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TEMPORAL VARIATIONS OF THE GLOBAL SEISMIC PARAMETERS OF HD 49933 OVER A MAGNETIC CYCLE

  • Kim, Ki-Beom (Department of Astronomy and Atmospheric Sciences, Kyungpook National University) ;
  • Chang, Heon-Young (Department of Astronomy and Atmospheric Sciences, Kyungpook National University)
  • Received : 2021.05.07
  • Accepted : 2021.07.01
  • Published : 2021.08.31

Abstract

It has been established that the acoustic mode parameters of the Sun and Sun-like stars vary over activity cycles. Since the observed variations are not consistent with an activity-related origin, even Sun-like stars showing out-of-phase changes of mode frequencies and amplitudes need to be carefully studied using other observational quantities. In order to test whether the presumed relations between the global seismic parameters are a signature of the stellar activity cycle, we analyze the photometric light curve of HD 49933 for which the first direct detection of an asteroseismic signature for activity-induced variations in a Sun-like star was made, using observations by the CoRoT space telescope. We find that the amplitude of the envelope significantly anti-correlates with both the maximum frequency of the envelope and the width of the envelope unless superflare-like events completely contaminate the light curve. However, even though the photometric proxy for stellar magnetic activity appears to show relations with the global asteroseismic parameters, they are statistically insignificant. Therefore, we conclude that the global asteroseismic parameters can be utilized in cross-checking asteroseismic detections of activity-related variations in Sun-like stars, and that it is probably less secure and effective to construct a photometric magnetic activity proxy to indirectly correlate the global asteroseismic parameters. Finally, we seismically estimate the mass of HD 49933 based on our determination of the large separation of HD 49933 with evolutionary tracks computed by the MESA code and find a value of about 1.2M and a sub-solar metallicity of Z = 0.008, which agrees with the current consensus and with asteroseismic and non-asteroseismic data.

Keywords

Acknowledgement

The authors thank the anonymous referees for critical comments and helpful suggestions which greatly improved the original version of the manuscript. This study was funded by the Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Science, ICT and Future Planning (2018R1A6A1A06024970). HYC was supported by a National Research Foundation of Korea Grant funded by the Korean government (NRF-2018R1D1A3B070421880).

References

  1. Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W. 2010, Asteroseismology, Astronomy and Astrophysics Library (Berlin: Springer)
  2. Ak, T., Ozkan, M. T., & Mattei, J. A. 2001, Solar-type Cycles of the Secondary Stars in Cataclysmic Variables, A&A, 369, 882 https://doi.org/10.1051/0004-6361:20010167
  3. Appourchaux, T., Michel, E., Auvergne, M., et al. 2008, CoRoT Sounds the Stars: p-mode Parameters of Sun-like Oscillations on HD 49933, A&A, 488, 705 https://doi.org/10.1051/0004-6361:200810297
  4. Appourchaux, T., Belkacem, K., Broomhall, A.-M., et al. 2010, The Quest for the Solar g Modes, A&ARv, 18, 197 https://doi.org/10.1007/s00159-009-0027-z
  5. Arentoft, T., Kjeldsen, H., Bedding, T. R., et al. 2008, A Multisite Campaign to Measure Solar-like Oscillations in Procyon. I. Observations, Data Reduction, and Slow Variations, ApJ, 687, 1180 https://doi.org/10.1086/592040
  6. Baglin, A., Auvergne, M., Barge, P., et al. 2006, Scientific Objectives for a Minisat: CoRoT, The CoRoT Mission Pre-Launch Status - Stellar Seismology and Planet Finding, ESA Special Publication, 1306, 33
  7. Baliunas, S. L., Donahue, R. A., Soon, W. H., et al. 1995, Chromospheric Variations in Main-Sequence Stars. II, ApJ, 438, 269 https://doi.org/10.1086/175072
  8. Balmforth, N. J. 1992, Solar Pulsational Stability - III. Acoustical Excitation by Turbulent Convection, MNRAS, 255, 639 https://doi.org/10.1093/mnras/255.4.639
  9. Basu, S., Christensen-Dalsgaard, J., Chaplin, W. J., et al. 1997, Solar Internal Sound Speed as Inferred from Combined BiSON and LOWL Oscillation Frequencies, MNRAS, 292, 243 https://doi.org/10.1093/mnras/292.2.243
  10. Bedding, T. R., Kjeldsen, H., Campante, T. L., et al. 2010, A Multi-Site Campaign to Measure Solar-Like Oscillations in Procyon. II. Mode Frequencies, ApJ, 713, 935 https://doi.org/10.1088/0004-637X/713/2/935
  11. Belkacem, K., Goupil, M. J., Dupret, M. A., et al. 2011, The Underlying Physical Meaning of the νmaxc Relation, A&A, 530, A142 https://doi.org/10.1051/0004-6361/201116490
  12. Belkacem, K., Samadi, R., Mosser, B., et al. 2013, Progress in Physics of the Sun and Stars: A New Era in Helio- and Asteroseismology, ASPC, 479, 61
  13. Benomar, O., Appourchaux, T., & Baudin, F. 2009, The Solar-like Oscillations of HD 49933: A Bayesian Approach, A&A, 506, 15 https://doi.org/10.1051/0004-6361/200911657
  14. Benomar, O., Baudin, F., Marques, J. P., et al. 2010, Spectrum Analysis and Seismic Interpretation of a Solar-like Pulsator (HD 49933) Observed by CoRoT, AN, 331, 956
  15. Bigot, L., Mourard, D., Berio, P., et al. 2011, The Diameter of the CoRoT Target HD 49933. Combining the 3D limb Darkening, Asteroseismology, and Interferometry, A&A, 534, L3 https://doi.org/10.1051/0004-6361/201117349
  16. Bohm-Vitense, E. 2007, Chromospheric Activity in G and K Main-Sequence Stars, and What It Tells Us about Stellar Dynamos, ApJ, 657, 486 https://doi.org/10.1086/510482
  17. Brown, T. M., & Gilliland, R. L. 1994, Asteroseismology, ARA&A, 32, 37 https://doi.org/10.1146/annurev.aa.32.090194.000345
  18. Bruntt, H., Bikmaev, I. F., Catala, C., et al. 2004, Abundance Analysis of Targets for the COROT/MONS Asteroseismology Missions. II. Abundance Analysis of the COROT Main Targets, A&A, 425, 683 https://doi.org/10.1051/0004-6361:20040464
  19. Bruntt, H., De Cat, P., & Aerts, C. 2008, A Spectroscopic Study of Southern (Candidate) γ Doradus Stars. II. Detailed Abundance Analysis and Fundamental Parameters, A&A, 478, 487 https://doi.org/10.1051/0004-6361:20078523
  20. Bruntt, H. 2009, Accurate Fundamental Parameters of CoRoT Asteroseismic Targets. The Solar-like Stars HD 49933, HD 175726, HD 181420, and HD 181906, A&A, 506, 235 https://doi.org/10.1051/0004-6361/200911925
  21. Cenarro, A. J., Peletier, R. F., Sanchez-Blazquez, P., et al., 2007, Medium-resolution Isaac Newton Telescope Library of Empirical Spectra - II. The Stellar Atmospheric Parameters, MNRAS, 374, 664 https://doi.org/10.1111/j.1365-2966.2006.11196.x
  22. Chaintreuil, S., Deru, A., Baudin, F., et al. 2016, The "Ready to Use" CoRoT Data, The CoRoT Legacy Book: The Adventure of the Ultra High Precision Photometry from Space, 61 (Les Ulis: EDP Sciences)
  23. Chaplin, W. J., Elsworth, Y., Isaak, G. R., et al. 1998, An Analysis of Solar p-mode Frequencies Extracted from BiSON Data: 1991-1996, MNRAS, 300, 1077 https://doi.org/10.1046/j.1365-8711.1998.01999.x
  24. Chaplin, W. J., Elsworth, Y., Isaak, G. R., et al. 2000, Variations in the Excitation and Damping of Low-l Solar p Modes over the Solar Activity Cycle, MNRAS, 313, 32 https://doi.org/10.1046/j.1365-8711.2000.03176.x
  25. Chaplin, W. J., Appourchaux, T., Elsworth, Y., et al. 2001, The Phenomenology of Solar-cycle-induced Acoustic Eigenfrequency Variations: A Comparative and Complementary Analysis of GONG, BiSON and VIRGO/LOI Data, MNRAS, 324, 910 https://doi.org/10.1046/j.1365-8711.2001.04357.x
  26. Chaplin, W. J., Elsworth, Y., Isaak, G. R., et al. 2004, The Solar Cycle as Seen by Low-l p-mode Frequencies: Comparison with Global and Decomposed Activity Proxies, MNRAS, 352, 1102 https://doi.org/10.1111/j.1365-2966.2004.07998.x
  27. Chaplin, W. J., Elsworth, Y., Houdek, G., et al. 2007, On Prospects for Sounding Activity Cycles of Sun-like Stars with Acoustic Modes, MNRAS, 377, 17 https://doi.org/10.1111/j.1365-2966.2007.11581.x
  28. Chaplin, W. J., Bedding, T. R., Bonanno, A., et al. 2011, Evidence for the Impact of Stellar Activity on the Detectability of Solar-like Oscillations Observed by Kepler, ApJL, 732, L5 https://doi.org/10.1088/2041-8205/732/1/L5
  29. Chaplin, W. J., & Miglio, A. 2013, Asteroseismology of Solar-Type and Red-Giant Stars, ARA&A, 51, 353 https://doi.org/10.1146/annurev-astro-082812-140938
  30. Christensen-Dalsgaard, J., Duvall, T. L., Gough, D. O., et al. 1985, Speed of Sound in the Solar Interior, Nature, 315, 378 https://doi.org/10.1038/315378a0
  31. Christensen-Dalsgaard, J. 1988, A Hertzsprung-Russell Diagram for Stellar Oscillations, Advances in Helio- and Asteroseismology, Proc. IAU, 123, 295
  32. Christensen-Dalsgaard, J. 2002, Helioseismology, RvMP, 74, 1073
  33. Christensen-Dalsgaard, J. 2008, ADIPLS - The Aarhus Adiabatic Oscillation Package, Ap&SS, 316, 113 https://doi.org/10.1007/s10509-007-9689-z
  34. Dall, T. H., Bruntt, H., Stello, D., et al. 2010, Solar-like Oscillations and Magnetic Activity of the Slow Rotator EK Eridani, A&A, 514, A25 https://doi.org/10.1051/0004-6361/200913710
  35. Elsworth, Y., Howe, R., Isaak, G. R., et al. 1990, Variation of Low-order Acoustic Solar Oscillations over the Solar Cycle, Nature, 345, 322 https://doi.org/10.1038/345322a0
  36. Elsworth, Y., Howe, R., Isaak, G. R., et al. 1993, The Variation in the Strength of Low-l Solar p-modes - 1981-92, MNRAS, 265, 888 https://doi.org/10.1093/mnras/265.4.888
  37. Elsworth, Y., Howe, R., Isaak, G. R., et al. 1994, Solar p-Mode Frequencies and Their Dependence on Solar Activity: Recent Results from the BISON Network, ApJ, 434, 801 https://doi.org/10.1086/174783
  38. Elsworth, Y., Howe, R., Isaak, G. R., et al. 1995, Slow Rotation of the Sun's Interior, Nature, 376, 669 https://doi.org/10.1038/376669a0
  39. Fossat, E., Gelly, B., Grec, G., et al. 1987, Search for Solar P-Mode Frequency Changes Between 1980 and 1985, A&A, 177, L47
  40. Palle, P. L., Regulo, C., & Roca Cortes, T. 1989, Solar Cycle Induced Variations of the Low L Solar Acoustic Spectrum, A&A, 224, 253
  41. Goldreich, P., & Keeley, D. A. 1977, Solar Seismology. II. The Stochastic Excitation of the Solar p-modes by Turbulent Convection, ApJ, 212, 243 https://doi.org/10.1086/155043
  42. Goldreich, P., & Kumar, P. 1988, The Interaction of Acoustic Radiation with Turbulence, ApJ, 326, 462 https://doi.org/10.1086/166108
  43. Goldreich, P., Murray, N., & Kumar, P. 1994, Excitation of Solar p-Modes, ApJ, 424, 466 https://doi.org/10.1086/173904
  44. Gontcharov, G. A., 2006, Pulkovo Compilation of Radial Velocities for 35 495 Hipparcos Stars in a Common System, AstL, 32, 759
  45. Gough, D. 1985, Inverting Helioseismic Data, Sol. Phys., 100, 65 https://doi.org/10.1007/BF00158422
  46. Gough, D. O. 1990, Comments on Helioseismic Inference, Progress of Seismology of the Sun and Stars, LNP, 367, 283
  47. Garcia, R. A., Mathur, S., Salabert, D., et al. 2010, CoRoT Reveals a Magnetic Activity Cycle in a Sun-Like Star, Science, 329, 1032 https://doi.org/10.1126/science.1191064
  48. Gelly, B., Lazrek, M., Grec, G., et al. 2002, Solar p-modes from 1979 Days of the GOLF Experiment, A&A, 394, 285 https://doi.org/10.1051/0004-6361:20021106
  49. Gillon, M., & Magain, P. 2006, High Precision Determination of the Atmospheric Parameters and Abundances of the COROT Main Targets, A&A, 448, 341 https://doi.org/10.1051/0004-6361:20053965
  50. Guggenberger, E., Hekker, S., Basu, S., et al. 2016, Significantly Improving Stellar Mass and Radius Estimates: A New Reference Function for the ∆ν Scaling Relation, MNRAS, 460, 4277 https://doi.org/10.1093/mnras/stw1326
  51. Hekker, S., Broomhall, A.-M., Chaplin, W. J., et al. 2010, The Octave (Birmingham-Sheffield Hallam) Automated Pipeline for Extracting Oscillation Parameters of Solarlike Main-sequence Stars, MNRAS, 402, 2049 https://doi.org/10.1111/j.1365-2966.2009.16030.x
  52. Hekker, S. 2020, Scaling Relations for Solar-like Oscillations: A Review, Front. Astron. Space Sci., 7, 3 https://doi.org/10.3389/fspas.2020.00003
  53. Henry, T. J., Soderblom, D. R., Donahue, R. A., et al. 1996, A Survey of Ca ii H and K Chromospheric Emission in Southern Solar-Type Stars, AJ, 111, 439 https://doi.org/10.1086/117796
  54. Howe, R., Komm, R. W., & Hill, F. 2002, Localizing the Solar Cycle Frequency Shifts in Global p-Modes, ApJ, 580, 1172 https://doi.org/10.1086/343892
  55. Howe, R., Chaplin, W. J., Basu, S., et al. 2020, Solar Cycle Variation of νmax in Helioseismic Data and Its Implications for Asteroseismology, MNRAS, 493, L49 https://doi.org/10.1093/mnrasl/slaa006
  56. Huber, D., Bedding, T. R., Stello, D., et al. 2011, Testing Scaling Relations for Solar-like Oscillations from the Main Sequence to Red Giants Using Kepler Data, ApJ, 743, 143 https://doi.org/10.1088/0004-637X/743/2/143
  57. Isaacson, H., & Fischer, D. 2010, Chromospheric Activity and Jitter Measurements for 2630 Stars on the California Planet Search, ApJ, 725, 875 https://doi.org/10.1088/0004-637X/725/1/875
  58. Jimenez, A., Garcia, R. A., & Palle, P. L. 2011, The Acoustic Cutoff Frequency of the Sun and the Solar Magnetic Activity Cycle, ApJ, 743, 99 https://doi.org/10.1088/0004-637X/743/2/99
  59. Jimenez-Reyes, S. J., Garcia, R. A., Jimenez, A., et al. 2003, Excitation and Damping of Low-Degree Solar p-Modes during Activity Cycle 23: Analysis of GOLF and VIRGO Sun Photometer Data, ApJ, 595, 446 https://doi.org/10.1086/377304
  60. Jo, Y. A., Kim, K., & Chang, H. Y. 2021, in preparation
  61. Kallinger, T., Gruberbauer, M., Guenther, D. B., et al. 2010, The Nature of p-modes and Granulation in HD 49933 Observed by CoRoT, A&A, 510, A106 https://doi.org/10.1051/0004-6361/200811438
  62. Karoff, C. 2008, Observational Asteroseismology, Ph.D. Thesis, Aarhus University
  63. Kim, K., & Chang, H. Y. 2021a, Scaling Relations for Width of the Power Excess of Stellar Oscillations, New Astron., 84, 101522 https://doi.org/10.1016/j.newast.2020.101522
  64. Kim, K., & Chang, H. Y. 2021b, Do the Observed Relations of the Global Seismic Parameters on the Magnetic Activity Level?, JKAS, 54, 121
  65. Kjeldsen, H. & Bedding, T. R. 1995, Amplitudes of Stellar Oscillations: The Implications for Asteroseismology, A&A, 293, 87
  66. Kjeldsen, H., Bedding, T. R., Arentoft, T., et al. 2008, The Amplitude of Solar Oscillations Using Stellar Techniques, ApJ, 682, 1370 https://doi.org/10.1086/589142
  67. Kjeldsen, H., & Bedding, T. R. 2011, Amplitudes of Solarlike oscillations: A New Scaling Relation, A&A, 529, L8 https://doi.org/10.1051/0004-6361/201116789
  68. Komm, R. W., Howe, R., & Hill, F. 2000, Solar-Cycle Changes in Gong P-Mode Widths and Amplitudes 1995-1998, ApJ, 531, 1094 https://doi.org/10.1086/308518
  69. Libbrecht, K. G., & Woodard, M. F. 1990, Solar-cycle Effects on Solar Oscillation Frequencies, Nature, 345, 779 https://doi.org/10.1038/345779a0
  70. Liu, Z., Yang, W., Bi, S., et al. 2014, Asteroseismic Analysis of the CoRoT Target HD 49933, ApJ, 780, 152
  71. Lomb, N. R. 1976, Least-Squares Frequency Analysis of Unequally Spaced Data, Ap&SS, 39, 447 https://doi.org/10.1007/BF00648343
  72. Mathur, S., Garcia, R. A., Regulo, C., et al. 2010, Determining Global Parameters of the Oscillations of Solar-like Stars, A&A, 511, A46 https://doi.org/10.1051/0004-6361/200913266
  73. Mathur, S., Garcia, R. A., Ballot, J., et al. 2014, Magnetic Activity of F Stars Observed by Kepler, A&A, 562, A124 https://doi.org/10.1051/0004-6361/201322707
  74. Metcalfe, T. S., Dziembowski, W. A., Judge, P. G., et al. 2007, Asteroseismic Signatures of Stellar Magnetic Activity Cycles, MNRAS, 379, L16 https://doi.org/10.1111/j.1745-3933.2007.00325.x
  75. Meunier, N., Lagrange, A.-M., & Cuzacq, S. 2019, Activity Time Series of Old Stars from Late F to Early K. IV. Limits of the Correction of Radial Velocities Using Chromospheric Emission, A&A, 632, A81 https://doi.org/10.1051/0004-6361/201935348
  76. Michel, E., Baglin, A., Weiss, W. W., et al. 2008, First Asteroseismic Results from CoRoT, Commun. Asteroseismol. 156, 73 https://doi.org/10.1553/cia156s73
  77. Miglio, A., Chiappini, C., Morel, T., et al. 2013, Galactic Archaeology: Mapping and Dating Stellar Populations with Asteroseismology of Red-giant Stars, MNRAS, 429, 423 https://doi.org/10.1093/mnras/sts345
  78. Mosser, B., Bouchy, F., Catala, C., et al. 2005, Seismology and Activity of the F Type Star HD 49933, A&A, 431, L13 https://doi.org/10.1051/0004-6361:200500003
  79. Mosser, B., Baudin, F., Lanza, A. F., et al. 2009, Short-lived Spots in Solar-like Stars as Observed by CoRoT, A&A, 506, 245 https://doi.org/10.1051/0004-6361/200911942
  80. Mosser, B., Michel, E., Belkacem, K., et al. 2013, Asymptotic and Measured Large Frequency Separations, A&A, 550, A126 https://doi.org/10.1051/0004-6361/201220435
  81. Ong, J. M. J., & Basu, S. 2019, Explaining Deviations from the Scaling Relationship of the Large Frequency Separation, ApJ, 870, 41 https://doi.org/10.3847/1538-4357/aaf1b5
  82. Paxton, B., Smolec, R., Schwab, J., et al. 2019, Modules for Experiments in Stellar Astrophysics (MESA): Pulsating Variable Stars, Rotation, Convective Boundaries, and Energy Conservation, ApJS, 243, 10 https://doi.org/10.3847/1538-4365/ab2241
  83. Piau, L., Turck-Chieze, S., Duez, V., et al. 2009, Impact of the Physical Processes in the Modeling of HD 49933, A&A, 506, 175 https://doi.org/10.1051/0004-6361/200911947
  84. Pires, S., Mathur, S., Garcia, R. A., et al. 2015, Gap Interpolation by Inpainting Methods: Application to Ground and Space-based Asteroseismic Data, A&A, 574, A18 https://doi.org/10.1051/0004-6361/201322361
  85. Radick, R. R., Lockwood, G. W., Skiff, B. A., et al. 1998, Patterns of Variation among Sun-like Stars, ApJS, 118, 239 https://doi.org/10.1086/313135
  86. Regulo, C., Garcia, R. A., & Ballot, J. 2016, Magnetic Activity Cycles in Solar-like Stars: The Cross-correlation Technique of p-mode Frequency Shifts, A&A, 589, A103 https://doi.org/10.1051/0004-6361/201425408
  87. Rodrigues, T. S., Bossini, D., Miglio, A., et al. 2017, Determining Stellar Parameters of Asteroseismic Targets: Going beyond the Use of Scaling Relations, MNRAS, 467, 1433
  88. Saar, S. H., & Brandenburg, A. 1999, Time Evolution of the Magnetic Activity Cycle Period. II. Results for an Expanded Stellar Sample, ApJ, 524, 295 https://doi.org/10.1086/307794
  89. Salabert, D., Jimenez-Reyes, S. J., & Tomczyk, S. 2003, Study of p-mode Excitation and Damping Rate Variations from IRIS++ Observations, A&A, 408, 729 https://doi.org/10.1051/0004-6361:20031014
  90. Salabert, D., Regulo, C., Ballot, J., et al. 2011, About the p-mode Frequency Shifts in HD 49933, A&A, 530, A127 https://doi.org/10.1051/0004-6361/201116633
  91. Santos, A. R. G., Campante, T. L., Chaplin, W. J., et al. 2018, Signatures of Magnetic Activity in the Seismic Data of Solar-type Stars Observed by Kepler, ApJS, 237, 17 https://doi.org/10.3847/1538-4365/aac9b6
  92. Santos, A. R. G., Campante, T. L., Chaplin, W. J., et al. 2019, Signatures of Magnetic Activity: On the Relation between Stellar Properties and p-mode Frequency Variations, ApJ, 883, 65 https://doi.org/10.3847/1538-4357/ab397a
  93. Scargle, J. D. 1982, Studies in Astronomical Time Series Analysis. II. Statistical Aspects of Spectral Analysis of Unevenly Spaced Data, ApJ, 263, 835 https://doi.org/10.1086/160554
  94. Sharma, S., Stello, D., Bland-Hawthorn, J., et al. 2016, Stellar Population Synthesis Based Modeling of the Milky Way Using Asteroseismology of 13,000 Kepler Red Giants, ApJ, 822, 15 https://doi.org/10.3847/0004-637X/822/1/15
  95. Stello, D., Huber, D., Kallinger, T., et al. 2011, Amplitudes of Solar-like Oscillations: Constraints from Red Giants in Open Clusters Observed by Kepler, ApJL, 737, L10 https://doi.org/10.1088/2041-8205/737/1/L10
  96. Tassoul, M. 1980, Asymptotic Approximations for Stellar Nonradial Pulsations, ApJS, 43, 469 https://doi.org/10.1086/190678
  97. Thomas, A. E. L., Chaplin, W. J., Basu, S., et al. 2021, Impact of Magnetic Activity on Inferred Stellar Properties of Main-sequence Sun-like Stars, MNRAS, 502, 5808 https://doi.org/10.1093/mnras/stab354
  98. Thompson, M. J., Toomre, J., Anderson, E. R., et al. 1996, Differential Rotation and Dynamics of the Solar Interior, Science, 272, 1300 https://doi.org/10.1126/science.272.5266.1300
  99. Townsend, R. H. D., & Teitler, S. A. 2013, GYRE: An Opensource Stellar Oscillation Code based on a New Magnus Multiple Shooting Scheme, MNRAS, 435, 3406 https://doi.org/10.1093/mnras/stt1533
  100. Ulrich, R. K. 1986, Determination of Stellar Ages from Asteroseismology, ApJL, 306, L37 https://doi.org/10.1086/184700
  101. van Leeuwen, F., 2007, Validation of the New Hipparcos Reduction, A&A, 474, 653 https://doi.org/10.1051/0004-6361:20078357
  102. Viani, L. S., Basu, S., Corsaro, E., et al. 2019, Determining the Best Method of Calculating the Large Frequency Separation For Stellar Models, ApJ, 879, 33 https://doi.org/10.3847/1538-4357/ab232e
  103. White, T. R., Bedding, T. R., Stello, D., et al. 2011, Calculating Asteroseismic Diagrams for Solar-like Oscillations, ApJ, 743, 161 https://doi.org/10.1088/0004-637X/743/2/161
  104. Woodard, M. F., & Noyes, R. W. 1985, Change of Solar Oscillation Eigenfrequencies with the Solar Cycle, Nature, 318, 449 https://doi.org/10.1038/318449a0