Anomalies in the Kepler Asteroseismic Legacy Project Data. A re-analysis of 16 Cyg A&B, KIC8379927 and 6 solar-like stars

TL;DR

This study re-analyzes data from the Kepler Asteroseismic Legacy Project, revealing significant inconsistencies and errors in the published frequencies, covariance matrices, and errors for several solar-like stars, questioning the reliability of the legacy data.

Contribution

It provides a detailed re-analysis of the Legacy data, identifying specific issues with covariance matrices and error estimates, and compares results with previous studies to highlight discrepancies.

Findings

Legacy covariance matrices often not positive definite

Legacy errors on separation ratios are significantly overestimated

Inconsistencies found in frequencies and errors for multiple stars

Abstract

I compare values of the frequencies, separation ratios, errors and covariance matrices from a new analysis of 9 solar-like stars with the Legacy project values reported by Lund et al and, for 16Cyg A&B and KIC8379927, with values derived by Davies et al. There is good agreement between my results (using Davies power spectra) and Davies's for these 3 stars, but no such agreement with the Legacy project results. My frequencies differ from the Legacy values, there are inconsistencies in the Legacy frequency covariance matrices which are not positive definite, and the Legacy errors on separation ratios are up to 40 times larger than mine and the values and upper limits derived from the Legacy frequency covariances. There are similar anomalies for 6 other solar-like stars: frequencies and separation ratio errors disagree and 2 have non positive definite covariance matrices. There are inconsistencies in the covariance matrices of 27 the 66 stars in the full Legacy set and problems with the ratio errors for the vast majority of these stars}

Tables

Star/KIC	kasoc power spectrum	Quarters
16CygA	kplr012069424_kasoc-wpsd_slc_v1.pow	Q6-17.2
16CygB	kplr012069449_kasoc-wpsd_slc_v2.pow	Q6-17.2
8379927	kplr008379927_kasoc-wpsd_slc_v2.pow	Q2-17.2
9098294	kplr009098294_kasoc-wpsd_slc_v1.pow	Q5-17.2
8760414	kplr008760414_kasoc-wpsd_slc_v1.pow	Q5-17.2
6603624	kplr006603624_kasoc-psd_slc_v1.pow	Q5-17.2
6225718	kplr006225718_kasoc-wpsd_slc_v1.pow	Q6-17.2
6116048	kplr006106415_kasoc-wpsd_slc_v2.pow	Q5-17.2
6106415	kplr006106415_kasoc-wpsd_slc_v2.pow	Q6-16.3

Table 2. Table 1: χ 2 superscript 𝜒 2 \chi^{2} of fits of Roxburgh ( ν L ​ R subscript 𝜈 𝐿 𝑅 \nu_{LR} ) to Legacy ( ν L subscript 𝜈 𝐿 \nu_{L} ) frequencies

Star	${\chi }_L^2$	${\chi }_{LSN}^2$		${\chi }_{LR}^2$	${\chi }_{LRSN}^2$
16 Cyg A	0.791	0.897		7.077	8.590
16 Cyg A∗	0.717	0.805		1.540	1.427
16 Cyg B	1.160	1.296		5.067	1.509
16 Cyg B†	1.155	1.296		1.412	1.509
8379927	1.121	0 .427		0.776	0 .499

Table 3. Table 2: χ 2 superscript 𝜒 2 \chi^{2} of fits of Roxburgh ( ν D ​ R subscript 𝜈 𝐷 𝑅 \nu_{DR} ) to Davies ( ν D subscript 𝜈 𝐷 \nu_{D} ) frequencies

Star	${\chi }_D^2$	${\chi }_{DSN}^2$		${\chi }_{DR}^2$	${\chi }_{DRSN}^2$
16 Cyg A	0.141	0 .045		0.284	0 .062
16 Cyg A∗	0.127	0 .023		0.263	0 .026
16 Cyg B	0.167	0 .022		4.227	0.032
16 Cyg B†	0.137	0 .022		0.376	0 .032
8379927	0.184	0 .033		0.506	0 .034

Table 4. Table 3: χ 2 superscript 𝜒 2 \chi^{2} of fits of Davies ( ν D subscript 𝜈 𝐷 \nu_{D} ) to Legacy ( ν L subscript 𝜈 𝐿 \nu_{L} ) frequencies

Star	${\chi }_L^2$	${\chi }_{LSN}^2$		${\chi }_D^2$	${\chi }_{DSN}^2$
16 Cyg A	0.882	0 .642		1.424	0 .873
16 Cyg A∗	0.784	0 .642		0.949	0 .873
16 Cyg B	1.617	1.496		1.786	1.910
16 Cyg B†	1.631	1.496		1.742	1.910
8379927	0.936	0 .663		0.586	0 .570

Table 5. Table 4: Fit for rotation Davies, Roxburgh (Dspec), Roxburgh (Lspec)

	Davies		RoxD		RoxL
Star/KIC	${\nu }_{\Omega }\sin i$	$i$	${\nu }_{\Omega }\sin i$	$i$	${\nu }_{\Omega }\sin i$	$i$
16 Cyg A	$0.41\pm 0.01$	$56\pm 6$	$0.40\pm 0.01$	$56\pm 4$	$0.40\pm 0.03$	$46\pm 3$
16 Cyg B	$0.27\pm 0.02$	$36\pm 12$	$0.27\pm 0.01$	$34\pm 3$	$0.26\pm 0.01$	$50\pm 5$
8379927	$1.11\pm 0.03$	$63\pm 6$	$1.11\pm 0.03$	$66\pm 5$	$1.11\pm 0.02$	$80\pm 10$

Table 6. Table 5: χ 2 superscript 𝜒 2 \chi^{2} of covariance fits of Roxburgh ( ν L ​ R subscript 𝜈 𝐿 𝑅 \nu_{LR} ) to Legacy ( ν L subscript 𝜈 𝐿 \nu_{L} )

Star	${\chi }_L^2$	${\chi }_{LSN}^2$		${\chi }_{LR}^2$	${\chi }_{LRSN}^2$
16CygA	-3.719	0.464		7.072	8.572
16CygB	-0.675	-0.432		5.071	1.511
8379927	1.293	0.211		0.797	0.504

Table 7. Table 8: Fit of Roxburgh to Legacy frequencies: 6 solar-like stars

Star		${\chi }_L^2$	${\chi }_{LSN}^2$		${\chi }_{LR}^2$	${\chi }_{LRSN}^2$
9098294	$\sigma$	0.535	0.505		0.861	0.502
	cov	0.523	0.485		0.814	0.487
8760414	$\sigma$	15.170	0.355		13.462	0.380
	cov	26.301	-0.041		16.141	0.383
6603624	$\sigma$	0.762	0.226		1.967	0.268
	cov	0.794	0.225		2.001	0.268
6225718	$\sigma$	0.852	0.518		1.141	0.615
	cov	1.094	0.643		1.136	0.615
6116048	$\sigma$	0.512	0.443		0.828	0.597
	cov	0.176	0.486		0.809	0.583
6106415	$\sigma$	1.479	1.100		1.671	1.169
	cov	2.106	1.294		1.592	1.166

Table 8. Table 9: Fit of Roxburgh to Legacy frequencies: KIC3427720

Star		${\chi }_L^2$	${\chi }_{LSN}^2$		${\chi }_R^2$	${\chi }_{RSN}^2$
3427720	$\sigma$	0.502	0.492		0.798	0.512
	cov	0.513	0.491		0.810	0.512

L	n		${\nu }_L$	${\sigma }_L$		${\nu }_{LR}$	${\sigma }_{LR}$	S/N		${\nu }_D$	${\sigma }_D$		${\nu }_{DR}$	${\sigma }_{DR}$	S/N
1	11		1334.285	1.006		1334.401	0.062	0.95		0.000	0.000		0.000	0.000	0.00
0	12		1390.808	0.863		1391.648	0.063	1.44		0.000	0.000		0.000	0.000	0.00
1	12		1437.385	0.447		1437.580	0.063	1.39		0.000	0.000		0.000	0.000	0.00
2	12		1487.831	0.707		1487.349	0.122	0.52		1488.237	0.515		1488.368	0.053	0.77
0	13		1495.053	0.235		1494.961	0.064	1.97		1495.002	0.073		1494.991	0.049	3.46
1	13		1542.060	0.140		1541.952	0.054	2.45		1541.922	0.065		1541.906	0.048	2.07
2	13		1590.366	0.387		1591.180	0.074	1.05		1591.291	0.187		1591.224	0.122	0.71
0	14		1598.690	0.072		1598.683	0.053	3.90		1598.694	0.070		1598.690	0.064	2.53
1	14		1645.140	0.109		1644.996	0.099	3.76		1645.063	0.086		1645.046	0.088	2.67
2	14		1693.937	0.186		1694.037	0.181	1.16		1694.167	0.170		1694.219	0.166	1.02
0	15		1700.952	0.101		1700.915	0.088	3.15		1700.911	0.083		1700.899	0.080	3.04
1	15		1747.199	0.085		1747.181	0.081	5.23		1747.149	0.076		1747.150	0.080	4.67
2	15		1795.843	0.131		1795.816	0.108	2.10		1795.747	0.107		1795.750	0.110	1.93
0	16		1802.351	0.079		1802.317	0.070	6.06		1802.310	0.068		1802.312	0.071	5.79
3	15		0.000	0.000		0.000	0.000	0.00		1838.516	0.669		1838.333	0.309	0.29
1	16		1849.009	0.056		1849.016	0.056	9.83		1848.976	0.053		1848.977	0.056	8.29
2	16		1898.399	0.098		1898.344	0.093	3.88		1898.262	0.099		1898.278	0.099	3.23
0	17		1904.521	0.058		1904.591	0.050	11.95		1904.609	0.058		1904.612	0.055	10.00
3	16		0.000	0.000		0.000	0.000	0.00		1941.223	0.562		1940.714	0.320	0.49
1	17		1952.008	0.050		1952.027	0.049	17.66		1951.996	0.049		1951.997	0.050	14.11
2	17		2001.588	0.082		2001.732	0.079	6.90		2001.673	0.077		2001.663	0.071	5.63
0	18		2007.538	0.045		2007.571	0.042	21.20		2007.576	0.046		2007.576	0.043	17.75
3	17		2045.851	0.368		2045.876	0.229	0.88		2045.976	0.365		2045.912	0.195	0.84
1	18		2055.493	0.047		2055.502	0.048	29.27		2055.524	0.047		2055.526	0.048	23.59
2	18		2105.374	0.056		2105.334	0.049	10.83		2105.312	0.055		2105.306	0.052	9.16
0	19		2110.949	0.041		2110.900	0.041	33.95		2110.909	0.039		2110.914	0.040	29.91
3	18		2150.057	0.204		2149.943	0.150	1.17		2149.936	0.134		2149.929	0.134	1.14
1	19		2159.149	0.049		2159.167	0.047	37.40		2159.151	0.044		2159.149	0.046	30.98
2	19		2208.928	0.072		2208.956	0.069	11.60		2208.900	0.064		2208.894	0.064	9.97
0	20		2214.225	0.054		2214.274	0.048	32.68		2214.224	0.048		2214.222	0.050	28.84
3	19		2253.796	0.250		2253.329	0.157	1.16		2253.535	0.163		2253.533	0.153	1.08
1	20		2262.562	0.051		2262.552	0.051	34.77		2262.537	0.048		2262.534	0.049	28.91
2	20		2312.505	0.079		2312.526	0.082	9.46		2312.536	0.087		2312.525	0.085	7.84
0	21		2317.282	0.057		2317.321	0.052	24.78		2317.322	0.051		2317.330	0.053	20.99
3	20		2357.497	0.227		2357.226	0.200	0.91		2357.392	0.189		2357.341	0.198	0.79
1	21		2366.245	0.060		2366.229	0.061	24.93		2366.248	0.057		2366.253	0.062	20.03
2	21		2416.249	0.123		2416.349	0.113	5.91		2416.249	0.127		2416.260	0.127	4.52
0	22		2420.937	0.080		2420.959	0.079	12.50		2420.897	0.081		2420.920	0.084	9.35
3	21		2461.452	0.358		2461.688	0.373	0.55		2462.078	0.385		2461.877	0.405	0.45
1	22		2470.227	0.091		2470.361	0.082	13.20		2470.305	0.077		2470.298	0.086	10.01
2	22		2520.734	0.199		2520.618	0.174	3.15		2520.459	0.212		2520.475	0.191	2.51
0	23		2524.950	0.156		2525.071	0.132	5.59		2525.071	0.158		2525.154	0.141	4.44
3	22		0.000	0.000		0.000	0.000	0.00		2566.969	0.608		2567.284	0.662	0.25
1	23		2574.660	0.121		2574.691	0.125	5.95		2574.784	0.126		2574.792	0.129	4.85
2	23		2624.636	0.362		2624.975	0.369	1.34		2624.322	0.324		2624.331	0.334	1.21
0	24		2628.930	0.259		2629.294	0.237	2.04		2629.204	0.178		2629.245	0.201	1.93
3	23		0.000	0.000		0.000	0.000	0.00		2669.765	1.036		2668.860	1.209	0.13
1	24		2679.726	0.201		2679.406	0.201	2.57		2679.872	0.188		2679.857	0.203	2.31
2	24		2730.024	0.756		2729.839	0.546	0.80		2730.233	0.886		2729.550	0.716	0.68
0	25		2733.571	0.420		2734.482	0.394	1.24		2733.615	0.463		2734.049	0.370	1.08
1	25		2783.816	0.335		2784.118	0.337	1.37		2784.222	0.354		2784.243	0.364	1.13
2	25		2836.088	0.798		2836.291	1.692	0.31		2835.339	1.147		2834.364	2.878	0.24
0	26		2840.148	0.944		2838.819	1.264	0.47		2838.398	0.779		2838.578	0.922	0.35
1	26		2890.198	0.692		2890.361	0.719	0.59		2891.270	0.740		2891.381	0.814	0.45
2	26		2940.393	1.103		2941.252	3.200	0.18		2941.479	1.538		2939.644	2.527	0.15
0	27		2944.937	0.792		2945.011	1.683	0.27		2945.321	1.179		2946.213	1.033	0.23
1	27		2994.840	1.013		2994.958	1.981	0.27		2996.375	1.191		2996.211	1.907	0.28

L	n		${\nu }_L$	${\sigma }_L$		${\nu }_{LR}$	${\sigma }_{LR}$	S/N		${\nu }_D$	${\sigma }_D$		${\nu }_{DR}$	${\sigma }_{DR}$	S/N
1	12		1631.088	0.286		1631.105	0.035	1.61		0.000	0.000		0.000	0.000	0.00
2	12		1685.793	0.664		1686.578	0.057	0.55		1686.419	0.313		1686.822	0.032	0.57
0	13		1695.023	0.126		1695.061	0.063	2.25		1695.069	0.087		1695.069	0.053	2.71
1	13		1749.253	0.183		1749.189	0.084	1.97		1749.214	0.101		1749.186	0.062	2.53
2	13		1804.243	0.587		1803.859	0.211	0.58		1804.168	0.273		1804.249	0.170	0.51
0	14		1812.444	0.133		1812.440	0.068	1.89		1812.428	0.097		1812.412	0.078	1.59
1	14		1866.483	0.117		1866.511	0.092	2.67		1866.523	0.118		1866.521	0.098	2.50
2	14		1921.246	0.181		1921.152	0.139	1.00		1921.206	0.160		1921.194	0.162	0.91
0	15		1928.886	0.103		1928.908	0.076	2.86		1928.901	0.072		1928.899	0.070	2.53
3	14		0.000	0.000		0.000	0.000	0.00		1970.959	5.137		1973.695	0.375	0.15
1	15		1982.607	0.084		1982.498	0.073	4.41		1982.592	0.071		1982.586	0.072	4.60
2	15		2037.203	0.177		2036.815	0.192	1.63		2036.667	0.137		2036.676	0.128	1.72
0	16		2044.357	0.069		2044.305	0.067	4.44		2044.278	0.060		2044.273	0.058	4.97
3	15		0.000	0.000		0.000	0.000	0.00		2085.370	1.498		2085.478	0.353	0.24
1	16		2098.163	0.064		2098.087	0.058	7.20		2098.084	0.057		2098.081	0.058	7.19
2	16		2152.517	0.109		2152.440	0.098	2.68		2152.420	0.102		2152.419	0.099	2.47
0	17		2159.503	0.058		2159.612	0.061	7.40		2159.581	0.057		2159.577	0.059	6.35
3	16		0.000	0.000		0.000	0.000	0.00		2200.579	1.224		2200.453	0.354	0.37
1	17		2214.334	0.069		2214.208	0.056	12.41		2214.166	0.056		2214.163	0.058	11.31
2	17		2269.112	0.094		2269.034	0.073	4.76		2268.956	0.083		2268.957	0.083	4.37
0	18		2275.949	0.054		2275.994	0.049	13.13		2275.948	0.048		2275.949	0.047	11.44
3	17		2318.958	0.290		2318.917	0.230	0.78		2319.120	0.374		2319.208	0.214	0.67
1	18		2331.163	0.041		2331.141	0.042	23.45		2331.138	0.040		2331.139	0.043	21.01
2	18		2386.252	0.070		2386.214	0.057	8.82		2386.263	0.061		2386.262	0.060	7.66
0	19		2392.645	0.042		2392.711	0.041	26.87		2392.711	0.043		2392.719	0.040	22.21
3	18		2436.781	0.255		2436.409	0.250	1.19		2436.656	0.299		2436.744	0.194	1.04
1	19		2448.181	0.048		2448.237	0.041	35.07		2448.253	0.041		2448.251	0.042	32.20
2	19		2503.411	0.066		2503.444	0.059	11.41		2503.498	0.060		2503.497	0.059	10.54
0	20		2509.678	0.042		2509.659	0.040	33.15		2509.667	0.041		2509.668	0.040	29.09
3	19		2554.181	0.188		2554.026	0.125	1.41		2554.146	0.147		2554.167	0.157	1.25
1	20		2565.426	0.043		2565.422	0.042	40.07		2565.403	0.042		2565.400	0.042	37.08
2	20		2620.562	0.066		2620.534	0.059	12.00		2620.564	0.066		2620.562	0.062	10.89
0	21		2626.458	0.050		2626.413	0.045	32.90		2626.397	0.045		2626.397	0.044	28.60
3	20		2671.592	0.260		2671.703	0.174	1.32		2671.722	0.168		2671.738	0.167	1.11
1	21		2682.247	0.047		2682.407	0.048	34.54		2682.402	0.048		2682.407	0.049	29.52
2	21		2737.707	0.075		2737.666	0.073	8.74		2737.744	0.079		2737.743	0.080	7.02
0	22		2743.322	0.062		2743.346	0.054	20.06		2743.329	0.058		2743.330	0.060	14.41
3	21		2789.000	0.365		2788.887	0.250	0.90		2789.155	0.276		2789.141	0.288	0.66
1	22		2799.613	0.072		2799.721	0.062	19.92		2799.734	0.063		2799.737	0.065	15.06
2	22		2855.507	0.121		2855.569	0.111	4.28		2855.631	0.124		2855.619	0.120	3.50
0	23		2860.680	0.098		2860.762	0.092	8.15		2860.720	0.101		2860.749	0.099	6.33
3	22		2906.905	0.490		2906.922	0.391	0.44		2906.865	0.435		2906.862	0.445	0.34
1	23		2917.890	0.110		2917.824	0.100	8.31		2917.793	0.097		2917.784	0.101	6.82
2	23		2973.400	0.302		2973.535	0.234	1.80		2973.564	0.235		2973.535	0.217	1.66
0	24		2978.180	0.175		2978.454	0.157	3.09		2978.504	0.151		2978.529	0.145	2.79
3	23		0.000	0.000		0.000	0.000	0.00		3025.061	1.128		3024.682	1.203	0.16
1	24		3035.810	0.174		3036.046	0.166	3.27		3036.058	0.155		3036.048	0.164	2.97
2	24		3092.492	0.577		3092.285	0.433	0.75		3093.036	0.507		3092.795	0.424	0.68
0	25		3097.170	0.419		3096.476	0.403	1.27		3096.850	0.419		3097.107	0.372	1.10
3	24		0.000	0.000		0.000	0.000	0.00		3144.035	1.415		3144.275	1.288	0.07
1	25		3154.703	0.300		3154.229	0.267	1.45		3154.307	0.290		3154.291	0.291	1.21
2	25		3210.654	1.187		3212.063	1.445	0.39		3213.398	1.536		3211.987	0.943	0.30
0	26		3216.451	0.482		3215.846	0.533	0.66		3214.925	1.040		3215.900	0.697	0.48
1	26		3273.587	0.473		3273.266	0.502	0.79		3273.168	0.643		3273.312	0.659	0.62
2	26		3330.030	2.226		3330.323	1.340	0.23		3333.060	2.743		3331.294	1.357	0.20
0	27		3336.009	1.060		3336.187	1.516	0.39		3334.219	1.903		3337.847	1.054	0.33
1	27		3391.761	1.090		3393.623	0.821	0.37		3393.448	0.768		3393.091	0.709	0.39

L	n		${\nu }_L$	${\sigma }_L$		${\nu }_{LR}$	${\sigma }_{LR}$	S/N		${\nu }_D$	${\sigma }_D$		${\nu }_{DR}$	${\sigma }_{DR}$	S/N
0	13		0.000	0.000		0.000	0.000	2.25		1728.138	0.435		1728.025	0.141	0.72
1	13		1783.395	0.441		1783.819	0.169	0.54		1783.342	0.273		1783.389	0.164	0.45
0	14		1847.244	0.243		1847.300	0.146	0.76		1847.636	0.854		1847.426	0.174	0.61
1	14		1903.592	0.282		1903.637	0.131	0.68		1904.672	0.846		1903.864	0.191	0.47
2	14		0.000	0.000		0.000	0.000	1.00		1954.857	0.725		1954.580	0.678	0.15
0	15		1967.982	0.255		1967.920	0.154	0.91		1968.190	0.217		1968.206	0.197	0.69
1	15		2023.838	0.297		2023.892	0.230	0.98		2023.666	0.310		2023.654	0.244	0.67
2	15		0.000	0.000		0.000	0.000	1.63		2075.323	0.590		2075.643	0.432	0.26
0	16		2087.937	0.198		2087.844	0.152	1.27		2087.990	0.156		2087.960	0.144	1.05
1	16		2143.133	0.180		2143.140	0.156	1.55		2143.237	0.178		2143.207	0.171	1.14
2	16		2195.244	0.469		2194.891	0.256	0.58		2195.355	0.299		2195.316	0.338	0.43
0	17		2206.506	0.150		2206.692	0.126	1.97		2206.657	0.126		2206.635	0.120	1.78
1	17		2261.245	0.136		2261.106	0.122	2.37		2261.245	0.116		2261.218	0.118	1.88
2	17		2312.707	0.370		2312.251	0.313	0.92		2312.901	0.276		2312.899	0.264	0.74
0	18		2324.439	0.106		2324.451	0.112	3.16		2324.322	0.111		2324.307	0.109	2.80
1	18		2379.779	0.099		2379.770	0.095	3.49		2379.939	0.108		2379.911	0.106	2.72
2	18		2432.197	0.204		2432.318	0.178	1.28		2432.294	0.223		2432.348	0.199	0.95
0	19		2443.152	0.101		2443.116	0.096	4.34		2443.150	0.103		2443.131	0.096	3.61
1	19		2499.437	0.092		2499.391	0.088	5.12		2499.398	0.104		2499.388	0.099	3.65
2	19		2552.415	0.153		2552.284	0.126	1.96		2552.244	0.166		2552.222	0.171	1.35
0	20		2563.543	0.086		2563.596	0.077	6.75		2563.605	0.082		2563.587	0.081	5.00
1	20		2619.926	0.090		2619.986	0.093	6.54		2619.991	0.086		2619.970	0.092	4.81
2	20		2673.136	0.145		2673.038	0.136	2.17		2673.135	0.145		2673.157	0.133	1.69
0	21		2683.948	0.099		2683.962	0.088	6.92		2684.018	0.097		2683.995	0.092	6.02
1	21		2740.437	0.087		2740.463	0.088	6.95		2740.526	0.089		2740.507	0.092	5.25
2	21		2793.385	0.159		2793.453	0.154	2.29		2793.482	0.185		2793.435	0.184	1.70
0	22		2804.566	0.087		2804.494	0.087	7.05		2804.495	0.093		2804.480	0.096	5.06
1	22		2860.993	0.095		2861.027	0.097	6.68		2861.002	0.107		2860.981	0.107	4.89
2	22		2913.836	0.177		2914.001	0.163	2.01		2914.039	0.219		2914.029	0.200	1.56
0	23		2924.530	0.094		2924.483	0.092	6.06		2924.469	0.112		2924.447	0.107	4.42
1	23		2981.323	0.120		2981.304	0.115	5.37		2981.241	0.123		2981.220	0.123	4.06
2	23		3034.225	0.276		3034.302	0.255	1.45		3033.920	0.271		3033.886	0.262	1.15
0	24		3044.841	0.130		3044.850	0.131	3.48		3044.862	0.129		3044.853	0.129	2.92
1	24		3102.033	0.157		3101.964	0.156	3.59		3102.002	0.155		3101.960	0.162	2.80
2	24		3155.043	0.376		3155.020	0.374	0.99		3154.846	0.359		3154.852	0.351	0.77
0	25		3165.482	0.243		3165.537	0.231	1.91		3165.555	0.233		3165.580	0.234	1.60
1	25		3223.052	0.226		3223.183	0.222	2.23		3223.302	0.246		3223.279	0.246	1.74
2	25		3275.789	0.587		3276.510	0.536	0.67		3275.749	0.585		3275.754	0.545	0.53
0	26		3286.718	0.298		3286.792	0.306	1.25		3286.935	0.365		3287.007	0.329	0.97
1	26		3344.137	0.294		3344.679	0.287	1.50		3344.479	0.386		3344.531	0.363	1.11
2	26		3397.777	0.684		3398.078	0.608	0.46		3397.302	0.668		3397.091	0.721	0.34
0	27		3408.187	0.477		3409.214	0.433	0.81		3408.663	0.512		3408.606	0.532	0.61
1	27		3465.693	0.433		3466.067	0.419	0.93		3466.404	0.472		3466.430	0.499	0.69
2	27		3518.572	0.951		3518.691	1.077	0.26		3520.401	1.085		3519.568	1.151	0.21
0	28		3531.333	0.741		3531.243	0.775	0.41		3531.755	0.927		3532.584	1.107	0.34
1	28		3587.270	0.657		3587.792	0.810	0.53		3587.318	1.097		3587.141	1.312	0.39
2	28		3640.416	1.853		3641.615	1.818	0.17		3641.946	1.253		3643.727	2.903	0.12
0	29		3651.161	0.840		3650.143	1.027	0.28		3650.679	0.795		3649.769	1.716	0.19
1	29		3710.839	0.840		3710.035	0.897	0.36		3710.678	0.938		3710.382	1.182	0.27
2	29		3762.282	1.692		3771.014	7.568	0.12		3762.746	1.892		3764.758	2.095	0.10
0	30		3769.722	1.096		3768.325	3.623	0.19		3770.309	1.537		3769.174	1.213	0.19
1	30		3836.226	1.273		3837.493	1.676	0.20		3835.787	1.141		3836.531	0.888	0.24

Equations

$$r_{101}(n)=\frac{d_{101}(n)}{{\Delta }_n},\text{leavevmode}\text{leavevmode}{r}_{02}(n)=\frac{d_{02}(n)}{{\Delta }_n}$$

$$d_{101}(n)=\frac{1}{8}\left[{\nu }_{n-1,0}-4{\nu }_{n-1,1}+6{\nu }_{n,0}-4{\nu }_{n,1}+{\nu }_{n+1,0}\right]$$

$$d_{02}(n)={\nu }_{n,0}-{\nu }_{n-1,2},\text{leavevmode}\text{leavevmode}\text{leavevmode}\mathrm{and}\text{leavevmode}\text{leavevmode}\text{leavevmode}{\Delta }_n={\nu }_{n,1}-{\nu }_{n-1.1}$$

$$cov(r_n,r_m)=\sum _j\sum _k{A}_j{B}_{k}cov({\nu }_j,{\nu }_k)$$

$${\sigma }_n^2=cov(r_n,r_n)=\sum _j\sum _k{A}_j{A}_{k}cov({\nu }_j,{\nu }_k)=$$

$$\sum _j\sum _k{A}_j{A}_{k}cor{r}_{jk}{\sigma }_j{\sigma }_k$$

$${\sigma }_n\le {\sigma }_L,\text{leavevmode}\text{leavevmode}\text{leavevmode}{\sigma }_L^2=\sum _j\sum _k|A_j{A}_k|{\sigma }_j{\sigma }_k$$

$$r_n=\frac{d_n}{{\Delta }_n}=\frac{d_n}{{\Delta }_0}-\frac{d_0{\Delta }_n}{{\Delta }_0^2}+\frac{d_0}{{\Delta }_0}=\frac{1}{{\Delta }_0}\left[\sum _k{D}_k{\nu }_k-r_0{\Delta }_n\right]+r_0$$

$$\{{\nu }_k\}=\{{\nu }_1,\text{leavevmode}{\nu }_2,\text{leavevmode}{\nu }_3,\text{leavevmode}{\nu }_4\}=\{{\nu }_{n-1,1},\text{leavevmode}{\nu }_{n-1,2},\text{leavevmode}{\nu }_{n,0},\text{leavevmode}{\nu }_{n,1}\}$$

$$\{A_k\}=\left\{\frac{r_0}{{\Delta }_0},\text{leavevmode}-\frac{1}{{\Delta }_0},\text{leavevmode}\frac{1}{{\Delta }_0},\text{leavevmode}-\frac{r_0}{{\Delta }_0}\right\}$$

$$\{{\nu }_k\}=\{{\nu }_1,{\nu }_2,{\nu }_3,{\nu }_4\text{leavevmode}{\nu }_5\}=\{{\nu }_{n-1,0},{\nu }_{n-1,1},{\nu }_{n,0},{\nu }_{n,1},{\nu }_{n+1,0}\}$$

$$\{A_k\}=\left\{\frac{1}{8{\Delta }_0},-\frac{1-2r_0}{2{\Delta }_0},\frac{3}{4{\Delta }_0},-\frac{1+2r_0}{2{\Delta }_0},\frac{1}{8{\Delta }_0}\right\}$$

Full text

1 Introduction

The Kepler Asteroseismic Legacy Project (Lund et al, 2017) analysed 66 Kepler main sequence targets providing 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 latest (robust) values.

In sections 3 to 7 I compare my results for 3 Kepler targets, 16 CygA&B and KIC8379927, with the Legacy values and results from independent analyses by Davies et al (2015a,b, 2016), using Davies’ power spectra. My results agree well with those of Davies et al, but do not agree with the Legacy project 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 $\chi^{2}$ when comparing frequency sets,

In section 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 section 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 16CygB for modes with heights greater than the background ($S/N>1$) - which are less sensitive to background modelling and misidentification of noise for signal than is the case modes with $S/N<1$. I also gives the $\chi^{2}$ of the fits using the different error estimates. The bottom panel compares errors on the separation ratios $r_{02}$ 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 $\sim 300\mu$Hz beyond both ends of the range of frequencies to be fitted, with unconstrained parameters $X_{k}$: frequencies $\nu_{n,\ell}$; mode heights $h_{n}$ and widths $w_{n}$ of the $\ell=0$ modes; mode height ratios $h_{10},h_{20},h_{30}$ of modes $\ell=1,2,3$ to the heights of modes with $\ell=0$ (with the geometrical constraint $1+h_{20}=h_{10}+h_{30}$), the same for all modes; rotational splitting $\nu_{\Omega}$ and inclination $i$ (the same for all modes); and 4 parameters of a Harvey-like model of the background ($A/[1+B\nu^{c}]+D$). The heights and widths of the $\ell=1,2,3$ modes are determined by (linear) interpolation in the values for the $\ell=0$ 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 $H(i,j)=\partial^{2}MLE/\partial X_{i}\partial X_{j}$ and the errors on the $X_{k}$ are given as $\sigma_{k}=[H^{-1}(k,k)]^{1/2}$.

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:16CygA&B, KIC 8379927

Tables 1 to 3 gives the $\chi^{2}$ 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 section 5 below).

Table 1 compares the fit of the Legacy frequencies and errors ($\nu_{L}\pm\sigma_{L}$) to those of Roxburgh ($\nu_{LR}\pm\sigma_{LR}$) (using the Legacy power spectra), $\chi^{2}_{L}$ is the value using Legacy errors and $\chi^{2}_{LR}$ using Roxburgh’s errors. $\chi^{2}_{LSN}$ is the value using Legacy errors but only comparing frequencies with S/N ¿1, and likewise $\chi^{2}_{LRSN}$. 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 $\nu_{DR}$ (using Davies’s power spectra) to Davies’s frequencies, $\nu_{D}$ 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 16CygB for all frequencies is strongly influenced by the misfit of the $\nu_{14,3}$ mode which has S/N=0.15 and is unreliable; the Roxburgh-Davies fit for 16CygA for modes with S/N ¡1 is strongly influenced by the $\nu_{25,0}$ mode which has $S/N=1.08$, if this is excluded $\chi^{2}_{DSN}=0.023,\,\chi^{2}_{DRSN}=0.026$.

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 both the Legacy and Davies power spectra; there is very good agreement for all 3 stars, the fits to the Legacy spectra yielding almost the same values as obtained in fitting the Davies spectra.

4 Fitting low frequency modes

As stated above and in the footnotes to the tables there are some problems in fitting some low frequency modes. For 16CygA Legacy fits (table 1) the problem is illustrated in Fig 2 which shows the kasoc power spectrum for 16CygA smoothed by a Gaussian smoother (with FWHM of $0.1\mu$Hz), and overlaid the Roxburgh fit to the full power spectrum and the location of the Legacy and Roxburgh frequencies for modes $\nu_{12,0}$ and the pair $\nu_{13.2},\nu_{14,0}$. The Legacy values for $\nu_{12,0}$ and $\nu_{13.2}$ 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 $\chi^{2}_{LR}$ and $\chi^{2}_{LRSN}$ from 7.077 and 8.590 to 1.540 and 1.427 respectively. A similar problem exists for the fit to $\nu_{12,2}$ for 16CygB; excluding this mode reduces the $\chi^{2}_{LR}$ of the fit using Roxburgh’s errors from 5.067to 1.412. The S/N values remain unchanged since this mode has S/N¡1.

Davies’s value of $\nu_{12,2}$ for 16CygB is also a poor fit to his power spectrum. $\nu_{14,3}$ (which has S/N=0.15) differs from my value by $\sim 3\mu$Hz so I determined the quality of fits to the section of the Davies power spectrum between $1982.6\pm 29\mu$Hz for a $100^{2}$ matrix of values of $\nu_{15,1},\nu_{14,3}$ and 10 values of height ratio $h_{31}$ between $0.01$ to $0.1$, with fitting parameters the $\ell=1$ mode height, one width for both $\ell=1$ and $3$ and a constant background; all with $\{\nu_{\Omega}\sin i,i\}=\{0.27,34\}$. Fig 3 shows the quality of fits ($MLE-MLE_{min}$) for 2 values of $h_{31}$; the best fits for all $h_{31}$ have $\nu_{14,3}=1973.71\mu$Hz; my full fit value is $1973.69\pm 0.37\mu$Hz.

5 Covariance matrices and frequency comparison

The $\chi^{2}$s of the fit of $N$ frequencies incorporating their correlations are given by $[D\,C^{-1}\,D^{T}]/N$ where $D$ is the vector of frequency differences and $C^{-1}$ the inverse of the frequency covariance matrix $C$. Tables 5,6,7 give the results of such fits for 16CygA&B and KIC8379927 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 $\chi^{2}$ for the Roxburgh-Davies fits are compatible (and small) and consistent with the values using frequency errors as given in tables 1 to 3, the $\chi^{2}$s 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 $\chi^{2}$.

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 $det(C-w_{j}U)$ and $det(C+w_{j}U)$, is zero, or closest to zero given rounding errors. [U is the unit matrix] All 3 Legacy covariance matrices have negative eigenvalues, 16CygA having 10, 16CygB 12, KIC8379927 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 16CygA [magnitude=size of points, black +ve, red -ve]. Something is clearly amiss with the Legacy evaluation of the covariance matrices from their MCMC analysis.

6 Frequency separation ratios

The ratios of small ($d$) to large ($\Delta$) 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)

[TABLE]

where

[TABLE]

[TABLE]

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 $r_{010}$ ratios but these do not contain any additional information since from 2N ($\ell=0,1$) frequencies one can only determine N surface layer independent quantities.

The values of the ratios $r_{101}$ and $r_{02}$ 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 $\sigma_{101}$ on the ratios $r_{101}$ 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 $\sigma_{02}$ on $r_{02}$. 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 $r_{n}(\nu_{j})=\sum A_{j}\nu_{j}$, and $r_{m}(\nu_{k})=\,\sum B_{k}\nu_{k}$ of variables $\nu_{i}$ is given by

[TABLE]

and the error estimate $\sigma_{n}$ on $r_{n}(\nu_{k})$ is given by the variance

[TABLE]

[TABLE]

where $corr_{jk}$ are the correlations and $\sigma_{i}$ the error estimates on $\nu_{i}$. Since $|corr_{jk}|\leq 1$ it follows that an upper bound on $\sigma_{n}$ is given by taking $corr_{jk}=+1$ if $A_{j}\,A_{k}>0$ and $-1$ if negative, hence

[TABLE]

The small separations $d_{n}$ [both $d_{101}(n)$ and $d_{02}(n)$] are linear functions of $\nu$,  $d_{n}=\sum D_{k}\nu_{k}$ (cf Eqn 1b, 1c), but the contribution of the large separation $\Delta_{n}$ introduces a small non linearity in the ratios. To a good approximation ($1\,{\rm in}\,10^{3}$ see below) this can be incorporated by expanding around the average value for the differences $d_{0}$, large separation $\Delta_{0}$ and ratios $r_{0}$ ,which gives

[TABLE]

which is a linear function of $\nu_{k}$ as $\Delta_{n}$ is a linear function of $\nu_{k}$.

The constant term $r_{0}$ makes no contribution to the covariances so, with the $A_{k}$ defined through Eqn 5 (given below), the error $\sigma_{n}$ on $r_{n}$ is given by Eqn 3 and the upper limit by Eqn 4.

**Coefficients $A_{k}$ for the errors $\sigma_{02}(n),\sigma_{101}$ on $r_{02}(n),r_{101}(n)$ ** For $r_{0}=r_{02}(n)$, $\Delta_{0}=\nu_{n,1}-\nu_{n-1,1}$

[TABLE]

[TABLE]

For $r_{0}=r_{101}(n)$, $\Delta_{0}=\nu_{n,1}-\nu_{n-1,1}$

[TABLE]

[TABLE]

Fig 6 shows the fractional differences between the $\sigma$’s given by Davies’s MCMC analysis of 16CygA&B and KIC 8379927, and the $\sigma_{Dcov}$ given by Eqns 3, 6 and 7, using Davies’s frequencies and frequency covariances; all but two are less than $10^{-3}$. The two are KIC8379927 $\sigma_{02}(n),\,n=14,15$, which have values $-2.5\,10^{-3},\,4.4\,10^{-3}$, and are derived from modes with S/N¡1.

Fig 7 shows the same comparison but between Legacy $\sigma$’s and the values $\sigma_{Lcov}$ derived using the Legacy frequencies and frequency covariances; here many of the differences are huge.

As shown in Fig 8 the Legacy $\sigma$’s also exceed the upper limits $\sigma_{u}$ given by Eqn 4, whereas Davies’s and Roxburgh’s values, and the re-derived Legacy values $\sigma_{Lcov}$, 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 $\Delta$ in the range $100-120\mu$Hz and $\nu_{max}$ in the range $2138-2470\mu$Hz, namely KIC9098294, 8760414, 6603624, 6225718, 6116048, 6106415. The fit of the Roxburgh to Legacy frequencies for KIC6225718 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 $\sigma$) and the inverse covariance matrices (labelled $cov$) both for all frequencies and for the subset with $S/N>1$.

For KIC 9098294, 6603264, there is good agreement between $\chi^{2}$ using the Legacy covariance matrices and uncorrelated errors, and reasonable agreement for 6225718 and 6106415, but the $\chi^{2}$ are still an order of magnitude larger than the 3 Roxburgh-Davies fits for S/N¿1. The fits for KIC8760414 and 6116048 are not so good: KIC 8760414 having a negative $\chi^{2}$ 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.

Fig 10 plots the Legacy error estimates $\sigma_{101}$ and $\sigma_{02}$ on the separation ratios $r_{101}$ and $r_{02}$ which show a similar behaviour to those of 16CygA&B and KIC 8379927 in that the Legacy estimates are all larger than Roxburgh’s for all 6 stars. For KIC9098294 this is only by a factor $\sim 2$ but for KIC6116048 the Legacy value is up to a factor $50$ larger then Roxburgh’s.

As was the case for 16CygA&B and KIC8379927, the Legacy ratio errors for all of these 6 stars also exceed the upper limits calculated as described in section 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.

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 2017) 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 eigevnalues, and whether they have anomalously large error estimates $\sigma_{101},\sigma_{02}$ for the separation ratios $r_{101},r_{02}$.

The eigenvalues of all 66 Legacy covariance matrices were determined by the same procedure as applied to 16CygA&B and KIC8379927, the absolute magnitudes $w$ from SVD, and the sign from the determinants $||C\pm wU||$. 27 have covariance matrices with negative eigenvalues and are therefore inconsistent, the remaining 39 stars have positive definite covariance matrices.

Next I compare the Legacy values for the error estimates $\sigma_{101},\sigma_{02}$ on the separation ratios $r_{101},r_{02}$ with the values re-derived from the frequency covariance matrices and the upper limits as determined by Eqns 3,4 6 and 7 in section 7. Fig 11 shows the Legacy data error estimates $\sigma_{101},\sigma_{02}$ divided by the upper limits and the re-derived values divided by the upper limits all averaged over 3 values around their $\nu_{max}$. The blue triangles are stars with inconsistent covariance matrices (negative eigenvalues). 7 stars have values of $\sigma_{101}$ and $\sigma_{02}$ less than their upper limits all of which have positive definite covariance matrices, of which KIC3427720, 8938364, 9353712 and 10079226 have Legacy values for ratio errors within 2% of the re-derived values from their covariance matrices.

I selected KIC3427720, 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 $\chi^{2}$s 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 !6CygA&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 16CygA, 16CygB 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 $\chi^{2}$ of the fits of Roxburgh to Davies’s frequencies are $\leq 0.062$, both for comparisons using only error estimates and using full covariance matrices ($\leq 0.035$ if one mode with S/N=1.08 is excluded).

The same code when applied to the Legacy power spectra for 16CygA, 16CygB and KIC 8379927 does not reproduce the Legacy values. Frequency comparison when using covariance matrices produces anomalous results including negative $\chi^{2}$; 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 16CygA&B and KIC8739927, for modes with S/N¿1 the best fit of Roxburgh to Legacy (KIC 6603624) has a $\chi^{2}<0.27$ (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 (KIC8760414) gave a negative $\chi^{2}$ 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; KIC9098294 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 there (positive definite) covariance matrices. On fitting the power spectrum of one of these, KIC3427720; my resulting frequencies still did not agree with the Legacy values, all the fits having a $\chi^{2}\sim 0.5$ 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.

Acknowledgements

The author thanks Dr G R Davies for supplying and giving permission to use detailed files of the results of his analyses of 16CygA, 16CygB, and KIC8379927; Dr M N Lund for supplying and giving permission to use the updated (robust) results of the Legacy analyses; and a referee for constructive comments. The author gratefully acknowledges support from the the UK Science and Technology Facilities Council (STFC) under grant ST/M000621/1.

Appendix A Frequency tables

The following tables give the frequencies for 16CygA, 16CygB and KIC8379927 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.