© ESO, 2017

1. Introduction

The Kepler Asteroseismic Legacy Project (Lund et al. 2017b) analysed 66 Kepler main sequence targets determining frequencies, separation ratios, error estimates and covariance matrices. From the outset of this project I queried the data (cf. Roxburgh 2015, 2016) so I developed my own mode fitting routine, applied this to the Legacy power spectra for 9 solar-like stars, and here compare my results with the Legacy project’s values.

In Sects. 3 to 7 I compare my results for 3 Kepler targets, 16 Cyg A & B and KIC 8379927, with the Legacy values and results from independent analyses by Davies et al. (2015), Davies (2015), using Davies’ power spectra. My results agree well with those of Davies et al., but do not agree with the Legacy project’s values.

The Legacy frequencies are different and the error estimates on separation ratios are up to a factor 40 larger and exceed upper limits derived from covariance matrices by a similar factor. The covariance matrices are inconsistent as they have negative eigenvalues and are therefore not positive semi-definite as they should be, giving negative χ² when comparing frequency sets.

Fig. 1 16 Cyg B: top 2 panels: frequency differences Legacy-Roxburgh, Davies-Roxburgh and χ2 of fits; bottom panel: error estimates σ02 on the frequency separation ratios r02, Legacy, Davies, Roxburgh.

In Sect. 8 I compare Legacy and my results for a further 6 solar-like Legacy stars; 2 have non positive definite covariance matrices, none give good agreement on frequencies or separation errors. In Sect. 9 I inspect the covariance matrices and errors on separation ratios for all 66 Legacy targets and find similar anomalies. Something is amiss with the Legacy data.

The differences between the Legacy results and those of Roxburgh and Davies are clearly shown in Fig. 1, which compares the different frequency sets for 16 Cyg B for modes with heights greater than the background (S/N> 1). I also gives the χ² of the fits using the different error estimates. The bottom panel compares errors on the separation ratios r₀₂ from all 4 analyses. The agreement between Roxburgh and Davies is up to 35 times better than between the Roxburgh and Legacy values.

2. Roxburgh’s mode fitting algorithm

My mode fitting algorithm searches for a minimum in the negative log likelihood (cf. Toutain & Appourchaux 1994) of a global fit of mode power + background to a section of the power spectrum that extends ~300 μHz beyond both ends of the range of frequencies to be fitted, with unconstrained parameters X_k: frequencies ν_n,ℓ; mode heights h_n and widths w_n of the ℓ = 0 modes; mode height ratios h₁₀,h₂₀,h₃₀ of modes ℓ = 1,2,3 to the heights of modes with ℓ = 0 (with the geometrical constraint 1 + h₂₀ = h₁₀ + h₃₀), the same for all modes; rotational splitting ν_Ω and inclination i (the same for all modes); and 4 parameters of a Harvey-like model of the background (A/ [1 + Bν^c] + D). The heights and widths of the ℓ = 1,2,3 modes are determined by (linear) interpolation in the values for the ℓ = 0 modes at the respective frequencies and, for mode heights, then multiplied by the mode height ratios. The modes are fitted with symmetric rotationally split Lorentzians. The covariance matrix is the inverse of the Hessian H(i,j) = ∂²MLE/∂X_i∂X_j and the errors on the X_k are given as .

Power spectra

For comparison with Davies’s results I used their power spectra kindly supplied to me by Guy Davies, and for comparison with the Legacy results I used the Legacy power spectra taken from the kasoc web site namely:

3. Results for frequencies: 16 Cyg A & B, KIC 8379927

Tables 1 to 3 gives the χ² of the fits of one set of frequencies to another both for all modes and just for modes with mode-height/background =S/N> 1 (as determined by my fits). I used frequency errors in the fits as I encountered severe problems when using Legacy covariance matrices (see Sect. 5 below).

Table 1 compares the fit of the Legacy frequencies and errors (ν_L ± σ_L) to those of Roxburgh (ν_LR ± σ_LR) (using the Legacy power spectra), is the value using Legacy errors and using Roxburgh’s errors. is the value using Legacy errors but only comparing frequencies with S/N> 1, and likewise . The first row is for the full frequency sets and the second for frequency sets with “misfits” (discussed below) removed. Table 2 gives the fit of Roxburgh’s frequencies ν_DR (using Davies’s power spectra) to Davies’s frequencies, ν_D and Table 3 compares the Legacy and Davies’s values.

The Roxburgh-Davies fit for modes with S/N> 1 is very good for all 3 stars, much better than that of Davies’s or Roxburgh’s fits to the Legacy values. The Roxburgh-Davies fit to 16 Cyg B for all frequencies is strongly influenced by the misfit of the ν_14,3 mode which has S/N = 0.15 and is unreliable; the Roxburgh-Davies fit for 16 Cyg A for modes with S/N< 1 is strongly influenced by the ν_25,0 mode which has S/N = 1.08, if this is excluded , .

The frequency sets obtained from my analysis for both the Legacy and Davies power spectra, the Legacy and Davies frequencies, and my S/N values, are given in the Appendix.

Table 4 compares the rotational parameters as determined by Davies et al. and as determined by Roxburgh’s fits to the Davies power spectra; there is very good agreement for all 3 stars, Roxburgh’s fits to the Davies spectra yielding almost the same values as those obtained by Davies.

4. Fitting low frequency modes

Fig. 2 16 Cyg A: legacy and Roxburgh fits to Legacy power spectrum.

As stated above and in the footnotes to the tables there are some problems in fitting some low frequency modes. For 16 Cyg A Legacy fits (Table 1) the problem is illustrated in Fig. 2 which shows the kasoc power spectrum for 16 Cyg A smoothed by a Gaussian smoother (with σ = 0.1 μHz), and overlaid the Roxburgh fit to the full power spectrum and the location of the Legacy and Roxburgh frequencies for modes ν_12,0 and the pair ν_13.2,ν_14,0. The Legacy values for ν_12,0 and ν_13.2 are poor fits and and Roxburgh’s error estimates (from the Hessian of the MLE fit) are considerably smaller than the Legacy values. Excluding these 2 modes reduces and from 7.077 and 8.590 to 1.540 and 1.427 respectively. A similar problem exists for the fit to ν_12,2 for 16 Cyg B; excluding this mode reduces the of the fit using Roxburgh’s errors from 5.067 to 1.412. The S/N values remain unchanged since this mode has S/N< 1.

Davies’s value of ν_12,2 for 16 Cyg B is also a poor fit to his power spectrum. ν_14,3 (which has S/N = 0.15) differs from my value by ~3 μHz so I determined the quality of fits to the section of the Davies power spectrum between 1982.6 ± 29 μHz for a 100² matrix of values of ν_15,1, ν_14,3 and 10 values of height ratio h₃₁ between 0.01 to 0.1, with fitting parameters the ℓ = 1 mode height, one width for both ℓ = 1 and 3 and a constant background; all with { ν_Ωsini,i } = { 0.27,34 }. Figure 3 shows the quality of fits (MLE−MLE_min) for 2 values of h₃₁; the best fits for all h₃₁ have ν_14,3 = 1973.71 μHz; my full fit value is 1973.69 ± 0.37 μHz.

Fig. 3 16 Cyg B: quality of free mode fits to Davies power spectrum.

5. Covariance matrices and frequency comparison

The χ²s of the fit of N frequencies incorporating their correlations are given by [DC^-1D^T] /N where D is the vector of frequency differences and C^-1 the inverse of the frequency covariance matrix C. Tables 5–7 give the results of such fits for 16 Cyg A & B and KIC 8379927 for both the full frequency sets and for modes with S/N> 1 using Legacy (L), Davies (D) and Roxburgh (R) inverse covariance matrices (determined using the SVD algorithm). Whilst the χ² for the Roxburgh-Davies fits are compatible (and small) and consistent with the values using frequency errors as given in Table 2, the χ²s using the Legacy covariance matrices give negative values, which should not be the case since covariance matrices and their inverses are necessarily positive semi-definite so should always give positive χ².

Since a symmetric matrix C is positive semi-definite if and only if all its eigenvalues are non-negative, I determined the eigenvalues for the Legacy covariance matrices for all 3 stars. The absolute value of the eigenvalues w_j is given by SVD and the sign from which of det(C−w_jU) and det(C + w_jU), is zero, or closest to zero given rounding errors [U is the unit matrix]. All 3 Legacy covariance matrices have negative eigenvalues, 16 Cyg A having 10, 16 Cyg B 12, KIC 8379927 10. The Roxburgh and Davies covariance matrices are all positive definite.

The stark difference between Legacy and Roxburgh matrices is illustrated in Fig. 4 which displays their inverse covariance matrices for 16 Cyg A [magnitude=size of points, black +ve, red -ve]. Something is clearly amiss with the Legacy evaluation of the covariance matrices from their Markov chain Monte Carlo (MCMC) analysis.

Fig. 4 16 Cyg A: inverse covariance matrices. Top: legacy; bottom: Roxburgh [black +ve, red -ve, magnitude = size of points].

6. Frequency separation ratios

The ratios of small (d) to large (Δ) frequency separations are widely used in model fitting since they are (almost) independent of the structure of the outer layers of a star. These ratios are defined as (Roxburgh & Vorontsov 2003, 2013, Roxburgh 2005) (1a)where The Legacy project and Davies give values of the ratios, errors and ratio covariance matrices for the 3 stars analysed here. They also give values for r₀₁₀ ratios but these do not contain any additional information since from 2N (ℓ = 0,1) frequencies one can only determine N surface layer independent quantities.

Fig. 5 Top panel: error estimates σ101 on ratios r101(n) from Legacy, Davies, and Roxburgh analyses of 16Cyg A&B, and KIC 8379927; bottom errors σ02 on ratios r02.

The values of the ratios r₁₀₁ and r₀₂ as determined by the different analyses are similar but, as shown in Fig. 5 the error estimates are wildly different. The top panel shows the 4 determinations of error estimates σ₁₀₁ on the ratios r₁₀₁ by Legacy, Davies, and Roxburgh using both the kasoc and Davies power spectra, all limited to modes with S/N> 1. The bottom panel shows the error estimates σ₀₂ on r₀₂. Davies’s and the two Roxburgh values are very close but the Legacy error estimates are very much larger than those of Davies and Roxburgh, by a factor of up to 40.

7. Error estimates and upper limits for separation ratios from frequency covariances

The covariance of two linear functions r_n(ν_j) = ∑ A_jν_j, and r_m(ν_k) = ∑ B_kν_k of variables ν_i is given by (2)and the error estimate σ_n on r_n(ν_k) is given by the variance (3)where corr_jk are the correlations and σ_i the error estimates on ν_i. Since | corr_jk | ≤ 1 it follows that an upper bound on σ_n is given by taking corr_jk = + 1 if A_jA_k> 0 and −1 if negative, hence (4)The small separations d_n [both d₁₀₁(n) and d₀₂(n)] are linear functions of ν, d_n = ∑ D_kν_k (cf Eqs. (1b), (1c)), but the contribution of the large separation Δ_n introduces a small non linearity in the ratios. To a good approximation (1 in 10³ see below) the variance of the separation ratios r_n(ν_k) is given by (5)and so, with the A_k defined through Eq. (5) (see below), the error estimate σ_n on r_n is given by Eq. (3) and the upper limit σ_L by Eq. (4).

Coefficients A _k for errors σ₀₂(n),σ₁₀₁(n) on r₀₂(n),r₁₀₁(n)

For r₀ = r₀₂(n), Δ₀ = ν_n,1−ν_n−1,1(6)For r₀ = r₁₀₁(n), Δ₀ = ν_n,1−ν_n−1,1(7)Figure 6 shows the fractional differences between the σ’s given by Davies’s analysis of 16 Cyg A & B and KIC 8379927, and the σ_Dcov given by Eqs. (3), (6) and (7), using Davies’s frequencies and frequency covariances; all but two are less than 10^-3. The two are KIC 8379927 σ₀₂(n), n = 14,15, which have values −2.5 × 10^-3, 4.4 × 10^-3, and are derived from modes with S/N < 1.

Fig. 6 Fractional difference between Davies’s MCMC values for the errors σ101, σ02 and the values from Eqs. (3), (6), (7). All but two <10-3.

Fig. 7 Fractional difference between the Legacy MCMC values for the errors σ101, σ02 and the σLcov values from Eqs. (3), (6), (7).

Fig. 8 Ratios of Legacy error estimates σ101, σ02 to the upper limits σL from Eq. (4) for 16 Cyg A & B and KIC 8379927, for modes with S/N> 1. The Legacy values exceed the upper limits by a factor of up to 30.

Figure 7 shows the same comparison but between Legacy σ’s and the values σ_Lcov derived using the Legacy frequencies and frequency covariances; here many of the differences are huge.

As shown in Fig. 8 the Legacy σ’s also exceed the upper limits σ_L given by Eq. (4), whereas Davies’s and Roxburgh’s values, and the re-derived Legacy values σ_Lcov, are less than their corresponding upper limits. Something seems to be amiss with the Legacy values.

8. Comparison of Legacy and Roxburgh results for a further 6 solar-like stars

Having verified that my code gives results in agreement with Davies et al., I then applied my analysis to the 6 other solar-like stars from the Legacy short list of 22 high priority targets which have large separations Δ in the range 100–120 μHz and ν_max in the range 2138–2470 μHz, namely KIC 9098294, 8760414, 6603624, 6225718, 6116048, 6106415. The fit of the Roxburgh to Legacy frequencies for KIC 6225718 is shown in Fig. 9.

Table 8 gives the fits of the Legacy frequencies to Roxburgh’s for all 6 stars using the frequency errors (rows labelled σ) and the inverse covariance matrices (labelled cov) both for all frequencies and for the subset with S/N> 1.

Fig. 9 KIC 6225718: frequency differences, Legacy-Roxburgh.

For KIC 9098294, 6603264, there is good agreement between χ² using the Legacy covariance matrices and uncorrelated errors, and reasonable agreement for 6225718 and 6106415, but the χ² are still an order of magnitude larger than the 3 Roxburgh-Davies fits for S/N> 1. The fits for KIC 8760414 and 6116048 are not so good: KIC 8760414 having a negative χ² and KIC 6116048 a factor 3 difference between values with the Legacy covariance matrix and uncorrelated errors. Analysis of the covariance matrices revealed that for the best 4 of the 6 stars the Legacy covariance matrices had no negative eigenvalues and are therefore positive definite, whilst the other 2 have negative eigenvalues and are therefore inconsistent.

Figure 10 plots the Legacy error estimates σ₁₀₁ and σ₀₂ on the separation ratios r₁₀₁ and r₀₂ which show a similar behaviour to those of 16 Cyg A & B and KIC 8379927 in that the Legacy estimates are all larger than Roxburgh’s for all 6 stars. For KIC 9098294 this is only by a factor ~2 but for KIC 6116048 the Legacy value is up to a factor 50 larger then Roxburgh’s.

As was the case for 16 Cyg A & B and KIC 8379927, the Legacy ratio errors for all of these 6 stars also exceed the upper limits calculated as described in Sect. 7 above, and likewise new values for errors on the Legacy ratios calculated using the Legacy covariance matrices gave lower values, all of which are less than the corresponding upper limits.

Fig. 10 Logarithm of the error estimates on the ratios r101,r02 for all 6 solar-like stars. The Legacy values exceed Roxburgh’s values by factors ranging from 2 to 50.

9. Covariance matrices and errors on separation ratios for all 66 Legacy target stars

The Legacy Project analysed a total of 66 main sequence stars (Lund et al. 2017b) only 9 of which have been analysed by my code and compared with the Legacy data. Whilst this may ultimately be expanded to all the Legacy targets, I here just examine the Legacy data on all 66 stars to see whether their covariance matrices are positive semi-definite or whether they have negative eigenvalues, and whether they have anomalously large error estimates σ₁₀₁,σ₀₂ for the separation ratios r₁₀₁,r₀₂.

The eigenvalues of all 66 Legacy covariance matrices were determined by the same procedure as applied to 16 Cyg A & B and KIC 8379927, the absolute magnitudes w from SVD, and the sign from the determinants | | C ± wU | |. 27 have covariance matrices with negative eigenvalues and are therefore inconsistent, the remaining 39 stars have positive definite covariance matrices.

Fig. 11 Logarithm of the ratio of the error estimates to the upper limits on the ratios r101,r02 for all 66 Legacy stars. The blue triangles are stars with inconsistent covariance matrices, the red stars the values re-computed from the covariance matrices. 59 of the Legacy values exceed one or both upper limits on σ.

Next I compare the Legacy values for the error estimates σ₁₀₁,σ₀₂ on the separation ratios r₁₀₁,r₀₂ with the values re-derived from the frequency covariance matrices and the upper limits as determined by Eqs. (3), (4), (6), and (7) in Sect. 7. Figure 11 shows the Legacy data error estimates σ₁₀₁,σ₀₂ divided by the upper limits and the re-derived values divided by the upper limits all averaged over 3 values around their ν_max. The blue triangles are stars with inconsistent covariance matrices (negative eigenvalues). 7 stars have values of σ₁₀₁ and σ₀₂ less than their upper limits all of which have positive definite covariance matrices, of which KIC 3427720, 8938364, 9353712 and 10079226 have Legacy values for ratio errors within 2% of the re-derived values from their covariance matrices.

I selected KIC 3427720, the brightest and most solar-like of the 4 to test if, for such a star, the Legacy frequencies agreed with values obtained on applying my mode fitting algorithm to the power spectrum ( kplr003427720_kasoc-wpsd_slc_v1.pow). The details of the fits are given in Table 9; as anticipated the χ²s of the fits using errors and those using covariance matrices are in good agreement, but the values for modes with S/N> 1 are still more than a factor 10 larger than those of the Roxburgh-Davies fits to 16 Cyg A & B and KIC 8379927.

10. Conclusions and discussion

1) I developed a new mode fitting code different from, and independent of, the codes used by Davies et al and the Legacy project which, when applied to the Davies et al power spectra for 16 Cyg A, 16 Cyg B and KIC 8379927, reproduces the frequencies, separation ratios, errors, rotational parameters and covariance matrices of Davies’s analysis to good accuracy, especially for modes with S/N = heights/background >1 which are least sensitive to differences in the modelling of the background and the possibility of misidentification of fluctuations in noise as signal. For modes with S/N> 1 the χ² of the fits of Roxburgh to Davies’s frequencies are ≤0.062, both for comparisons using only error estimates and using full covariance matrices (≤0.035 if one mode with S/N = 1.08 is excluded).

The same code when applied to the Legacy power spectra for 16 Cyg A, 16 Cyg B and KIC 8379927 does not reproduce the Legacy values. Frequency comparison when using covariance matrices produces anomalous results including negative χ²; all 3 covariance matrices are inconsistent as they have negative eigenvalues and are therefore not positive semi-definite as any covariance matrix should be.

The Legacy errors on separation ratios are up to 40 times larger than my values and exceed values and upper limits derived from the Legacy frequency covariances by a similar factor.

2) I then fitted the power spectra for 6 additional solar-like stars taken from the Legacy high priority list. Here the agreement is not as bad as for 16 Cyg A & B and KIC 8739927, for modes with S/N> 1 the best fit of Roxburgh to Legacy (KIC 6603624) has a χ²< 0.27 (still an order of magnitude larger than the Roxburgh-Davies fits), and good agreement between fits using errors and fits using covariance matrices; the worst fit (KIC 8760414) gave a negative χ² on fitting with the Legacy covariance matrix. The 4 best fits have positive definitive covariance matrices, the 2 worst fits do not. For all 6 stars the Legacy error estimates on the separation ratios exceed the values and upper limits derived using their covariances; KIC 9098294 is the one for which the Legacy values are closest to the values obtained using the frequency covariances.

3) Finally I examined all 66 Legacy targets both to test if their covariance matrices were positive semi-definite, and whether or not the errors on separation ratios satisfied their upper limits. The covariance matrices of 27 stars have negative eigenvalues and are therefore inconsistent, 39 have positive definite covariance matrices. 59 did not satisfy the upper limits on their separation ratio errors, and 4 had ratio errors consistent to within 2% of the re-derived values from their (positive definite) covariance matrices. On fitting the power spectrum of one of these, KIC 3427720; my resulting frequencies still did not agree with the Legacy values, all the fits having a χ² ~ 0.5 whether using Roxburgh or Legacy errors or covariance matrices.

To summarise: results using my mode fitting code agree with those of Davies et al.; results from my code do not agree with the Legacy values; many of the Legacy covariance matrices are inconsistent having negative eigenvalues and therefore are not positive semi-definite; almost all of the Legacy values for errors on the separation ratios do not agree with values and upper limits derived using the Legacy covariance matrices.

It is difficult to escape the conclusion that there is something amiss with the Legacy analysis.

Note added in proof. Lund et al. (2017a) have now produced a draft addendum to their original paper in which they have identified the reason for the overestimation of uncertainties on separation ratios as a missing trimming in their post-processing of the MCMC chains, which reduces their estimates to values comparable to the values obtained by the analysis given in this paper. They will also provide covariance matrices for the mode frequencies, separation ratios and second differences.

Lund M. N., Silva-Aguirre V., Davies G. R., et al. 2017a, manuscript published on the KASOC web site 12/06/2017

Acknowledgments

The author thanks Dr. G. R. Davies for supplying and giving permission to use detailed files of the results of his analyses of 16 Cyg A, 16 Cyg B, and KIC 8379927, and Dr M. N. Lund for supplying and giving permission to use the updated (robust) results of the Legacy analyses including the unpublished covariance matrices. The author gratefully acknowledges support from the UK Science and Technology Facilities Council (STFC) under grant ST/M000621/1.

References

Appendix A: Frequency tables

The following tables give the frequencies for 16 Cyg A, 16 Cyg B and KIC 8379927 as determined by the Legacy project, Roxburgh (Legacy), Davies, and Roxburgh (Davies), and the values of S/N from my analyses where S/N is defined as the maximum height of a rotationally split mode divided by the local background.

All Tables

All Figures

	Fig. 1 16 Cyg B: top 2 panels: frequency differences Legacy-Roxburgh, Davies-Roxburgh and χ2 of fits; bottom panel: error estimates σ02 on the frequency separation ratios r02, Legacy, Davies, Roxburgh.
In the text

	Fig. 2 16 Cyg A: legacy and Roxburgh fits to Legacy power spectrum.
In the text

	Fig. 3 16 Cyg B: quality of free mode fits to Davies power spectrum.
In the text

	Fig. 4 16 Cyg A: inverse covariance matrices. Top: legacy; bottom: Roxburgh [black +ve, red -ve, magnitude = size of points].
In the text

	Fig. 5 Top panel: error estimates σ101 on ratios r101(n) from Legacy, Davies, and Roxburgh analyses of 16Cyg A&B, and KIC 8379927; bottom errors σ02 on ratios r02.
In the text

	Fig. 6 Fractional difference between Davies’s MCMC values for the errors σ101, σ02 and the values from Eqs. (3), (6), (7). All but two <10-3.
In the text

	Fig. 7 Fractional difference between the Legacy MCMC values for the errors σ101, σ02 and the σLcov values from Eqs. (3), (6), (7).
In the text

	Fig. 8 Ratios of Legacy error estimates σ101, σ02 to the upper limits σL from Eq. (4) for 16 Cyg A & B and KIC 8379927, for modes with S/N> 1. The Legacy values exceed the upper limits by a factor of up to 30.
In the text

	Fig. 9 KIC 6225718: frequency differences, Legacy-Roxburgh.
In the text

	Fig. 10 Logarithm of the error estimates on the ratios r101,r02 for all 6 solar-like stars. The Legacy values exceed Roxburgh’s values by factors ranging from 2 to 50.
In the text

	Fig. 11 Logarithm of the ratio of the error estimates to the upper limits on the ratios r101,r02 for all 66 Legacy stars. The blue triangles are stars with inconsistent covariance matrices, the red stars the values re-computed from the covariance matrices. 59 of the Legacy values exceed one or both upper limits on σ.
In the text