from a faintcompanion to E243, but a contribution of several percent cannotbe ruled out. Four instruments have been used for the CfA veloci-ties reported here for the first time (see Appendix A): three almostidentical versions of the CfA Digital Speedometers (Latham 1992)on the 1.5-m Wyeth Reflector at the Oak Ridge Observatory in theTown of Harvard, Massachusetts, and on the MMT and 1.5-m Till-inghast Reflector at the Fred Lawrence Whipple Observatory onMount Hopkins, Arizona; and more recently the Tillinghast Re-flector Echelle Spectrograph (TRES; Szentgyorgyi & Furész 2007;Fürész 2008), a modern fiber-fed CCD echelle spectrograph onMount Hopkins. Both stars show substantial line broadening dueto rotation, ∼
22 km s − for E167 and ∼
67 km s − for E243, so theline profiles are heavily oversampled at the instrumental resolu-tions of about 6 . − . Velocities were derived using the mostappropriate rotationally broadened templates from the CfA libraryof synthetic spectra and are reported here on the native CfA sys-tem, which is about 0 .
14 km s − more negative than the IAU system.Thus 0 .
14 km s − should be added to the velocities reported in Ta-ble A1 and plotted in Figure A1 to put them onto the IAU system.Mean radial velocities in the IAU system of 38 . ± .
63 km s − (E167) and 39 . ± .
83 km s − (E243) are obtained, where the un-certainties are given by the root-mean-square ( rms ) values of the in-dividual velocities. These mean velocities agree well with the meanradial velocity of 39 . ± .
36 km s − derived by Madsen, Dravins& Lindegren (2002) for the Hyades from a moving-cluster analysis,and thus stipulate to the Hyades membership of the stars.There are a few additional historical velocities for E167 andE243 in the literature, extending the time coverage to 82 and 101years, respectively. Unfortunately the precision for the earliest ve-locities is poor and the systematic o ff set of the zero points is notwell established. The historical velocities do strengthen the impres-sion that E167 has been constant, and the velocity of E243 waslower 100 years ago. We have obtained spectroscopic parameters for the targets fromseveral sources: (1) Values are available from the Geneva-Copenhagen Survey (GCS; Nordström et al. 2004) in their re-derived version by Casagrande et al. (2011); (2) Spectroscopic datawere collected using the TRES spectrograph on the 1.5-m Till-inghast telescope at the F. L. Whipple Observatory; atmosphericparameters were derived using the Stellar Parameter Classifica-tion pipeline (SPC; Buchhave et al. 2012). Following Torres et al.(2012) we added in quadrature uncertainties of 59 K and 0 .
062 dexto the T e ff and [M / H] from SPC; (3) We also estimated T e ff us-ing the Infra-Red Flux Method (IRFM; Casagrande et al. 2014).This method also gives a measure of the stellar angular diameter θ which combined with the parallax provides an independent es-timate of the stellar radius (Silva Aguirre et al. 2012). A redden-ing of E ( B − V ) = . ± .
002 (Taylor 1980) was adopted forthe IRFM derivation. Reddening was neglected in the derivationof the GCS values, but the low value for E ( B − V ) has virtuallyno impact on the derived stellar parameters. Final SPC parameters c (cid:13) , 1–10 steroseismology of Hyades stars Frequency (µHz) P S D ( pp m / µ H z ) Frequency spacing (µHz) 0 800 1600 2400 3200 4000
Frequency (µHz) P S D ( pp m / µ H z ) Frequency spacing (µHz)
Figure 1.
Power density spectra of E167 (left) and E243 (right), smoothed by a 3 µ Hz Epanechnikov filter. The inserts show power-of-power spectra (PS ⊗ PS)from the region of the power spectra in black. The (full) vertical red lines indicate the spectroscopically estimated values for ν max of 1695 ± µ Hz (E167)and 1568 ± µ Hz (E243), with uncertainties given by the shaded green region; the corresponding lines in the PS ⊗ PS inserts give the expected values for ∆ ν/ ν max estimates following Huber et al. (2011). Dashed vertical lines indicate the measured values for ν max and ∆ ν (see Section 4.1). The fullred lines show the obtained fits to the granulation backgrounds, with the red dashed line showing the fitted Gaussian envelope used to estimate ν max . were obtained after iterating with a log g fixed at the asteroseis-mic value determined from ν max and T e ff from g ∝ ν max √ T e ff , andwith a fixed metallicity of [M / H] = . ± .
08. The metallic-ity is obtained from the average of the recent spectroscopic analy-sis results of Hyades members by Liu et al. (2016). We note thatthe adopted value from Liu et al. (2016) agrees well, within theadopted uncertainty of 0 .
08 dex, with previous average estimatesfrom, for instance, Cayrel, Cayrel de Strobel & Campbell (1985);Boesgaard & Budge (1988); Boesgaard & Friel (1990); Perrymanet al. (1998); Paulson, Sneden & Cochran (2003); Takeda et al.(2013); and Dutra-Ferreira et al. (2016). A fixed metallicity wasadopted because the SPC pipeline has di ffi culties with stars witha value of v sin i (cid:63) as high as that inferred for E243 (Table 1); thisis because high rotation leads to rotational broadening that mightcause blending of lines. The overall agreement between the dif-ferent parameter sets does, however, lend credibility to the SPCvalues. Final parameters are given in Table 1 — T e ff and [M / H]will serve as constraints in the asteroseismic modelling presentedin Section 4.3.From the spectroscopic parameters we can predict values for ν max using scaling relations (Kjeldsen & Bedding 1995). Masseswere estimated from the IRFM T e ff via the Hyades isochronefrom Pinsonneault et al. (2004), radii from L and T e ff , and using ν max , (cid:12) = ± µ Hz, and T e ff , (cid:12) = ν max .For the above prediction we estimated luminosities from kinemati-cally improved parallaxes by Madsen, Dravins & Lindegren (2002)and V -band magnitudes from Joner et al. (2006). The relations ofFlower (1996) as presented in Torres (2010) were used for the bolo-metric correction. Such a comparison is valuable, because it allowsus a check of our predictions against the estimated seismic observ-ables, and thus our ability to securely propose targets for future K2campaigns. We first determined the global asteroseismic properties ∆ ν and ν max .Here ∆ ν is defined as the frequency spacing between consecutiveradial orders ( n ) of modes with a given angular degree ( l ), and ν max as the frequency where the modes show their maximum amplitudes.To estimate ν max we fit the stellar noise background following theprocedure described in Lund et al. (2014). For the background weadopt a model given by a sum of power laws with free exponents,one for each phenomenon contributing to the background (see, e. g.,Kallinger et al. 2014, and references therein), and include a Gaus-sian envelope to account for the power excess from oscillations.The obtained background fits are shown in Figure 1. We estimate ∆ ν from fit of a squared Gaussian function including a backgroundto a narrow range of the power-of-power spectrum (PS ⊗ PS) centredon the ∆ ν/ δν — given by the frequency di ff erence between adjacent l = l = δν = ν n , − ν n − , — could not be estimated fromthe data. See Table 1 for extracted parameters. The extracted val-ues for ν max and ∆ ν agree within uncertainties with the ∆ ν ∝ βν α max scaling by Huber et al. (2011).In Figure 2 we show the background corrected échelle dia-gram (Grec, Fossat & Pomerantz 1983) for E167, smoothed to aresolution of 10 µ Hz. Over-plotted is the scaled échelle diagram(see Bedding & Kjeldsen 2010) of frequencies for KIC 3733735 ,which in terms of fundamental parameters is similar to E167, es-pecially the similar age estimated at 800 ±
400 Myr (Chaplin et al.2014). It is noteworthy how well the structure in the ridges of KIC3733735 match that of E167, which indicates that the targets areindeed very similar. KIC 3733735, which is also a fast rotator, hasbeen studied in relation to activity and rotation by Mathur et al.(2014a) and Keifer et al. (2016; submitted). The ridge identifica-tion from this scaling matches that obtained using the (cid:15) -method byWhite et al. (2011). a.k.a. Shere-Khan in the KASC working group 1 CATalogue.c (cid:13) , 1–10
M. N. Lund et al.
Table 1.
Spectroscopic parameters and common identifications for Hyades targets with detected oscillations. We give values obtained from the Stellar Parameter Classification pipeline (SPC; Buchhave et al.2012), the Geneva-Copenhagen Survey (GCS; Nordström et al. 2004) in their re-derived version by Casagrande et al. (2011), and the InfraRed Flux Method (IRFM; see Casagrande et al. 2014). Angular diameters( θ ) are from the IRFM. Systematic uncertainties of 59 K and 0 .
062 dex were added in quadrature to the SPC T e ff and [M / H] following Torres et al. (2012). We have highlighted in bold face the measured seismicvalues of ∆ ν and ν max . SPC values were iterated with a log g fixed to the seismic estimate and a fixed metallicity of [M / H] = .
164 (Liu et al. 2016).
EPIC HIP HD Kp θ ν max ∆ ν Source T e ff [M / H] log g v sin i (cid:63) (mag) (mas) ( µ Hz) ( µ Hz) (K) (dex) (cgs; dex) (km s − )210444167 a . ± . ±
47 86 . ± . SPC 6761 ±
77 0 . ± .
080 4 . ± . . ± . ±
102 0 .
07 4 . ± b . ± . ±
58 79 . ± . SPC 6901 ±
77 0 . ± .
080 4 . ± . . ± . ±
80 0.17 4.18IRFM 6771 ± a Also known as Cl Melotte 25 37, vB 37; b also known as 48 Tau, V1099 Tau, HR 1319, Cl Mellote 25 20, vB 20. F r e q u e n c y ( µ H z ) Frequency modulo 86.47 µHz
Figure 2.
Grey scale échelle diagram for E167, smoothed horizontally to aresolution of 10 µ Hz. The lines show the frequency-ridges of KIC 3733735(red l =
0; green l =
1; blue l =
2) after multiplying by 0.9361 and forcing(by eye) the (cid:15) to match that of E167 by shifting the ridges by 6 µ Hz. Thetop panel show the vertically collapsed échelle giving the combined signalof the ridges.
The determination of ∆ ν for E243 is more uncertain than thatfor E167, as also seen from the PS ⊗ PS in Figure 1. We believe thereason for this can be found in the combination of the rotation rate,which for both stars is high, and stellar inclination — as describedin Section 4.2 below, E167 is likely seen at a low inclination angle,while E243 seems to be observed edge-on. For E167 this wouldgreatly decrease the visibility of rotationally split ( m (cid:44)
0) modecomponents, leaving with highest visibility the zonal ( m =
0) com-ponents (Gizon & Solanki 2003). First of all, this would explainthe distinguishable ridges in the échelle diagram (Figure 2). Addi-tionally, the value of δ ≈ µ Hz obtained from the ridge averagesshown in Figure 2 explains the strong signal in the PS ⊗ PS at ∆ ν/
2— with δ given as the o ff set of l = l = δ = ( ν n , + ν n + , ) − ν n , (see, e. g., Bedding 2011). On the other hand, the i (cid:63) ≈ ◦ con-figuration for E243 would maximise the rotational confusion ofthe power spectrum, and the di ffi culty in extracting ∆ ν . Concern-ing the estimation of ∆ ν we tested the e ff ect of adding rotation onthe PS ⊗ PS and found that this had a negligible impact on thecentral position of the ∆ ν and ∆ ν/ i (cid:63) = ◦ and 90 ◦ . The main e ff ect observed was achange in the relative heights between the peaks. This suggests thatthat PS ⊗ PS provides a robust estimate of ∆ ν even in the case ofhigh rotation, in agreement with Mosser & Appourchaux (2009) who obtained good ∆ ν estimates for some fast rotators observed byCoRoT (Mosser et al. 2009). We checked for needed line-of-sightcorrections to ν max following Davies et al. (2014) — For E167 thisamounts to δν s ≈ . µ Hz at ν max ; for E243 δν s ≈ . µ Hz, bothwell within our adopted uncertainties on ν max . We were unable to obtain a good estimate of the stellar rotationfrom the rotational splittings of mode frequencies. The surface rota-tion could, however, be studied using K2 light curves, which showsigns of spot modulation. We estimated the surface rotation pe-riod in three complimentary ways (Aigrain et al. 2015), namely,from the imprint in the power spectrum at low frequencies (see,e. g., Nielsen et al. 2013), from the autocorrelation function (ACF)(see, e. g., McQuillan, Aigrain & Mazeh 2013), and from a Morletwavelet analysis of the light curve (see Mathur et al. 2010; Garcíaet al. 2014; Ceillier et al. 2016). All three methods agree for bothstars. In Figure 3 we show the ACFs of the light curves where onlythe correction from the apparent movement on the CCD has beenremoved, i. e., any long term trends will be retained.For E167 we see the presence of two periodicities (see Fig-ure 3), viz., ∼ . ∼ ∼ . P rot ≈ . ∼
18% of break-up and have an inclination of ∼ ± ◦ . Concerning the projected rota-tion rate used to obtain the inclination Mermilliod, Mayor & Udry(2009) gives an independent and corroborating value of v sin i (cid:63) = . ± . − . The longer period signal might be interpreted asthe e ff ect of beating of close frequencies from di ff erential rotation— indeed, signal is seen in the power spectrum in a group of peaksat ∼ . µ Hz ( ∼ .
84 days), at ∼ . µ Hz (close to sum of groupedpeaks), and at ∼ . − . µ Hz (close to di ff erences of grouped peaks)as would be expected for a beating signal (see, e. g., Mathur et al.2014a). Removing the ∼ ∼
10 days (see McQuillan, Aigrain & Mazeh 2013,their Figure 4). In terms of ( B − V ) , or mass, the fast scenario issupported by the proximity to the Kraft break (Kraft 1967), whichmarks where the surface convection zone becomes too shallow toproduce a significant braking through a magnetised wind and theobserved rotation rate thus becomes highly dependent on the initialrotation rate (see, e. g., van Saders & Pinsonneault 2013). Giventhat we see solar-like oscillations and possible signals from spots computed as v crit ≈ (cid:112) GM / R p and assuming that the polar radii R p canbe approximated by the non-rotating radius (Maeder 2009).c (cid:13)000
10 days (see McQuillan, Aigrain & Mazeh 2013,their Figure 4). In terms of ( B − V ) , or mass, the fast scenario issupported by the proximity to the Kraft break (Kraft 1967), whichmarks where the surface convection zone becomes too shallow toproduce a significant braking through a magnetised wind and theobserved rotation rate thus becomes highly dependent on the initialrotation rate (see, e. g., van Saders & Pinsonneault 2013). Giventhat we see solar-like oscillations and possible signals from spots computed as v crit ≈ (cid:112) GM / R p and assuming that the polar radii R p canbe approximated by the non-rotating radius (Maeder 2009).c (cid:13)000 , 1–10 steroseismology of Hyades stars Lag (Days) −0.3−0.2−0.10.00.10.20.30.4 A C F Lag (Days) −0.8−0.6−0.4−0.20.00.20.40.60.8 A C F Figure 3.
Autocorrelation functions for E167 (left) and E243 (right), where only the systematics from the apparent stellar motion on the CCD and a 30 dayEpanechnikov (Hastie, Tibshirani & Friedman 2009) filter have been removed. Vertical broken lines indicate the first four maxima of a given periodicity.For E167 we have in red added a 1 .
23 day Epanechnikov smoothed version to highlight the underlying ∼ the star must have a convective envelope, but given the relativelyyoung age of the star (compared to its MS lifetime) it is conceiv-able that spin-down has not had time to take e ff ect. The slow sce-nario of P rot ≈ P rot ≈ P rot − ( J − K s ) relation for the Hyades by De-lorme et al. (2011), but we note that the ( J − K s ) colour of ourtarget is at the limit of the calibration for this relation. In terms ofother gyrochronology relations from, for instance, Barnes (2007)and Mamajek & Hillenbrand (2008) the fast scenario is supported.If the rotation rate follows the fast scenario higher order ef-fects should perturb the oscillation frequencies (see, e. g., Kjeldsenet al. 1998; Reese, Lignières & Rieutord 2006; Ouazzani & Goupil2012). Suárez et al. (2010) describe the e ff ect on the ridges of theéchelle diagram from including rotation in a perturbative mannerand near-degeneracy e ff ects, and find among other e ff ects a shiftin the δ spacing between ridges. As mentioned in Section 4.1 wefind δ ≈ µ Hz from the ridge averages shown in Figure 2; from arange of models matching the star in terms of mass, age, and metal-licity we derive a median δ ≈ . ± . µ Hz, where the uncertaintyis given by the median-absolute-deviation of the individual modelvalues. This di ff erence could potentially be caused by rotation, butan in-depth analysis of such higher order e ff ects is beyond the scopeof the current paper.For E243 only a single period of ∼ .
28 days is seen in the ACF(right panel of Figure 3), which corresponds to ∼
12% of break-up.Comparing the measured v sin i (cid:63) to the estimated rotational veloc-ity suggests that the star is seen at an angle of i (cid:63) ≈ ◦ , that is, edge-on. Concerning the projected rotation rate used to obtain the incli-nation Gunn & Kraft (1963), Kraft (1965), and Ho ffl eit & Jaschek(1982) (see Ho ffl eit & Warren 1995), give independent and largelycorroborating values of v sin i (cid:63) = ,
55 , and 53 km s − . E243 wasstudied by Krisciunas et al. (1995) in a search for γ -Doradus Typevariables in the Hyades, where the authors postulate that the de-tected variability is likely due to spot modulation . Curiously, these according to the SIMBAD database this study is the reason why the staris listed in Samus, Durlevich & et al. (2009) (and hence SIMBAD) as anellipsoidal variable star, which it is not. authors find a periodicity of 1 . ROSAT
X-ray hardness ratio measurements in the0 . − . R X = − . ± .
08, with R X = L X / L bol . Here we used the conversion between ROSAT countsand hardness ratio to flux by Fleming et al. (1995) and Schmitt,Fleming & Giampapa (1995), and the luminosity estimated in Sec-tion 3. The above value corresponds largely to those from the0 . − . Einstein Observatory and
ROSAT
All-Sky Sur-vey (RASS) measurements by, respectively, Stern et al. (1981) andStern, Schmitt & Kahabka (1995) after converting when appropri-ate to the
ROSAT . − . . For E243we derive from measurements by Voges et al. (1999) a value oflog R X ≈ − . ± .
16; Coronal X-ray measurements from theRASS by Huensch, Schmitt & Voges (1998) and
ROSAT measure-ments from Stern, Schmitt & Kahabka (1995) largely agree withthis estimate. Wright et al. (2011) o ff ers a relation between R X andthe Rossby Ro number (see also Pizzolato et al. 2003; Douglaset al. 2014), where Ro is defined as P rot /τ c with τ c being the mass-dependent convective turnover time-scale. In the following we usethe τ c ( M ) relation from Wright et al. (2011) to determine Ro , withthe mass from the seismic modelling (Section 4.3).For E167 the two di ff erent scenarios for the rotation rate cor-responds to Rossby numbers of Ro ∼ . ± .
01 (fast) and Ro ∼ . ± .
06 (slow). From Wright et al. (2011) one should for Ro ∼ .
83 expect a level of R X ≈ − . ± .
24, and for Ro ∼ .
10 the starshould fall in the saturated regime with log R X ≈ − . ± .
08. ForE243 one would expect a value of log R X ≈ − . ± .
28. As seenthe measured levels disagree with those expected for E243 and thefast scenario for E167. The ( B − V ) colours of the stars, with valuesof ( B − V ) = .
42 (E167) and ( B − V ) = .
41 (E243), do, how-ever, also place the stars outside the calibration range adopted by The Chandra Portable Interactive Multi-Mission Simulator, c (cid:13) , 1–10 M. N. Lund et al.
Wright et al. (2011). Comparing instead to Vilhu & Walter (1987)who include hotter stars we find that the two stars conform withan expected range of R X ≈ − . − .
5. The relatively low levelsof chromospheric activity also agrees with the results of Simon &Landsman (1991) and Schrijver (1993), who both include E243 intheir analysis. These authors find that activity is reduced for starsearlier than ∼ F5, likely due to an ine ffi cient dynamo operating inthe shallow convection zone of such early-type stars. In a study ofF5-type stars in the Hyades Böhm-Vitense et al. (2002) finds that( B − V ) ≈ . − .
43 marks a transition region in the dependenceof X-ray flux with v sin i (cid:63) (with a decreasing X-ray flux with in-creasing v sin i (cid:63) ), and in the onset of an e ffi cient magnetic braking.Both stars thus seem to be in a very interesting region in terms ofrotation and activity.For both stars we further assessed the mean activity level us-ing the activity proxy (cid:104) S ph , k = (cid:105) as defined in García et al. (2010) andMathur et al. (2014a,b). For E167 we adopted the fast scenario forthe period used in calculating the activity proxy. We obtained val-ues of (cid:104) S ph , k = (cid:105) = ± (cid:104) S ph , k = (cid:105) = ± P rot – (cid:104) S ph , k = (cid:105) space that is unexplored with data from the nominal mission —this likely stems from the sparsity in the number of young, hot,stars that were suggested for observations for the sake of detectingsolar-like oscillations. The two targets analysed here only provide us with limited seis-mic information, that is, only the average seismic parameters ∆ ν and ν max . These were used together with estimates of the two stars’metallicity and e ff ective temperatures to determine the global pa-rameters of the stars using grid based searches. Three pipelineswere used in the modelling — the Yale-Birmingham pipeline (YB;Basu, Chaplin & Elsworth 2010; Basu et al. 2012; Gai et al.2011), the Bellaterra Stellar Parameters Pipeline (BeSPP; Serenelliet al. 2013, Serenelli (in prep.)), and the Bayesian Stellar Algo-rithm pipeline (BASTA; Silva Aguirre et al. 2015). Three di ff er-ent grids of models were used in the case of YB, with modelsfrom the Dartmouth group (Dotter et al. 2008), the Yonsei-Yale( Y ) isochrones (Demarque et al. 2004), and the Yale Stellar Evo-lution Code (YREC; Demarque et al. 2008) as described by Basuet al. (2012) (YREC2). In all cases ∆ ν for the YB models werecalculated using the simple scaling relation between ∆ ν and den-sity (i. e., ∆ ν ∝ (cid:112) M / R ). BeSPP and BASTA used grids of modelscalculated using the Garching Stellar Evolution Code (GARSTEC;Weiss & Schlattl 2008). For BeSPP and BASTA model values for ∆ ν were calculated using both the scaling relation and individualfrequencies of radial modes. In cases where the ∆ ν scaling rela-tion was used, the corrections given in White et al. (2011) (forYB) and Serenelli et al. (2016, in prep.) (for BeSPP and BASTA)were applied to correct for the deviations of ∆ ν values from theusual scaling relations. The value of ν max was computed using theusual scaling scaling relation ( ν max ∝ g/ √ T e ff ). Further details ofthe pipelines, grids, and scaling relations are described in Chaplinet al. (2014) and Silva Aguirre et al. (2015).From the grid-based modelling (GBM) we obtain for E167values of M = . ± .
06 M (cid:12) , R = . ± .
03 R (cid:12) , ρ = . ± .
15 g cm − , and t = ±
387 Myr; for E243 we obtain M = . ± .
06 M (cid:12) , R = . ± .
03 R (cid:12) , ρ = . ± .
13 g cm − ,and t = ±
304 Myr. The reported values are those obtainedfrom the BASTA pipeline using the SPC T e ff and [Fe / H]. We have added in quadrature to the formal uncertainties a systematic un-certainty given by the root-mean-square di ff erence between the re-ported BASTA values and those obtained from the other pipelinesand spectroscopic inputs.Both grid-based age estimates are seen to be slightly higherthan those normally derived from isochrone fitting to the fullcolour-magnitude diagram (CMD). This di ff erence is not com-pletely unexpected, and has its origins in the limited nature of thedata available to us here, as we now go on to explain. Neverthe-less, as we shall also see, including the asteroseismic parametersfor these main-sequence stars gives much better constraints on thefundamental properties than would be possible from CMD fittingof main-sequence stars alone.We begin by recalling that CMD fitting of clusters works wellonly when data are available that span a range of evolutionarystates, i.e, including turn-o ff stars and also red-giant-branch stars.The left-hand panel of Figure 4 shows the CMD of the Hyadesusing data from Stern, Schmitt & Kahabka (1995). The plottedisochrones are from GARSTEC, with colour indicating age (see thesidebar) and linestyle indicating metallicity (full-line isochroneshave [Fe / H] = .
2, while dashed-line isochrones have [Fe / H] = . m − M ) = .
25. We used E ( B − V ) = . ± . R V ≡ A V / E ( B − V ) = . χ surfaces for two fits: one where we limited data tothe main-sequence only (6 < V <
11, right) and another wherewe used all the available data (stars with V <
11, left). For bothcases shown we adopted [Fe / H] = .
2. We see that limiting to themain-sequence only provides no discernible constraint on age. Theconstraints are of course even weaker if we perform CMD fits us-ing the two asteroseismic stars only (again with no seismic data). Incontrast, we obtain good constraints on the age, and optimal valuesthat agree with the canonical literature values, when we include tar-gets beyond the main-sequence (see also Perryman et al. 1998 andPinsonneault et al. 2004). Unfortunately, stars close to the turn o ff are likely to be too hot to show solar-like oscillations. Nevertheless,we see that the asteroseismic results obtained on the two stars – al-beit using average asteroseismic parameters only – give much bet-ter constraints than those provided by the CMD fits to non-seismicdata on main-sequence stars alone.That the age constraints from the asteroseismic results are nottighter still reflects the nature of the average asteroseismic param-eters. Both depend (in whole or large part) on di ff erent combina-tions of ratios of mass and radius — they thus lack explicit infor-mation on core properties and this has an impact on age estimatesfor stars in the relatively slow MS phase (Gai et al. 2011; Chaplinet al. 2014). Much tighter constraints are possible on the low-agepart of the MS when using individual oscillation frequencies (SilvaAguirre et al. 2015).Nevertheless, we still see a bias in the asteroseismic age esti-mates, and some of this arises from the way in which ages are esti-mated using a probabilistic approach when matching to isochrones(or grids) of stellar models. If a star lies equally close to twoisochrones in terms of input parameters the most likely will be cho-sen based on evolutionary speed. Therefore, without prior knowl-edge, one is much more likely to find a star that belongs to an older(say over 1 Gyr) isochrone because evolution is slower than for a c (cid:13) , 1–10 steroseismology of Hyades stars (B − V) V A g e ( G y r ) Distance modulus A g e ( G y r ) Distance modulus
Figure 4.
Left: colour-magnitude diagram (CMD) of Hyades stars, with parameters adopted from Stern, Schmitt & Kahabka (1995). The colour of theGARSTEC plotted isochrones indicate to age; full-line isochrones have [Fe / H] = .
2, while dashed-lined ones have [Fe / H] = .
15. For all isochrones weadopted for this plot a distance modulus of ( m − M ) = .
25. We adopted E ( B − V ) = . ± .
002 (Taylor 1980) and R V ≡ A V / E ( B − V ) = . χ surfaces for CMD fits to stars with V <
11 (left panel) and 6 < V <
11 (right panel) as a function of age and distance modulus, goingfrom high χ -values in dark blue to low values in light blue. For both cases shown [Fe / H] = . sub 1-Gyr isochrone. Two of our pipelines (BASTA and BeSPP)use Bayesian schemes when computing the posterior parameter dis-tributions, and here correct for the density of points in the adoptedgrids to make a proper marginalisation — this correction explicitlyintroduces the e ff ect of evolutionary speed (see, e. g., Pont & Eyer(2004) and Jørgensen & Lindegren (2005) for examples and furtherdiscussion).There are two main reasons for the bias, one easy to removeand one more fundamental. The first reason is that at low ages, thedistribution function of ages for a given star cuts o ff abruptly atzero, biasing the result to higher ages. This e ff ect can be mitigatedto some extent by using the logarithm of the ages, but this does notremove the bias completely. The second reason for the bias is morefundamental, and has to do with the fact that on the main-sequence,stars within a small metallicity range can have many di ff erent agesfor a given range of temperature and luminosity (or in the astero-seismic context a given range of ν max ). In other words, isochronesof many di ff erent ages can pass through the error-box. As describedabove, evolutionary speed makes it much more likely to encounteran older than a younger star, and therefore the results of any grid-based modelling will have a fundamental bias towards higher agesif no prior on age is adopted. Figure B1 shows this clearly. Thebias can be reduced if e ff ective temperature and metallicity can bemeasured to a much better precision. In the case of clusters, havingdata on more stars in di ff erent evolutionary phases of course helpsgreatly because we can apply the condition that all stars must havethe same age, which is essentially the assumption made in deter-mining ages by fitting isochrones to cluster colour-magnitude dia-grams. There are also other factors to note. The two stars here aredi ff erent to many of the stars analysed for asteroseismology in Ke-pler , in terms of being relatively hot, massive, young, and rapidlyrotating. This of course raises the question of whether assumptionsmade regarding the mapping of the average asteroseismic param-eters to stellar properties are incorrect for these stars? The resultssuggest there is not a significant bias. First, the relationship of ∆ ν to ν max follows that shown by the asteroseismic cohort of Kepler stars. We also examined the potential impact of rotational mixing on ∆ ν , which is unaccounted for in the models we used, by look-ing at di ff erences in stellar MESA models (Paxton et al. 2011) withand without convective core overshoot. We found no appreciablechange in ∆ ν from varying the amount of overshoot, which con-forms with the results reported by Eggenberger et al. (2010) whostudied the e ff ect of adding rotational mixing to a 1M (cid:12) model. Wetherefore adopt the assumption that the model values of ∆ ν and ν max are representative of what would be found for slowly rotating stars.We also remind the reader that in Section 4.1 it was found that ro-tation should not a ff ect our ability to extract a good estimate of ∆ ν .The above of course also goes to the issue of the physicsused in our stellar models. Might missing physics be a cause ofthe bias? The obvious question we can answer in relation to this iswhether, when we use our adopted models, we are able to recoverthe canonical age estimate when presented with the usual observa-tional CMD data as input (i. e., colours and an assumed distancemodulus and metallicity as input). As discussed above (Figure 4),we have demonstrated that when BeSPP is coupled to GARSTEC,we recover a satisfactory age. That does not of course say that thephysics is indeed correct.Recently, Brandt & Huang (2015a,b,c) performed anisochrone analysis which included rotation via the models of Ek-ström et al. (2012) and Georgy et al. (2013). By adding rotation,which in the adopted models had the e ff ect of lengthening the MSlifetime, these authors find a slightly higher age than the consensus,namely, t ∼ ±
100 Myr. This result rests on the same handfulof upper MS ( M > . (cid:12) ) turn-o ff stars that guided the isochronefittings by Perryman et al. (1998). With the seismic solution for the stellar radii and an angular diam-eter from the IRFM, we can estimate the seismic distance to thecluster as follows: d seis = C R seis θ IRFM , (4.1) c (cid:13) , 1–10 M. N. Lund et al.
Hip07 Hip97 MadTyc MadHip deBTyc deBHip−8−6−4−202 d s e i s . − d li t . ( p c ) E167 E243
Figure 5.
Comparison of distances obtained from the seismic radii (averagefrom di ff erent spectroscopic inputs and pipelines) and the IRFM angulardiameter with those determined from parallaxes in the literature. where C is a conversion factor to parsec (see Silva Aguirre et al.2012; Rodrigues et al. 2014). We find seismic distances of d seis = . ± . d seis = . ± . Hipparcos by van Leeuwen (2007) (Hip07), vanLeeuwen et al. (1997) (Hip97), those from de Bruijne, Hoogerw-erf & de Zeeuw (2001) using secular parallaxes from Tycho-2 (Høget al. 2000) (deBTyc) or
Hipparcos (van Leeuwen et al. 1997) (deB-Hip), and those from Madsen, Dravins & Lindegren (2002) usingsecular parallaxes as above (MadTyc / MadHip). We find that all par-allax distances for E243 match the seismic ones reasonably well;for E167 all distances are > σ larger than the seismic ones. We have presented the asteroseismic results on two cool main-sequence stars belonging to the Hyades open cluster. These are thefirst ever detections of solar-like oscillations in main-sequence starsin an open cluster. Both stars are very likely fast rotators ( P rot < ff erent from the older, moreslowly rotating cool main-sequence stars that dominated the aster-oseismic cohort from the nominal Kepler mission.The K2 mission is scheduled to re-observe the Hyades clus-ter in C13, providing an unprecedented opportunity to expand theasteroseismic cohort, potentially to stars for which we can do de-tailed modelling on individual frequncies (something that is verychallenging for the two stars reported here). We have indicatedin Figure 6 the stars from C13 for which we predict a detectionof solar-like oscillations (including predicted marginal detections).The estimates of L used here were computed from Hipparcos par-allaxes (van Leeuwen 2007), while T e ff values were computed fromthe colour- T e ff relations of Casagrande et al. (2010).Unfortunately neither of the targets analysed in this paper ispredicted to lie on active silicon in C13 . We find, however, that55 of the Hyades members from Perryman et al. (1998) will be on using the K2FOV tool; T eff (K) L / L ⊙ E167 E243 0.7Gyr
Figure 6.
BaSTI isochrone (Pietrinferni et al. 2004) with an age of 700Myr. For this isochrone standard BaSTI input physics was adopted, but withovershoot on the MS and with adopted heavy and Helium mass fractions of Z = . Y = . T e ff given by the SPC and with the di ff erenceto the IRFM value added in quadrature to the uncertainty. The luminosi-ties for the two targets are given by these T e ff values and the seismic radii.Circular markers indicate stars that could be observed in Campaign 13 forwhich we predict detectable solar-like oscillations. The stars that are furthermarked with a cross will be amenable to interferometric observations withPAVO@CHARA (ten Brummelaar et al. 2005; Ireland et al. 2008). Thedashed line gives the red edge of the classical instability strip, as definedby Pamyatnykh (2000). The parts of the isochrone overlaid with a solidblack line indicate where ν max values computed from the standard scalingby Kjeldsen & Bedding (1995) drops below the long-cadence Nyquist fre-quency of ∼ µ Hz. silicon in C13; of these we estimate ∼
22 will have
Kepler magni-tudes in the range Kp = − . T e ff (cid:46) P rot ≈ −
15 days. Based on knowledge of K2noise properties asteroseismic analysis of these targets should befeasible (see Stello et al. 2015; Van Cleve et al. 2015; Lund et al.,in press). A joint analysis may provide constraints on the clusterage, especially if individual frequencies or even just an estimate ofthe core-sensitive small frequency separation δν can be obtainedin some stars (Christensen-Dalsgaard 1993). Moreover, for severalstars independent constraints may be obtained from interferometrywith PAVO@CHARA (ten Brummelaar et al. 2005; Ireland et al.2008). One of the C13 targets is a giant and can comfortably be ob-served in long-cadence (LC; ∆ t ≈
30 min) — this star (HIP 20885;77 Tau) would be valuable to constrain the cluster age, especiallyif combined with the MS targets and the two C4 giants analysed byWhite et al. (in prep.).
ACKNOWLEDGMENTS
We acknowledge the dedicated team behind the
Kepler and K2 missions,without whom this work would not have been possible. We thank DanielHuber and Benoit Mosser for useful comments on an earlier version of thepaper. M.N.L. acknowledges the support of The Danish Council for Inde-pendent Research | Natural Science (Grant DFF-4181-00415). M.N.L. waspartly supported by the European Community’s Seventh Framework Pro-gramme (FP7 / (cid:13)000
Kepler and K2 missions,without whom this work would not have been possible. We thank DanielHuber and Benoit Mosser for useful comments on an earlier version of thepaper. M.N.L. acknowledges the support of The Danish Council for Inde-pendent Research | Natural Science (Grant DFF-4181-00415). M.N.L. waspartly supported by the European Community’s Seventh Framework Pro-gramme (FP7 / (cid:13)000 , 1–10 steroseismology of Hyades stars (ASTERoseismic Investigations with SONG and Kepler ) funded by the Eu-ropean Research Council (Grant agreement no.: 267864). W.J.C., G.R.D,and A.M. acknowledge the support of the UK Science and TechnologyFacilities Council (STFC). S.B. acknowledges partial support from NASAgrant NNX13AE70G and NSF grant AST-1514676. A.M.S. is partially sup-ported by grants ESP2014-56003-R and ESP2015-66134-R (MINECO).V.S.A. acknowledges support from VILLUM FONDEN (research grant10118). R.A.G. acknowledges the support from the ANR program IDEE(n. ANR-12-BS05-0008) and the CNES. D.W.L acknowledges partial sup-port from the
Kepler mission under Cooperative Agreement NNX13AB58Bwith the Smithsonian Astrophysical Observatory. This research has madeuse of the Washington Double Star Catalog maintained at the U.S. NavalObservatory; the WEBDA database, operated at the Department of Theoret-ical Physics and Astrophysics of the Masaryk University; and the SIMBADdatabase, operated at CDS, Strasbourg, France.
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Table A1 gives the CfA radial velocities obtained over a period of over 35years. The dates are given in heliocentric Julian date (minus 2400000), andthe radial velocities are on the native CfA system in km s − . To put thesevelocities onto the IAU system, add 0 .
14 km s − . Concerning the estimatedinternal errors, these are for the CfA Digital Speedometers estimated fromthe anti-symmetric noise in the correlation function as described in Tonry& Davis (1979); for TRES it is an educated guess based on extensive expe-rience with dozens of hot rapidly rotating stars. The velocities are plotted inFigure A1. See Section 2 for additional info on the data. APPENDIX B: BASTA MODEL DISTRIBUTIONS
Figure B1 Presents the posterior distributions from the GBM of E167 withBASTA. As seen from the models with an age in the interval between500 −
800 Myr (marked in green) a higher metallicity is preferred for agood reconciliation with isochrone-based ages. The higher [Fe / H] gives acorresponding increase in T e ff and mass, and with it a decrease in age. It isalso clear that the models that provide an age as expected for the Hyadesstill match the average seismic parameters for the star, indicating that theseonly contribute with a mild constraint on age.c (cid:13)000
800 Myr (marked in green) a higher metallicity is preferred for agood reconciliation with isochrone-based ages. The higher [Fe / H] gives acorresponding increase in T e ff and mass, and with it a decrease in age. It isalso clear that the models that provide an age as expected for the Hyadesstill match the average seismic parameters for the star, indicating that theseonly contribute with a mild constraint on age.c (cid:13)000 , 1–10 steroseismology of Hyades stars Table A1.
Radial velocity data for E167 (28 measurements) and E243 (24 measurements). The four columns give the heliocentric Julian date (minus 2400000), the radial velocity on the native CfA systemin km s − , the estimated internal error, and the source of the data. To put these velocities onto the IAU system, add 0 . − . For the CfA Digital Speedometers the error is estimated from the anti-symmetricnoise in the correlation function as described in the Tonry & Davis (1979); for TRES it is an educated guess based on extensive experience with dozens of hot rapidly rotating stars. E167 E243Date RV Uncertainty Source † Date RV Uncertainty Source † (HJD-2400000) (km s − ) (km s − ) (HJD-2400000) (km s − ) (km s − )44560.8212 38.67 0.74 2 44560.7928 38.53 2.72 244887.8537 38.48 0.67 2 44954.7683 35.84 1.73 244954.8595 36.36 0.85 2 45241.9557 39.47 1.58 245339.8980 37.66 0.61 2 45604.9773 40.01 1.52 245725.5179 39.35 0.53 3 45694.6675 41.38 1.83 346777.6586 39.60 0.57 3 45710.5961 38.37 1.79 147084.8265 38.84 0.33 3 45717.7013 38.95 1.33 249004.6987 38.43 1.14 3 45721.6767 39.09 1.83 349015.5712 38.15 0.40 3 45723.5431 40.07 1.68 349023.6034 38.26 0.68 3 48284.6245 45.81 2.19 349033.6385 38.40 0.78 3 49004.6898 39.93 1.80 349085.5137 38.18 0.69 3 49018.5060 40.15 2.23 349259.7909 38.59 0.51 3 49261.7933 39.00 1.20 349314.8078 38.67 0.63 3 49435.5377 40.36 1.43 349352.6751 39.17 0.52 3 49614.9038 38.30 1.26 349640.8260 39.12 1.02 3 49768.5359 40.25 1.12 350421.7596 38.45 0.77 3 49965.8878 40.30 1.57 350470.5632 39.74 0.53 3 53013.6234 41.71 1.50 350797.6904 38.37 0.80 3 53040.5089 41.33 1.06 351146.7703 38.65 0.47 3 53043.5878 41.17 1.23 352706.5335 38.95 0.83 3 56673.7786 38.54 0.50 456308.7281 38.81 0.10 4 56675.8239 38.52 0.50 456309.6904 38.82 0.10 4 56703.5994 38.39 0.50 456310.7357 38.85 0.10 4 57410.7730 38.01 0.50 456323.6397 39.19 0.10 456324.7170 38.85 0.10 456677.6733 38.85 0.10 457385.8825 38.90 0.10 4 †
1: MMT Digital Speedometer; 2: Tillinghast Reflector Digital Speedometer; 3: Wyeth Reflector Digital Speedometer; 4: Tillinghast Reflector Echelle Spectrograph
HJD − 2400000 (days) R a d i a l v e l o c i t y ( k m s − ) Mean = 38.66 kms −1 RMS = 0.63 kms −1 HJD − 2400000 (days) R a d i a l v e l o c i t y ( k m s − ) Mean = 39.73 kms −1 RMS = 1.83 kms −1 Figure A1.
Radial velocity data for E167 (left) and E243 (right) from CfA spanning a period of ∼
35 years, see Table A1 for the individual data values. Thedashed and dotted lines indicate the mean and rms values of the velocities, with the values given in the plots. The markers indicate the di ff erent instrumentsused in obtaining the data, specifically, the MMT Digital Speedometer ( (cid:67) ); the Tillinghast Reflector Digital Speedometer ( (cid:3) ); the Wyeth Reflector DigitalSpeedometer ( ◦ ); the Tillinghast Reflector Echelle Spectrograph ( (cid:52) ).c (cid:13) , 1–10 M. N. Lund et al.
Figure B1.
Posterior distributions from the GBM of E167 with BASTA. The di ff erent panels give the distributions for di ff erent model quantities of interest,with observed average seismic parameters and spectroscopic inputs from SPC indicated by vertical red lines, and final model values from BASTA indicatedby vertical cyan lines; dashed lines indicate the 1 − σ values on the parameters. For the luminosity the indicated value is calculated from the SPC T e ff and thedistance from the parallax of Madsen, Dravins & Lindegren (2002). The models shown in green are the ones falling in the age interval between 500 − (cid:13)000