The K2 Galactic Archaeology Program Data Release 2: Asteroseismic Results from Campaigns 4, 6, and 7

In the context of the GAP, analyses of the campaigns presented here were prioritized due to their coverage of the sky: the Galactic center (C7), the Galactic anticenter (C4), and out of the Galactic plane (C6). These results will ultimately be joined with a forthcoming analysis of the rest of the K2 campaigns for which GAP targets have been observed. The GAP targets red giants because they are bright (probing far into the Galaxy) and because their oscillations are detectable from the K2 long-cadence data, which has a Nyquist frequency of $\sim 280\,\mu \mathrm{Hz}$. All GAP targets for campaigns 4, 6, and 7 were selected from the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006) to have J − K > 0.5 and good photometric quality based on 2MASS flags. ²¹ The proposed targets passing these selection criteria were prioritized based on a rank ordering in V-band magnitude from bright to faint. ²² C4 and C6 targets were chosen to have 9 < V < 15, and C7 targets were chosen to have 9 < V < 14.5, with some exceptions to the prioritization on a campaign-to-campaign basis, as follows: One giant with existing RAVE data was prioritized in C4. In C6, priority was given to 129 giants with existing APOGEE (Majewski et al. 2010) spectra, 607 with existing RAVE spectra, and 5 low-metallicity giants chosen from the literature to be giants with [Fe/H] < −3. The highest priority for GAP targets in C7 was given to 222 known giants with existing spectroscopic data from APOGEE, and 23 targets in NGC 6717 (but see below).

Because observed targets were selected in a linear way from the target priority list, the selection functions for each of the campaigns are well-defined. Nearly all of the C6 targets were observed, and so the observed targets conform to the selection function 9 < V < 15. Our C4 targets were observed down to V = 13.447, and therefore follow the selection function down to that magnitude limit (this magnitude limit approximately corresponds to the top 5000 GAP targets). In addition to these GAP-selected targets, there are fainter stars on the GAP target list that were serendipitously observed by K2 through other non-GAP target proposals. Those stars do not follow the GAP selection function but are still analyzed in this paper. However, we caution their use for population studies. The observed C7 targets were accidentally chosen from an inverted priority list during the mission-wide target list consolidation before upload to the spacecraft. As a result, observations were made of only two of the higher-priority APOGEE C7 targets and none from NGC 6717. As discussed in Sharma et al. (2019), the resulting effective selection function is one of either 9 < V < 14.5 or 14.276 < V < 14.5, depending on the position on the sky. Approximately, the 3500 lowest-priority GAP targets follow the full selection function. There is, however, an additional population of ∼600 stars in C7 that can be used for scientific purposes, with the understanding that its selection function of 14.276 < V < 14.5 is different than that of the rest of the campaign (9 < V < 14.5). For additional details, see Sharma et al. (2019). We account for these selection functions when comparing to models in our analysis by making sure to compare to the subset of observed stars that follow reproducible GAP target criteria. An accounting of observed and targeted stars is given in Table 1. The approximate lines of sight for these campaigns are shown in Figure 1. An on-sky map of the campaigns is shown in Figure 2.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We used the light curves generated by the EVEREST pipeline (Luger et al. 2018), which uses campaign-specific basis vectors to remove noise correlated across pixels. Though the K2 GAP DR1 (Stello et al. 2017) used Vanderburg & Johnson (2014) (K2SFF) light curves, we found that the EVEREST pipeline removes at least as much of the sawtooth-like systematic flux variation induced by the K2 six-hour thruster firings as K2SFF does, while furthermore generating lower levels of white noise in the giant spectra than K2SFF. Because the K2 GAP DR1 used K2SFF light curves, we attempted to reconcile systematic differences in the amplitudes of flux variations in the EVEREST and K2SFF light curves for consistency between this release and DR1. To do so, for each star, we multiplied the EVEREST light-curve flux by a scalar factor such that its power is equal to the K2SFF power. We then applied a boxcar high-pass filter with a width of 4 days to the light curve to remove most of the non-oscillation variability (Stello et al. 2015), and 4σ outliers in the time series were removed. Seven targets were labeled as extended in the EPIC and were not included in the present analysis. ²³

In this data release, we focus on two asteroseismic quantities, ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$, which we describe in turn.

The frequency at maximum acoustic power, ${\nu }_{\max }$, scales with the acoustic cutoff frequency at the stellar atmosphere (Brown et al. 1991; Kjeldsen & Bedding 1995; Chaplin et al. 2008; Belkacem et al. 2011), such that

$$\begin{eqnarray}&&\displaystyle \frac{{\nu }_{\max }}{{\nu }_{\max ,\odot }}\approx \displaystyle \frac{M/{M}_{\odot }}{{\left(R/{R}_{\odot }\right)}^{2}\sqrt{({T}_{\mathrm{eff}}/{T}_{\mathrm{eff},\odot })}}.\end{eqnarray} \tag{ 1 }$$

The frequency separation between modes of the same degree but consecutive radial order, ${\rm{\Delta }}\nu$, scales with the stellar density (Ulrich 1986; Kjeldsen & Bedding 1995) according to

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}\nu }{{\rm{\Delta }}{\nu }_{\odot }}\approx \sqrt{\displaystyle \frac{M/{M}_{\odot }}{{\left(R/{R}_{\odot }\right)}^{3}}}.\end{eqnarray} \tag{ 2 }$$

The ${\nu }_{\max ,\odot }$ and ${\rm{\Delta }}{\nu }_{\odot }$ solar reference values are not set in stone, and are themselves measured quantities from asteroseismic data of the Sun. All the pipelines have an internal set of recommended solar reference values, ν_max,⊙,PIP and Δν_⊙,PIP. In addition to those pipeline-specific solar reference values, we adopt a solar reference temperature of ${T}_{\mathrm{eff},\odot }=5772\,{\rm{K}}$ (Mamajek et al. 2015).

Like Stello et al. (2017), we analyze the data using multiple asteroseismic pipelines, descriptions of which are found therein. We briefly revisit each pipeline below.

A2Z+ (hereafter A2Z) is based on the A2Z pipeline described in Mathur et al. (2010). Two models are used to compute Δν: the autocorrelation function of the time series and the power spectrum of the power spectrum. The results from both methods are then compared, and a ${\rm{\Delta }}\nu$ value is only kept when both methods agree to within 10%. The power density spectra (PDS) of those stars are checked to select the ones where modes are present to high confidence. The FliPer metric (Bugnet et al. 2018) is used to check stars where the amplitude of the convective background does not agree with the seismic detection. Finally, the Δν value is refined by cross-correlating a template of the radial modes around the region of the modes where Δν is varied. The value reported corresponds to the one obtained for the highest correlation coefficient. After fitting the convective background with two Harvey laws (Mathur et al. 2011; Kallinger et al. 2014) and subtracting it from the PDS, a Gaussian function is fit to the modes in order to estimate the frequency of maximum power, ν_max.

BHM (Elsworth et al. 2020) is based on the OCT pipeline (Hekker et al. 2010), and performs hypothesis testing for solar-like oscillations above the granulation background. If there is a frequency window that has significant signal, ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ are computed according to Hekker et al. (2010), with increased, K2-specific quality control to ensure ${\rm{\Delta }}\nu$ is not an alias of the true value.

CAN returns ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ for stars whose autocorrelation functions have characteristic timescales and rms variability that accord with a relation expected from solar-like oscillators (Kallinger et al. 2016). A Bayesian evidence is computed to determine whether there are solar-like oscillations near the detected autocorrelation timescale (and therefore near ${\nu }_{\max }$). A threshold for the evidence is determined based on visual inspection of a test set of power spectra. The value of ${\rm{\Delta }}\nu$ is then calculated by fitting individual radial modes.

COR first fits for ${\rm{\Delta }}\nu$ using the autocorrelation of the light curve. COR (Mosser & Appourchaux 2009) returns results for stars that have ${\rm{\Delta }}\nu$, FWHM, amplitude of power, and the granulation at ${\nu }_{\max }$ that follow relations from Mosser et al. (2010), with stricter, K2-specific requirements for stars with a possible ${\nu }_{\max }$ detected near the K2 thruster firing frequency (or alias thereof).

SYD computes ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ according to Huber et al. (2009). Results are returned only for stars that are classified as having solar-like oscillations by a machine-learning algorithm described in Hon et al. (2018b). Results for ${\rm{\Delta }}\nu$ are provided only for stars that are classified as having reliable ${\rm{\Delta }}\nu$ values by a machine-learning algorithm. The ${\rm{\Delta }}\nu$-vetting algorithm is a convolutional neural network trained on K2 C1 data that were classified by eye as having detectable ${\rm{\Delta }}\nu$ values. The algorithm takes as an input the autocorrelation of a granulation background-corrected spectrum in a window with a width of $\pm 0.59\times {\nu }_{\max }^{0.9}$ centered around ${\nu }_{\max }$. This latter window size is taken from the observed width of detected stellar oscillations in giants (Mosser et al. 2010). The algorithm takes the autocorrelation, passes it through a convolutional neural network, and outputs a score varying from 0 (not a valid ${\rm{\Delta }}\nu$ detection) to 1 (100% certain valid ${\rm{\Delta }}\nu$ detection), but which is not a linear mapping to percent confidence in the detection. For this reason, the score was calibrated using visually verified ${\rm{\Delta }}\nu$ detections, which showed that a score of 0.8 corresponds to a completeness of about 70% (i.e., this score rejects 30% stars that have valid ${\rm{\Delta }}\nu$ detections) and near total purity (i.e., all stars have visually verified ${\rm{\Delta }}\nu$ detections); ${\rm{\Delta }}\nu$ detections were considered valid for stars with a score above this 0.8 value.

BAM calculates ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ according to Zinn et al. (2019c). The ${\rm{\Delta }}\nu$ values for this data release are provided based on the autocorrelation method described therein, which, in turn, is based on that of SYD. Stars are classified as oscillators or not based on the Bayesian evidence for the presence of solar-like oscillations (for details, see Zinn et al. (2019c)), and results in this data release are returned for stars with $3.5\,\mu \mathrm{Hz}\lt {\nu }_{\max }\lt 250\,\mu \mathrm{Hz}$. Values of ${\rm{\Delta }}\nu$ are only provided for those that satisfy the SYD ${\rm{\Delta }}\nu$-vetting algorithm described above.

Any pipeline may return a ${\nu }_{\max }$ and/or a ${\rm{\Delta }}\nu$ for a given star. In what follows, however, we consider a "detection" for a given pipeline to be a target for which a ${\nu }_{\max }$ is returned, unless otherwise noted. A fraction of ${\nu }_{\max }$-detected stars will not have a reliable ${\rm{\Delta }}\nu$ measurement reported. The raw ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ and the uncertainties reported by each pipeline are provided in Table 2, while the number of ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ values returned are provided in Table 3.

Table 2. Raw Asteroseismic ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ Values for K2 GAP DR2 for Each Pipeline, with Evolutionary States

EPIC	Campaign	Priority	Evo. State	νmax,A2Z	σνmax,A2Z	ΔνA2Z	${\sigma }_{{\rm{\Delta }}\nu ,{\rm{A}}2{\rm{Z}}}$	νmax,BAM	σνmax,BAM	${\rm{\Delta }}{\nu }_{\mathrm{BAM}}$	${\sigma }_{{\rm{\Delta }}\nu ,\mathrm{BAM}}$	νmax,BHM	σνmax,BHM	${\rm{\Delta }}{\nu }_{\mathrm{BHM}}$
				(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)
210306475	4	903	RGB	28.550	2.25	3.620	0.010	30.135	0.808	3.277	0.197	26.900	1.4	3.470
210307958	4	2771	RGB	27.940	2.47	3.890	0.000	30.316	0.723	3.723	0.124	⋯	⋯	⋯
210314854	4	1141	RGB	29.030	2.03	4.100	0.120	31.863	0.743	4.115	0.083	31.200	1.0	4.090
210315825	4	1651	RGB	59.580	3.63	5.910	0.160	59.253	1.379	5.899	0.060	60.000	1.3	6.010
210318976	4	988	RGB	24.800	1.88	3.270	0.040	25.080	0.841	⋯	⋯	21.700	1.1	3.080

${\sigma }_{{\rm{\Delta }}\nu ,\mathrm{BHM}}$	${\nu }_{\max ,\mathrm{CAN}}$	${\sigma }_{\nu \max ,\mathrm{CAN}}$	${\rm{\Delta }}{\nu }_{\mathrm{CAN}}$	${\sigma }_{{\rm{\Delta }}\nu ,\mathrm{CAN}}$	${\nu }_{\max ,\mathrm{COR}}$	${\sigma }_{\nu \max ,\mathrm{COR}}$	${\rm{\Delta }}{\nu }_{\mathrm{COR}}$	${\sigma }_{{\rm{\Delta }}\nu ,\mathrm{COR}}$	${\nu }_{\max ,\mathrm{SYD}}$	${\sigma }_{\nu \max ,\mathrm{SYD}}$	${\rm{\Delta }}{\nu }_{\mathrm{SYD}}$	${\sigma }_{{\rm{\Delta }}\nu ,\mathrm{SYD}}$
(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)
0.160	27.670	1.27	⋯	⋯	⋯	⋯	⋯	⋯	29.016	0.63245	3.450	0.170
⋯	29.260	1.03	4.004	0.094	28.350	0.83	3.805	0.083	28.789	0.62378	3.740	0.203
0.190	30.600	1.49	4.066	0.056	31.640	0.89	4.204	0.079	31.044	0.88666	4.033	0.238
0.190	59.390	1.28	5.960	0.084	58.960	1.26	5.975	0.087	59.213	1.21725	5.873	0.058
0.100	23.660	1.22	3.482	0.090	24.540	0.73	3.402	0.076	24.144	0.77871	⋯	⋯

Notes. These are the parameters returned by a given pipeline, along with their uncertainties, without any of the rescaling described in Section 3.2.1 applied. Evolutionary states, which have been derived in this work (see Section 3.2.2), are also included in this table. If classified, a star's evolutionary state is assigned as either "RGB," "RGB/AGB," or "RC." "Priority" refers to the K2 GAP target priority discussed in Section 2 (a smaller numerical value corresponds to higher priority); serendipitous targets do not have a populated priority entry.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

Download table as:  DataTypeset image

Table 3. Asteroseismic Yields across Pipeline and Campaign

		${\nu }_{\max }$	${\nu }_{\max }^{{\prime} }$	${\rm{\Delta }}\nu$	${\rm{\Delta }}\nu ^{\prime}$	${\kappa }_{R}^{{\prime} }$	${\kappa }_{M}^{{\prime} }$
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

Note. Numbers of stars with raw asteroseismic values (${\nu }_{\max }$, ${\rm{\Delta }}\nu$), rescaled asteroseismic values (${\nu }_{\max }^{{\prime} }$, ${\rm{\Delta }}\nu ^{\prime}$), and radius and mass coefficients (${\kappa }_{R}^{{\prime} }$, ${\kappa }_{M}^{{\prime} }$), as a function of pipeline and campaign.

Download table as:  ASCIITypeset image

3.2.1. Mean Asteroseismic Parameters, $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$

Consolidating results from multiple pipelines, as we do here, has three primary benefits: outliers can be rejected (e.g., stars that only one pipeline identifies as an oscillator can be considered dubious); the accuracy and precision of the parameters can be improved by averaging results for the same star from different pipelines; and the spread in values for a given star can be translated into an uncertainty estimate. To perform this averaging, however, the possibility that there are systematic differences in pipeline results needs to be taken into account. Pinsonneault et al. (2018) demonstrated that such systematics existed in their multipipeline analysis of Kepler data, finding that the relative zero points of the measurements returned by different pipelines do not scale in the same way as their solar measurements do (Pinsonneault et al. 2018). Figures 3 and 4 demonstrate these systematics in our K2 ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ values. In what follows, we describe an approach that rescales each pipeline's ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ values such that they are on average on the same scale as the other pipelines. These rescaled values are used to perform outlier rejection with sigma clipping, and the values are averaged for each star in order to compute more accurate and precise asteroseismic values, as well as to define empirical uncertainties on ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

To reduce the systematic uncertainty across pipelines for a single star, we follow the approach taken by Pinsonneault et al. (2018): under the assumption that each pipeline's ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ values are distributed around the true values, we can fit for near-unity scale factors that effectively modify each pipeline's solar reference values so that the mean ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ returned by pipelines are all on the same scale. This therefore reduces the systematic uncertainty due to the choice of a single pipeline value on a single star's ${\nu }_{\max }$ or ${\rm{\Delta }}\nu$, and it allows for averaging values across pipelines on a star-by-star basis.

The above procedure to rescale the pipeline-specific solar reference values is performed by initially assuming that the pipeline values are already on the same system, so that the rescaled ${\nu }_{\max }$, ${\nu }_{\max ,{\rm{s}},{\rm{p}}}^{{\prime} }$, of a star, s, for pipeline, p, equals the raw value returned by the pipeline: ${\nu }_{\max ,{\rm{s}},{\rm{p}}}^{{\prime} }={\nu }_{\max ,{\rm{s}},{\rm{p}}}$. An average value $\langle {\nu }_{\max ,{\rm{s}}}^{{\prime} }\rangle \equiv {\sum }_{p}{\nu }_{\max ,{\rm{s}},{\rm{p}}}^{{\prime} }/{N}_{p}$ is then calculated for each star that has at least two pipelines returning a value. The sum is over those N_p reporting pipelines. A scalar factor for each pipeline, ${X}_{{\nu }_{\max },p}\equiv {\sum }_{s}\left[{\nu }_{\max ,{\rm{s}},{\rm{p}}}/\langle {\nu }_{\max ,{\rm{s}}}^{{\prime} }\rangle \right]/{N}_{s}$, is calculated using the N_s stars for which the pipeline returned a raw value, ν_max,s,p, and which had a defined mean value $\langle {\nu }_{\max ,{\rm{s}}}^{{\prime} }\rangle$. The pipeline values are rescaled by this factor so that ${\nu }_{\max ,{\rm{s}},{\rm{p}}}^{{\prime} }={\nu }_{\max ,{\rm{s}},{\rm{p}}}/{X}_{{\nu }_{\max },p}$. For each star, a 3σ clipping is performed to reject rescaled values returned by a pipeline that are highly discordant with the results from other pipelines. This whole process is repeated until convergence in the rescaled value, ${\nu }_{\max ,{\rm{s}},{\rm{p}}}^{{\prime} }$. The same procedure is done for ${\rm{\Delta }}\nu$. Following the observation by Pinsonneault et al. (2018) that the RC exhibits significantly different structure in the pipeline ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ zero-point differences, the same procedure is done separately for RGB and RGB/AGB stars and RC stars. In the end, four scale factors are derived for each pipeline: a ${\nu }_{\max }$ scale factor for RGB stars, a ${\rm{\Delta }}\nu$ scale factor for RGB stars, a ${\nu }_{\max }$ scale factor for RC stars, and a ${\rm{\Delta }}\nu$ scale factor for RC stars. The resulting factors are at the percent level or below. For each pipeline, we provide the rescaled values, ${\nu }_{\max }^{{\prime} }$ and ${\rm{\Delta }}\nu ^{\prime}$, in addition to the raw values. We also provide mean values $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$, which have been averaged across pipelines. The root mean squares across all pipelines for each star are taken as the uncertainties on the $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ values, ${\sigma }_{\langle {\nu }_{\max }^{{\prime} }\rangle }$ and ${\sigma }_{\langle {\rm{\Delta }}\nu ^{\prime} \rangle }$ (see Section 3.2.3). The sample with $\langle {\nu }_{\max }^{{\prime} }\rangle$ is shown in Figure 5.

Zoom In Zoom Out Reset image size

Unless otherwise noted, the mean ${\rm{\Delta }}\nu$ values for each star, $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$, as well as the pipeline-specific rescaled ${\rm{\Delta }}\nu$ values, ${\rm{\Delta }}\nu ^{\prime}$, have been multiplied by a factor as described in Sharma et al. (2016) that depends on the star's temperature, metallicity, surface gravity, mass, and evolutionary state. This factor is provided in Table 4 as X_Sharma, and is a theoretically motivated factor that improves the homology assumption in the ${\rm{\Delta }}\nu$ scaling relation (Equation (2)). We calculated these factors using temperatures and metallicities from the EPIC (Huber et al. 2016); surface gravities from Equation (1) using ${\nu }_{\max };$ mass estimates from Equation (6) using ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$, before the factor in question has been applied; and evolutionary states derived from the neural network approach laid out in Section 3.2.2. The correction is applied multiplicatively such that ${\rm{\Delta }}\nu ^{\prime}$ is multiplied by X_Sharma. Pipeline values rejected by the above sigma-clipping process will not have a corrected pipeline value populated in Table 4.

Table 4. Derived Asteroseismic ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ Values for K2 GAP DR2

EPIC	$\langle {\nu }_{\max }^{{\prime} }\rangle$	${\sigma }_{\langle {\nu }_{\max }^{{\prime} }\rangle }$	Nνmax	$\langle {\rm{\Delta }}\nu ^{\prime} \rangle$	${\sigma }_{\langle {\rm{\Delta }}\nu ^{\prime} \rangle }$	${N}_{{\rm{\Delta }}\nu }$	XSharma	σXSharma	$\langle {\rm{\Delta }}\nu \rangle$	${\nu }_{\max ,{\rm{A}}2{\rm{Z}}}^{{\prime} }$	${\rm{\Delta }}{\nu }_{{\rm{A}}2{\rm{Z}}}^{{\prime} }$
	(μHz)	(μHz)		(μHz)	(μHz)				(μHz)	(μHz)	(μHz)
210306475	28.487	1.235	5	3.610	0.099	3	1.027	0.010	3.517	28.502	3.723
210307958	28.974	0.935	5	3.957	0.123	5	1.032	0.013	3.834	27.893	4.090
210314854	30.907	0.990	6	4.170	0.057	6	1.016	0.018	4.102	28.981	4.167
210315825	59.422	0.463	6	6.088	0.045	6	1.025	0.010	5.939	59.479	6.071
210318976	24.478	0.414	5	3.487	0.094	3	1.031	0.013	3.381	24.758	3.381

${\nu }_{\max ,\mathrm{BAM}}^{{\prime} }$	${\rm{\Delta }}{\nu }_{\mathrm{BAM}}^{{\prime} }$	${\nu }_{\max ,\mathrm{BHM}}^{{\prime} }$	${\rm{\Delta }}{\nu }_{\mathrm{BHM}}^{{\prime} }$	${\nu }_{\max ,\mathrm{CAN}}^{{\prime} }$	${\rm{\Delta }}{\nu }_{\mathrm{CAN}}^{{\prime} }$	${\nu }_{\max ,\mathrm{COR}}^{{\prime} }$	${\rm{\Delta }}{\nu }_{\mathrm{COR}}^{{\prime} }$	${\nu }_{\max ,\mathrm{SYD}}^{{\prime} }$	${\rm{\Delta }}{\nu }_{\mathrm{SYD}}^{{\prime} }$	EPIC Teff	σT	EPIC [Fe/H]	σ[Fe/H]
(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(μHz)	(K)	(K)
30.004	⋯	26.764	3.558	27.948	⋯	⋯	⋯	29.218	3.547	4797	134	−0.27	0.30
30.184	3.853	⋯	⋯	29.554	4.110	28.249	3.926	28.990	3.867	4750	138	−0.36	0.26
31.724	4.202	31.042	4.157	30.907	4.118	31.527	4.268	31.260	4.115	4953	174	−0.51	0.33
58.995	6.067	59.696	6.156	59.986	6.077	58.749	6.125	59.626	6.029	4827	180	−0.30	0.30
24.971	⋯	⋯	⋯	23.897	3.565	24.452	3.508	24.312	⋯	4680	140	−0.20	0.26

Notes. Asteroseismic values rescaled for scalar offsets among pipelines are denoted by a prime (the pipeline-specific solar reference scale factors are listed in Table 5); mean ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ values for each star across all pipelines are denoted by $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle ;$ the standard deviation of these values for each star across all pipelines are denoted by ${\sigma }_{\langle \nu {\max }^{{\prime} }\rangle }$ and ${\sigma }_{\langle {\rm{\Delta }}\nu ^{\prime} \rangle }$. The value of $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ is adjusted using theoretically motivated correction factors, X_Sharma (Sharma et al. 2016), for use in asteroseismic scaling relations; an uncorrected version of $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ for each star is provided, $\langle {\rm{\Delta }}\nu \rangle =\langle {\rm{\Delta }}\nu ^{\prime} \rangle /{X}_{\mathrm{Sharma}}$, should the user wish to compute custom ${\rm{\Delta }}\nu$ corrections. EPIC temperatures and metallicities are provided for this purpose, though these are relatively uncertain estimates of the true temperatures and metallicities (these uncertainties are also provided for convenience). The uncertainties in X_Sharma and σ_XSharma are computed by perturbing the EPIC temperature and metallicities in a Monte Carlo procedure. Pipeline-specific rescaled values, ${\nu }_{\max }^{{\prime} }$ and ${\rm{\Delta }}\nu ^{\prime}$, are only provided for targets for which at least two pipelines returned concordant results, and otherwise have a blank entry; the numbers of pipelines returning valid results for ${\nu }_{\max }$ or ${\rm{\Delta }}\nu$ are denoted by ${N}_{\nu \max }$ and N_Δν . See text for details.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

Download table as:  DataTypeset image

Table 5 shows the resulting solar reference scale factors compared to those from Pinsonneault et al. (2018). The agreement is mixed, depending on the pipeline. As noted in Section 3.1, the pipelines as implemented for this analysis have been modified to account for K2 data, and so may well differ slightly in the way they perform compared to their implementation for Pinsonneault et al. (2018). Additionally, we consider results from BAM (Zinn et al. 2019c), which the Pinsonneault et al. (2018) analysis did not consider. 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 ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ derived with K2 data compared to Kepler data. These uncertainties are calculated assuming ${\sigma }_{{\nu }_{\max }}$ and ${\sigma }_{{\rm{\Delta }}\nu }$ are exact (i.e., with no uncertainty on the standard deviation used to calculate them (see Section 3.2.3)). Given this assumption, the uncertainties on the scale factors are indicative and not definitive. We also provide the rescaled, pipeline-specific solar reference values themselves in Table 5 (attained by multiplying the pipeline-specific solar reference value by the pipeline's solar reference scale factor derived here).

Table 5. Derived Solar Reference Value Scale Factors and Solar Reference Values

	A2Z	CAN	COR	SYD	BAM	BHM
${X}_{{\nu }_{\max },\mathrm{RGB},\mathrm{APOKASC}2}$	1.0023 ± 0.00002	1.0082 ± 0.00002	0.9989 ± 0.00002	1.0006 ± 0.00002	⋯	⋯
${X}_{{\nu }_{\max },\mathrm{RGB}}$	1.0017 ± 0.00150	0.9901 ± 0.00072	1.0036 ± 0.00062	0.9931 ± 0.00072	1.0044 ± 0.00056	1.0051 ± 0.00063
${\nu }_{\max ,{\odot }_{\mathrm{RGB}}}$	3103 ± 5	3109 ± 2	3061 ± 2	3069 ± 2	3108 ± 2	3066 ±2
${X}_{{\rm{\Delta }}\nu ,\mathrm{RGB},\mathrm{APOKASC}2}$	0.9993 ± 0.00001	1.0007 ± 0.00001	1.0051 ± 0.00001	0.9995 ± 0.00001	⋯	⋯
${X}_{{\rm{\Delta }}\nu ,\mathrm{RGB}}$	0.9977 ± 0.00090	1.0050 ± 0.00048	1.0001 ± 0.00061	0.9982 ± 0.00120	0.9966 ± 0.00159	1.0008 ± 0.00099
${\rm{\Delta }}{\nu }_{{\odot }_{\mathrm{RGB}}}$	134.6 ± 0.1	135.6 ± 0.1	134.9 ± 0.1	134.9 ± 0.2	134.4 ± 0.2	135.0 ± 0.1
${X}_{{\nu }_{\max },\mathrm{RC},\mathrm{APOKASC}2}$	1.0035 ± 0.00003	1.0067 ± 0.00002	0.9909 ± 0.00002	1.0010 ± 0.00003	⋯	⋯
${X}_{{\nu }_{\max },\mathrm{RC}}$	0.9931 ± 0.00260	0.9830 ± 0.00115	1.0056 ± 0.00090	0.9971 ± 0.00113	1.0160 ± 0.00083	0.9989 ± 0.00101
${\nu }_{\max ,{\odot }_{\mathrm{RC}}}$	3076 ± 8	3086 ± 4	3067 ± 3	3081 ± 3	3143 ± 3	3047 ± 3
${X}_{{\rm{\Delta }}\nu ,\mathrm{RC},\mathrm{APOKASC}2}$	0.9965 ± 0.00003	1.0108 ± 0.00002	0.9960 ± 0.00001	1.0032 ± 0.00002	⋯	⋯
${X}_{{\rm{\Delta }}\nu ,\mathrm{RC}}$	0.9960 ± 0.00151	1.0070 ± 0.00094	1.0006 ± 0.00086	0.9927 ± 0.00390	0.9945 ± 0.00328	1.0032 ± 0.00154
${\rm{\Delta }}{\nu }_{{\odot }_{\mathrm{RC}}}$	134.4 ± 0.2	135.9 ± 0.1	135.0 ± 0.1	134.1 ± 0.5	134.1 ± 0.4	135.3 ± 0.2

Note. Factors and values from this work (see Section 3.2.1), compared to those computed for some of the same pipelines using a similar method with Kepler data (Pinsonneault et al. 2018; "APOKASC-2").

Download table as:  ASCIITypeset image

Even with these rescalings, residual differences between pipelines as a function of ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ will remain. These trends are shown in the bottom panels of Figures 6–9, where fractional differences between $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ and individual pipeline rescaled values, ${\nu }_{\max }^{{\prime} }$ and ${\rm{\Delta }}\nu ^{\prime}$ are shown. By definition, the means of ${\nu }_{\max }^{{\prime} }$ and ${\rm{\Delta }}\nu ^{\prime}$ across the pipelines are $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$, though what these figures demonstrate is substructure as a function of ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$, indicating ${\nu }_{\max }$- and ${\rm{\Delta }}\nu$-dependent systematic errors, not zero-point errors. We turn to a more robust estimate of the systematic errors on $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ in Section 4.2.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

After these rescalings are applied for each pipeline ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$, the absolute value of the solar reference values are free to be chosen, which we take to be ${\nu }_{\max ,\odot }=3076\,\mu \mathrm{Hz}$ and ${\rm{\Delta }}{\nu }_{\odot }=135.146\,\mu \mathrm{Hz}$ (Pinsonneault et al. 2018). The effect of this choice is evaluated in Section 4.2.

The main K2 GAP DR2 sample is defined to be that with a valid $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}{\nu }^{{\prime} }\rangle$ (i.e., stars for which at least two pipelines returned both ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ that agree to within 3σ), and its contents are summarized in Table 4. The sample contains 4395 stars.

3.2.2. Evolutionary States

3.2.3. Uncertainties on $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$, ${\sigma }_{\langle {\nu }_{\max }^{{\prime} }\rangle }$, and ${\sigma }_{\langle {\rm{\Delta }}\nu ^{\prime} \rangle }$

3.2.4. Radius and Mass Coefficients, $\langle {\kappa }_{R}^{{\prime} }\rangle$ and $\langle {\kappa }_{M}^{{\prime} }\rangle$

Given Equations (1) and (2), the radius of a star may be derived according to

$$\begin{eqnarray}&&\displaystyle \frac{R}{{R}_{\odot }}\approx \left(\displaystyle \frac{{\nu }_{\max }}{{\nu }_{\max ,\odot }}\right){\left(\displaystyle \frac{{\rm{\Delta }}\nu }{{\rm{\Delta }}{\nu }_{\odot }}\right)}^{-2}{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{T}_{\mathrm{eff},\odot }}\right)}^{1/2}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\equiv \ {\kappa }_{R}{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{T}_{\mathrm{eff},\odot }}\right)}^{1/2},\end{eqnarray} \tag{ 4 }$$

and its mass according to

$$\begin{eqnarray}&&\displaystyle \frac{M}{{M}_{\odot }}\approx {\left(\displaystyle \frac{{\nu }_{\max }}{{\nu }_{\max ,\odot }}\right)}^{3}{\left(\displaystyle \frac{{\rm{\Delta }}\nu }{{\rm{\Delta }}{\nu }_{\odot }}\right)}^{-4}{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{T}_{\mathrm{eff},\odot }}\right)}^{3/2}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\equiv \ {\kappa }_{M}{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{T}_{\mathrm{eff},\odot }}\right)}^{3/2}.\end{eqnarray} \tag{ 6 }$$

It is our wish not to impose a choice of temperature on the user when providing asteroseismic radii and masses, which is why we provide κ_R and κ_M instead of direct radius and mass. With the rescaled asteroseismic values, ${\nu }_{\max }^{{\prime} }$ and ${\rm{\Delta }}\nu ^{\prime}$, in hand, we can construct the radius and mass coefficients, ${\kappa }_{R}^{{\prime} }$ and ${\kappa }_{M}^{{\prime} }$, for each star and each pipeline, which correspond to the asteroseismic component of the scaling relations for stellar mass and radius in solar units. We show the pairwise comparisons between pipeline values of ${\kappa }_{R}^{{\prime} }$ and ${\kappa }_{M}^{{\prime} }$ for all stars in the DR2 sample in Figures 10 and 11. As with the pairwise ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ comparisons (Figures 3 and 4), we see systematic differences between pipeline ${\kappa }_{R}^{{\prime} }$ and ${\kappa }_{M}^{{\prime} }$ values, which we quantify in Section 3.4. We also construct the average for each star, $\langle {\kappa }_{R}^{{\prime} }\rangle$ and $\langle {\kappa }_{M}^{{\prime} }\rangle$, based on $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$. Uncertainties on $\langle {\kappa }_{R}^{{\prime} }\rangle$ and $\langle {\kappa }_{M}^{{\prime} }\rangle$ are calculated according to standard propagation of error, using ${\sigma }_{\langle {\nu }_{\max }^{{\prime} }\rangle }$ and ${\sigma }_{\langle {\rm{\Delta }}\nu ^{\prime} \rangle }$. All of these coefficients are reported in Table 6.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Table 6. Radius and Mass Coefficients

EPIC	$\langle {\kappa }_{R}^{{\prime} }\rangle$	${\sigma }_{\langle {\kappa }_{R}^{{\prime} }\rangle }$	$\langle {\kappa }_{M}^{{\prime} }\rangle$	${\sigma }_{\langle {\kappa }_{M}^{{\prime} }\rangle }$	${\kappa }_{R,{\rm{A}}2{\rm{Z}}}^{{\prime} }$	${\sigma }_{\kappa R^{\prime} ,{\rm{A}}2{\rm{Z}}}$	${\kappa }_{M,{\rm{A}}2{\rm{Z}}}^{{\prime} }$	${\sigma }_{\kappa M^{\prime} ,{\rm{A}}2{\rm{Z}}}$	${\kappa }_{R,\mathrm{BAM}}^{{\prime} }$	${\sigma }_{\kappa R^{\prime} ,\mathrm{BAM}}$	${\kappa }_{M,\mathrm{BAM}}^{{\prime} }$	${\sigma }_{\kappa M^{\prime} ,\mathrm{BAM}}$	${\kappa }_{R,\mathrm{BHM}}^{{\prime} }$
210306475	12.976	0.906	1.559	0.265	12.209	0.964	1.381	0.327	⋯	⋯	⋯	⋯	12.556
210307958	10.987	0.767	1.137	0.179	9.900	⋯	0.889	⋯	12.072	0.834	1.430	0.212	⋯
210314854	10.555	0.446	1.119	0.124	9.909	0.898	0.925	0.222	10.668	0.492	1.174	0.124	10.668
210315825	9.521	0.160	1.751	0.066	9.582	0.773	1.776	0.375	9.517	0.291	1.737	0.139	9.354
210318976	11.953	0.677	1.137	0.136	12.862	1.022	1.331	0.309	⋯	⋯	⋯	⋯	⋯

${\sigma }_{\kappa R^{\prime} ,\mathrm{BHM}}$	${\kappa }_{M,\mathrm{BHM}}^{{\prime} }$	${\sigma }_{\kappa M^{\prime} ,\mathrm{BHM}}$	${\kappa }_{R,\mathrm{CAN}}^{{\prime} }$	${\sigma }_{\kappa R^{\prime} ,\mathrm{CAN}}$	${\kappa }_{M,\mathrm{CAN}}^{{\prime} }$	${\sigma }_{\kappa M^{\prime} ,\mathrm{CAN}}$	${\kappa }_{R,\mathrm{COR}}^{{\prime} }$	${\sigma }_{\kappa R^{\prime} ,\mathrm{COR}}$	${\kappa }_{M,\mathrm{COR}}^{{\prime} }$	${\sigma }_{\kappa M^{\prime} ,\mathrm{COR}}$	${\kappa }_{R,\mathrm{SYD}}^{{\prime} }$	${\sigma }_{\kappa R^{\prime} ,\mathrm{SYD}}$	${\kappa }_{M,\mathrm{SYD}}^{{\prime} }$	${\sigma }_{\kappa M^{\prime} ,\mathrm{SYD}}$
1.304	1.372	0.327	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	13.793	1.360	1.807	0.367
⋯	⋯	⋯	10.390	0.598	1.037	0.145	10.883	0.560	1.088	0.133	11.509	1.234	1.248	0.274
1.033	1.148	0.237	10.823	0.603	1.177	0.183	10.277	0.478	1.083	0.122	10.961	1.309	1.221	0.302
0.612	1.698	0.237	9.644	0.337	1.814	0.154	9.299	0.331	1.652	0.141	9.740	0.275	1.839	0.134
⋯	⋯	⋯	11.166	0.804	0.969	0.179	11.797	0.620	1.106	0.138	⋯	⋯	⋯	⋯

Notes. Here, $\langle {\kappa }_{R}^{{\prime} }\rangle$ and $\langle {\kappa }_{M}^{{\prime} }\rangle$, as well as their uncertainties, are computed based on $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ and $\langle {\nu }_{\max }^{{\prime} }\rangle$, according to Equations (4) and (6), and represent pipeline-averaged radius and mass coefficients. Pipeline-specific radius and mass coefficients, ${\kappa }_{R}^{{\prime} }$ and ${\kappa }_{M}^{{\prime} }$, are computed with pipeline-specific asteroseismic parameters, ${\rm{\Delta }}\nu ^{\prime}$ and ${\nu }_{\max }^{{\prime} }$. See Section 3.2.4 for details.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

Download table as:  DataTypeset image

We show in Figures 20–21 the distribution of the fractional uncertainties, ${\sigma }_{\langle {\nu }_{\max }^{{\prime} }\rangle }/\langle {\nu }_{\max }^{{\prime} }\rangle$ and ${\sigma }_{\langle {\rm{\Delta }}\nu ^{\prime} \rangle }/\langle {\rm{\Delta }}\nu ^{\prime} \rangle$, as a function of evolutionary state (RGB or RGB/AGB, RC). The curves overplotted on the distributions represent models of the uncertainty distributions assuming Gaussian statistics. Under the assumption that all stars in K2 have the same, true fractional uncertainty, σ, these distributions would be described by generalized gamma distributions with probability density function $f(x,a,d,p)=\tfrac{\left(p/{a}^{d}\right){x}^{d-1}{e}^{-{\left(x/a\right)}^{p}}}{{\rm{\Gamma }}(d/p)}$, where Γ(z) denotes the gamma function, p = 2, d = dof − 1, and $a=\sqrt{2{\sigma }^{2}/\left(\mathrm{dof}-1\right)}$, where dof (degrees of freedom) corresponds to the number of pipelines contributing results for a star. We consider two models: one with σ fixed to be the observed median fractional uncertainty for each dof, and one with two generalized gamma distributions for which both σ and dof are allowed to vary. The gray (black) curves in Figures 20–21 are weighted sums of the best-fitting single-component (two-component) models across all dof (see the Appendix for details). It is clear that the observed distributions of ${\sigma }_{\langle {\nu }_{\max }^{{\prime} }\rangle }/\langle {\nu }_{\max }^{{\prime} }\rangle$ and ${\sigma }_{\langle {\rm{\Delta }}\nu ^{\prime} \rangle }/\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ are not perfectly described by a generalized gamma distribution with a unique σ, as they would be if all stars had the same fractional uncertainty in $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$. Nevertheless, allowing σ to be a function of the number of reporting pipelines, dof, appears to be a surprisingly good approximation. This indicates that (1) the fractional uncertainties for all stars are not a strong function of ${\nu }_{\max }$ or ${\rm{\Delta }}\nu$ (but rather have a fractional uncertainty that varies modestly according to the number of reporting pipelines) and (2) our uncertainty estimates are distributed like they should be according to χ statistics. As we note in the Appendix, the fractional uncertainties vary mostly according to the evolutionary state, with RC parameters being less precisely measured, and it happens that the typical uncertainties are the same for ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ for RGB stars. Our typical uncertainties for $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ are listed in Table 7, along with the corresponding median fractional uncertainties from APOKASC-2 (Pinsonneault et al. 2018) ("APOKASC-2" in the table). We also include a comparison to the median fractional uncertainties from the analysis of Yu et al. (2018) ("Y18" in the table). The latter analysis uses only the SYD pipeline, as opposed to APOKASC-2, which reports parameters averaged across five different asteroseismic pipelines. The methodology in this work is much the same as that of Pinsonneault et al. (2018), but we include a comparison to Yu et al. (2018) to give an indication of the variation in uncertainties resulting from an aggregated pipeline versus individual pipeline approach. The most significant difference between the uncertainties of the two Kepler analyses is in RC ${\nu }_{\max }$, for which uncertainties from Yu et al. (2018) are larger than those from APOKASC-2; the larger ${\nu }_{\max }$ uncertainties from Yu et al. (2018) map into correspondingly larger RC κ_R and κ_M uncertainties. Comparing our results to those of APOKASC-2, we see that the uncertainties in ${\nu }_{\max }$ in K2 are up to a factor of two larger than in Kepler, and the uncertainties in ${\rm{\Delta }}\nu$ are larger by up to a factor of four for RGB stars. These differences between Kepler and K2 come from differences in photometric precision and the differences in dwell time, and will be further explored in the next K2 GAP data release.

Table 7. Median Fractional Uncertainties of Kepler and K2 Asteroseismic Quantities (in Percent)

		RGB or	RGB/AGB		RC
	APOKASC-2	Y18	K2	APOKASC-2	Y18	K2
${\sigma }_{{\nu }_{\max }}$	0.9	1.0	1.7	1.3	2.1	2.4
σΔν	0.4	0.3	1.7	1.1	1.1	2.3
${\sigma }_{{\kappa }_{R}}$	1.3	1.1	3.3	2.7	3.3	5.0
${\sigma }_{{\kappa }_{M}}$	3.4	3.1	7.7	6.2	8.4	10.5

Notes. "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).

Download table as:  ASCIITypeset image

The analogous uncertainty distributions for $\langle {\kappa }_{R}^{{\prime} }\rangle$ and $\langle {\kappa }_{M}^{{\prime} }\rangle$ are shown in Figures 22–23. Note that the number of stars plotted, N, is not 4395 (the total number of stars we provide with $\langle {\kappa }_{R}^{{\prime} }\rangle$ and $\langle {\kappa }_{M}^{{\prime} }\rangle$). This is because, in this treatment, we require the number of pipelines reporting values for ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ be the same in order to fulfill the generalized gamma distribution requirements; most stars do not have the same number of ${\nu }_{\max }$ measurements as ${\rm{\Delta }}\nu$: there are 1030 such RGB stars and 257 such RC stars. Considering the excellent match of the fitted generalized gamma distributions (black), we adopt the fitted σ as statistical uncertainties for $\langle {\kappa }_{R}^{{\prime} }\rangle$ and $\langle {\kappa }_{M}^{{\prime} }\rangle$ as listed in Table 7. We also list the corresponding median fractional uncertainties in Kepler, computed using the evolutionary states and ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ values from Pinsonneault et al. (2018) ("APOKASC-2") or those from Yu et al. (2018) ("Y18"). We find that the K2 radius (mass) uncertainties are larger by up to a factor of three (two) compared to that of Kepler for RGB stars from APOKASC-2. The uncertainties in RC stellar parameters are more comparable between the two data sets, with the increase in uncertainty from Kepler to K2 not being larger than a factor of two. The uncertainties are more comparable between Yu et al. (2018) results and K2 because of the larger uncertainty in RC ${\nu }_{\max }$ from Yu et al. (2018) compared to APOKASC-2.

We now turn to systematic uncertainties in $\langle {\nu }_{\max }^{{\prime} }\rangle$, $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$, $\langle {\kappa }_{R}^{{\prime} }\rangle$, and $\langle {\kappa }_{M}^{{\prime} }\rangle$ that take the form of ${\nu }_{\max }$- and ${\rm{\Delta }}\nu$-dependent offsets among the pipelines. By definition, the rescaling process described in Section 3.2.1 removes offsets among the pipelines by averaging over all ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$. However, there are ${\nu }_{\max }$- and ${\rm{\Delta }}\nu$-dependent trends seen in the fractional differences in $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ shown in Figures 6–9. Our approach to account for these systematics is to adopt the largest excursion of any pipeline from $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ for each bin plotted in the bottom panel of Figures 6–9, add to that the uncertainty on the median, and adopt the result as 2σ systematic uncertainties. The resulting (1σ) uncertainties are listed in Table 8. For most stars, the typical systematic uncertainty is less than the statistical uncertainty: for a typical RGB star with $({\nu }_{\max },{\rm{\Delta }}\nu )\sim (75,7.5\,\mu \mathrm{Hz})$ or RC star with $({\nu }_{\max },{\rm{\Delta }}\nu )\sim (30,4.0\,\mu \mathrm{Hz})$, the systematic uncertainties are similar, at ∼0.6% in $\langle {\nu }_{\max }^{{\prime} }\rangle$ and ∼0.3% in $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$. We adopt these numbers as typical systematic uncertainties for $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$, though they are clearly a function of ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$. Generally, though, the uncertainty is larger at smaller ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$. This is a result both of having intrinsically fewer stars in this regime (as they are more evolved and therefore shorter-lived) as well as there being difficulties in measuring low-frequency ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$ due to the K2 frequency resolution. We take the systematic uncertainties in $\langle {\kappa }_{R}^{{\prime} }\rangle$ and $\langle {\kappa }_{M}^{{\prime} }\rangle$ to be 1% and 2%, from propagation of the systematic uncertainties in $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$. As with $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$, in detail, these systematic uncertainties are a function of $\langle {\kappa }_{R}^{{\prime} }\rangle$ and $\langle {\kappa }_{M}^{{\prime} }\rangle$ (see Figures 10 and 11).

Table 8. Systematic Uncertainties of $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$, as a Function of $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$

$\langle {\nu }_{\max }^{{\prime} }{\rangle }_{\mathrm{RGB}}$	${\sigma }_{\langle {\nu }_{\max }^{{\prime} }\rangle ,\mathrm{RGB}}$	$\langle {\nu }_{\max }^{{\prime} }{\rangle }_{\mathrm{RC}}$	${\sigma }_{\langle {\nu }_{\max }^{{\prime} }\rangle ,\mathrm{RC}}$	$\langle {\rm{\Delta }}\nu ^{\prime} {\rangle }_{\mathrm{RGB}}$	${\sigma }_{\langle {\rm{\Delta }}\nu ^{\prime} \rangle ,\mathrm{RGB}}$	$\langle {\rm{\Delta }}\nu ^{\prime} {\rangle }_{\mathrm{RC}}$	${\sigma }_{\langle {\rm{\Delta }}\nu ^{\prime} \rangle ,\mathrm{RC}}$
($\mu \mathrm{Hz}$)	(%)	($\mu \mathrm{Hz}$)	(%)	($\mu \mathrm{Hz}$)	(%)	($\mu \mathrm{Hz}$)	(%)
12	0.67	23	1.2	1.7	1.3	3.4	0.85
17	0.91	28	0.83	2.3	1.0	3.7	0.52
23	0.64	31	0.24	2.9	0.68	4.1	0.24
30	0.40	38	0.68	3.8	0.31	4.5	0.47
42	0.29	43	0.74	4.9	0.28	5.1	0.66
56	0.55	53	1.1	5.8	0.21	5.7	0.75
76	0.68	62	0.69	7.1	0.24	6.3	0.59
110	0.73	73	0.92	9.3	0.40	7.0	0.68
140	0.70	84	1.2	11	0.43	7.7	0.92
190	0.50	93	1.4	15	0.81	8.7	0.43

Notes. Systematic uncertainties of $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$, listed as percentages. The binned medians of the fractional difference between an individual pipeline's asteroseismic values and the mean values, $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$, shown in the bottom panels of Figures 6–9 are taken to be indications of systematic uncertainty in $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$; see Section 3.4 for details.

Download table as:  ASCIITypeset image

We note that the ${\rm{\Delta }}\nu$ correction applied to $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ and therefore $\langle {\kappa }_{R}^{{\prime} }\rangle$ and $\langle {\kappa }_{M}^{{\prime} }\rangle$, X_Sharma, is computed using the EPIC temperature and metallicity scale. We acknowledge that the user may wish to use their own temperatures, and therefore we caution that using a different temperature scale will introduce systematics. For example, using a temperature scale 100 K hotter (cooler) than the EPIC temperature will make radii 1% lower (higher) if X_Sharma is not also recomputed with the user's adopted temperatures. In order to give the user as much convenience as possible, we provide EPIC temperatures in Table 4, should the user wish to compute consistent radii/masses; we also include the EPIC metallicities used for computing X_Sharma. In the event the user wishes to use a different temperature scale and does not wish to sustain additional ∼1% systematic uncertainties, we encourage recomputing X_Sharma with the user's own temperatures and/or metallicities using asfgrid (Sharma & Stello 2016; Sharma et al. 2016), which is available at http://www.physics.usyd.edu.au/k2gap/Asfgrid/.

We start by comparing properties of the K2 GAP DR2 sample to those of a Galaxia simulation (Sharma et al. 2011), with corrections made to the simulated metallicity scale described in Sharma et al. (2019). Each campaign has been modeled separately because they each probe different regions of the Galaxy. To make the simulated populations comparable to the data, we select only the simulated stars that would be seismically detected. We defined the detectable sample to be stars with $3\,\mu \mathrm{Hz}\lt {\nu }_{\max }\lt 280\,\mu \mathrm{Hz}$ and signal-to-noise ratio yielding a probability of detection greater than 95% (calculated according to the procedure used in Chaplin et al. (2011)). We impose the same selection of the simulated and observed stars (see Section 2.1 and also Sharma et al. (2019)), using a synthetic V-band magnitude that depends on J − K_s color according to $V={K}_{{\rm{s}}}+2.0((J-{K}_{{\rm{s}}})+0.14)\,+0.382{e}^{2(J-{K}_{{\rm{s}}}-0.2)}$ (Sharma et al. 2018).

Galaxia models of the magnitude–${\nu }_{\max }$ distributions show good agreement with the observations, as shown in Figures 12–14. The most obvious feature in these plots is the RC, which, because red giants spend a relatively long amount of time in this phase, results in a "clump" of stars at ${\nu }_{\max }\sim 30\,\mu \mathrm{Hz}$. Lower-gravity (lower-${\nu }_{\max }$) red giants oscillate with larger power than higher-gravity red giants, and so at a fixed magnitude, it is easier to measure oscillations in lower-gravity red giants. The amplitude of oscillations is primarily a function of surface gravity (Kallinger et al. 2014), and so the diagonal cutoff in the top right corner of both Galaxia predictions and (for the most part) observations is a result of the signal-to-noise ratio from the surface gravity–dependent oscillation amplitude compared to the magnitude-dependent white noise. To demonstrate this ${\nu }_{\max }$-dependent white noise limit, we assume a detectability threshold of ${\nu }_{\max ,\mathrm{detect}}\lt 5\times {10}^{4}{1.6}^{-H}$ in Figures 12–14 (dashed lines). This detectability threshold describes the observed data well, and scales like a flux-dependent white noise would. In Kp-band space, this threshold would mean a detectability limit of Kp ≈ 15, fainter than which the white noise is too large to detect high-${\nu }_{\max }$ oscillators. Compared to the detectability threshold of C1 described in Stello et al. (2017), the dependence on magnitude is less steep, and likely reflects the improved noise qualities following improved pointing control starting with C3. The reason the faint and bright limits do not form straight, vertical trends in Figures 12–14, which show the H band, is because we selected stars in the V band (see Section 2.1). For convenience, we also show the detection distributions as a function of only H-band magnitude in Figure 15.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We condense the comparison between observed and simulated asteroseismic values to just the ${\nu }_{\max }$ dimension for C4 and C7 in Figures 16 and 17. The agreement is generally good. There are two main discrepancies, however. First, the number of predicted oscillators in C4 does not agree with the observations (Figure 16(a)). However, plotting the normalized distribution such that each bin is divided by the bin size (representing probability density) results in agreement, except for the low-${\nu }_{\max }$ regime (Figure 16(b)). We discuss the discrepancy in Figure 16(a) below. Second, there are fewer observed low-${\nu }_{\max }$ stars (${\nu }_{\max }\lesssim 10\mbox{--}20\,\mu \mathrm{Hz}$) than predicted in both C4 and C7. Although Stello et al. (2017) found this same bias in K2 GAP DR1, we attempt to verify that it is not due to a bias in the Galaxia models themselves by manually inspecting all K2 GAP spectra in C6 for evidence of solar-like oscillations. Two experts did the exercise separately, classifying the K2 spectra according to whether or not they had a detectable ${\nu }_{\max }$ (yes/maybe/no), additionally assigning a visual estimate of ${\nu }_{\max }$. Objects that showed evidence of solar-like oscillations below the effective high-pass cutoff of $3\,\mu \mathrm{Hz}$ were classified as "no." Note that this exercise did not require the detection of ${\rm{\Delta }}\nu$ for classification as a solar-like oscillator, which is consistent with our definition of a pipeline detection as a valid ${\nu }_{\max }$ measurement (see Section 3.1). The results from each person were nearly identical, with any contested classifications discussed individually and a final consensus classification agreed upon.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

As evident in Figure 18, the predicted ${\nu }_{\max }$ distribution from Galaxia is in good agreement with the visually confirmed distribution (classifications of "yes" are plotted, but those of "maybe" are not), though they are formally inconsistent with being drawn from the same distribution, according to a Kolmogorov–Smirnov test. By going through all of the observed targets by eye and not just ones that were returned by pipelines as being red giants, we are able to be more confident that the Galaxia predictions are robust and not subject to obvious biases. The distributions of ${\nu }_{\max }$ returned by individual pipelines all fall short of being formally consistent with either the visually confirmed distribution or the Galaxia distribution, though some of the pipeline ${\nu }_{\max }$ distributions qualitatively show good agreement with the predicted and visually confirmed distributions.

Zoom In Zoom Out Reset image size

As noted above, there is nonetheless a bias against detecting stars with ${\nu }_{\max }$below $\sim 10\mbox{--}20\,\mu \mathrm{Hz}$. This is true for all three of the campaigns, which indicates that this detection bias was not solely a function of the particular DR1 sample, which were all in C1. With the data we have in hand, we cannot definitively say what causes this bias. At these low frequencies, there are relatively few modes observable. This may hinder detecting these oscillations for any pipeline that relies on finding ${\rm{\Delta }}\nu$ as part of its ${\nu }_{\max }$ detection step, and can also hamper fits to the power excess using a Gaussian, since a Gaussian does not perform well at describing a few discrete modes. It is also possible that low-${\nu }_{\max }$ oscillators are harder to recover at lower frequencies because of the relatively short K2 dwell time, which leads to a smaller ratio of the frequency resolution to the mode width; the result is that the spectra of these stars can sometimes be hard to distinguish from a pure granulation background. The degree to which these oscillators are detected or not is highly pipeline-dependent, as one can see from Figures 16–18.

Keeping this low-${\nu }_{\max }$ detection bias in mind, we can go on to test the detection rate for ${\nu }_{\max }\gt 20\,\mu \mathrm{Hz}$, using Galaxia and visual inspection as ground truths. In C7, there are 1740 such stars that have asteroseismic values from at least one pipeline. This number is consistent with the 1758 expected stars from the Galaxia simulation for C7. The number of stars (2312) recovered by at least one pipeline in C6 is between the number found by visual inspection (2214) and that predicted by Galaxia (2511). In C4, however, there are significantly fewer stars observed by at least one pipeline than predicted by Galaxia (2177 versus 2670). These "missing" stars are at magnitudes Kp < 13, and therefore should yield observable asteroseismic parameters, given our empirical detectability threshold. The Galaxia model seems to underpredict reddening in this campaign, which may explain this discrepancy. Indeed, the predicted fraction of dwarfs (and therefore the fraction of stars that would not be detected as oscillators in long-cadence K2 data) depends on the assumed reddening. For K2 GAP, the selection "draws a line" in J − K_s color space to separate the sample of predominantly blue dwarfs from the sample containing the red giants (and some red dwarfs), which are the targets with J − K_s > 0.5. The same cut is applied to the synthetic stellar population in the Galaxia simulation, using an assumed reddening. An underestimated reddening in Galaxia means that fewer of the blue dwarfs are reddened enough to fall on the giant side of the J − K_s > 0.5 dwarf/giant dividing line, increasing the number of predicted oscillators compared to reality.

The derived $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ and $\langle {\nu }_{\max }^{{\prime} }\rangle$ have not necessarily been placed on an absolute scale. While we scaled the asteroseismic values to a common mean scale, we were free to impose solar reference values for $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ to be ${\nu }_{\max ,\odot }=3076\,\mu \mathrm{Hz}$ and ${\rm{\Delta }}{\nu }_{\odot }=135.146\,\mu \mathrm{Hz}$, which are relatively close to the average pipeline-specific solar reference values, and which are the same as determined by the absolute asteroseismology rescaling done in the APOKASC-2 analysis. Were we working with Kepler data and using the same pipelines as in the APOKASC-2 analysis, this choice of solar reference values would be valid and would put the asteroseismic radii on a scale that is consistent with open cluster masses and Gaia radii (Pinsonneault et al. 2018; Zinn et al. 2019b). There are at least two reasons why choosing our solar reference values as ${\nu }_{\max ,\odot }=3076\,\mu \mathrm{Hz}$ and ${\rm{\Delta }}{\nu }_{\odot }=135.146\,\mu \mathrm{Hz}$ may not result in $\langle {\kappa }_{R}^{{\prime} }\rangle$ and $\langle {\kappa }_{M}^{{\prime} }\rangle$ being on an absolute scale. First, although our rescaling procedure to derive $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ and $\langle {\nu }_{\max }^{{\prime} }\rangle$ is nearly the same as in Pinsonneault et al. (2018), we have added BAM as one of the pipelines that contributes to the rescaling procedure: APOKASC-2 used results from A2Z, CAN, COR, SYD, and OCT, and in this work, we have used results from A2Z, CAN, COR, SYD, BHM (based on OCT), and BAM (see Section 3.1 for summaries of the pipeline methodologies). The addition of BAM in this work to the pipelines used for aggregating asteroseismic results may result in a slightly different mean scale for $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$. This is because the APOAKSC-2 solar reference values were chosen such that the aggregated stellar masses agreed with open cluster masses—using a different set of pipelines to average over may have required a different set of solar reference values to achieve agreement with open cluster masses. We see this in the differences between the rescaling values from our analysis and from that of Pinsonneault et al. (2018), shown in Table 5. Second, there is evidence to suggest that there are systematic biases in asteroseismic parameters based on the dwell time of the data (J. C. Zinn et al. 2020, in preparation), which could be suggestive of a need to modify the ${\nu }_{\max ,\odot }$ and/or ${\rm{\Delta }}{\nu }_{\odot }$ for K2 data compared to Kepler data. We now test this choice of zero points by comparing the derived radii to radii using parallaxes from Gaia Data Release 2 (Gaia Collaboration et al. 2018; Lindegren et al. 2018).

We populated the overlap sample of stars in K2 GAP DR2 with both $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ and $\langle {\nu }_{\max }^{{\prime} }\rangle$ and Gaia DR2 by matching on 2MASS ID using the Gaia Archive. ²⁴ We also required APOGEE metallicities and temperatures from DR16 (Ahumada et al. 2020). Because of a known, position-, magnitude-, and color-dependent zero point in the Gaia parallaxes (e.g., Lindegren et al. 2018; Zinn et al. 2019a), we did not work directly with the Gaia DR2 parallaxes. Instead, we followed the methodology of Schönrich et al. (2019) to derive distance estimates for stars in our K2 GAP sample. We did this separately for RC and RGB stars, with the understanding that RGB and RC populations will have different selection functions, which is an important consideration in the Bayesian distance estimates in the Schönrich et al. (2019) framework (see also Schönrich & Aumer 2017).

For the purposes of establishing a Gaia calibration of ${\nu }_{\max ,\odot }$ and ${\rm{\Delta }}{\nu }_{\odot }$, we used only stars with more than two pipelines returning results for ${\rm{\Delta }}\nu$, and only considered stars with π > 0.4 and Gaia G-band <13 mag in order to ensure that the results are less sensitive to any residual Gaia parallax zero points that may not be accounted for in the Schönrich et al. (2019) method; [Fe/H] > −1 to ensure that there are no metallicity-dependent asteroseismic radius systematics (see Zinn et al. 2019b); and $R\lt 30{R}_{\odot }$ to ensure there are no radius-dependent asteroseismic radius systematics (see Zinn et al. 2019b). For this sample, which has spectroscopic information, we recomputed ${\rm{\Delta }}\nu$ correction factors using metallicities that are adjusted to account for nonsolar alpha abundances according to the Salaris et al. (1993) prescription: ${[\mathrm{Fe}/{\rm{H}}]}^{{\prime} }=[\mathrm{Fe}/{\rm{H}}]+{\mathrm{log}}_{10}(0.638\times {10}^{[\alpha /{\rm{M}}]}+0.362)$. We computed a Gaia radius for the 261 resulting stars, following a Monte Carlo procedure of the sort detailed in Zinn et al. (2017). The method uses the Stefan-Boltzmann law to translate the flux, temperature, and distance of a star into a radius. To do so, we computed bolometric fluxes with a K_s-band bolometric correction (González Hernández & Bonifacio 2009), APOGEE effective temperatures, and metallicities, combined with asteroseismic surface gravities from $\langle {\nu }_{\max }^{{\prime} }\rangle$. Extinctions were computed using the three-dimensional dust map of Green et al. (2015), as implemented in mwdust ²⁵ (Bovy et al. 2016). Five stars with asteroseismic and Gaia radii discrepant at more than the 3σ level were removed from subsequent analysis.

We compare the Gaia radius scale to our K2 asteroseismic radius scale in Figure 19. The top panel shows the points colored by evolutionary state (RGB in red and RC in blue), and the bottom panel shows the fractional agreement of the two radius scales. Figure 19 indicates that the radius scale of $\langle {\kappa }_{R}^{{\prime} }\rangle$ is consistent with the Gaia radius scale for both RC and RGB stars to within ∼3%. We find that the RGB stars are in more disagreement than the RC stars: while the median agreement is 3.0% ± 0.4% for RGB stars, it is 1.6% ± 0.8% for RC stars. This median statistic is computed as the median radius ratio for all RGB or RC stars, with the uncertainty in the median taken to be ${\sigma }_{\mathrm{med}}=\sqrt{\tfrac{\pi }{2}\sum {\sigma }_{R}^{2}/{N}^{2}}$, where ${\sigma }_{R}=\tfrac{{R}_{\mathrm{seis}}}{{R}_{\mathrm{Gaia}}}\sqrt{{\left(\tfrac{{\sigma }_{R,\mathrm{Gaia}}}{{R}_{\mathrm{Gaia}}}\right)}^{2}+{\left(\tfrac{{\sigma }_{R,\mathrm{seis}}}{{R}_{\mathrm{seis}}}\right)}^{2}}$. Using the EPIC extinctions instead of those from Green et al. (2015) leads to insignificant variations in the radius agreement. However, the agreement is discrepant at the ∼3σ level between stars in C4 versus those in C7. This could be an indication of Gaia parallax zero-point issues, given that we do not expect such variations in the asteroseismic data by campaign. We therefore also consider the Gaia zero point from Khan et al. (2019), who compared asteroseismic distances to Gaia distances in K2 C3 and C6. For this exercise, we restricted our sample to the stars in C6, and adopted their derived zero point of −17 μas. The result is that the asteroseismic radii are consistent with Gaia radii to within ≈1% (RGB) and ≈5% (RC).

Zoom In Zoom Out Reset image size

Because the median agreement between the radius scales could be biased by underlying skewed distributions of the individual radii, we finally evaluate the agreement using a weighted mean: $\langle {{R}_{\mathrm{seis}}/R}_{\mathrm{Gaia}}\rangle =\tfrac{\sum \tfrac{{R}_{\mathrm{seis}}}{{R}_{\mathrm{Gaia}}}/{\sigma }_{R}^{2}}{\sum 1/{\sigma }_{R}^{2}}$, where ${\sigma }_{R}\,=\tfrac{{R}_{\mathrm{seis}}}{{R}_{\mathrm{Gaia}}}\sqrt{{\left(\tfrac{{\sigma }_{R,\mathrm{Gaia}}}{{R}_{\mathrm{Gaia}}}\right)}^{2}+{\left(\tfrac{{\sigma }_{R,\mathrm{seis}}}{{R}_{\mathrm{seis}}}\right)}^{2}}$. We calculate this for those stars that have fractional parallax uncertainties less than 10%, in order to mitigate potential biases due to parallax systematics. According to this metric, the agreement becomes 2.2% ±0.3% for RGB stars and 2.0% ± 0.6% for RC stars.

Due to the variation in this agreement based on the tests described above, we opt not to rescale our $\langle {\nu }_{\max }^{{\prime} }\rangle$ or $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ values, and instead allow for a systematic zero-point uncertainty in our derived $\langle {\kappa }_{R}^{{\prime} }\rangle$ values. Acknowledging these uncertainties in the K2–Gaia agreement, we take the weighted mean estimate of the radius scales using the Schönrich et al. (2019) Gaia distances. Our asteroseismic radius coefficients could therefore be overestimated by up to 2.2% ± 0.3% for RGB stars and up to 2.0% ± 0.6% for RC stars. The uncertainty on this agreement is solely due to the standard uncertainty on the mean, and does not account for intrinsic scatter or trends in the radius agreement. Indeed, this should be thought of as being in addition to the systematic uncertainty from pipeline-to-pipeline variation in $\langle {\kappa }_{R}^{{\prime} }\rangle$ as a function of ${\nu }_{\max }$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ discussed in Section 3.4. We also note that this agreement does not account for systematic variation in the radius ratio due to choice of temperature, bolometric correction, or Gaia zero point, which may contribute to a systematic uncertainty of about ±2% (Zinn et al. 2019b).

Broadly speaking, the excellent level of agreement between asteroseismology and Gaia corroborates findings of the accuracy of the scaling relations in this regime from previous work based on asteroseismology–Gaia comparisons (Huber et al. 2017; Zinn et al. 2019b). In detail, there do appear to be trends with radius evident in Figure 19: the RGB radii appear to inflate compared to Gaia radii at around $R\sim 7.5{R}_{\odot }$ (red error bars), while the red clump radii (blue error bars) seem to deflate compared to the Gaia radii with increasing radius at all radii. At least for the RC stars, systematics in the tracks used to generate the ${\rm{\Delta }}\nu$ corrections could be to blame, particularly given the disagreement among RC models from the literature (An et al. 2019). Indeed, it appears that the RC radius trend is mostly a trend in ${\rm{\Delta }}\nu$, with some metallicity dependence as well. At the population level, the RC radii have been found to agree within 5% with Gaia radii (Hall et al. 2019). However, to our knowledge, the scaling relations for RC stars have not been tested as a function of radius as we do here. We note also that a ≈1% relative difference between the zero point for RGB versus RC stars is not ruled out by Pinsonneault et al. (2018) (see also Khan et al. 2019).

Regarding the mean agreement of RGB and RC stars, the most likely culprit for the (small) radius disagreement is a systematic in the solar reference value combination ${\nu }_{\max ,\odot }/{\rm{\Delta }}{\nu }_{\odot }^{2}$. That the absolute K2 asteroseismic radius scale is consistent with the Gaia radius scale to within ∼2% naively implies that our $\langle {\kappa }_{M}^{{\prime} }\rangle$ are within ∼6% of an absolute mass scale, according to standard propagation of error from Equations (4) and (6). However, this is only approximate, because we can test only ${\nu }_{\max ,\odot }/{\rm{\Delta }}{\nu }_{\odot }^{2}$ against Gaia, whereas the mass coefficient goes as ${\nu }_{\max ,\odot }^{3}/{\rm{\Delta }}{\nu }_{\odot }^{4}$.

In summary, because we are delivering asteroseismic values $\langle {\nu }_{\max }^{{\prime} }\rangle$ and $\langle {\rm{\Delta }}\nu ^{\prime} \rangle$ that are averaged values from several pipelines, we needed to evaluate to what extent the resulting asteroseismic scale is on an absolute scale. We did this by comparing to Gaia radii, and we found a systematic offset between the K2 GAP DR2 asteroseismic and Gaia radius scales of about 2%. This translates roughly to a 20% systematic uncertainty in age. Given the typical uncertainty in mass listed in Table 7, ages based on our RGB asteroseismic masses would be expected to have statistical uncertainties of ≈20%, with potential scale shifts by ≈20% due to the level of systematics we identify in this section. This anticipated 20% statistical age uncertainty makes the data particularly interesting for potentially identifying the history of minor mergers in the Galaxy based on their impact on the age–velocity dispersion relation (Martig et al. 2014). For this and other Galactic archeology applications (e.g., age–abundance patterns), the 20% systematic uncertainty should not be significant, given that the differential age relationship between stellar populations would be preserved. Regarding mass- or magnitude-dependent systematics among RGB asteroseismic parameters, the small inflation of RGB asteroseismic radii at $R\sim 7.5{R}_{\odot }$ seems to map onto a corresponding trend in ${\nu }_{\max }$, and so this may introduce an inflation in the RGB mass scale by perhaps up to 15% for the minority of stars with $50\,\mu \mathrm{Hz}\lesssim \langle {\nu }_{\max }^{{\prime} }\rangle \lesssim 80\,\mu \mathrm{Hz}$. RC stars in our sample, on the other hand, appear to suffer from strong radius-dependent trends that seem to be related to the ${\rm{\Delta }}\nu$ and not ${\nu }_{\max }$ scaling relation: there is a strong trend of RC agreement with ${\rm{\Delta }}\nu$, which suggests that the ${\rm{\Delta }}\nu$ scaling relation for RC stars is not well-calibrated using our ${\rm{\Delta }}\nu$ corrections. Despite the concerning magnitude of the RC systematic, RC ages are not in popular use because of uncertainties in modeling mass loss (Casagrande et al. 2016). We will nonetheless explore the RC systematic further in the next and final K2 GAP data release, as having accurate red clump masses and radii is important for reckoning red clump models with observed red clump properties (e.g., An et al. 2019).