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The K2 Galactic Archaeology Program Data Release 2 Zinn, Joel C.; Stello, Dennis; Elsworth, Yvonne; García, Rafael A.; Kallinger, Thomas; Mathur, Savita; Mosser, Benoît; Bugnet, Lisa; Jones, Caitlin; Hon, Marc; Sharma, Sanjib; Schönrich, Ralph; Warfield, Jack T.; Luger, Rodrigo; Pinsonneault, Marc H.; Johnson, Jennifer A.; Huber, Daniel; Aguirre, Victor Silva; Chaplin, William J.; Davies, Guy R. DOI: 10.3847/1538-4365/abbee3

Citation for published version (Harvard): Zinn, JC, Stello, D, Elsworth, Y, García, RA, Kallinger, T, Mathur, S, Mosser, B, Bugnet, L, Jones, C, Hon, M, Sharma, S, Schönrich, R, Warfield, JT, Luger, R, Pinsonneault, MH, Johnson, JA, Huber, D, Aguirre, VS, Chaplin, WJ, Davies, GR & Miglio, A 2020, 'The K2 Galactic Archaeology Program Data Release 2: Asteroseismic results from campaigns 4, 6, & 7', The Astrophysical Journal Supplement Series, Volume, vol. 251, no. 2. https://doi.org/10.3847/1538-4365/abbee3

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THE K2 GALACTIC ARCHAEOLOGY PROGRAM DATA RELEASE 2: ASTEROSEISMIC RESULTS FROM CAMPAIGNS 4, 6, & 7

Joel C. Zinn,1, 2, 3 Dennis Stello,2, 4, 5, 6 Yvonne Elsworth,7, 5 Rafael A. Garc´ıa,8 Thomas Kallinger,9 Savita Mathur,10, 11, 12 Benoˆıt Mosser,13 Lisa Bugnet,14, 8 Caitlin Jones,7 Marc Hon,15 Sanjib Sharma,4, 6 Ralph Schonrich,¨ 16 Jack T. Warfield,17, 18 Rodrigo Luger,19, 20 Marc H. Pinsonneault,17 Jennifer A. Johnson,17 Daniel Huber,21 Victor Silva Aguirre,5 William J. Chaplin,7, 5 Guy R. Davies,7, 5 and Andrea Miglio5, 7

1Department of Astrophysics, American Museum of Natural History, Central Park West at 79th Street, New York, NY 10024, USA 2School of Physics, University of New South Wales, Barker Street, Sydney, NSW 2052, Australia 3Department of Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus, OH 43210, USA 4Sydney Institute for Astronomy (SIfA), School of Physics, University of Sydney, NSW 2006, Australia 5Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark 6Center of Excellence for Astrophysics in Three Dimensions (ASTRO-3D), Australia 7School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK 8AIM, CEA, CNRS, Universit´eParis-Saclay, Universit´eParis Diderot, Sorbonne Paris Cit´e, F-91191 Gif-sur-Yvette, France 9Institute of Astrophysics, University of Vienna, Turkenschanzstrasse¨ 17, Vienna 1180, Austria 10Space Science Institute, 4750 Walnut Street Suite #205, Boulder, CO 80301, USA 11Instituto de Astrof´ısica de Canarias, La Laguna, Tenerife, Spain 12Dpto. de Astrof´ısica, Universidad de La Laguna, La Laguna, Tenerife, Spain 13LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universit´e, Universit´ede Paris Diderot, 92195 Meudon, France 14IRFU, CEA, Universit´eParis-Saclay, 91191 Gif-sur-Yvette, France 15School of Physics, University of New South Whales, Barker Street, Sydney, NSW 2052, Australia 16Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking RH5 6NT, UK 17Department of Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus OH 43210, USA 18Department of Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus OH 43210 19Center for Computational Astrophysics, Flatiron Institute, New York, NY, USA 20Virtual Planetary Laboratory, University of Washington, Seattle, WA, USA 21 Institute for Astronomy, University of Hawai‘i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

ABSTRACT Studies of Galactic structure and evolution have benefitted enormously from Gaia kinematic infor- mation, though additional, intrinsic stellar parameters like age are required to best constrain Galactic models. Asteroseismology is the most precise method of providing such information for field star pop- ulations en masse, but existing samples for the most part have been limited to a few narrow fields of view by the CoRoT and Kepler missions. In an effort to provide well-characterized stellar parameters across a wide range in Galactic position, we present the second data release of red giant asteroseismic parameters for the K2 Galactic Archaeology Program (GAP). We provide νmax and ∆ν based on six independent pipeline analyses; first-ascent red giant branch (RGB) and red clump (RC) evolutionary arXiv:2012.04051v1 [astro-ph.SR] 7 Dec 2020 state classifications from machine learning; and ready-to-use radius & mass coefficients, κR & κM , which, when appropriately multiplied by a solar-scaled effective temperature factor, yield physical stellar radii and masses. In total, we report 4395 radius and mass coefficients, with typical uncertain- ties of 3.3% (stat.) ± 1% (syst.) for κR and 7.7% (stat.) ± 2% (syst.) for κM among RGB stars, and 5.0% (stat.) ± 1% (syst.) for κR and 10.5% (stat.) ± 2% (syst.) for κM among RC stars. We verify that the sample is nearly complete — except for a dearth of stars with νmax . 10 − 20µHz — by comparing

Corresponding author: Joel C. Zinn [email protected] 2

to Galactic models and visual inspection. Our asteroseismic radii agree with radii derived from Gaia Data Release 2 parallaxes to within 2.2 ± 0.3% for RGB stars and 2.0 ± 0.6% for RC stars.

1. INTRODUCTION We have previously released a collection of νmax and The Galactic Archaeology Program (GAP; Stello et al. ∆ν values for 1210 K2 GAP red giants in Stello et al. 2015) has taken advantage of the multidirectional view (2017). The present release covers campaigns 4, 6, & of the Galaxy offered by the re-purposed Kepler mission, 7, and comprises 4395 stars. In addition to the global K2. With hundreds of thousands of stars observed, K2’s asteroseismic parameters, νmax and ∆ν, we also provide potential for studying the Galaxy is significant. Instead scaling-relation quantities that, when combined with an of a single snapshot of the Galaxy with Kepler (Borucki effective temperature, yield radii and masses. We also et al. 2008), K2 (Howell et al. 2014) observed along the provide estimates of systematic and statistical errors on ecliptic, including the local disk, the bulge, and even dis- the asteroseismic quantities, and establish the complete- tant regions of the halo. Importantly for this work, the ness of observed targets in order to ensure a well-defined K2 mission has delivered the quality of data necessary selection function. for asteroseismic analysis. The K2 GAP aims to provide fundamental stellar pa- 2. DATA rameters for red giants across the Galaxy. In combina- 2.1. Target selection tion with temperature and metallicity information, as- In the context of the GAP, analysis of the campaigns teroseismology can provide stellar radii, masses, and, presented here were prioritized due to their coverage when combined with stellar models, ages. Kepler red gi- of the sky: the Galactic center (C7) the Galactic anti- ant asteroseismology has yielded important findings for center (C4) and out of the Galactic plane (C6). These Galactic archaeology, including verifying the presence results will ultimately be joined with a forthcoming anal- of a vertical age gradient in the Galactic disc (Miglio ysis of the rest of the K2 campaigns for which GAP tar- et al. 2013; Casagrande et al. 2016); testing Galactic gets have been observed. The GAP targets red giants chemical evolution models (e.g., Spitoni et al. 2020); because they are bright (probing far into the Galaxy) and confirming an age difference between chemically- and because their oscillations are detectable from the and kinematically-defined thin and thick discs (Silva K2 long-cadence data, which has a Nyquist frequency Aguirre et al. 2018). Nevertheless, the Kepler aster- of ∼ 280µHz. All GAP targets for campaigns 4, 6 & oseismic sample was not curated for Galactic studies, 7 were selected from 2MASS (Skrutskie et al. 2006) to and so GAP’s deliberate and well-understood target se- have J − K > 0.5 and good photometric quality based lection for Galactic archaeology purposes sets up K2 to on 2MASS flags.1 The proposed targets passing these be a more useful tool for Galactic archaeology, particu- selection criteria were prioritized based on a rank or- larly in light of its expanded view of the Galaxy. Indeed, dering in V -band magnitude from bright to faint.2 C4 the K2 data is providing interesting insights into the rel- and C6 targets were chosen to have 9 < V < 15, and ative ages of chemically-defined stellar populations be- C7 targets were chosen to have 9 < V < 14.5, with yond the solar vicinity (Rendle et al. 2019, Warfield et some exceptions to the prioritization on a campaign- al., in preparation). to-campaign basis, as follows: One giant with existing K2 ’s potential is tempered, however, by a decreased RAVE data was prioritized in C4. In C6, priority was photometric precision compared to Kepler and a ∼ 80d given to 129 giants with existing APOGEE (Majewski dwell time per campaign instead of up to ∼ 4yr for Ke- et al. 2010) spectra, 607 with existing RAVE spectra, pler. These two limitations mean that K2 is mostly suited for giants with logg above ∼ 1.4; probes one to two magnitudes ‘shallower’ than Kepler; and yields less 1 The 2MASS qflg photometric quality flag was required to be A or precise asteroseismic measurements compared to Kepler B for J, H, and Ks, which ensures, among other things, that the signal-to-noise ratio is greater than 7. Additional flags ensured (Stello et al. 2017). And, although the accuracy of stel- that the photometry did not suffer from confusion from nearby lar parameters derived through asteroseismology is at objects (cflg == 0); was a single, unblended source (bflg == 1); the percent level (e.g., Silva Aguirre et al. 2012; Huber was not extended (xflg == 0); was not a known solar system object (aflg == 0); and had no neighbors within 6” (prox > et al. 2012; Zinn et al. 2019b), at this level, there are 6.0). See http:vizier.u-strasbg.frcgi-binVizieR?-source=B2mass measurement systematics that need to be corrected for for more details. (Pinsonneault et al. 2018). We therefore devote special 2 At the time of targeting, it was typically not well-known which attention in what follows to understanding the statis- stars were giants or dwarfs, but generally the giant fraction was expected to be close to 100% at the bright end and down to as tical and systematic uncertainties in our asteroseismic low as 20% ,depending on the campaign, at the faint end of the quantities. selection. 3 and 5 low-metallicity giants chosen from the literature Table 1. Targeted and observed stars as a function of campaign to be giants with [Fe/H]< −3. The highest priority to Campaign K2 GAP targeted K2 GAP observed GAP selected observed GAP targets in C7 was given to 222 known giants with existing spectroscopic data from APOGEE, and 23 tar- C4 17410 6357 5000 C6 8371 8313 8300 gets in NGC 6717 (but see below). C7 18698 4362 3500 Because observed targets were selected in a linear way from the target priority list, the selection functions forNote—‘K2 GAP targeted’ refers to the number of proposed targets and ‘K2 GAP observed’ are those that were observed, including targets not following the GAP each of the campaigns are well-defined. Nearly all of the selection function (see text). ‘GAP selected observed’ refers to the approximate C6 targets were observed, and so the observed targets number of GAP observed stars that follow the GAP target selection function conform to the selection function 9 < V < 15. Our C4 from the bright end of V = 9 to a limiting magnitude of 13.447, 15, and 14.5 targets were observed down to V = 13.447, and there- for C4, C6, and C7, respectively. fore follow the selection function down to that magni- tude limit (this magnitude limit approximately corre- sponds to the top 5000 GAP targets). In addition to these GAP-selected targets, there are fainter stars on K2SFF does, while furthermore generating lower levels the GAP target list that were serendipitously observed of white noise in the giant spectra than K2SFF. Be- by K2 through other non-GAP target proposals. Those cause the K2 GAP DR1 used K2SFF light curves, we stars do not follow the GAP selection function but are attempted to reconcile systematic differences in the am- still analyzed in this paper. However, we caution their plitudes of flux variations in the EVEREST and K2SFF use for population studies. The observed C7 targets light curves for consistency between this release and were accidentally chosen from an inverted priority list DR1. To do so, for each star, we multiply the EVEREST during the mission-wide target list consolidation before light curve flux by a scalar factor such that its power is upload to the spacecraft. As a result, only two of the equal to the K2SFF power. We then applied a boxcar higher priority APOGEE C7 targets were observed and high-pass filter with a width of 4 days to the light curve none from NGC 6717. As discussed in Sharma et al. to remove most of the non-oscillation variability (Stello (2019), the resulting effective selection function is one et al. 2015), and outliers 4σ in the time series were re- of either 9 < V < 14.5 or 14.276 < V < 14.5, de- moved. Seven targets were labeled as extended in the pending on the position on the sky. Approximately, the EPIC, and were not included in the present analysis.3 3500 lowest-priority GAP targets follow the full selec- tion function. There is, however, an additional popu- 3. METHODS lation of ∼ 600 stars in C7 that can be used for scien- 3.1. Extraction of asteroseismic parameters tific purposes, with the understanding that its selection function of 14.276 < V < 14.5 is different than that of In this data release, we focus on two asteroseismic the rest of the campaign (9 < V < 14.5). For addi- quantities, νmax and ∆ν, which we describe in turn. tional details, see Sharma et al.(2019). We account for The frequency at maximum acoustic power, νmax, these selection functions when comparing to models in scales with the acoustic cutoff frequency at the stellar at- our analysis by making sure to compare to the subset mosphere (Brown et al. 1991; Kjeldsen & Bedding 1995; of observed stars that follow reproducible GAP target Chaplin et al. 2008; Belkacem et al. 2011), so that criteria. An accounting of observed and targeted stars ν M/M is given in Table1. The approximate lines of sight for max ≈ . (1) ν 2p these campaigns are shown in Figure1. An on-sky map max, (R/R ) (Teff /Teff, ) of the campaigns is shown in Figure2. The frequency separation between modes of the same degree but consecutive radial order, ∆ν, scales with the 2.2. K2 light curve pre-processing stellar density (Ulrich 1986; Kjeldsen & Bedding 1995) We used the light curves generated by the EVER- according to EST pipeline (Luger et al. 2018), which uses campaign- s specific basis vectors to remove noise correlated across ∆ν M/M ≈ 3 . (2) pixels. Though the K2 GAP DR1 (Stello et al. 2017) ∆ν (R/R ) used Vanderburg & Johnson(2014) (K2SFF) light curves, we found that the EVEREST pipeline removes 3 The following EPIC IDs are affected in C4: 210344244, at least as much of the sawtooth-like systematic flux 210489346, 210766860, 210497173, 210608879; and in C6: variation induced by the K2 six-hour thruster firings as 212680904, 212708542. 4

6 10 Campaign 6 Solar circle ~ 8 kpc 4 5 ] 2 ] c c p p G.C. k k 0 [ [

Sun Y Z 0 Campaign 7 Campaign 4 5

2 10

4 2 4 6 8 10 12 10 5 0 5 10 R [kpc] X [kpc]

Figure 1. Left: approximate Z-R distribution of stars in K2 GAP DR2, for each campaign. Right: approximate X-Y distribution of stars in K2 GAP DR2, for each campaign.

Figure 2. Positions of observed K2 GAP stars in C4, C6, and C7, relative to the Galactic plane. Background image modified from ESA/Gaia/DPAC.

The νmax, and ∆ν solar reference values are not set Like Stello et al.(2017), we analyze the data using in stone, and are themselves measured quantities from multiple asteroseismic pipelines, whose descriptions are asteroseismic data of the Sun. All the pipelines have found therein. We briefly revisit each pipeline below. an internal set of recommended solar reference values, A2Z+ (hereafter A2Z) is based on the A2Z pipeline νmax, ,PIP and ∆ν ,PIP. In addition to those pipeline- described in Mathur et al.(2010). ∆ ν is computed using specific solar reference values, we adopt a solar reference two methods: the autocorrelation function of the time temperature of Teff, = 5772K (Mamajek et al. 2015). series and the Power Spectrum of the Power Spectrum. Then, the results from both methods are compared, and 5 a ∆ν value is only kept when both methods agree to of detected stellar oscillations in giants (Mosser et al. within 10%. The Power Density Spectra (PDS) of those 2010). The algorithm takes the autocorrelation, passes stars are checked to select the ones where modes are it through a convolutional neural network, and outputs present to high confidence. The FliPer metric (Bugnet a score varying from 0 (not a valid ∆ν detection) to 1 et al. 2018) is used to check stars where the amplitude (100% certain valid ∆ν detection), but which is not a of the convective background does not agree with the linear mapping to percent confidence in the detection. seismic detection. Finally, the ∆ν value is refined by For this reason, the score was calibrated using visually- cross-correlating a template of the radial modes around verified ∆ν detections, which showed that a score of 0.8 the region of the modes where ∆ν is varied. The value corresponds to a completeness of about 70% (i.e., this reported corresponds to the one obtained for the high- score rejects 30% stars that have valid ∆ν detections) est correlation coefficient. After fitting the convective and near total purity (i.e., all stars have visually-verified background with two Harvey laws (Mathur et al. 2011; ∆ν detections); ∆ν detections were considered valid for Kallinger et al. 2014) and subtracting it from the PDS, stars with a score above this 0.8 value. a Gaussian function is fit to the modes to estimate the BAM calculates νmax and ∆ν according to Zinn et al. frequency of maximum power, νmax. (2019c). ∆ν values for this data release are provided BHM is based on the OCT pipeline (Hekker et al. based on the autocorrelation method described therein, 2010), and performs hypothesis testing for solar-like os- which, in turn, is based on that of SYD. Stars are classi- cillations above the granulation background. If there is fied as oscillators or not based on the Bayesian evidence a frequency window that has significant signal, a νmax for the presence of solar-like oscillations (see Zinn et al. and ∆ν is computed according to Hekker et al.(2010), 2019c for details), and results in this data release are with increased, K2-specific quality control to ensure ∆ν returned for stars with 3.5µHz < νmax < 250µHz. ∆ν is not an alias of the true value. values are only provided for those that satisfy the SYD CAN returns νmax and ∆ν for stars whose autocorre- ∆ν-vetting algorithm described above. lation functions have characteristic time-scales and rms Any pipeline may return a νmax and/or a ∆ν for a variability that accord with a relation expected from given star. In what follows, however, we consider a “de- solar-like oscillators (Kallinger et al. 2016). A Bayesian tection” for a given pipeline to be a target for which a evidence is computed to determine whether there are νmax is returned, unless otherwise noted. A fraction of solar-like oscillations near the detected autocorrelation νmax-detected stars will not have a reliable ∆ν measure- time-scale (and therefore near νmax). A threshold for ment reported. The raw νmax and ∆ν and the uncertain- the evidence is determined based on visual inspection of ties reported by each pipeline are provided in Table2, a test set of power spectra. ∆ν is then calculated by while the number of νmax and ∆ν values returned are fitting individual radial modes. provided in Table3. COR first fits for ∆ν using the autocorrelation of the light curve. COR (Mosser & Appourchaux 2009) re- 3.2. Derived quantities turns results for stars that have ∆ν, FWHM, amplitude 0 0 3.2.1. Mean asteroseismic parameters, hνmaxi and h∆ν i of power, and the granulation at ν that follow rela- max Consolidating results from multiple pipelines, as we tions from Mosser et al.(2010), with stricter, K2-specific do here, has three primary benefits: outliers can be re- requirements for stars with a possible ν detected near max jected (e.g., stars that only one pipeline identifies as an the K2 thruster firing frequency, or alias thereof. oscillator can be considered dubious); the accuracy and SYD computes ν and ∆ν according to Huber et al. max precision of the parameters can be improved by averag- (2009). Results are returned only for stars that are ing results for the same star from different pipelines; and classified as having solar-like oscillations by a machine- the spread in values for a given star can be translated learning algorithm described in Hon et al.(2018b). ∆ ν into an uncertainty estimate. To perform this averag- results are provided only for stars that are classified ing, however, the possibility that there are systematic as having reliable ∆ν values by a machine-learning al- differences in pipeline results needs to be taken into ac- gorithm. The ∆ν-vetting algorithm is a convolutional count. Pinsonneault et al.(2018) demonstrated that neural network trained on K2 C1 data that were classi- such systematics existed in their multi-pipeline analy- fied by eye as having detectable ∆ν values. The algo- sis of Kepler data, finding that the relative zero-points rithm takes as an input the autocorrelation of a granula- of the measurements returned by different pipelines do tion background–corrected spectrum in a window with not scale in the same way as their solar measurements a width of ±0.59 × ν0.9 centered around ν . This max max do Pinsonneault et al.(2018). Figures3&4 demon- latter window size is taken from the observed width strate these systematics in our K2 νmax and ∆ν values. 6

Figure 3. Pairwise comparisons between asteroseismology pipeline νmax values for K2 C4. The red error bars represent binned medians and uncertainty on those binned medians. Nx and Ny indicate the number of stars with asteroseismic values returned for pipeline x and pipeline y, with the number of stars returned by both pipelines indicated by Nxy. Trends seen here are present in C6 and C7, as well. In what follows, we describe an approach that re-scales true values, we can fit for near-unity scale factors that ef- each pipeline’s νmax and ∆ν values such that they are on fectively modify each pipeline’s solar reference values so average on the same scale as the other pipelines. These that the mean νmax and ∆ν returned by pipelines are all rescaled values are used to perform outlier rejection with on the same scale. This therefore reduces the systematic sigma clipping, and the values are averaged for each star uncertainty due to the choice of a single pipeline value to compute more accurate and precise asteroseismic val- on a single star’s νmax or ∆ν, and it allows for averaging ues, and to define empirical uncertainties on νmax and values across pipelines on a star-by-star basis. ∆ν. The above procedure to re-scale the pipeline-specific To reduce the systematic uncertainty across pipelines solar reference values is performed by initially as- for a single star, we follow the approach taken by Pin- suming that the pipeline values are already on the 0 sonneault et al.(2018): under the assumption that each same system, so that the re-scaled νmax, νmax,s,p, of pipeline’s νmax and ∆ν values are distributed around the a star, s, for pipeline, p, equals the raw value re- 7

Figure 4. Pairwise comparisons between asteroseismology pipeline ∆ν values for K2 C4. The red error bars represent binned medians and uncertainty on those binned medians. Nx and Ny indicate the number of stars with asteroseismic values returned for pipeline x and pipeline y, with the number of stars returned by both pipelines indicated by Nxy. Trends seen here are present in C6 and C7, as well.

0 turned by the pipeline: νmax,s,p = νmax,s,p. An av- from other pipelines. This whole process is repeated un- 0 P 0 0 erage value hνmax,si ≡ p νmax,s,p/Np is then cal- til convergence in the re-scaled value, νmax,s,p. The same culated for each star that has at least two pipelines procedure is done for ∆ν. Following the observation by returning a value. The sum is over those Np re- Pinsonneault et al.(2018) that the RC exhibits signif- porting pipelines. A scalar factor for each pipeline, icantly different structure in the pipeline νmax and ∆ν P  0  Xνmax,p ≡ s νmax,s,p/hνmax,si /Ns, is calculated us- zero-point differences, the same procedure is done for ing the Ns stars for which the pipeline returned a raw RGB & RGB/AGB stars and RC stars, separately. In value, νmax,s,p, and which had a defined mean value the end, four scale factors are derived for each pipeline: 0 hνmax,si. The pipeline values are re-scaled by this factor a νmax scale factor for RGB stars; a ∆ν scale factor for 0 so that νmax,s,p = νmax,s,p/Xνmax,p. For each star, a 3σ RGB stars; a νmax scale factor for RC stars; and a ∆ν clipping is performed to reject re-scaled values returned scale factor for RC stars. The resulting factors are at by a pipeline that are highly discordant with the results the per cent level or below. For each pipeline, we pro- 8

(Zinn et al. 2019c), which was not considered for the Pinsonneault et al.(2018) analysis. The uncertainties on the scale factors derived in this work are also larger than the ones found by Pinsonneault et al.(2018), in part due to the increased scatter in the νmax and ∆ν derived with K2 data compared to Kepler data. These uncer-

tainties are calculated assuming σνmax and σ∆ν are exact (i.e., with no uncertainty on the standard deviation used to calculate them [see §3.2.3]). Given this assumption, the uncertainties on the scale factors are indicative and not definitive. We also provide the re-scaled, pipeline- specific solar reference values themselves in Table5 (at- tained by multiplying the pipeline-specific solar refer- Figure 5. Kiel diagram for the C4, C6, and C7 observed K2 ence value by the pipeline’s solar reference scale factor GAP DR2 stars, colored by evolutionary state, and using derived here). EPIC effective temperatures. Surface gravities are computed 0 Even with these re-scalings, residual differences be- using hνmaxi, and so only stars with consensus values from multiple asteroseismic pipelines are shown. tween pipelines as a function of νmax and ∆ν will re- main. These trends are shown in the bottom panels vide the re-scaled values, ν0 and ∆ν0, in addition to of Figures6-9, where fractional differences between max 0 0 the raw values. We also provide mean values hν0 i and hνmaxi & h∆ν i and individual pipeline re-scaled val- max 0 0 h∆ν0i, which have been averaged across pipelines. The ues, νmax & ∆ν are shown. By definition, the mean of 0 0 0 0 root mean square across all pipelines for each star are νmax and ∆ν across the pipelines are hνmaxi and h∆ν i, 0 0 though what these figures demonstrate is sub-structure taken as the uncertainty on the hνmaxi and h∆ν i values, 0 as a function of νmax and ∆ν, indicating νmax- and ∆ν- σ 0 and σ 0 (see §3.2.3). The sample with hν i hνmaxi h∆ν i max is shown in Figure5. dependent systematic errors, not zero-point errors. We Unless otherwise noted, the mean ∆ν values for each turn to a more robust estimate of the systematic errors 0 0 star, h∆ν0i, as well as the pipeline-specific re-scaled ∆ν on hνmaxi and h∆ν i in §4.2. values, ∆ν0, have been multiplied by a factor as de- After these re-scalings are applied for each pipeline scribed in Sharma et al.(2016) that depends on the νmax and ∆ν, the absolute value of the solar refer- star’s temperature, metallicity, surface gravity, mass, ence values are free to be chosen, which we take to be and evolutionary state. This factor is provided in Ta- νmax, = 3076µHz and ∆ν = 135.146µHz (Pinson- neault et al. 2018). The effect of this choice is evaluated ble4 as XSharma, and is a theoretically-motivated fac- tor that improves the homology assumption in the ∆ν in § 4.2. scaling relation (Equation2). We calculated these fac- The main K2 GAP DR2 sample is defined to be that 0 0 tors using temperatures and metallicities from the EPIC with a valid hνmaxi and h∆ν i (i.e., stars for which at (Huber et al. 2016); surface gravity from Equation1 us- least two pipelines returned both νmax and ∆ν that agree to within 3σ), and its contents are summarized ing νmax; mass estimates from Equation6 using νmax and ∆ν, before the factor in question has been applied; in Table4. The sample contains 4395 stars. and evolutionary states derived from the neural network approach laid out in §3.2.2. The correction is applied 3.2.2. Evolutionary states 0 multiplicatively such that ∆ν is multiplied by XSharma. Recent results have shown that highly accurate evolu- Pipeline values rejected by the above sigma-clipping pro- tionary state classification between RGB and RC can be cess will not have a corrected pipeline value populated achieved even on short (K2 -like) time series using ma- in Table4. chine learning approaches (Hon et al. 2017, 2018a; Kus- Table5 shows the resulting solar reference scale fac- zlewicz et al. 2020) where ‘classical’ asteroseismic-based tors compared to those from Pinsonneault et al.(2018). classification is not possible (Bedding et al. 2011; Mosser The agreement is mixed, depending on the pipeline. As et al. 2019; Elsworth et al. 2019). We note also that de- noted in §3.1, the pipelines as implemented for this anal- termining evolutionary state for short times series may ysis have been modified to account for K2 data, and so be possible by measuring c, the radial order position may well differ slightly in the way they perform com- for the central radial mode of a power spectrum, to the pared to their implementation for Pinsonneault et al. extent that mode identification is possible with K2 light (2018). Additionally, we consider results from BAM curves; see Kallinger et al.(2012). In this work, we de- 9

0 0 Figure 6. Left: comparison of RGB νmax among pipelines, showing the re-scaled νmax from each pipeline versus the mean hνmaxi 0 0 across pipelines in the top panel, and the fractional difference between hνmaxi and νmax in the bottom panel, with error bars 0 showing binned errors on the median fractional difference, assuming the uncertainty on hνmaxi to be the standard deviation 0 among the re-scaled pipeline ν values, σ 0 (this quantity is described in §3.2.1& §3.2.3). Right: σ 0 is plotted against max hνmaxi hνmaxi 0 the reported uncertainty on νmax for each pipeline. The fractional difference between the two uncertainties is shown in the bottom panel.

Figure 7. Same as Figure6, but for RC stars. termine evolutionary state using the method described low-frequency modes, as can happen when smoothing in Hon et al.(2017, 2018a), which has a similar accu- with a window with fixed size in linear frequency, by racy to that of the recent machine learning approach using the fact that solar-like oscillators with different from Kuszlewicz et al.(2020) (95% and 91% for K2 -like νmax have similar granulation background shapes in log data, respectively). The chosen technique requires that space. We then apply the machine-learning technique the granulation background of the stellar power spec- from Hon et al.(2017, 2018a) using the background- 0 0 trum be removed, and that νmax and ∆ν be provided subtracted power spectrum, hνmaxi, and h∆ν i as in- in order to search the appropriate part of the spectrum puts. For details on how the neural network is trained, for evolutionary state diagnostics. We remove the back- including how the probability of being RGB or RC is cal- ground by subtracting a smoothed version of each power ibrated, see Hon et al.(2017, 2018a). Because the ma- spectrum in log space. This approach avoids removing chine learning algorithm was not trained on stars with 10

∆ν < 3.2µHz (corresponding to stars with radii larger which is why we provide κR and κM instead of direct than those in the RC), we assign stars in this ∆ν regime radius and mass. With the re-scaled asteroseismic val- 0 0 to be ambiguous RGB/AGB stars. The spectra of these ues, νmax and ∆ν , in hand, we can construct the radius 0 0 stars from K2 cannot be used to distinguish between and mass coefficients, κR and κM for each star and each RGB and AGB because both types of stars are shell pipeline, which correspond to the asteroseismic compo- burning stars, and so we consider them for the purposes nent of the scaling relations for stellar mass and radius in of the following analysis to be equivalent to RGB stars. solar units. We show the pairwise comparisons between pipeline values of κ0 and κ0 for all stars in the DR2 0 0 R M 3.2.3. Uncertainties on hνmaxi and h∆ν i, σhν0 i and max sample in Figures 10& 11. As with the pairwise νmax σ 0 h∆ν i and ∆ν comparisons (Figures3&4), we see systematic The root mean square of the re-scaled ν0 and ∆ν0 0 0 max differences between pipeline κR and κM values, which values for each star across pipelines can be thought of we quantify in §3.4. We also construct the average for as — and in this work are taken to be — the statisti- 0 0 0 0 each star, hκRi and hκM i, based on hνmaxi and h∆ν i. 0 0 cal uncertainties on hν i and h∆ν i, denoted σ 0 0 0 max hνmaxi Uncertainties on hκRi and hκM i are calculated accord- and σh∆ν0i. These values are listed in Table4. We ing to standard propagation of error, using σ 0 and hνmaxi compare σ 0 and σ 0 to the reported statisti- hνmaxi h∆ν i σh∆ν0i. All of these coefficients are reported in Table6. cal uncertainties on the parameters returned by each pipeline in the bottom panels of Figures6-9. These 0 0 0 3.3. Statistical uncertainties in hνmaxi, h∆ν i, hκRi, panels show the fractional differences in the pipeline- and hκ0 i 0 M reported uncertainty and the root mean square of νmax 0 We show in Figures 20-21 the distribution of the frac- and ∆ν across pipelines, and indicates whether or not 0 0 tional uncertainties, σhν0 i/hν i and σh∆ν0i/h∆ν i as the pipeline values are over- (above the grey dashed max max a function of evolutionary state (RGB or RGB/AGB, line) or under-estimated (below the grey dashed line), RC). The curves over-plotted on the distributions rep- assuming the pipeline values are distributed like normal resent models of the uncertainty distributions assum- variables around hν0 i and h∆ν0i. The uncertainties max ing Gaussian statistics. Under the assumption that all in ν and ∆ν appear to be over-estimated for stars max stars in K2 have the same, true fractional uncertainty, with smaller σhν0 i and σh∆ν0i. This over-estimation max σ, these distributions would be described by general- worsens with decreasing σhν0 i and σh∆ν0i, and does max ized gamma distributions with probability density func- so more rapidly for RC stars than RGB stars. Part of p (p/ad)xd−1e−(x/a) this over-estimation trend may well be a selection effect: tion f(x, a, d, p) = Γ(d/p) , where Γ(z) de- notes the gamma function, p = 2, d = dof − 1, and σhν0 i and σh∆ν0i are computed after a sigma-clipping max p 2 procedure, which will tend to make a smaller root mean a = 2σ /(dof − 1). We consider two models: one square. We evaluate the accuracy of these uncertainties with σ fixed to be the observed median fractional un- further in §3.3. certainty for each dof, and one with two generalized gamma distributions for which both σ and dof are al- 0 0 3.2.4. Radius and mass coefficients, hκRi and hκM i lowed to vary. The grey (black) curves in Figures 20-21 Given Equations1&2, the radius of a star may be are weighted sums of the best-fitting single-component derived according to: (two-component) models across all dof (see the Ap- pendix for details). It is clear that the observed dis-    −2  1/2 0 0 R νmax ∆ν Teff tributions of σhν0 i/hν i and σh∆ν0i/h∆ν i are not ≈ (3) max max R νmax, ∆ν Teff, perfectly described by a generalized gamma distribution with a unique σ, as they would be if all stars had the  T 1/2 ≡ κ eff , (4) same fractional uncertainty in hν0 i and h∆ν0i. Nev- R T max eff, ertheless, allowing σ to be a function of the number of and its radius from reporting pipelines appears to be a surprisingly good approximation. This indicates that 1) the fractional M  ν 3  ∆ν −4  T 3/2 ≈ max eff (5) uncertainties for all stars are not a strong function of M νmax, ∆ν Teff, νmax or ∆ν (but rather have a fractional uncertainty  3/2 that varies modestly according to the number of report- Teff ≡ κM . (6) ing pipelines), and 2) our uncertainty estimates are dis- Teff, tributed like they should be according to χ statistics. It is our wish not to impose a choice of temperature on As we note in the Appendix, the fractional uncertainties the user when providing asteroseismic radii and masses, vary mostly according to the evolutionary state, with RC 11

Figure 8. Same as Figure6, but for ∆ ν.

Figure 9. Same as Figure8, but for RC stars. parameters less precisely measured, and it happens that aggregated pipeline versus individual pipeline approach. the typical uncertainties are the same for νmax and ∆ν The most significance difference between the uncertain- 0 for RGB stars. Our typical uncertainties for hνmaxi and ties of the two Kepler analyses is in RC νmax, for which h∆ν0i are listed in Table7, along with the correspond- uncertainties from Yu et al.(2018) are larger than those ing median fractional uncertainties from APOKASC-2 from APOKASC-2; the larger νmax uncertainties from (Pinsonneault et al. 2018) (“APOKASC-2” in the ta- Yu et al.(2018) map into correspondingly larger RC κR ble). We also include a comparison to the median frac- and κM uncertainties. Comparing our results to those tional uncertainties from the analysis of Yu et al.(2018) of APOKASC-2, we see that the uncertainties in νmax (“Y18” in the table). The latter analysis uses only the in K2 are up to a factor of two larger than in Kepler, SYD pipeline, as opposed to APOKASC-2, which re- and the uncertainties in ∆ν are larger by up to a factor ports parameters averaged across five different astero- of four for RGB stars. These differences between Kepler seismic pipelines. The methodology in this work is much and K2 come from differences in photometric precision the same as that of Pinsonneault et al.(2018), but we and the differences in dwell time, and will be further include a comparison to Yu et al.(2018) to give an indi- explored in the next K2 GAP data release. cation of the variation in uncertainties resulting from an 12

0 The analogous uncertainty distributions for hκRi and they are clearly a function of νmax and ∆ν. Generally, 0 hκM i are shown in Figures 22-23. Note that the num- though, the uncertainty is larger at smaller νmax and ∆ν. ber of stars plotted, N, is not 4395 (the total number of This is a result of both intrinsically fewer stars in this 0 0 stars we provide with hκRi and hκM i). This is because regime (as they are more evolved, and therefore shorter- in this treatment, we require the number of pipelines re- lived) as well as difficulties in measuring low-frequency porting values for νmax and ∆ν be the same in order to νmax and ∆ν due to the K2 frequency resolution. We 0 0 fulfill the generalized gamma distribution requirements; take the systematic uncertainties in hκRi and hκM i to be most stars do not have the same number of νmax mea- 1% and 2%, from propagation of the systematic uncer- 0 0 0 0 surements as ∆ν: there are 1030 such RGB stars and tainties in hνmaxi and h∆ν i. As with hνmaxi and h∆ν i, 257 such RC stars. Considering the excellent match in detail, these systematic uncertainties are a function 0 0 of the fitted generalized gamma distributions (black), of hκRi and hκM i (see Figures 10& 11). 0 0 0 we adopt statistical uncertainties for hκRi and hκM i as We note that the ∆ν correction applied to h∆ν i and 0 0 listed in Table7. We also list the corresponding me- therefore hκRi and hκM i, XSharma, is computed using dian fractional uncertainties in Kepler, computed using the EPIC temperature and metallicity scale. We ac- the evolutionary states and νmax & ∆ν values from Pin- knowledge the user may wish to use their own temper- sonneault et al.(2018) (“APOKASC-2”) or those from atures, and therefore we caution that using a different Yu et al.(2018) (“Y18”). We find that the K2 radius temperature scale will introduce systematics. For ex- (mass) uncertainties are larger by up to a factor of three ample, using a temperature scale 100K hotter (cooler) (two) compared to that of Kepler for RGB stars from than the EPIC temperature will make radii 1% lower APOKASC-2. The uncertainties in RC stellar param- (higher) if XSharma is not also recomputed with the eters are more comparable between the two data sets, user’s adopted temperatures. In order to give the user with the increase in uncertainty from Kepler to K2 not as much convenience as possible, we provide EPIC tem- being larger than a factor of two. The uncertainties are peratures in Table4, should the user wish to compute more comparable between Yu et al.(2018) results and consistent radii/masses; we also include the EPIC metal- K2 because of the larger uncertainty in RC νmax from licities used for computing XSharma. In the event the Yu et al.(2018) compared to APOKASC-2. user wishes to use a different temperature scale and does not wish to sustain additional ∼ 1% systematic 0 0 0 uncertainties, we encourage re-computing XSharma with 3.4. Systematic uncertainties in hνmaxi, h∆ν i, hκRi, 0 the user’s own temperatures and/or metallicities using and hκM i asfgrid (Sharma et al. 2016; Sharma & Stello 2016), We now turn to systematic uncertainties in hν0 i, max which is available at http://www.physics.usyd.edu.au/ h∆ν0i, hκ0 i, and hκ0 i that take the form of ν - and R M max k2gap/Asfgrid/. ∆ν-dependent offsets among the pipelines. By defini- tion, the rescaling process described in §3.2.1 removes offsets among the pipelines by averaging over all νmax 4. RESULTS and ∆ν. However, there are νmax- and ∆ν-dependent 0 4.1. Comparison to Galaxia trends seen in the fractional differences in hνmaxi and h∆ν0i shown in Figures6-9. These trends do not affect We start by comparing properties of the K2 GAP DR2 the statistical uncertainty discussion above, as they are sample to those of a Galaxia simulation (Sharma et al. removed when computing σ 0 and σ 0 . Our ap- 2011), with corrections made to the simulated metal- hνmaxi h∆ν i proach to account for these systematics is therefore to licity scale described in Sharma et al.(2019). Each 0 adopt the largest excursion of any pipeline from hνmaxi campaign has been modeled separately because they and h∆ν0i for each bin plotted in the bottom panel of each probe different regions of the Galaxy. To make Figures6-9, add to that the uncertainty on the me- the simulated populations comparable to the data, we dian, and adopting the result as 2σ systematic uncer- select only the simulated stars that would be seismi- tainties. The resulting (1σ) uncertainties are listed in cally detected. We defined the detectable sample to Table8. For most stars, the typical systematic uncer- be stars with 3µHz < νmax < 280µHz and signal-to- tainty is less than the statistical uncertainty: for a typi- noise ratio yielding a probability of detection greater cal RGB star with (νmax, ∆ν) ∼ (75µHz, 7.5µHz) or RC than 95% (calculated according to the procedure used star with (νmax, ∆ν) ∼ (30µHz, 4.0µHz), the system- in Chaplin et al.(2011)). We impose the same se- 0 atic uncertainties are similar, at ∼ 0.6% in hνmaxi and lection of the simulated and observed stars (see §2.1 ∼ 0.3% in h∆ν0i. We adopt these numbers as typical and also Sharma et al. 2019), using a synthetic V -band 0 0 systematic uncertainties for hνmaxi and h∆ν i, though magnitude that depends on J − Ks color according to 13

0 Figure 10. Pairwise comparisons between asteroseismology pipeline κR values.

2(J−Ks−0.2) V = Ks + 2.0((J − Ks) + 0.14) + 0.382e higher-gravity red giants, and so at a fixed magnitude, (Sharma et al. 2018). it is easier to measure oscillations in lower-gravity red Galaxia models of the magnitude-νmax distributions giants. The amplitude of oscillations is primarily a show good agreement with the observations, as shown function of surface gravity (Kallinger et al. 2014), and in Figures 12–14. The most obvious feature in these so the diagonal cutoff in the top right corner of both plots is the RC, which, because red giants spend a rel- Galaxia predictions and (for the most part) observa- atively long amount of time in this phase, results in tions is a result of the signal-to-noise ratio from the sur- a “clump” of stars at νmax ∼ 30µHz. Lower-gravity face gravity–dependent oscillation amplitude compared (lower-νmax) red giants oscillate with larger power than to the magnitude-dependent white noise. To demon- 14

0 Figure 11. Pairwise comparisons between asteroseismology pipeline κM values. strate this νmax-dependent white noise limit, we assume (2017), the dependence on magnitude is less steep, and 4 −H a detectability threshold of νmax,detect < 5 × 10 1.6 likely reflects the improved noise qualities following im- in Figures 12–14 (dashed lines). This detectability proved pointing control starting with C3. The reason threshold describes the observed data well, and scales the faint and bright limits do not form straight, vertical like a flux-dependent white noise would. In Kp-band trends in Figures 12–14, which show H-band, is because space, this threshold would mean a detectability limit we selected stars in V -band (see §2.1). For convenience, of Kp ≈ 15, fainter than which the white noise is too we also show the detection distributions as a function of large to detect high-νmax oscillators. Compared to the only H-band magnitude in Figure 15. detectability threshold of C1 described in Stello et al. 15

Figure 12. Distributions of νmax as a function of magnitude, as predicted by Galaxia (black) and observed (green), for K2 C4. 4 −H The dashed lines represent our adopted detectability threshold of νmax,detect < 5 × 10 1.6 . We condense the comparison between observed and ∆ν for classification as a solar-like oscillator, which is simulated asteroseismic values to just the νmax dimen- consistent with our definition of a pipeline detection as sion for C4 and C7 in Figures 16& 17. The agreement a valid νmax measurement (see §3.1). The results from is generally good. There are two main discrepancies, each person were nearly identical, with any contested however. First, the number of predicted oscillators in classifications discussed individually and a final consen- C4 does not agree with the observations (Figure 16a). sus classification agreed upon. However, plotting the normalized distribution such that As evident in Figure 18, the predicted νmax distribu- each bin is divided by the bin size (representing proba- tion from Galaxia is in good agreement with the vi- bility density) results in agreement, except for the low- sually confirmed distribution (“yes”s are plotted, not νmax regime (Figure 16b). We discuss the discrepancy in “maybe”s), though they are formally inconsistent with Figure 16a below. Second, there are fewer observed low being drawn from the same distribution, according to νmax stars (νmax . 10 − 20µHz) than predicted in both a Kolmogorov-Smirnov test. By going through all of C4 and C7. Although Stello et al.(2017) found this same the observed targets by eye and not just ones that were bias in K2 GAP DR1, we attempt to verify that it is not returned by pipelines as being red giants, we are able due to a bias in the Galaxia models themselves by man- to be more confident that the Galaxia predictions are ually inspecting all K2 GAP spectra in C6 for evidence of robust and not subject to obvious biases. The dis- solar-like oscillations. Two experts did the exercise sep- tributions of νmax returned by individual pipelines all arately, classifying the K2 spectra according to whether fall short of being formally consistent with either the or not they had a detectable νmax (yes/maybe/no), ad- visually-confirmed distribution or the Galaxia distri- ditionally assigning a visual estimate of νmax. Objects bution, though some of the pipeline νmax distributions that showed evidence of solar-like oscillations below the qualitatively show good agreement with the predicted effective high-pass cutoff of 3µHz were classified as no. and visually-confirmed distributions. Note that this exercise did not require the detection of 16

Figure 13. Distributions of νmax as a function of magnitude, as predicted by Galaxia (black) and observed (green), for K2 C6. 4 −H The dashed lines represent our adopted detectability threshold of νmax,detect < 5 × 10 1.6 . As noted above, there is nonetheless a bias against Galaxia and visual inspection as ground truths. In C7, detecting stars with νmax, below ∼ 10 − 20µHz. This there are 1740 such stars that have asteroseismic values is true for all three of the campaigns, which indicates from at least one pipeline. This number is consistent that this detection bias was not solely a function of the with the 1758 expected stars from the Galaxia simu- particular DR1 sample, which were all in C1. With the lation for C7. The 2312 stars recovered by at least one data we have in hand, we cannot definitively say what pipeline in C6 is between the number found by visual in- causes this bias. At these low frequencies, there are rela- spection, 2214, and that predicted by Galaxia, 2511. In tively few modes observable. This may hinder detecting C4, however, there are significantly fewer stars observed these oscillations for pipelines that rely on finding ∆ν by at least one pipeline than predicted by Galaxia (2177 as part of its νmax detection step, and can also hamper versus 2670). These ‘missing’ stars are at magnitudes fits to the power excess using a Gaussian since a Gaus- Kp < 13, and therefore should yield observable aster- sian does not very well describe a few discrete modes. oseismic parameters, given our empirical detectability It is also possible that low-νmax oscillators are harder threshold. The Galaxia model seems to under-predict to recover at lower frequencies because of the relatively reddening in this campaign, which may explain this dis- short K2 dwell time, which leads to a smaller ratio of crepancy. Indeed, the predicted fraction of dwarfs (and the frequency resolution to the mode width; the result is therefore the fraction of stars that would not be detected that the spectra of these stars can sometimes be hard to as oscillators in long-cadence K2 data) depends on the distinguish from a pure granulation background. The assumed reddening. For K2 GAP, the selection ‘draws degree to which these oscillators are detected or not a line’ in J − Ks color space to separate the sample of is highly pipeline-dependent, as one can see from Fig- predominantly blue dwarfs from the sample containing ures 16-18. the red giants (and some red dwarfs), which are the tar- Keeping this low-νmax detection bias in mind, we can gets with J − Ks > 0.5. The same cut is applied to the go on to test the detection rate for νmax > 20µHz, using synthetic stellar population in the Galaxia simulation, 17

Figure 14. Distributions of νmax as a function of magnitude, as predicted by Galaxia (black) and observed (green), for K2 C7. 4 −H The dashed lines represent our adopted detectability threshold of νmax,detect < 5 × 10 1.6 .

0 0 using an assumed reddening. An under-estimated red- may not result in hκRi and hκM i being on an absolute dening in Galaxia means that fewer of the blue dwarfs scale. First, although our re-scaling procedure to de- 0 0 are reddened enough to fall on the giant side of the rive h∆ν i and hνmaxi is nearly the same as in Pinson- J − Ks > 0.5 dwarf/giant dividing line, increasing the neault et al.(2018), we have added BAM as one of the number of predicted oscillators compared to reality. pipelines that contributes to the re-scaling procedure: APOKASC-2 used results from A2Z, CAN, COR, SYD, 4.2. Absolute radius calibration and OCT, and in this work, we have used results from 0 0 A2Z, CAN, COR, SYD, BHM (based on OCT), and The derived h∆ν i and hνmaxi have not necessarily been placed on an absolute scale. While we scaled the BAM (see §3.1 for summaries of the pipeline methodolo- asteroseismic values to a common mean scale, we were gies). The addition of BAM in this work to the pipelines 0 0 used for aggregating asteroseismic results may result in a free to impose solar reference values for hνmaxi and h∆ν i slightly different mean scale for hν0 i and h∆ν0i. This to be νmax, = 3076µHz and ∆ν = 135.146µHz, which max are relatively close to the average pipeline-specific solar is because the APOAKSC-2 solar reference values were reference values, and which are the same as determined chosen such that the aggregated stellar masses agreed by the absolute asteroseismology re-scaling done in the with open cluster masses — using a different set of APOKASC-2 analysis. Were we working with Kepler pipelines to average over may have required a different data and using the same pipelines as in the APOKASC- set of solar reference values to achieve agreement with 2 analysis, this choice of solar reference values would be open cluster masses. We see this in the differences be- valid and would put the asteroseismic radii on a scale tween the re-scaling values from our analysis and from that is consistent with open cluster masses and Gaia that of Pinsonneault et al.(2018), shown in Table5. radii (Pinsonneault et al. 2018; Zinn et al. 2019b). There Second, there is evidence to suggest that there are sys- are at least two reasons why choosing our solar reference tematic biases in asteroseismic parameters based on the dwell time of the data (Zinn et al., in prep.), which could values as νmax, = 3076µHz and ∆ν = 135.146µHz 18 be suggestive of a need to modify the νmax, and/or ∆ν et al. 2016). Five stars with asteroseismic and Gaia radii for K2 data compared to Kepler data. We now test this discrepant at more than the 3σ level were removed from choice of zero-points by comparing the derived radii to subsequent analysis. radii using parallaxes from Gaia Data Release 2 (Gaia We compare the Gaia radius scale to our K2 astero- Collaboration et al. 2018; Lindegren et al. 2018). seismic radius scale in Figure 19. The top panel shows We populated the overlap sample of stars in K2 GAP the points colored by evolutionary state (RGB in red 0 0 DR2 with both h∆ν i and hνmaxi and Gaia DR2 by and RC in blue), and the bottom panels shows the frac- matching on 2MASS ID using the Gaia Archive4. We tional agreement of the two radius scales. Figure 19 indi- 0 also required APOGEE metallicities and temperatures cates that the radius scale of hκRi is consistent with the from DR16 (Ahumada et al. 2020). Because of a known, Gaia radius scale for both RC and RGB stars to within position, magnitude, and color-dependent zero-point in ∼ 3%. We find that the RGB stars are in more disagree- the Gaia parallaxes (e.g., Lindegren et al. 2018; Zinn ment than the RC stars: while the median agreement is et al. 2019a), we did not work directly with the Gaia 3.0%±0.4% for RGB stars, it is 1.6%±0.8% for RC stars. DR2 parallaxes. Instead, we followed the methodology This median statistic is computed as the median radius of Sch¨onrich et al.(2019) to derive distance estimates ratio for all RGB or RC stars, with the uncertainty in q for stars in our K2 GAP sample. We did this sepa- π P 2 2 the median taken to be σmed = 2 σR/N , where rately for RC and RGB stars, with the understanding r 2  2 Rseis σR,Gaia σR,seis that RGB and RC populations will have different selec- σR = + . Using the EPIC RGaia RGaia Rseis tion functions, which is an important consideration in extinctions instead of those from Green et al.(2015) the Bayesian distance estimates in the Schonrich et al. ¨ leads to insignificant variations in the radius agreement. (2019) framework (see also Schonrich & Aumer 2017). ¨ However, the agreement is discrepant at the ∼ 3σ level For the purposes of establishing a Gaia calibration of between stars in C4 versus those in C7. This could be ν and ∆ν , we used only stars with more than two max, an indication of Gaia parallax zero-point issues, given pipelines returning results for ∆ν, and only considered we do not expect such variations in the asteroseismic stars with π > 0.4 and Gaia G-band < 13mag to en- data by campaign. We therefore also consider the Gaia sure that the results are less sensitive to any residual zero-point from Khan et al.(2019), who compared aster- Gaia parallax zero-points that may not be accounted oseismic distances to Gaia distances in K2 C3 and C6. for in the Schonrich et al.(2019) method; [Fe/H] > −1 ¨ For this exercise, we restricted our sample to the stars to ensure that there are no metallicity-dependent aster- in C6, and adopted their derived zero-point of −17µas. oseismic radius systematics (see Zinn et al. 2019b); and The result is that the asteroseismic radii are consistent R < 30R to ensure there are no radius-dependent as- with Gaia radii to within ≈ 1% (RGB) and ≈ 5% (RC). teroseismic radius systematics (see Zinn et al. 2019b). Because the median agreement between the For this sample, which has spectroscopic information, radius scales could be biased by underlying we re-compute ∆ν correction factors using metallicities skewed distributions of the individual radii, we that are adjusted to account for non-solar alpha abun- finally evaluate the agreement using a weighted dances according to the Salaris et al.(1993) prescrip- P Rseis /σ2 0 [α/M] RGaia R tion: [Fe/H] = [Fe/H] + log (0.638 × 10 + 0.362). mean: hRseis/RGaiai = P 2 , where σR = 10 1/σR r 2 2 We computed a Gaia radius for the 261 resulting stars  σ   σ  Rseis R,Gaia + R,seis . We calculate this for following a Monte Carlo procedure of the sort detailed RGaia RGaia Rseis in Zinn et al.(2017). The method uses the Stefan- those stars that have fractional parallax uncertainties Boltzmann law to translate the flux, temperature, and less than 10%, in order to mitigate potential biases distance of a star into a radius. To do so, we computed due to parallax systematics. According to this metric, bolometric fluxes with a Ks-band bolometric correction the agreement becomes 2.2 ± 0.3% for RGB stars and (Gonz´alez Hern´andez & Bonifacio 2009), APOGEE ef- 2.0 ± 0.6% for RC stars. fective temperatures, and metallicities, combined with Due to the variation in this agreement based on the 0 0 asteroseismic surface gravities from hνmaxi. Extinctions tests described above, we opt not to re-scale our hνmaxi were computed using the three-dimensional dust map of or h∆ν0i values, and instead allow for a systematic zero- 5 0 Green et al.(2015), as implemented in mwdust (Bovy point uncertainty in our derived hκRi values. Acknowl- edging these uncertainties in the K2 -Gaia agreement, we use the weighted mean estimate of the radius scales 4 https://gea.esac.esa.int/archive/ using the Schonrich et al.(2019) Gaia distances. Our 5 ¨ https://github.com/jobovy/mwdust asteroseismic radius coefficients could therefore be over- 19 estimated by up to 2.2 ± 0.3% for RGB stars and up We did this by comparing to Gaia radii, and we found a to 2.0 ± 0.6% for RC stars. The uncertainty on this systematic offset between the K2 GAP DR2 asteroseis- agreement is solely due to the standard uncertainty on mic and Gaia radius scales of about 2%. This translates the mean, and does not account for intrinsic scatter or roughly to a 20% systematic uncertainty in age. Given trends in the radius agreement. Indeed, this should be the typical uncertainty in mass listed in Table7, ages thought of as being in addition to the systematic un- based on our RGB asteroseismic masses would be ex- 0 certainty from pipeline-to-pipeline variation in hκRi as pected to have statistical uncertainties of ≈ 20%, with 0 a function of νmax and h∆ν i discussed in §3.4. We also potential scale shifts by ≈ 20% due to the level of sys- note that this agreement does not account for systematic tematics we identify in this section. This anticipated variation in the radius ratio due to choice of tempera- 20% statistical age uncertainty makes the data particu- ture, bolometric correction, or Gaia zero-point, which larly interesting for potentially identifying the history of may contribute to a systematic uncertainty of about minor mergers in the Galaxy based on their impact on ±2% (Zinn et al. 2019b). the age–velocity dispersion relation (Martig et al. 2014). Broadly speaking, the excellent level of agreement For this and other Galactic archaeology applications between asteroseismology and Gaia corroborates find- (e.g., age-abundance patterns), the 20% systematic un- ings of the accuracy of the scaling relations in this certainty should not be significant, given that the differ- regime from previous work based on asteroseismology- ential age relationship between stellar populations would Gaia comparisons (Huber et al. 2017; Zinn et al. 2019b). be preserved. Regarding mass- or magnitude-dependent In detail, there do appear to be trends with radius evi- systematics among RGB asteroseismic parameters, the dent in Figure 19: the RGB radii appear to inflate com- small inflation of RGB asteroseismic radii at R ∼ 7.5R pared to Gaia radii at around R ∼ 7.5R (red error seems to map onto a corresponding trend in νmax, and so bars), while the red clump radii (blue error bars) seem this may introduce an inflation in the RGB mass scale to deflate compared to the Gaia radii with increasing by perhaps up to 15% for the minority of stars with 0 radius at all radii. At least for the RC stars, system- 50µHz . hνmaxi . 80µHz. RC stars in our sample, atics in the tracks used to generate the ∆ν corrections on the other hand, appear to suffer from strong radius- could be to blame, particularly given the disagreement dependent trends that seem to be related to the ∆ν and among RC models from the literature (An et al. 2019). not νmax scaling relation: there is a strong trend of RC Indeed, it appears that the RC radius trend is mostly a agreement with ∆ν, which suggests that the ∆ν scaling trend in ∆ν, with some metallicity dependence, as well. relation for RC stars is not well-calibrated using our ∆ν At the population level, the RC radii have been found corrections. Despite the concerning magnitude of the to agree within 5% with Gaia radii (Hall et al. 2019). RC systematic, RC ages are not in popular use because However, to our knowledge, the scaling relations for RC of uncertainties in modeling mass loss (Casagrande et al. stars have not been tested as a function of radius as we 2016). We will nonetheless explore the RC systematic do here. We note also that a ≈ 1% relative difference further in the next and final K2 GAP data release, as between the zero-point for RGB versus RC stars is not having accurate red clump masses and radii is impor- ruled out by Pinsonneault et al.(2018) (see also Khan tant for reckoning red clump models with observed red et al. 2019). clump properties (e.g., An et al. 2019). Regarding the mean agreement of RGB and RC stars, the most likely culprit for the (small) radius disagree- 5. CONCLUSION ment is a systematic in the solar reference value com- We have described the second data release of K2 GAP, 2 bination νmax, /∆ν . That the absolute K2 astero- containing red giants for campaigns 4, 6, and 7. We seismic radius scale is consistent to within ∼ 2% with have derived evolutionary state classifications for our 0 the Gaia radius scale naively implies that our hκM i are sample, and have placed the raw asteroseismic obser- within ∼ 6% of an absolute mass scale, according to vations on a self-consistent scale, resulting in 4395 stars standard propagation of error from Equation4 to6. with mean asteroseismic parameters. We have also pro- 0 0 However, this is only approximate, because we can test vided ready-to-use derived quantities, hκM i and hκRi, 2 only νmax, /∆ν against Gaia, whereas the mass coef- for these stars, which yield masses and radii when com- 3 4 ficient goes as νmax, /∆ν . bined with a weakly temperature-dependent factor that In summary, because we are delivering asteroseismic users may compute with their preferred effective tem- 0 0 values hνmaxi and h∆ν i that are averaged values from perature. We conclude the following: several pipelines, we needed to evaluate to what extent 1. The observed K2 GAP targets in campaigns 4, 6, the resulting asteroseismic scale is on an absolute scale. and 7 have reproducible selection functions, which 20

enable them for use in Galactic archaeology stud- Future work will focus on calibrating the mass coeffi- ies. cients, which at this point cannot be definitively placed on an absolute scale, for lack of convenient mass calibra- 2. νmax- and ∆ν- dependent trends among pipelines tors in the sample. In the next K2 GAP data release, have been improved by bringing the pipelines onto we may be able to place masses on an absolute scale, as a common scale. This re-scaling process effectively we have done for radii in this work, by appealing to the changes pipeline-specific solar reference values at red giant branch mass of open clusters observed by K2. or below the 1% level, in different measures, de- pending on the pipeline and the evolutionary state of the star.

3. We provide empirical uncertainties in νmax and ∆ν values for stars that have results from at least two pipelines, which have statistically reasonable dis- tributions, and which indicate that fractional un- certainties are not strong functions of νmax or ∆ν or the number of pipelines reporting, but rather vary mostly according to evolutionary state: RGB stars have better-measured parameters than do RC stars. Systematic uncertainties for νmax and ∆ν values are similar across evolutionary state, at ∼ 0.6% and ∼ 0.3% for both RGB and RC stars.

4. The distributions of our mean νmax are in good agreement with those predicted by theoretical stellar population synthesis models. Crucially, both the observed and predicted νmax distribu- tions globally agree with an unbiased estimate of the νmax distributions from manual inspection of the data, which indicates our model predictions are accurate, and our observed samples are largely complete. A notable exception to the asteroseis- mic detection completeness is for red giants with νmax . 10 − 20µHz, where pipelines may report lower-than-expected numbers of oscillating stars.

5. The radius and mass coefficients that we pro- 0 0 vide, hκRi and hκM i, have typical uncertain- ties of σ 0 = 3.3% (RGB or RGB/AGB) & hκRi σ 0 = 5.0% (RC) and σ 0 = 7.7% (RGB or hκRi hκM i RGB/AGB) & σ 0 = 10.5% (RC). These uncer- hκM i tainties are a factor of two to three higher than the uncertainties from Kepler radii and masses.

6. Our asteroseismic radii have been validated to be on the Gaia radius scale, to within 2.2 ± 0.3% for RGB stars and 2.0% ± 0.6% for RC stars.

K2 GAP Data Release 2 successfully builds upon K2 GAP DR1 in providing evolutionary state informa- tion; re-scaled asteroseismic parameters and uncertain- ties that take advantage of the information from multi- ple asteroseismic pipelines; radius and mass coefficients; and placing the radius coefficients on an absolute scale. 21

Kp [mag] Kp [mag] Kp [mag] 8 10 12 14 16 8 10 12 14 16 8 10 12 14 16 0.5 COR COR COR A2Z A2Z 0.7 A2Z BAM 0.35 BAM BAM CAN CAN CAN BHM BHM 0.6 BHM 0.4 SYD 0.30 SYD SYD Galaxia Eye Galaxia ALL Galaxia ALL 0.25 ALL 0.5 0.3 0.20 0.4

0.2 0.15 0.3

Normalized count per bin Normalized count0 per bin .10 Normalized count0 per bin .2 0.1 0.05 0.1

0.0 0.00 0.0 6 8 10 12 14 6 8 10 12 14 6 8 10 12 14 H [mag] H [mag] H [mag]

Figure 15. Distributions of H-band magnitude for C4 (left), C6 (middle), and C7 (right), as predicted by Galaxia (red) and 0 observed for each pipeline, according to the legend. The distribution of stars with hνmaxi is labelled as “ALL” — this is not the 0 same as summing the individual pipeline histograms because not every star will have a hνmaxi since that requires at least two pipelines reporting values. The approximate Kp-band scale is indicated on the top x-axis.

log g log g 1.4 1.8 2.4 2.9 3.3 1.4 1.8 2.4 2.9 3.3

COR 0.06 COR A2Z a) A2Z b) 700 BAM BAM CAN CAN BHM 0.05 BHM 600 SYD SYD Galaxia Galaxia ALL ALL 500 0.04

400 0.03

Count per300 bin 0.02

200 Normalized count per bin 0.01 100

0 0.00 101 102 101 102 νmax [µHz] νmax [µHz]

Figure 16. Left: Distributions of νmax for stars that have been recovered by a particular pipeline (i.e., observed; colored according to the legend) compared to the Galaxia simulation of predicted detections for K2 C4 (red). The approximate asteroseismic surface gravity scale was computed with scaling relations according to Equation1, and assuming a temperature of 4500K. Right: Same as left, but showing normalized counts such that the distributions represent probability density. 22

log g log g 1.4 1.8 2.4 2.9 3.3 1.4 1.8 2.4 2.9 3.3

COR COR A2Z a) A2Z b) 0.08 600 BAM BAM CAN CAN BHM 0.07 BHM SYD SYD 500 Galaxia Galaxia ALL 0.06 ALL

400 0.05

300 0.04

Count per bin 0.03 200 Normalized count0 per bin .02 100 0.01

0 0.00 101 102 101 102 νmax [µHz] νmax [µHz]

Figure 17. Same as Figure 16, but for C7. 23

log g 1.4 1.8 2.4 2.9 3.3

COR A2Z BAM 600 CAN BHM SYD 500 Eye Galaxia ALL 400

300 Count per bin

200

100

0 101 102 νmax [µHz]

Figure 18. Distributions of νmax for stars that have been recovered by a particular pipeline (i.e., observed; colored according to the legend); that have a visible νmax as determined by eye (black); and that are expected to have a detectable νmax according to the Galaxia simulation (red) for K2 C6. The approximate asteroseismic surface gravity scale was computed with scaling relations according to Equation1, and assuming a temperature of 4500K. 24 Hz BHM ν µ ∆ BHM , Hz SYD µ ν, max Hz ν ∆ σ µ σ 2 (a § SYD Hz BHM ν , µ Hz µ ∆ max ν SYD , Hz µ BAM Hz max ν, ν µ σ ∆ σ ).3.2.2 If classified, a star’s § SYD , Hz Hz BAM µ ν µ max ∆ ν GAP target priority discussed in K2 BAM COR , Hz Hz ν, µ µ ∆ max σ ν σ Hz COR ν µ BAM , ∆ Hz µ max ν COR , GAP DR2 for each pipeline, with evolutionary states Hz µ A2Z max Hz ν ν, K2 µ σ ∆ σ COR , A2Z Hz Hz ν µ µ values for ∆ max ν ν A2Z , and ∆ CAN Hz Hz µ ν, µ max ∆ ν max σ σ ν A2Z Hz , CAN Hz ν µ µ ∆ max ν CAN , Hz µ max Raw asteroseismic ν σ CAN Table 2. , Hz µ —These are the parameters returned by a given pipeline, along with their uncertainties, without any of the re-scaling described in max ν 3.2.1 applied. Evolutionary states are also included in this table, which have been derived in this work (see § evolutionary state is assigned as either “RGB”, “RGB/AGB”, or “RC”. “Priority” refers to the smaller numerical value correspondsthis table to is higher available priority); in serendipitous the targets online journal. do not have a populated priority entry. A full version of Note BHM Hz — 29.260 1.03 4.004 0.094 28.350 0.83 3.805 0.083 28.789 0.62378 3.740 0.203 ν, µ 0.160 27.670 1.27 — — — — — — 29.016 0.63245 3.450 0.170 0.1900.1900.100 30.600 59.390 23.660 1.49 1.28 1.22 4.066 5.960 0.056 3.482 0.084 0.090 31.640 58.960 24.540 0.89 1.26 0.73 4.204 5.975 0.079 3.402 0.087 31.044 0.076 59.213 0.88666 24.144 1.21725 4.033 0.77871 5.873 0.238 0.058 — — ∆ σ 210306475210307958210314854 4210315825 4210318976 4 4 903 2771 4 1141 RGB 1651 RGB 988 RGB 28.550 RGB 27.940 29.030 RGB 2.25 59.580 2.47 2.03 24.800 3.620 3.63 3.890 0.010 4.100 1.88 0.000 5.910 0.120 30.135 3.270 30.316 0.160 31.863 0.040 0.808 59.253 0.723 25.080 0.743 3.277 1.379 3.723 4.115 0.841 0.197 0.124 5.899 0.083 26.900 — 0.060 — 31.200 60.000 1.4 — 1.0 — 1.3 3.470 21.700 4.090 — 6.010 1.1 3.080 EPIC Campaign Priority Evo. State 25

Table 3. Numbers of stars with raw asteroseismic values 0 0 (νmax, ∆ν), re-scaled asteroseismic values (νmax, ∆ν ), and ra- 0 0 dius & mass coefficients (κR, κM ), as a function of pipeline and campaign

0 0 0 0 νmax νmax ∆ν ∆ν κR κM C4 A2Z 1536 1375 1536 1331 1331 1331 C6 A2Z 1086 1018 1086 985 985 985 C7 A2Z 993 912 293 279 279 279 Total A2Z 3615 3305 2915 2595 2595 2595 C4 BAM 2478 1480 844 741 741 741 C6 BAM 2529 1515 955 844 844 844 C7 BAM 2315 1267 677 589 589 589 Total BAM 7322 4262 2476 2174 2174 2174 C4 BHM 1984 1414 1529 1189 1189 1189 C6 BHM 2275 1482 1702 1229 1229 1229 C7 BHM 1803 1231 1238 1019 1019 1019 Total BHM 6062 4127 4469 3437 3437 3437 C4 CAN 1897 1395 968 788 788 788 C6 CAN 1956 1420 1455 1189 1189 1189 C7 CAN 1564 1137 1048 889 889 889 Total CAN 5417 3952 3471 2866 2866 2866 C4 COR 1803 1374 1803 1304 1304 1304 C6 COR 1443 1118 1443 1043 1043 1043 C7 COR 1561 1188 1561 1149 1149 1149 Total COR 4807 3680 4807 3496 3496 3496 C4 SYD 2136 1416 853 695 695 695 C6 SYD 2207 1335 868 727 727 727 C7 SYD 1675 1089 584 503 503 503 Total SYD 6018 3840 2305 1925 1925 1925 26 H] / [Fe σ i 0 ν ∆ h ; the standard EPIC [Fe/H] 0 A2Z Hz i 0 ν µ ν ∆ T ∆ h σ A2Z and , eff Hz i T µ 0 max ν 0 max , are only provided for targets 0 ν is adjusted using theoretically- h ν i EPIC i 0 ν ν Hz ∆ ∆ µ h h corrections. EPIC temperatures and and ∆ . ν 0 , are computed by perturbing the EPIC SYD Hz K K > ν µ 0 0 max ν ∆ GAP DR2 ν ∆ < Sharma Sharma σ K2 X X σ σ SYD , , Hz and µ 0 > max 0 ν Sharma values for max Sharma X ν X <ν 0 Hz COR σ ν µ ν ∆ ∆ and ∆ N i max COR 0 , ν Hz ν Hz µ ∆ h µ 0 max σ ν i . See text for details. A full version of this table is available in the online journal. 0 ν ν Hz ∆ 0 Hz CAN ∆ µ , should the user wish to compute custom ∆ h N ν µ values for each star across all pipelines are denoted by ∆ ν and max Sharma ν max and ∆ CAN N /X , ν Hz i Sharma et), al. ( 2016 for use in asteroseismic scaling relations; an uncorrected version of 0 Derived asteroseismic µ N ν i 0 max max ∆ ν ν h 0 max Hz ν µ = h Sharma σ i X ν Table 4. Hz 0 BHM ∆ ν µ i h ∆ Hz 0 max are denoted by µ ν h ν BHM , Hz or ∆ µ 0 max ν max ν 210306475 28.487210307958 28.974210314854 1.235 30.907210315825 0.935 59.422210318976 0.990 5 24.478 0.463 5 3.610 0.414 6 3.957 6 0.099 4.170 5 0.123 3 6.088 0.057 5 3.487 0.045 1.027 6 0.094 1.032 6 1.016 3 0.010 1.025 0.013 3.517 1.031 0.018 3.834 28.502 0.010 4.102 27.893 0.013 3.723 5.939 28.981 4.090 3.381 59.479 4.167 24.758 6.071 3.381 EPIC Hz 0 BAM ν µ —Asteroseismic values re-scaled for scalar offsets among pipelines are denoted by a prime (the pipeline-specific solar reference scale ∆ factors are listed in5 ); Table mean deviation of these values for each star across all pipelines are denoted by motivated correction factors, for each starmetallicities is are provided, provided(these for uncertainties this are also purpose, provided though for these convenience).for The are which uncertainties relatively at in uncertain leastresults two estimates for pipelines of returned the concordant results, true and temperatures otherwise and have a metallicities blank entry; the numbers of pipelines returning valid temperature and metallicities in a Monte Carlo procedure. Pipeline-specific re-scaled values, Note BAM , Hz µ 30.00430.18431.724 — 3.85358.995 4.20224.971 26.764 6.067 — 31.042 — 3.558 59.696 4.157 27.948 — 6.156 — 30.907 59.986 — 29.554 4.118 6.077 — 4.110 31.527 — 58.749 28.249 23.897 4.268 6.125 3.926 3.565 — 31.260 59.626 28.990 24.452 4.115 29.218 6.029 3.867 3.508 4953 3.547 4827 4750 24.312 174 4797 180 138 — 134 -0.51 -0.30 -0.36 -0.27 0.33 4680 0.30 0.26 140 0.30 -0.20 0.26 0 max ν 27 0.00154 0.00101 0.00099 0.00063 ± ± ± ± 0.00328 1.0032 0.00083 0.9989 0.00159 1.0008 0.00056 1.0051 ± ± ± ± ),3.2.1 compared to those computed for § 0.000020.00390 0.9945 — — 0.000030.00113 1.0160 — — 0.000010.00120 0.9966 — — 0.000020.00072 1.0044 — — ± ± ± ± ± ± ± ± 0.00001 1.0032 0.00086 0.9927 0.00002 1.0010 0.00090 0.9971 0.00001 0.9995 0.00061 0.9982 0.00002 1.0006 0.00062 0.9931 ± ± ± ± ± ± ± ± data ( Pinsonneault et al. ;2018 “APOKASC-2”). Kepler 0.00002 0.9960 0.00094 1.0006 0.00002 0.9909 0.00115 1.0056 0.00001 1.0051 0.00048 1.0001 0.00002 0.9989 0.00072 1.0036 ± ± ± ± ± ± ± ± 0.00003 1.0108 0.00151 1.0070 0.00003 1.0067 0.00260 0.9830 0.00001 1.0007 0.00090 1.0050 0.00002 1.0082 0.00150 0.9901 ± ± ± ± ± ± ± ± A2Z CAN COR SYD BAM BHM 134.38 135.86 135.00 134.11 134.09 135.35 3075.9 3086.5 3067.1 3080.9 3143.4 3046.8 134.61 135.59 134.93 134.86 134.38 135.03 3102.6 3108.8 3061.0 3068.6 3107.5 3065.6 0.9960 0.9965 0.9931 1.0035 0.9993 0.9977 1.0023 1.0017 RC RC RGB RC RGB APOKASC2 , RC RGB APOKASC2

APOKASC2 , RGB , ,

APOKASC2

, ν, , ,

ν, , ν max ∆ ν max ∆ ν RC ∆ max ν RGB X RC ∆ max ν , X RGB X ν , X ν, ν, max ∆ . Derived solar reference value scale factors and solar reference values from this work (see max ∆ ν ν X X X X Table 5 some of the same pipelines using a similar method with 28 SYD , 0 κM σ SYD BHM 0 M, 0 R, κ κ SYD , BAM 0 , 0 κR σ κM σ SYD 0 R, BAM κ 0 M, κ , are computed with pipeline- 0 M COR , κ 0 BAM , 0 κM and σ κR 0 R σ κ COR , according to4 & 6 , Equations and represent i 0 M, BAM κ 0 R, 0 max κ ν h COR , 0 and A2Z , 0 i 0 κR ν σ κM ∆ σ h COR A2Z 0 R, κ 0 M, κ CAN Radius and mass coefficients , A2Z 0 , 3.2.4 for details. A full version of this table is available in the online journal. 0 § κM κR σ σ . See Table 6. 0 max CAN A2Z ν 0 R, 0 M, κ κ and 0 i ν 0 M κ CAN h , 0 σ κR i σ 0 M κ h , and their uncertainties, are computed based on i CAN i 0 M 0 R 0 R, κ κ κ h h σ and i i BHM 0 R , 0 R 0 κ κ h h κM — σ pipeline-averaged radius and massspecific asteroseismic coefficients. parameters, Pipeline-specific ∆ radius and mass coefficients, Note BHM 0 M, EPIC 210306475 12.976210307958 0.906 10.987210314854 1.559 0.767 10.555210315825 1.137 0.446 0.265210318976 9.521 1.119 0.179 11.953 12.209 0.160 0.677 0.124 1.751 9.900 0.964 1.137 9.909 0.066 0.136 — 1.381 9.582 0.898 12.862 0.773 0.327 1.022 0.889 0.925 1.776 1.331 0.222 — — 0.375 0.309 10.668 12.072 — 9.517 0.492 — 0.834 0.291 1.174 — 1.430 — 1.737 0.124 0.212 — 0.139 10.668 — — 12.556 9.354 — — κ BHM , 0 —— — — — 10.390 — 0.598 11.166 1.037 0.804 0.145 0.969 10.883 0.179 0.560 11.797 1.088 0.620 0.133 1.106 11.509 0.138 1.234 — 1.248 0.274 — — — 1.3041.033 1.3720.612 1.148 0.327 1.698 0.237 — 0.237 10.823 9.644 — 0.603 0.337 1.177 — 1.814 0.183 0.154 — 10.277 9.299 0.478 — 0.331 1.083 1.652 — 0.122 0.141 10.961 — 1.309 9.740 1.221 0.275 — 0.302 1.839 13.793 0.134 1.360 1.807 0.367 κR σ 29

Table 7. Median fractional uncertainties of Kepler and K2 asteroseismic quan- tities (in percent)

RGB or RGB/AGB RC APOKASC-2 Y18 K2 APOKASC-2 Y18 K2

σνmax 0.9 1.0 1.7 1.3 2.1 2.4

σ∆ν 0.4 0.3 1.7 1.1 1.1 2.3

σκR 1.3 1.1 3.3 2.7 3.3 5.0

σκM 3.4 3.1 7.7 6.2 8.4 10.5 Note—“APOKASC-2” indicates median fractional uncertainties from the analysis of Pinsonneault et al.(2018), while “Y18” refers to the analysis of Yu et al.(2018). 30 . i 0 ν bin, ∆ i shown 0 h ν i RC , 0 i ∆ i ν 0 h 0 and ν ν ∆ i h ∆ or h ∆ h σ i 0 max ν and h 0 max i and ν RC h i i 0 0 Hz % max ν ν µ h 0 max ∆ h ν h RGB , i 0 ν ∆ h σ , as a function of RGB i 0 i Hz % 0 ν µ ν ∆ ∆ h h and RC , listed as per cent. The binned medians of the fractional , i i i 0 ν 0 max 0 max ∆ ν ν h h h σ and i RC i 0 max Hz % ν µ 0 h max ν h RGB , i 0 max ν h σ . Systematic uncertainties of RGB i Hz % 12172330 0.6742 0.9156 0.6476 0.40 23 0.29 28 0.55 31 1.2 0.68 38 0.83 43 0.24 53 0.68 1.7 62 2.3 0.74 2.9 1.1 3.8 0.69 1.3 1.0 4.9 0.68 5.8 0.31 7.1 3.4 3.7 0.28 4.1 0.21 4.5 0.85 0.24 0.52 5.1 0.24 5.7 0.47 6.3 0.66 0.75 0.59 110140190 0.73 0.70 0.50 73 84 93 0.92 1.2 1.4 9.3 11 15 0.40 0.43 0.81 7.0 7.7 0.68 8.7 0.92 0.43 µ Table 8 0 max 3.4 for details. Entries that are not populated contain ten or fewer stars in a given ν § —Systematic uncertainties of h difference between an individual pipeline’s asteroseismic values and the mean values, in the bottom panels of6 - 9 Figures are taken to be indications of systematic uncertainty in See and are also not plotted in6 - 9 . Figures Note 31

ACKNOWLEDGMENTS Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. JCZ and MHP acknowledge support from NASA Department of Energy Office of Science, and the Partici- grants 80NSSC18K0391 and NNX17AJ40G. YE and CJ pating Institutions. SDSS-IV acknowledges support and acknowledge the support of the UK Science and Tech- resources from the Center for High-Performance Com- nology Facilities Council (STFC). SM would like to puting at the University of Utah. The SDSS web site is acknowledge support from the Spanish Ministry with www.sdss.org. the Ramon y Cajal fellowship number RYC-2015-17697. SDSS-IV is managed by the Astrophysical Research RAG acknowledges funding received from the PLATO Consortium for the Participating Institutions of the CNES grant. RS acknowledges funding via a Royal So- SDSS Collaboration including the Brazilian Partici- ciety University Research Fellowship. D.H. acknowl- pation Group, the Carnegie Institution for Science, edges support from the Alfred P. Sloan Foundation Carnegie Mellon University, the Chilean Participa- and the National Aeronautics and Space Administra- tion Group, the French Participation Group, Harvard- tion (80NSSC19K0108). V.S.A. acknowledges support Smithsonian Center for Astrophysics, Instituto de As- from the Independent Research Fund Denmark (Re- trof´ısica de Canarias, The Johns Hopkins University, search grant 7027-00096B), and the Carlsberg founda- Kavli Institute for the Physics and Mathematics of tion (grant agreement CF19-0649). This research was the Universe (IPMU) / University of Tokyo, the Ko- supported in part by the National Science Foundation rean Participation Group, Lawrence Berkeley National under Grant No. NSF PHY-1748958. Laboratory, Leibniz Institut fur¨ Astrophysik Potsdam Funding for the Stellar Astrophysics Centre (SAC) is (AIP), Max-Planck-Institut fur¨ Astronomie (MPIA Hei- provided by The Danish National Research Foundation delberg), Max-Planck-Institut fur¨ Astrophysik (MPA (Grant agreement no. DNRF106). Garching), Max-Planck-Institut fur¨ Extraterrestrische The K2 Galactic Archaeology Program is supported Physik (MPE), National Astronomical Observatories of by the National Aeronautics and Space Administra- China, New Mexico State University, New York Uni- tion under Grant NNX16AJ17G issued through the K2 versity, University of Notre Dame, Observat´ario Na- Guest Observer Program. This publication makes use cional / MCTI, The Ohio State University, Pennsylva- of data products from the Two Micron All Sky Sur- nia State University, Shanghai Astronomical Observa- vey, which is a joint project of the University of Mas- tory, United Kingdom Participation Group, Universidad sachusetts and the Infrared Processing and Analysis Nacional Aut´onoma de M´exico, University of Arizona, Center/California Institute of Technology, funded by the University of Colorado Boulder, University of Oxford, National Aeronautics and Space Administration and the University of Portsmouth, University of Utah, Univer- National Science Foundation. sity of Virginia, University of Washington, University of This work has made use of data from the Euro- Wisconsin, Vanderbilt University, and Yale University. pean Space Agency (ESA) mission Gaia (https://www. cosmos.esa.int/gaia), processed by the Gaia Data Pro- Software: asfgrid (Sharma & Stello 2016), emcee cessing and Analysis Consortium (DPAC, https://www. (Foreman-Mackey et al. 2013), NumPy (Walt et al. cosmos.esa.int/web/gaia/dpac/consortium). Funding 2011), pandas (McKinney 2010), Matplotlib (Hunter for the DPAC has been provided by national institu- 2007), IPython (P´erez & Granger 2007), SciPy (Virta- tions, in particular the institutions participating in the nen et al. 2020) Gaia Multilateral Agreement.

APPENDIX

0 0 We model the distribution of the fractional uncertainties, σ 0 /hν i and σ 0 /h∆ν i as a function of the hνmaxi max h∆ν i number of pipelines used to compute the uncertainties, dof, and evolutionary state (RGB or RGB/AGB versus RC). As described in the text, we use two generalized gamma distributions to model each distribution of stars with a given dof (either 2, 3, 4, 5, or 6). When summed, the two fitted generalized gamma distributions are solutions to a least-squares minimization problem to describe the data, with Poisson uncertainties assumed for each bin in the observed distribution. To arrive at the fit, each component is weighted using a free parameter to describe the relative contribution of each component, and the degrees of freedom for each component are required to be less than or equal to the nominal degrees of freedom for the observed distribution (i.e., the number of reporting pipelines, dof). The solution is found using the Trust Region Reflective method as implemented in the scipy function, curve fit. As a reference, we 32

0 Figure 19. Comparison of asteroseismic scaling relation radii (Eq.4) using K2 GAP DR2 hκRi in combination with APOGEE effective temperatures and Gaia DR2 radii derived based on corrected Gaia parallaxes (see text). Grey dashed lines show one- to-one relations. In the bottom panel, blue error bars indicate weighted averages and standard errors on the weighted averages for RC stars (blue) and RGB stars (red). also fit each of the six distributions for both RGB or RGB/AGB and RC stars using a generalized gamma distribution with the uncertainty taken to be the median observed fractional uncertainty, and with the dof fixed to be the number 0 0 of reporting pipelines. We do the same for the fractional uncertainty distributions for hκRi and hκM i. 0 For hνmaxi among RGB or RGB/AGB stars, the fractional uncertainties inferred from fitting the two-component generalized gamma distribution model vary according to the number of pipelines that contribute to the scatter estimate, from 1.1 − 2.1%. For RC stars, the fitted uncertainties are larger, and have a range of 2.2 − 3.9%. The fitted fractional uncertainties on h∆ν0i have marginally smaller fractional uncertainties, and can range from 1.4% to 1.9% for RGB or 0 0 RGB/AGB stars and from 1.9% to 2.8% for RC stars. For both hνmaxi and h∆ν i, therefore, the uncertainties vary more as a function of evolutionary state than number of pipelines reporting. The uncertainty distributions for all of the RGB and RC stars are shown in Figures 20-21. We show both the expected distribution according to the observed median uncertainties using a fixed dof (grey curve), as well as the expected distribution from the two-component model (black curve), where we sum the uncertainties from each dof, weighting by the number of stars with a given dof. The agreement between the model for the uncertainties and the observed uncertainty distributions indicates that the uncertainties are largely not a function of νmax or ∆ν. Nevertheless, the approximation is not completely accurate: lower values of νmax and ∆ν tend to have marginally larger fractional uncertainties — up to 1% larger across the entire observed parameter range for νmax and ∆ν among RGB stars, and up to 3% larger for νmax among dof = 2 and dof = 3 RC stars with νmax < 30µHz. (These latter low-νmax RC 0 stars contribute to the extra bump at σ 0 /hν i ≈ 3 in Figure 20.) The uncertainties are not strong functions of hνmaxi max magnitude, which reflects the fact that the uncertainties are not dominated by white noise, but rather the length of the light curve, intrinsic properties of the star (e.g., evolutionary state), and pipeline agreement. 0 hκM i fractional uncertainties reinforce the trend for fractional uncertainties to vary more as a function of evolutionary state than a function of number of pipelines reporting: for RGB and RGB/AGB, the range is 6.2 − 9.0% and for RC 0 0 stars, the range is 9.9 − 12.1%. The uncertainty distributions of hκRi and hκM i for all of the RGB and RC stars are shown in Figures 22-23, where the grey and black curves correspond to the expected distribution according to 33

0 Figure 20. Left: The distribution of uncertainties in hνmaxi for RGB stars, with curves showing models for the distributions assuming the median (grey) and best-fitting uncertainties (black) — the characteristic uncertainty according to each of these two models is shown in grey and black in the legend. These distributions and models are the results of summing the distributions and models for stars with dof = {2, 3, 4, 5, 6} pipelines reporting (see Appendix for details). The number of stars contributing to the observed distribution is listed as N. Right: Same as Left, but for RC stars.

Figure 21. Left: Same as Figure 20, but for h∆ν0i. Right: Same as Left, but for RC stars. the sample median uncertainties and the best-fitting uncertainties, and which are obtained by appropriately weighted sums of the uncertainties at each dof.

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