Standing on the Shoulders of Dwarfs: the Kepler Asteroseismic LEGACY Sample. I. Oscillation Mode Parameters

Before the peak-bagging can commence, initial guesses must be defined for the mode-frequencies to include in Equation (3), and the modes must further be identified in terms of their angular degree l. For acoustic modes of high radial order n and low angular degree l the frequencies may be approximated by the asymptotic relation (Tassoul 1980; Scherrer et al. 1983):

$$\begin{eqnarray}&&{\nu }_{{nl}}\approx \left(n+\displaystyle \frac{l}{2}+\epsilon \right){\rm{\Delta }}\nu -l(l+1){D}_{0}.\end{eqnarray} \tag{ 10 }$$

Here ${\rm{\Delta }}\nu$ is the large separation, given by the average frequency spacing between consecutive overtones n for modes of a given l; is a dimensionless offset sensitive to the surface layers (see, e.g., Gough 1986; Pérez Hernández & Christensen-Dalsgaard 1998; Roxburgh 2016); D₀ is sensitive to the sound-speed gradient near the stellar core (Scherrer et al. 1983; Christensen-Dalsgaard 1993). Mode identification was then, by and large, achieved via visual inspection of échelle diagrams (Grec et al. 1983; Bedding 2011). Here, modes of a given l will form vertical ridges for the correct average large separation. The identification of l and radial order n was checked against the relation for as a function of ${T}_{\mathrm{eff}}$ (Figure 3), where can be found from the échelle diagram (Figure 4) by the vertical position of the radial degree (l = 0) ridge (see White et al. 2011a, 2011b).

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For this study, consisting of high S/N oscillation signals, the identification was relatively simple. Initial guesses for mode frequencies were primarily defined by hand from smoothed versions of the power density spectra. These were checked against frequencies returned from applying the pseudo-global fitting method of Fletcher et al. (2009).

The power spectrum was fitted in the range of ${f}_{\min }-5{\rm{\Delta }}\nu$ to ${f}_{\max }+5{\rm{\Delta }}\nu$, where ${f}_{\min }$ and ${f}_{\max }$ denote the minimum and maximum mode frequency included in the peak-bagging. Before the peak-bagging fit, a background-only fit was performed in the range from 5 μHz to the SC Nyquist frequency ${\nu }_{\mathrm{nq}}$ (∼8496 μHz). In this fit, the power from solar-like oscillations was accounted for by a Gaussian envelope centered at ${\nu }_{\max }.$ Using the posterior distributions from the background-only fit as priors in the peak-bagging allowed us to constrain the background in the relatively narrow frequency range included.

For mode frequencies, we adopted 14 μHz wide top-hat priors centered on the initial guesses of the frequencies. Top-hat priors were also adopted for the rotational frequency splitting ${\nu }_{s}$ and inclination i_⋆, with the inclination sampled from the range of −90° to 180°. The reason for the extended range in inclination is that if the solution is close to either ${i}_{\star }=0^\circ$ or ${i}_{\star }=90^\circ$ any sharp truncation from a prior at these values will make it difficult to properly sample these extreme values. The final posterior on the inclination was then obtained from folding the samples onto the range from 0° to 90°.

For the amplitudes and line widths, we adopted a modified Jeffrey's prior, given as (see, e.g., Handberg & Campante 2011)

$$\begin{eqnarray}{ \mathcal F }(\theta )=\left\{\begin{array}{ll}\tfrac{1}{(\theta +{\theta }_{\mathrm{uni}})+\mathrm{ln}[({\theta }_{\mathrm{uni}}+{\theta }_{\max })/{\theta }_{\mathrm{uni}}]}, & 0\leqslant \theta \leqslant {\theta }_{\max }\\ 0, & \mathrm{otherwise}\end{array}\right.\end{eqnarray} \tag{ 11 }$$

which behaves as a uniform prior when $\theta \ll {\theta }_{\mathrm{uni}}$ and a standard scale invariant Jeffrey's prior when $\theta \gg {\theta }_{\mathrm{uni}}$. The maximum of the prior occurs at ${\theta }_{\max }.$

For mode visibilities, we adopted truncated Gaussian functions ${ \mathcal N }({\theta }_{0},\sigma ,{\theta }_{\min },{\theta }_{\max })$ as priors, defined as

$$\begin{eqnarray}{ \mathcal F }(\theta )=\left\{\begin{array}{ll}\tfrac{1}{D\,\sqrt{2\pi }\sigma }\exp \left(\tfrac{-{(\theta -{\theta }_{0})}^{2}}{2{\sigma }^{2}}\right), & {\theta }_{\min }\leqslant \theta \leqslant {\theta }_{\max }\\ 0, & \mathrm{otherwise}\end{array}\right.\end{eqnarray} \tag{ 12 }$$

with D given as:

$$\begin{eqnarray}&&D=\displaystyle \frac{\mathrm{erf}\left(\tfrac{{\theta }_{\max }-{\theta }_{0}}{\sqrt{2}\sigma }\right)-\mathrm{erf}\left(\tfrac{{\theta }_{\min }-{\theta }_{0}}{\sqrt{2}\sigma }\right)}{2}.\end{eqnarray} \tag{ 13 }$$

Here, erf denotes the error function, ${\theta }_{0}$ and σ give the chosen mode value and width of the Gaussian, and ${\theta }_{\min }$ and ${\theta }_{\max }$ give the lower and upper truncations of the Gaussian. We specifically adopted ${ \mathcal N }(1.5,1.5,0,3)$ for l = 1, ${ \mathcal N }(0.5,0.5,0,1)$ for l = 2, and ${ \mathcal N }(0.05,0.05,0,0.5)$ for l = 3. Similarly, we adopted truncated Gaussian priors for the parameters of the background. Here we used as the Gaussian mode value (${\theta }_{0}$) the median of the posteriors from the background-only fit (see Section 3.2.1), and we adopted $\sigma =0.1\,{\theta }_{0}$, ${\theta }_{\min }=0.1\,{\theta }_{0}$, and ${\theta }_{\max }=10\,{\theta }_{0}$.

To ensure that the mode identification did not swap for neighboring l = 0 and 2 modes, which is a risk especially at high frequency where the small frequency separation $\delta {\nu }_{02}={\nu }_{n,0}-{\nu }_{n-\mathrm{1,2}}$ is small compared to the mode line width, we added the prior constraint that on $\delta {\nu }_{02}$ that it must be positive. In principle, $\delta {\nu }_{02}$ could be negative in the event of bumped l = 2 modes; however, because the stars were screened for bumped l = 1 modes and the strength of an avoided l = 2 crossing is expected to be lower than that of a l = 1 mode due to the larger evanescent region (see, e.g., Aerts et al. 2010; Deheuvels & Michel 2011), we do not expect values of $\delta {\nu }_{02}\lt 0$ for these stars.

For each fitted mode, we computed a metric for the quality of the fit in the same manner as detailed in Davies et al. (2016), see also Appourchaux (2004) and Appourchaux et al. (2012b). Briefly, we first ran a fast null hypothesis (H₀) test to identify which modes had unambiguous detections, and for which the probability of detection $p({\mathrm{Det}}_{n,l}| D)$ conditioned on the data D needed to be explicitly determined and evaluated. This was done because the explicit determination of $p({\mathrm{Det}}_{n,l}| D)$ is computationally expensive. In the fast H₀ test it was assessed whether the S/N in the background corrected power spectrum (D) for a given proposed mode, with the power binned across a number of frequencies to account for the spread in power from the mode line width, was consistent with a pure noise spectrum or whether the H₀ hypothesis could be rejected at the $p(D| {H}_{0})=0.001$ level (Appourchaux 2003a, 2004; Lund et al. 2012). When the high S/N modes had been identified in this manner the probability of detection $p({\mathrm{Det}}_{n,l}| D)$ was computed for the remainder low-S/N modes.

In the computation of $p({\mathrm{Det}}_{n,l}| D)$, both the probability of D assuming H₀, $p(D| {H}_{0})$, and the probability of the alternative hypothesis H₁ of a detected mode, $p(D| {H}_{1})$, need to be estimated. The latter was assessed by integrating the probability of measuring the data given a model over a range of mode parameters ${\boldsymbol{\theta }}$—this integration was achieved by marginalizing over $p(D| {H}_{1},{\boldsymbol{\theta }})$, with the parameter space sampled over the posteriors from the peak-bagging using emcee. Specifically, a mixture-model was used in which both $p(D| {H}_{0})$ and $p(D| {H}_{1})$ were optimized simultaneously to give $p(D| {\boldsymbol{\theta }},{p}_{a})$, the probability of observing the data given the model of a given set of modes with parameters ${\boldsymbol{\theta }}$:

$$\begin{eqnarray}&&p(D| {\boldsymbol{\theta }},{p}_{a})=(1-{p}_{a})p(D| {H}_{0})+{p}_{a}p(D| {H}_{1}).\end{eqnarray} \tag{ 14 }$$

Here, the parameter p_a, ranging between 0 and 1, then gives the probability of the detection $p(D| {\mathrm{Det}}_{l})$ of the given set of modes. This probability was kept free in the emcee run, and in the end was assessed from the posterior distribution of p_a (Hogg et al. 2010; Farr et al. 2015). We finally report the Bayes factor K, given as the median of the posterior probability distributions of the natural logarithm of the ratio of $p(D| {\mathrm{Det}}_{l})$ over $p(D| {H}_{0})$, as

$$\begin{eqnarray}&&\mathrm{ln}K=\mathrm{ln}p(D| {\mathrm{Det}}_{l})-\mathrm{ln}p(D| {H}_{0}).\end{eqnarray} \tag{ 15 }$$

The value of $\mathrm{ln}K$ can then be assessed qualitatively on the Kass & Raftery (1995) scale as follows.

$$\begin{eqnarray*}\mathrm{ln}K=\left\{\begin{array}{ll}\lt 0 & \mathrm{favors}\,{H}_{0}\\ 0\,\mathrm{to}\,1 & \mathrm{not}\ \mathrm{worth}\ \mathrm{more}\ \mathrm{than}\ {\rm{a}}\ \mathrm{bare}\ \mathrm{mention}\\ 1\,\mathrm{to}\,3 & \mathrm{positive}\\ 3\,\mathrm{to}\,5 & \mathrm{strong}\\ \gt 5 & \mathrm{very}\ \mathrm{strong}\end{array}\right.\end{eqnarray*}$$

For a detailed account of the quality control we refer to Davies et al. (2016).

A proper understanding of the uncertainties on mode frequencies is important, because they will ultimately limit the precision with which stellar parameters can be estimated from modeling the individual frequencies. We may compare the frequency uncertainties from the peak-bagging with expectations from an analytical maximum likelihood (ML) approach (see, e.g., Libbrecht 1992; Toutain & Appourchaux 1994; Ballot et al. 2008). It should be noted that the ML estimator (if unbiased) reaches the minimum variance bound, in accordance with the Cramér-Rao theorem (Cramér 1946; Rao 1945). Thus one should expect uncertainties at least as large as those from the ML estimator (MLE). The standard way of obtaining uncertainties on an ML estimator is from inverting the negative Hessian matrix, and therefore the standard parameters come from the diagonal elements of the resulting variance-covariance matrix. The Hessian itself is obtained from the matrix of second derivatives of the log-likelihood function with respect to the parameters. Assuming an isolated mode as described by Equation (1) and a likelihood function as given in Equation (9), one may follow Libbrecht (1992) and Toutain & Appourchaux (1994) in defining a theoretical Hessian corresponding to the average of a large number of realizations. From this, the predicted frequency uncertainty for an isolated mode of a given l is given as (Ballot et al. 2008)

$$\begin{eqnarray}&&{\sigma }_{{\nu }_{{nl}}}=\sqrt{\displaystyle \frac{1}{4\pi }\displaystyle \frac{{{\rm{\Gamma }}}_{{nl}}}{{T}_{\mathrm{obs}}}\,{f}_{l}(\beta ,{x}_{s},{i}_{\star })},\end{eqnarray} \tag{ 24 }$$

where $\beta =B/H$ (the signal-to-noise ratio) is the level of the background divided by the mode height; i_⋆ gives the stellar inclination; x_s is the reduced splitting, given by ${x}_{s}=2{\nu }_{s}/{\rm{\Gamma }};$ and ${T}_{\mathrm{obs}}$ is the observing duration. For radial modes (l = 0), the factor f_l only depends on β and is given by (Libbrecht 1992)

$$\begin{eqnarray}&&{f}_{0}(\beta )=\sqrt{\beta +1}{(\sqrt{\beta +1}+\sqrt{\beta })}^{3}.\end{eqnarray} \tag{ 25 }$$

For the general version of ${f}_{l}(\beta ,{x}_{s},{i}_{\star })$, see Ballot et al. (2008).

In Figure 7, we show the individual frequency uncertainties obtained from the peak-bagging as a function of ${\beta }^{-1}$, i.e., the signal-to-noise ratio of the mode. The left panel shows the behavior for the radial (l = 0) modes. We see that the uncertainties for a given value of $\sqrt{{\rm{\Gamma }}/{T}_{\mathrm{obs}}}$ depend on β as expected from Equation (25), and for a given β a clear dependence is seen as a function of $\sqrt{{\rm{\Gamma }}/{T}_{\mathrm{obs}}}$. Given the relatively small spread in ${T}_{\mathrm{obs}}$, the vertical gradient largely depicts the gradient in Γ and, as expected, the largest values of $\sqrt{{\rm{\Gamma }}/{T}_{\mathrm{obs}}}$ are seen toward low ${\beta }^{-1}$ values because of the correlation between Γ and mode heights. Comparing with Equation (24), we find that the median uncertainties from our MCMC peak-bagging are ∼1.2 times larger than those predicted from the ML estimator.

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The right panel of Figure 7 shows the uncertainties for all extracted modes. The uncertainties for each angular degree again follow the expected trend against β. Some extra scatter is expected from i_⋆ and x_s, but overall the l > 0 modes are seen to follow the trend of the l = 0 ones. The kernel density estimates of the uncertainties for a given l, obtained by representing each sample with a Gaussian kernel, show that the uncertainties of dipole (l = 1) modes are overall the lowest, followed by l = 0 and l = 2. This is expected because ${\tilde{V}}_{1}^{2}\gt {\tilde{V}}_{0}^{2}\gt {\tilde{V}}_{2}^{2}$ (Equation (2)), hence l = 1 modes will typically have the highest S/N. The l = 3 are seen not to follow this trend, probably because only the very highest amplitude l = 3 were selected for fitting.

It is reassuring to see that the measured uncertainties follow expectations to this level, also considering that Equation (24) is constructed for an isolated mode without factoring in potential contributions from close-by neighboring modes. We note that the demonstration of the agreement is useful for predicting uncertainty yields for future missions like TESS (Ricker et al. 2014) and PLATO (Rauer et al. 2013), and could potentially be used to adjust uncertainties derived from a much faster MLE fitting (see, e.g., Appourchaux et al. 2012b) in cases where the sheer number of stars investigated would render the relatively slow MCMC approach impractical.

For each star, we computed the average seismic parameters, including ${\nu }_{\max }$ and ${\rm{\Delta }}\nu$, from the mode frequencies (see Table 2). The value of ${\nu }_{\max }$ was obtained by fitting a Gaussian function to the extracted amplitudes as a function of frequency. The value found for the solar ${\nu }_{\max }$ of 3078 ± 13 μHz is in agreement with the value of ${\nu }_{\max ,\odot }=3090\pm 30\,\mu \mathrm{Hz}$ by Huber et al. (2011). We obtained ${\rm{\Delta }}\nu$ and from a fit to the extracted mode frequencies with an extended version of the asymptotic relation, as in Lund et al. (2014a; see also Mosser et al. 2011, 2013). This fit was made in a Bayesian manner using emcee, with a likelihood function assuming Gaussian frequency errors, and with final parameter values and uncertainties given by the posterior medians and HPDs. We initially adopted the following formula:

$$\begin{eqnarray}{\nu }_{{nl}} & \simeq & \left(n+\displaystyle \frac{l}{2}+\epsilon \right){\rm{\Delta }}{\nu }_{0}-l(l+1){D}_{0}\\ & & -l(l+1)\displaystyle \frac{{{dD}}_{0}}{{dn}}(n-{n}_{{\nu }_{\max },l})\\ & & +\displaystyle \frac{d{\rm{\Delta }}\nu /{dn}}{2}{(n-{n}_{{\nu }_{\max },l})}^{2}.\end{eqnarray} \tag{ 26 }$$

Here ${\rm{\Delta }}{\nu }_{0}$ gives the value of the large separation at ${\nu }_{\max }.$ The latter corresponds to the radial order ${n}_{{\nu }_{\max }},$ which may take on a non-integer value. Specifically, ${n}_{{\nu }_{\max },l}$ is obtained for a given l from interpolating the measured mode frequencies (${\nu }_{l}$) against radial order (n_l) to the frequency of ${\nu }_{\max }.$ This description gives a direct estimate of, for instance, the small frequency separations $\delta {\nu }_{01}$, given as the amount by which l = 1 modes are offset from the neighboring l = 0 modes ($\delta {\nu }_{01}=({\nu }_{n,0}\ -{\nu }_{n+\mathrm{1,0}})/2-{\nu }_{n,1}=2{D}_{0}$), and $\delta {\nu }_{02}={\nu }_{n,l=0}-{\nu }_{n-1,l=2}=6{D}_{0}$ (see, e.g., Bedding 2011). The value of $\delta {\nu }_{02}$ is a good probe of the evolutionary state of the star because it is sensitive to the sound-speed gradient of the core, which in turn varies with the composition. The frequency dependence of these frequency separations is captured by the change in D₀, assumed to change linearly with n (or frequency), as found, for instance, for the Sun by Elsworth et al. (1990; see also Anguera Gubau et al. 1992; Toutain & Fröehlich 1992). Lastly, the change in the large separation, assumed quadratic in n, is included and mimics the overall curvature of the ridges in the échelle diagram.

Table 2.  Values from the Fit of Equation (27) to the Mode Frequencies (see Figure 8)

KIC	${\nu }_{\max }$	${\rm{\Delta }}\nu$	$d{\rm{\Delta }}\nu /{dn}$		$\delta {\nu }_{01}$	$d\delta {\nu }_{01}/{dn}$	$\delta {\nu }_{02}$	$d\delta {\nu }_{02}/{dn}$
	(μHz)	(μHz)			(μHz)		(μHz)
1435467	1406.7${}_{-8.4}^{+6.3}$	70.369${}_{-0.033}^{+0.034}$	0.223${}_{-0.009}^{+0.010}$	1.114${}_{-0.009}^{+0.009}$	2.907${}_{-0.125}^{+0.125}$	−0.126${}_{-0.021}^{+0.020}$	5.682${}_{-0.233}^{+0.252}$	−0.116${}_{-0.040}^{+0.040}$
2837475	1557.6${}_{-9.2}^{+8.2}$	75.729${}_{-0.042}^{+0.041}$	0.179${}_{-0.012}^{+0.012}$	0.911${}_{-0.011}^{+0.011}$	2.235${}_{-0.188}^{+0.177}$	−0.017${}_{-0.028}^{+0.028}$	6.417${}_{-0.392}^{+0.408}$	−0.126${}_{-0.048}^{+0.049}$
3427720	2737.0${}_{-17.7}^{+10.7}$	120.068${}_{-0.032}^{+0.031}$	0.287${}_{-0.008}^{+0.010}$	1.356${}_{-0.006}^{+0.006}$	3.487${}_{-0.080}^{+0.079}$	−0.053${}_{-0.014}^{+0.014}$	10.186${}_{-0.112}^{+0.100}$	−0.124${}_{-0.022}^{+0.022}$
3456181	970.0${}_{-5.9}^{+8.3}$	52.264${}_{-0.039}^{+0.041}$	0.216${}_{-0.011}^{+0.011}$	0.988${}_{-0.014}^{+0.013}$	3.551${}_{-0.167}^{+0.158}$	−0.075${}_{-0.023}^{+0.024}$	4.370${}_{-0.248}^{+0.251}$	−0.030${}_{-0.043}^{+0.043}$
3632418	1166.8${}_{-3.8}^{+3.0}$	60.704${}_{-0.018}^{+0.019}$	0.232${}_{-0.004}^{+0.004}$	1.114${}_{-0.006}^{+0.005}$	3.697${}_{-0.062}^{+0.064}$	−0.041${}_{-0.010}^{+0.009}$	4.189${}_{-0.115}^{+0.102}$	−0.053${}_{-0.019}^{+0.017}$
3656476	1925.0${}_{-6.3}^{+7.0}$	93.194${}_{-0.020}^{+0.018}$	0.207${}_{-0.008}^{+0.009}$	1.445${}_{-0.004}^{+0.004}$	4.608${}_{-0.042}^{+0.041}$	−0.095${}_{-0.011}^{+0.010}$	4.554${}_{-0.051}^{+0.055}$	−0.157${}_{-0.014}^{+0.016}$
3735871	2862.6${}_{-26.5}^{+16.6}$	123.049${}_{-0.046}^{+0.047}$	0.260${}_{-0.019}^{+0.019}$	1.325${}_{-0.009}^{+0.008}$	3.823${}_{-0.124}^{+0.125}$	−0.004${}_{-0.029}^{+0.027}$	11.028${}_{-0.208}^{+0.202}$	−0.077${}_{-0.053}^{+0.047}$
4914923	1817.0${}_{-5.2}^{+6.3}$	88.531${}_{-0.019}^{+0.019}$	0.233${}_{-0.008}^{+0.007}$	1.377${}_{-0.004}^{+0.004}$	4.534${}_{-0.042}^{+0.048}$	−0.119${}_{-0.010}^{+0.009}$	5.378${}_{-0.074}^{+0.068}$	−0.118${}_{-0.021}^{+0.022}$
5184732	2089.3${}_{-4.1}^{+4.4}$	95.545${}_{-0.023}^{+0.024}$	0.216${}_{-0.006}^{+0.005}$	1.374${}_{-0.005}^{+0.005}$	2.772${}_{-0.047}^{+0.044}$	−0.104${}_{-0.009}^{+0.009}$	5.863${}_{-0.065}^{+0.063}$	−0.097${}_{-0.015}^{+0.015}$
5773345	1101.2${}_{-6.6}^{+5.7}$	57.303${}_{-0.027}^{+0.030}$	0.257${}_{-0.007}^{+0.007}$	1.077${}_{-0.010}^{+0.009}$	1.754${}_{-0.101}^{+0.122}$	−0.093${}_{-0.016}^{+0.016}$	4.131${}_{-0.261}^{+0.268}$	−0.047${}_{-0.046}^{+0.047}$
5950854	1926.7${}_{-20.4}^{+21.9}$	96.629${}_{-0.107}^{+0.102}$	0.208${}_{-0.029}^{+0.032}$	1.431${}_{-0.020}^{+0.023}$	4.988${}_{-0.178}^{+0.215}$	−0.212${}_{-0.038}^{+0.038}$	4.713${}_{-0.235}^{+0.273}$	−0.232${}_{-0.074}^{+0.078}$
6106415	2248.6${}_{-3.9}^{+4.6}$	104.074${}_{-0.026}^{+0.023}$	0.254${}_{-0.005}^{+0.005}$	1.343${}_{-0.005}^{+0.005}$	3.422${}_{-0.047}^{+0.049}$	−0.114${}_{-0.008}^{+0.008}$	6.881${}_{-0.070}^{+0.066}$	−0.118${}_{-0.013}^{+0.014}$
6116048	2126.9${}_{-5.0}^{+5.5}$	100.754${}_{-0.017}^{+0.017}$	0.258${}_{-0.005}^{+0.005}$	1.336${}_{-0.003}^{+0.003}$	3.687${}_{-0.052}^{+0.048}$	−0.143${}_{-0.009}^{+0.009}$	6.034${}_{-0.070}^{+0.068}$	−0.155${}_{-0.016}^{+0.014}$
6225718	2364.2${}_{-4.6}^{+4.9}$	105.695${}_{-0.017}^{+0.018}$	0.274${}_{-0.005}^{+0.005}$	1.225${}_{-0.004}^{+0.004}$	3.207${}_{-0.060}^{+0.061}$	−0.053${}_{-0.010}^{+0.011}$	8.741${}_{-0.088}^{+0.085}$	−0.062${}_{-0.017}^{+0.016}$
6508366	958.3${}_{-3.6}^{+4.6}$	51.553${}_{-0.047}^{+0.046}$	0.223${}_{-0.009}^{+0.009}$	1.006${}_{-0.017}^{+0.017}$	2.880${}_{-0.138}^{+0.132}$	−0.088${}_{-0.019}^{+0.019}$	2.535${}_{-0.199}^{+0.205}$	−0.108${}_{-0.028}^{+0.030}$
6603624	2384.0${}_{-5.6}^{+5.4}$	110.128${}_{-0.012}^{+0.012}$	0.201${}_{-0.004}^{+0.004}$	1.492${}_{-0.002}^{+0.002}$	2.801${}_{-0.027}^{+0.029}$	−0.159${}_{-0.006}^{+0.006}$	4.944${}_{-0.034}^{+0.031}$	−0.201${}_{-0.008}^{+0.008}$
6679371	941.8${}_{-5.0}^{+5.1}$	50.601${}_{-0.029}^{+0.029}$	0.181${}_{-0.007}^{+0.008}$	0.880${}_{-0.010}^{+0.011}$	2.861${}_{-0.116}^{+0.123}$	−0.016${}_{-0.016}^{+0.017}$	3.143${}_{-0.266}^{+0.265}$	0.044${}_{-0.034}^{+0.034}$
6933899	1389.9${}_{-3.6}^{+3.9}$	72.135${}_{-0.018}^{+0.018}$	0.255${}_{-0.005}^{+0.005}$	1.319${}_{-0.005}^{+0.004}$	5.314${}_{-0.041}^{+0.042}$	0.013${}_{-0.009}^{+0.008}$	4.910${}_{-0.054}^{+0.054}$	−0.063${}_{-0.015}^{+0.014}$
7103006	1167.9${}_{-6.9}^{+7.2}$	59.658${}_{-0.030}^{+0.029}$	0.211${}_{-0.008}^{+0.007}$	0.978${}_{-0.009}^{+0.010}$	2.504${}_{-0.107}^{+0.140}$	−0.045${}_{-0.018}^{+0.019}$	4.471${}_{-0.295}^{+0.354}$	0.031${}_{-0.048}^{+0.045}$
7106245	2397.9${}_{-28.7}^{+24.0}$	111.376${}_{-0.061}^{+0.063}$	0.246${}_{-0.032}^{+0.025}$	1.392${}_{-0.012}^{+0.011}$	3.489${}_{-0.118}^{+0.110}$	−0.113${}_{-0.031}^{+0.032}$	6.529${}_{-0.167}^{+0.189}$	−0.265${}_{-0.068}^{+0.069}$
7206837	1652.5${}_{-11.7}^{+10.6}$	79.131${}_{-0.039}^{+0.037}$	0.253${}_{-0.011}^{+0.010}$	1.054${}_{-0.010}^{+0.011}$	2.106${}_{-0.147}^{+0.140}$	−0.054${}_{-0.023}^{+0.023}$	6.619${}_{-0.417}^{+0.419}$	−0.094${}_{-0.073}^{+0.079}$
7296438	1847.8${}_{-12.6}^{+8.5}$	88.698${}_{-0.036}^{+0.040}$	0.242${}_{-0.015}^{+0.015}$	1.358${}_{-0.008}^{+0.009}$	4.505${}_{-0.081}^{+0.073}$	−0.055${}_{-0.022}^{+0.020}$	5.079${}_{-0.098}^{+0.088}$	−0.135${}_{-0.028}^{+0.029}$
7510397	1189.1${}_{-4.4}^{+3.4}$	62.249${}_{-0.020}^{+0.020}$	0.258${}_{-0.004}^{+0.004}$	1.112${}_{-0.006}^{+0.006}$	3.971${}_{-0.059}^{+0.072}$	0.003${}_{-0.009}^{+0.009}$	4.370${}_{-0.086}^{+0.086}$	−0.016${}_{-0.017}^{+0.015}$
7680114	1709.1${}_{-6.5}^{+7.1}$	85.145${}_{-0.043}^{+0.039}$	0.238${}_{-0.007}^{+0.007}$	1.368${}_{-0.009}^{+0.010}$	5.039${}_{-0.054}^{+0.050}$	−0.043${}_{-0.011}^{+0.011}$	4.980${}_{-0.072}^{+0.074}$	−0.108${}_{-0.016}^{+0.018}$
7771282	1465.1${}_{-18.7}^{+27.0}$	72.463${}_{-0.079}^{+0.069}$	0.368${}_{-0.039}^{+0.042}$	1.117${}_{-0.019}^{+0.021}$	3.527${}_{-0.238}^{+0.241}$	−0.118${}_{-0.050}^{+0.049}$	5.058${}_{-0.445}^{+0.484}$	−0.020${}_{-0.104}^{+0.089}$
7871531	3455.9${}_{-26.5}^{+19.3}$	151.329${}_{-0.023}^{+0.025}$	0.285${}_{-0.008}^{+0.009}$	1.504${}_{-0.004}^{+0.003}$	2.142${}_{-0.068}^{+0.071}$	−0.149${}_{-0.014}^{+0.013}$	7.350${}_{-0.169}^{+0.155}$	−0.143${}_{-0.049}^{+0.044}$
7940546	1116.6${}_{-3.6}^{+3.3}$	58.762${}_{-0.029}^{+0.029}$	0.217${}_{-0.005}^{+0.004}$	1.075${}_{-0.009}^{+0.009}$	3.985${}_{-0.071}^{+0.072}$	−0.002${}_{-0.011}^{+0.011}$	4.346${}_{-0.123}^{+0.133}$	0.012${}_{-0.019}^{+0.019}$
7970740	4197.4${}_{-18.4}^{+21.2}$	173.541${}_{-0.068}^{+0.060}$	0.272${}_{-0.005}^{+0.005}$	1.455${}_{-0.008}^{+0.010}$	2.356${}_{-0.084}^{+0.083}$	−0.097${}_{-0.010}^{+0.011}$	7.901${}_{-0.165}^{+0.169}$	−0.268${}_{-0.026}^{+0.025}$
8006161	3574.7${}_{-10.5}^{+11.4}$	149.427${}_{-0.014}^{+0.015}$	0.195${}_{-0.005}^{+0.005}$	1.547${}_{-0.002}^{+0.002}$	3.061${}_{-0.046}^{+0.041}$	−0.084${}_{-0.007}^{+0.008}$	9.680${}_{-0.063}^{+0.070}$	−0.150${}_{-0.012}^{+0.012}$
8150065	1876.9${}_{-32.4}^{+38.1}$	89.264${}_{-0.121}^{+0.134}$	0.403${}_{-0.047}^{+0.048}$	1.163${}_{-0.030}^{+0.029}$	3.027${}_{-0.203}^{+0.198}$	0.036${}_{-0.063}^{+0.079}$	6.357${}_{-0.348}^{+0.342}$	0.012${}_{-0.128}^{+0.135}$
8179536	2074.9${}_{-12.0}^{+13.8}$	95.090${}_{-0.054}^{+0.058}$	0.277${}_{-0.019}^{+0.019}$	1.153${}_{-0.013}^{+0.012}$	3.137${}_{-0.164}^{+0.171}$	−0.082${}_{-0.032}^{+0.034}$	8.245${}_{-0.315}^{+0.352}$	−0.041${}_{-0.070}^{+0.073}$
8228742	1190.5${}_{-3.7}^{+3.4}$	62.071${}_{-0.021}^{+0.022}$	0.244${}_{-0.005}^{+0.005}$	1.158${}_{-0.007}^{+0.006}$	4.371${}_{-0.061}^{+0.057}$	0.007${}_{-0.010}^{+0.010}$	4.517${}_{-0.087}^{+0.082}$	−0.048${}_{-0.020}^{+0.019}$
8379927	2795.3${}_{-5.7}^{+6.0}$	120.288${}_{-0.018}^{+0.017}$	0.232${}_{-0.005}^{+0.005}$	1.311${}_{-0.003}^{+0.003}$	3.676${}_{-0.062}^{+0.062}$	−0.058${}_{-0.011}^{+0.011}$	10.932${}_{-0.083}^{+0.096}$	−0.062${}_{-0.020}^{+0.022}$
8394589	2396.7${}_{-9.4}^{+10.5}$	109.488${}_{-0.035}^{+0.034}$	0.234${}_{-0.011}^{+0.011}$	1.267${}_{-0.007}^{+0.006}$	3.382${}_{-0.091}^{+0.089}$	−0.061${}_{-0.021}^{+0.022}$	7.979${}_{-0.161}^{+0.164}$	−0.106${}_{-0.045}^{+0.045}$
8424992	2533.7${}_{-28.1}^{+27.0}$	120.584${}_{-0.064}^{+0.062}$	0.120${}_{-0.038}^{+0.035}$	1.517${}_{-0.010}^{+0.012}$	2.678${}_{-0.103}^{+0.100}$	−0.179${}_{-0.032}^{+0.033}$	5.190${}_{-0.143}^{+0.122}$	−0.282${}_{-0.055}^{+0.046}$
8694723	1470.5${}_{-4.1}^{+3.7}$	75.112${}_{-0.021}^{+0.019}$	0.296${}_{-0.005}^{+0.005}$	1.113${}_{-0.005}^{+0.005}$	5.339${}_{-0.067}^{+0.068}$	0.005${}_{-0.009}^{+0.010}$	5.879${}_{-0.108}^{+0.111}$	0.012${}_{-0.021}^{+0.020}$
8760414	2455.3${}_{-8.3}^{+9.1}$	117.230${}_{-0.018}^{+0.022}$	0.295${}_{-0.007}^{+0.007}$	1.400${}_{-0.004}^{+0.003}$	4.403${}_{-0.053}^{+0.045}$	−0.280${}_{-0.012}^{+0.009}$	5.132${}_{-0.059}^{+0.063}$	−0.291${}_{-0.015}^{+0.014}$
8938364	1675.1${}_{-5.8}^{+5.2}$	85.684${}_{-0.020}^{+0.018}$	0.235${}_{-0.006}^{+0.007}$	1.444${}_{-0.004}^{+0.004}$	6.491${}_{-0.035}^{+0.044}$	−0.046${}_{-0.009}^{+0.010}$	5.184${}_{-0.052}^{+0.048}$	−0.119${}_{-0.014}^{+0.013}$
9025370	2988.6${}_{-16.9}^{+20.0}$	132.628${}_{-0.024}^{+0.030}$	0.205${}_{-0.015}^{+0.016}$	1.475${}_{-0.005}^{+0.004}$	3.099${}_{-0.062}^{+0.065}$	−0.066${}_{-0.017}^{+0.018}$	9.141${}_{-0.119}^{+0.113}$	−0.126${}_{-0.030}^{+0.037}$
9098294	2314.7${}_{-10.4}^{+9.2}$	108.894${}_{-0.022}^{+0.023}$	0.251${}_{-0.008}^{+0.009}$	1.439${}_{-0.005}^{+0.004}$	3.331${}_{-0.057}^{+0.053}$	−0.185${}_{-0.011}^{+0.012}$	5.265${}_{-0.088}^{+0.086}$	−0.228${}_{-0.025}^{+0.024}$
9139151	2690.4${}_{-9.0}^{+14.5}$	117.294${}_{-0.032}^{+0.031}$	0.240${}_{-0.010}^{+0.011}$	1.337${}_{-0.006}^{+0.006}$	3.557${}_{-0.077}^{+0.083}$	−0.019${}_{-0.020}^{+0.017}$	10.050${}_{-0.158}^{+0.162}$	−0.093${}_{-0.057}^{+0.055}$
9139163	1729.8${}_{-5.9}^{+6.2}$	81.170${}_{-0.036}^{+0.042}$	0.241${}_{-0.005}^{+0.005}$	1.007${}_{-0.010}^{+0.010}$	2.079${}_{-0.098}^{+0.109}$	−0.024${}_{-0.010}^{+0.012}$	6.213${}_{-0.215}^{+0.218}$	0.040${}_{-0.028}^{+0.026}$
9206432	1866.4${}_{-14.9}^{+10.3}$	84.926${}_{-0.051}^{+0.046}$	0.135${}_{-0.013}^{+0.013}$	0.958${}_{-0.012}^{+0.012}$	3.235${}_{-0.210}^{+0.207}$	0.001${}_{-0.031}^{+0.027}$	7.115${}_{-0.411}^{+0.388}$	0.004${}_{-0.055}^{+0.056}$
9353712	934.3${}_{-8.3}^{+11.1}$	51.467${}_{-0.104}^{+0.091}$	0.254${}_{-0.011}^{+0.012}$	1.095${}_{-0.031}^{+0.038}$	3.536${}_{-0.159}^{+0.146}$	−0.038${}_{-0.018}^{+0.018}$	3.907${}_{-0.250}^{+0.236}$	−0.090${}_{-0.031}^{+0.031}$
9410862	2278.8${}_{-16.6}^{+31.2}$	107.390${}_{-0.053}^{+0.050}$	0.223${}_{-0.021}^{+0.020}$	1.343${}_{-0.010}^{+0.009}$	3.625${}_{-0.121}^{+0.116}$	−0.181${}_{-0.032}^{+0.027}$	6.098${}_{-0.204}^{+0.201}$	−0.239${}_{-0.086}^{+0.081}$
9414417	1155.3${}_{-4.6}^{+6.1}$	60.115${}_{-0.024}^{+0.024}$	0.237${}_{-0.007}^{+0.006}$	1.045${}_{-0.008}^{+0.008}$	3.572${}_{-0.092}^{+0.106}$	−0.029${}_{-0.014}^{+0.015}$	4.648${}_{-0.200}^{+0.211}$	−0.006${}_{-0.030}^{+0.031}$
9812850	1255.2${}_{-7.0}^{+9.1}$	64.746${}_{-0.068}^{+0.067}$	0.240${}_{-0.011}^{+0.012}$	1.067${}_{-0.020}^{+0.021}$	2.654${}_{-0.166}^{+0.156}$	−0.134${}_{-0.025}^{+0.025}$	4.418${}_{-0.307}^{+0.297}$	−0.032${}_{-0.045}^{+0.048}$
9955598	3616.8${}_{-29.6}^{+21.2}$	153.283${}_{-0.032}^{+0.029}$	0.195${}_{-0.011}^{+0.011}$	1.529${}_{-0.004}^{+0.005}$	2.796${}_{-0.076}^{+0.073}$	−0.095${}_{-0.017}^{+0.016}$	8.941${}_{-0.126}^{+0.143}$	−0.172${}_{-0.030}^{+0.029}$
9965715	2079.3${}_{-10.4}^{+9.2}$	97.236${}_{-0.042}^{+0.041}$	0.373${}_{-0.015}^{+0.016}$	1.139${}_{-0.009}^{+0.009}$	3.685${}_{-0.129}^{+0.132}$	−0.102${}_{-0.027}^{+0.024}$	7.958${}_{-0.269}^{+0.254}$	−0.105${}_{-0.058}^{+0.063}$
10068307	995.1${}_{-2.7}^{+2.8}$	53.945${}_{-0.020}^{+0.019}$	0.247${}_{-0.003}^{+0.004}$	1.131${}_{-0.006}^{+0.007}$	4.220${}_{-0.058}^{+0.055}$	0.014${}_{-0.008}^{+0.008}$	3.799${}_{-0.088}^{+0.083}$	−0.041${}_{-0.014}^{+0.014}$
10079226	2653.0${}_{-44.3}^{+47.7}$	116.345${}_{-0.052}^{+0.059}$	0.264${}_{-0.028}^{+0.027}$	1.350${}_{-0.011}^{+0.010}$	3.236${}_{-0.199}^{+0.205}$	−0.014${}_{-0.046}^{+0.043}$	9.387${}_{-0.371}^{+0.401}$	0.098${}_{-0.098}^{+0.098}$
10162436	1052.0${}_{-4.2}^{+4.0}$	55.725${}_{-0.039}^{+0.035}$	0.242${}_{-0.004}^{+0.004}$	1.106${}_{-0.012}^{+0.013}$	3.746${}_{-0.080}^{+0.078}$	−0.033${}_{-0.009}^{+0.009}$	3.706${}_{-0.117}^{+0.115}$	−0.031${}_{-0.018}^{+0.017}$
10454113	2357.2${}_{-9.1}^{+8.2}$	105.063${}_{-0.033}^{+0.031}$	0.283${}_{-0.009}^{+0.009}$	1.206${}_{-0.007}^{+0.007}$	3.059${}_{-0.110}^{+0.095}$	−0.063${}_{-0.020}^{+0.016}$	9.426${}_{-0.165}^{+0.178}$	−0.079${}_{-0.033}^{+0.033}$
10516096	1689.8${}_{-5.8}^{+4.6}$	84.424${}_{-0.025}^{+0.022}$	0.257${}_{-0.008}^{+0.009}$	1.318${}_{-0.006}^{+0.005}$	5.001${}_{-0.061}^{+0.054}$	−0.058${}_{-0.012}^{+0.013}$	5.248${}_{-0.083}^{+0.084}$	−0.089${}_{-0.022}^{+0.023}$
10644253	2899.7${}_{-22.8}^{+21.3}$	123.080${}_{-0.055}^{+0.056}$	0.250${}_{-0.016}^{+0.016}$	1.313${}_{-0.011}^{+0.010}$	3.863${}_{-0.139}^{+0.140}$	0.004${}_{-0.030}^{+0.029}$	11.378${}_{-0.155}^{+0.192}$	−0.138${}_{-0.049}^{+0.049}$
10730618	1282.1${}_{-12.7}^{+14.6}$	66.333${}_{-0.064}^{+0.061}$	0.239${}_{-0.017}^{+0.016}$	1.032${}_{-0.018}^{+0.018}$	2.651${}_{-0.218}^{+0.199}$	0.019${}_{-0.037}^{+0.040}$	4.556${}_{-0.454}^{+0.428}$	0.113${}_{-0.080}^{+0.073}$
10963065	2203.7${}_{-6.3}^{+6.7}$	103.179${}_{-0.027}^{+0.027}$	0.297${}_{-0.008}^{+0.008}$	1.275${}_{-0.005}^{+0.005}$	3.567${}_{-0.071}^{+0.072}$	−0.081${}_{-0.014}^{+0.014}$	7.083${}_{-0.096}^{+0.103}$	−0.058${}_{-0.020}^{+0.022}$
11081729	1968.3${}_{-12.6}^{+11.0}$	90.116${}_{-0.047}^{+0.048}$	0.242${}_{-0.017}^{+0.016}$	1.020${}_{-0.011}^{+0.011}$	3.056${}_{-0.264}^{+0.251}$	−0.103${}_{-0.026}^{+0.031}$	6.602${}_{-0.664}^{+0.605}$	0.010${}_{-0.084}^{+0.087}$
11253226	1590.6${}_{-6.8}^{+10.6}$	76.858${}_{-0.030}^{+0.026}$	0.183${}_{-0.008}^{+0.008}$	0.920${}_{-0.008}^{+0.008}$	1.748${}_{-0.136}^{+0.155}$	−0.132${}_{-0.018}^{+0.020}$	6.973${}_{-0.396}^{+0.435}$	0.039${}_{-0.049}^{+0.048}$
11772920	3674.7${}_{-36.1}^{+55.1}$	157.746${}_{-0.033}^{+0.032}$	0.238${}_{-0.015}^{+0.015}$	1.516${}_{-0.005}^{+0.004}$	2.366${}_{-0.084}^{+0.078}$	−0.082${}_{-0.019}^{+0.017}$	7.849${}_{-0.209}^{+0.200}$	−0.131${}_{-0.063}^{+0.059}$
12009504	1865.6${}_{-6.2}^{+7.7}$	88.217${}_{-0.025}^{+0.026}$	0.289${}_{-0.007}^{+0.008}$	1.200${}_{-0.006}^{+0.006}$	3.533${}_{-0.079}^{+0.078}$	−0.072${}_{-0.014}^{+0.014}$	6.117${}_{-0.134}^{+0.133}$	−0.066${}_{-0.034}^{+0.032}$
12069127	884.7${}_{-8.0}^{+10.1}$	48.400${}_{-0.048}^{+0.048}$	0.204${}_{-0.014}^{+0.012}$	1.061${}_{-0.018}^{+0.018}$	3.399${}_{-0.173}^{+0.153}$	−0.037${}_{-0.025}^{+0.023}$	3.650${}_{-0.291}^{+0.244}$	−0.102${}_{-0.039}^{+0.039}$
12069424	2188.5${}_{-3.0}^{+4.6}$	103.277${}_{-0.020}^{+0.021}$	0.246${}_{-0.004}^{+0.004}$	1.437${}_{-0.004}^{+0.004}$	3.392${}_{-0.039}^{+0.039}$	−0.152${}_{-0.006}^{+0.007}$	5.274${}_{-0.047}^{+0.051}$	−0.181${}_{-0.010}^{+0.010}$
12069449	2561.3${}_{-5.6}^{+5.0}$	116.929${}_{-0.013}^{+0.012}$	0.201${}_{-0.004}^{+0.004}$	1.461${}_{-0.002}^{+0.002}$	2.707${}_{-0.031}^{+0.032}$	−0.117${}_{-0.006}^{+0.006}$	6.045${}_{-0.044}^{+0.037}$	−0.133${}_{-0.010}^{+0.009}$
12258514	1512.7${}_{-2.9}^{+3.3}$	74.799${}_{-0.015}^{+0.016}$	0.209${}_{-0.004}^{+0.004}$	1.281${}_{-0.004}^{+0.004}$	4.227${}_{-0.043}^{+0.042}$	−0.061${}_{-0.008}^{+0.007}$	4.827${}_{-0.052}^{+0.061}$	−0.056${}_{-0.011}^{+0.011}$
12317678	1212.4${}_{-4.9}^{+5.5}$	63.464${}_{-0.024}^{+0.025}$	0.231${}_{-0.005}^{+0.005}$	0.928${}_{-0.008}^{+0.006}$	3.883${}_{-0.115}^{+0.112}$	−0.032${}_{-0.013}^{+0.013}$	5.273${}_{-0.194}^{+0.188}$	−0.061${}_{-0.025}^{+0.023}$

Note.  ${\nu }_{\max }$ is obtained from the fit to the extracted modes. Plots of the fitted parameters are shown in Figures 9–12; the values for are shown in Figure 3.

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We found this description to perform poorly across the range of stars in the sample, which is to be expected for more evolved stars (see, e.g., Gabriel 1989; Christensen-Dalsgaard 1991). To that end, we modified the formula as follows.

$$\begin{eqnarray}{\nu }_{{nl}} & \simeq & \left(n+\displaystyle \frac{l}{2}+\epsilon \right){\rm{\Delta }}{\nu }_{0}-\delta {\nu }_{0l}\\ & & -\displaystyle \frac{d\delta {\nu }_{0l}}{{dn}}(n-{n}_{{\nu }_{\max },l})\\ & & +\displaystyle \frac{d{\rm{\Delta }}\nu /{dn}}{2}{(n-{n}_{{\nu }_{\max },l})}^{2},\end{eqnarray} \tag{ 27 }$$

where the term $l(l+1){D}_{0}$ has been replaced by $\delta {\nu }_{0l}$, which takes on independent values for l > 0. Thereby, we optimize independently for the separations $\delta {\nu }_{02}$ and $\delta {\nu }_{01}$. In the fit, we also included ${\nu }_{\max }$ with a Gaussian prior from the fit to the mode amplitudes; the values of ${\nu }_{\max },$ ${\rm{\Delta }}\nu$, and correlate strongly, but we include ${\nu }_{\max }$ to properly marginalize over the uncertainty of the pivoting ${n}_{{\nu }_{\max },l}$. Figure 8 gives an example of the fit of Equation (27) to KIC 8228742 (Horace). All parameters from the fits are listed in Table 2.

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Figure 9 shows the estimated values for ${\rm{\Delta }}\nu$ versus ${\nu }_{\max },$ together with the empirical relation by Huber et al. (2011), with which we find an excellent agreement. We can even see the expected mass gradient in the residuals, with higher mass stars having slightly lower ${\rm{\Delta }}\nu$ for a given ${\nu }_{\max }.$ Also shown is the change in the large separation, which is found to be near-constant at $d{\rm{\Delta }}\nu /{dn}\,\approx \,0.25\,\mu \mathrm{Hz}$ and thus always with the same concavity sign, corresponding to a positive gradient of the large separation with frequency. Here one should remember that a constant value of $d{\rm{\Delta }}\nu /{dn}$ against ${\rm{\Delta }}{\nu }_{0}$ would correspond to a linear change in ${\rm{\Delta }}\nu$ with frequency.

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For the Sun, we obtain a value of ${\rm{\Delta }}{\nu }_{\odot }$ of 134.91 ± 0.02 μHz; comparing this to the value of ${\rm{\Delta }}{\nu }_{\odot }=135.1\pm 0.1\,\mu \mathrm{Hz}$ by Huber et al. (2011), we would expect a slightly smaller value, because the curvature is incorporated in our estimation. However, the value of ${\rm{\Delta }}{\nu }_{\odot }=134.92\pm 0.02\,\mu \mathrm{Hz}$ by Toutain & Fröehlich (1992), who also used a second-order version of the asymptotic relation, compares very well to our estimate.

The estimated values of have already been shown in Figure 3. For the estimation of , it should be noted that this value is strongly anti-correlated with ${\rm{\Delta }}\nu$, and a small change in ${\rm{\Delta }}\nu$ can thus induce a significant change in (White et al. 2011b). This will especially take effect if the estimate of ${\nu }_{\max }$ is off, because ${\nu }_{\max }$ defines the pivoting point for the change in $\delta {\nu }_{01}$, $\delta {\nu }_{02}$, and the curvatures. Like White et al. (2011b), we see an offset between the estimated and those obtained from model tracks, which is ascribed to the effects of the incorrect modeling of the stellar surface layers. In Figure 3, we show the -tracks adopted from White et al. (2011a), which were computed from evolutionary tracks from the Aarhus STellar Evolution Code (ASTEC; Christensen-Dalsgaard 2008), neglecting diffusion and core overshoot and with Z₀ = 0.017. It should also be noted that the from models were derived in a slightly different manner than used here, as described by White et al. (2011a). We therefore also estimated the values for ${\rm{\Delta }}\nu$ and using the White et al. (2011a) method, namely from a weighted fit of the asymptotic function in Equation (10) to the radial mode frequencies as a function of radial order n, and with weights given by a Gaussian with an FWHM of 0.25 ${\nu }_{\max }.$ Comparing the values for ${\rm{\Delta }}\nu$ and from the fit of Equation (10) versus Equation (27), we only find minor differences in estimated values. For ${\rm{\Delta }}\nu$ the maximum absolute difference was ∼0.36 μHz, with no systematic differences; for a maximum difference of ∼0.14 was found, and again with no systematic differences.

In Figures 10 and 11, we show the estimated values for the separations $\delta {\nu }_{01}$ and $\delta {\nu }_{02}$, together with their gradients in n (or frequency); the results for $\delta {\nu }_{02}$ in Figure 11 are given in the form of a modified C–D diagram (see Christensen-Dalsgaard 1993; White et al. 2011a, 2011b, 2012). Because all values for the changes in $\delta {\nu }_{01}$ and $\delta {\nu }_{02}$ are either zero or negative, we see that the small separations are virtually all decreasing functions of frequency. In Figure 11, we show again the ASTEC tracks from White et al. (2011a). It is interesting to see the degree to which the asymptotic relations to first (Equation (10)) and second (Equation (26)) order are satisfied. From these, one would expect a ratio of $\delta {\nu }_{02}/\delta {\nu }_{01}=3$ from assuming $\delta {\nu }_{0l}=l(l+1){D}_{0}$—in Figure 12, we show this ratio for the sample. As seen, only a few stars, including the Sun, actually adhere to the expectation from the asymptotic relation. This was similarly found by Christensen-Dalsgaard & Houdek (2010)¹⁹ for α Cen A (Bedding et al. 2004), while α Cen B (Kjeldsen et al. 2005) was found to fulfill the asymptotic relation. As explained by Christensen-Dalsgaard & Houdek (2010), this reflects a rapid variation of the sound speed in the core of α Cen A from its more evolved state due to its higher mass compared to the B component. Similarly, Verma et al. (2014) found an increasing departure from the asymptotic description of the oscillation frequencies while fitting the signatures of the acoustic glitches. The departure becomes noticeable when a peak in the Brunt–Väisälä frequency N develops just outside the stellar core toward the end of the MS and becomes comparable or higher than the lowest frequency fitted. To check this evolutionary explanation, we plot in the right panel of Figure 12 the $\delta {\nu }_{02}/\delta {\nu }_{01}$ against the central hydrogen content X_c, because this serves as a good probe for the evolutionary state. As seen, stars with high X_c, i.e., stars that are less evolved, adhere better to the asymptotic regime than the evolved stars with low X_c. The values of X_c were obtained as part of the modeling with the BASTA pipeline; we refer to Paper II for further details. While a difference from a ratio of $\delta {\nu }_{02}/\delta {\nu }_{01}=3$ can be explained from an evolutionary viewpoint, it is interesting to note that the majority of the stars analyzed indeed do not follow the asymptotic relation. This is a cautionary note to the use of Equation (10) and/or Equation (26) for extracting average seismic parameters as here, or if used for predicting the location of oscillation modes frequencies in the power spectrum from ${\rm{\Delta }}{\nu }_{0}$ measured in an independent manner. For details on the physics behind the curvatures, we refer to Tassoul (1980), Houdek & Gough (2007), Cunha & Metcalfe (2007), and Mosser et al. (2013) and references therein.

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