Skip to main content Accessibility help
×
Hostname: page-component-7479d7b7d-qs9v7 Total loading time: 0 Render date: 2024-07-08T16:20:19.677Z Has data issue: false hasContentIssue false

4 - Studying stars through frequency inversions

Published online by Cambridge University Press:  05 December 2013

Sarbani Basu
Affiliation:
Yale University
Pere L. Pallé
Affiliation:
Instituto de Astrofísica de Canarias, Tenerife
César Esteban
Affiliation:
Instituto de Astrofísica de Canarias, Tenerife
Get access

Summary

Introduction

Helioseismology, the study of the Sun using solar oscillations, has provided us with the means to probe the solar interior. Since the discovery of the oscillations in 1962 (Leighton et al., 1962) and their interpretation as global oscillation modes by Ulrich (1970) and Leibacher and Stein (1971), helioseismology has been used extensively to study the interior of the Sun, mainly through inversions of solar frequencies. With space missions such as CoRoT (Baglin et al., 2006) and Kepler (Borucki et al., 2010) now observing oscillations of other stars, inversions of stellar frequencies may soon be feasible. There are two ways by which we could use seismic data to make inferences about the stars. The first way involves trying to find models whose frequencies match the observed frequencies, usually referred to as “forward modeling.” This is essentially what is done in most fields of astronomy. The end result of the process is a model that is the best match to the observations. The second way is to invert the data. Inversions use the data directly to make inferences about the star. In the case of inversions, we can make a distinction between the structure of the star and the structure of the best-fit model. These days inversions are used to study the solar interior, while forward modeling is used to study other stars.

It is not possible to do an inverse analysis unless we can do the forward analysis.

Type
Chapter
Information
Asteroseismology , pp. 87 - 122
Publisher: Cambridge University Press
Print publication year: 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Antia, H. M. and Basu, S. 1994. Nonasymptotic helioseismic inversion for solar structure. A&AS, 107(Nov.), 421–44.Google Scholar
Antia, H. M. and Basu, S. 2011. Are recent solar heavy element abundances consistent with helioseismology?Journal of Physics Conference Series, 271 (Jan.), 012034.Google Scholar
Antia, H.M., Basu, S., and Chitre, S. M. 1998. Solar internal rotation rate and the latitudinal variation of the tachocline. MNRAS, 298(Aug.), 543–56.Google Scholar
Asplund, M., Grevesse, N., and Sauvai, A. J. 2005. The solar chemical composition. Cosmic Abundances as Records of Stellar Evolution and Nucleosynthesis, 336(Sep.), 25.Google Scholar
Backus, G. E. and Gilbert, J. F. 1968. The resolving power of gross earth data. Geophysical Journal, 16, 169–205.Google Scholar
Backus, G. E. and Gilbert, J. F. 1970. Uniqueness in the inversion of inaccurate gross Earth data. Royal Society of London Philosophical Transactions Series A, 266(Mar.), 123–92.Google Scholar
Baglin, A., Auvergne, M., Barge, P., Deleuil, M., Catala, C., Michel, E., Weiss, W., and the COROT team 2006. Scientific objectives for a Minisat: CoRoT. ESA Special Publication, 1306(Nov.), 33.Google Scholar
Bahcall, J.N., Basu, S., Pinsonneault, M., and Serenelli, A. M. 2005. Helioseismological implications of recent solar abundance determinations. ApJ, 618(Jan.), 1049–56.Google Scholar
Bahcall, J.N., Serenelli, A.M., and Basu, S. 2006. 10,000 standard solar models: a monte carlo simulation. ApJS, 165(Jul.), 400–31.Google Scholar
Basu, S. 1998. Effects of errors in the solar radius on helioseismic inferences. MNRAS, 298(Aug.), 719–28.Google Scholar
Basu, S. 2003. Stellar Inversions. Ap&SS, 284, 153–64.Google Scholar
Basu, S. and Antia, H. M. 1997. Seismic measurement of the depth of the solar convection zone. MNRAS, 287(May.), 189–98.Google Scholar
Basu, S. and Antia, H. M. 2008. Helioseismology and solar abundances. Phys. Rep., 457(Mar.), 217–83.Google Scholar
Basu, S. and Christensen-Dalsgaard, J. 1997. Equation of state and helioseismic inversions. A&A, 322 (Jun.), L5–L8.Google Scholar
Basu, S. and Thompson, M. J. 1996. On constructing seismic models of the Sun. A&A, 305(Jan.), 631.Google Scholar
Basu, S., Dappen, W., and Nayfonov, A. 1999. Helioseismic analysis of the hydrogen partition function in the solar interior. ApJ, 518(Jun.), 985–93.Google Scholar
Basu, S., Pinsonneault, M.H., and Bahcall, J. N. 2000. How much do helioseismological inferences depend on the assumed reference model?ApJ, 529(Feb.), 1084–1100.Google Scholar
Basu, S., Christensen-Dalsgaard, J., Monteiro, M. J. P. F. G., and Thompson, M. J. 2001. Seismology of solar-type stars. SOHO 10/GONG 2000 Workshop: Helio- and Asteroseismology at the Dawn of the Millennium, 464(Jan.), 407–10.Google Scholar
Basu, S., Christensen-Dalsgaard, J., and Thompson, M. J. 2002. SOLA inversions for the core structure of solar-type stars. Stellar Structure and Habitable Planet Finding, 485(Jan.), 249–52.Google Scholar
Basu, S., Chaplin, W.J., Elsworth, Y., New, R., and Serenelli, A. M. 2009. Fresh Insights on the Structure of the Solar Core. ApJ, 699(Jul.), 1403–17.Google Scholar
Bertello, L., Varadi, F., Ulrich, R.K., Henney, C.J., Kosovichev, A.G., Garcia, R.A., and Turck-Chieze, S. 2000. Identification of solar acoustic modes of low angular degree and low radial order. ApJ, 537(Jul.), L143–L146.Google Scholar
Borucki, W.J., and 70 colleagues. 2010. Kepler planet-detection mission: introduction and first results. Science, 327(Feb.), 977.Google Scholar
Canuto, V. M. and Mazzitelli, I. 1991. Stellar turbulent convection: a new model and applications. ApJ, 370(Mar.), 295–311.Google Scholar
Chandrasekhar, S. 1964. A general variational principle governing the radial and the non-radial oscillations of gaseous masses. ApJ, 139(Feb.), 664.Google Scholar
Christensen-Dalsgaard, J. 2002. Helioseismology. Reviews ofModern Physics, 74(Nov.), 1073–1129.Google Scholar
Christensen-Dalsgaard, J. 2003. Lecture notes on stellar oscillations. http://users-phys.au.dk/jcd/oscilnotes/.
Christensen-Dalsgaard, J. and Berthomieu, G. 1991. Theory of solar oscillations. Solar interior and atmosphere (A92-36201 14-92). Tucson, AZ, University of Arizona Press. Research supported by SNFO and CNRS. 401–78.
Christensen-Dalsgaard, J. and Daeppen, W. 1992. Solar oscillations and the equation of state. A&A Rev., 4, 267–361.Google Scholar
Christensen-Dalsgaard, J., Gough, D.O., and Thompson, M. J. 1991. The depth of the solar convection zone. ApJ, 378(Sep.), 413–37.Google Scholar
Christensen-Dalsgaard, J., Proffitt, C.R., and Thompson, M. J. 1993. Effects of diffusion on solar models and their oscillation frequencies. ApJ, 403(Feb.), L75–8.Google Scholar
Christensen-Dalsgaard, J., and 32 colleagues 1996. The current state of solar modeling. Science, 272(May), 1286–92.Google Scholar
Cox, J. P. 1980. Theory of stellar pulsation. Research supported by the National Science Foundation. Princeton, NJ. Princeton University Press.
Craig, I. J. D. and Brown, J. C. 1986. Inverse problems in astronomy: A guide to inversion strategies for remotely sensed data. Research supported by SERC. Bristol, England and Boston, MA. Adam Hilger, Ltd.
Daeppen, W., Anderson, L., and Mihalas, D. 1987. Statistical mechanics of partially ionized stellar plasma: the Planck-Larkin partition function, polarization shifts, and simulations of optical spectra. ApJ, 319(Aug.), 195–206.Google Scholar
Daeppen, W., Mihalas, D., Hummer, D.G., and Mihalas, B. W. 1988. The equation of state for stellar envelopes. III - Thermodynamic quantities. ApJ, 332(Sep.), 261–70.Google Scholar
de Boor, C. 2001. A practical guide to Splines. New York: Springer Verlag.
Demarque, P., Guenther, D.B., Li, L.H., Mazumdar, A., and Straka, C. W. 2008. YREC: the Yale rotating stellar evolution code. Non-rotating version, seismology applications. Ap&SS, 316(Aug.), 31–41.Google Scholar
Dziembowski, W.A., Pamyatnykh, A.A., and Sienkiewicz, R. 1990. Solar model from helioseis-mology and the neutrino flux problem. MNRAS, 244(Jun.), 542–50.Google Scholar
Eggleton, P.P., Faulkner, J., and Flannery, B. P. 1973. An approximate equation of state for stellar material. A&A, 23(Mar.), 325.Google Scholar
Elliott, J. R. 1996. Equation of state in the solar convection zone and the implications of helioseismology. MNRAS, 280(May), 1244–56.Google Scholar
Elliott, J. R. and Kosovichev, A. G. 1998. The adiabatic exponent in the solar core. ApJ, 500(Jun.), L199.Google Scholar
Gong, Z., Dappen, W., and Zejda, L. 2001. MHD equation of state with relativistic electrons. ApJ, 546(Jan.), 1178–82.Google Scholar
Gough, D. O. 1984. On the rotation of the sun. Royal Society of London Philosophical Transactions Series A, 313(Nov.), 27–38.Google Scholar
Gough, D. O. 1985. Inverting helioseismic data. Sol. Phys., 100(Oct.), 65–99.Google Scholar
Gough, D. O. 1990. Comments on helioseismic inference. Progress of Seismology of the Sun and Stars, 367, 283.Google Scholar
Gough, D. O. 1993. Linear adiabatic stellar pulsation. Astrophysical Fluid Dynamics – Les Houches 1987, 399–560.Google Scholar
Gough, D. O. and Kosovichev, A. G. 1993. Initial asteroseismic inversions. IAU Colloq. 137: Inside the Stars, 40(Jan.), 541.Google Scholar
Gough, D. O. and Thompson, M. J. 1991. The inversion problem. Solar interior and atmosphere (A92-36201 14-92). Tucson, AZ. University of Arizona Press, 519–61.
Grevesse, N. and Sauval, A. J. 1998. Standard Solar Composition. Space Sci. Rev., 85(May), 161–74.Google Scholar
Hansen, P. C. 1992. Numerical tools for analysis and solution of Fredholm integral equations of the first kind. Inverse Problems, 8(Dec.), 849–72.Google Scholar
Howe, R. and Thompson, M. J. 1996. On the use of the error correlation function in helioseismic inversions. MNRAS, 281(Aug.), 1385.Google Scholar
Hummer, D. G. and Mihalas, D. 1988. The equation of state for stellar envelopes. I. An occupation probability formalism for the truncation of internal partition functions. ApJ, 331 (Aug.), 794–814.Google Scholar
Lefebvre, S., Kosovichev, A.G., and Rozelot, J. P. 2007. Helioseismic test of nonhomologous solar radius changes with the 11 year activity cycle. ApJ, 658(Apr.), L135–L138.Google Scholar
Leighton, R.B., Noyes, R.W., and Simon, G. W. 1962. Velocity fields in the solar atmosphere. I. Preliminary report. ApJ, 135(Mar.), 474.Google Scholar
Leibacher, J. W. and Stein, R. F. 1971. A new description of the solar five-minute oscillation. Astrophys. Lett., 7, 191–2.Google Scholar
Mihalas, D., Dappen, W., and Hummer, D. G. 1988. The equation of state for stellar envelopes. II. Algorithm and selected results. ApJ, 331(Aug.), 815–25.Google Scholar
Miller, I., Freund, J. E. 1965. Probability and statistics for engineers. Prentice-Hall Mathematics Series, Englewoods Cliffs.
Pijpers, F. P. and Thompson, M. J. 1992. Faster formulations of the optimally localized averages method for helioseismic inversions. A&A, 262(Sep.), L33–L36.Google Scholar
Pijpers, F. P. and Thompson, M. J. 1994. The SOLA method for helioseismic inversion. A&A, 281 (Jan.), 231–40.Google Scholar
Rabello-Soares, M.C., Basu, S., and Christensen-Dalsgaard, J. 1999. On the choice ofparameters in solar-structure inversion. MNRAS, 309(Oct.), 35–47.Google Scholar
Robinson, F.J., Demarque, P., Li, L.H., Sofia, S., Kim, Y.-C., Chan, K.L., and Guenther, D. B. 2003. Three-dimensional convection simulations of the outer layers of the Sun using realistic physics. MNRAS, 340(Apr.), 923–36.Google Scholar
Rogers, F. J. and Nayfonov, A. 2002. Updated and Expanded OPAL Equation-of-State Tables: Implications for Helioseismology. ApJ, 576(Sep.), 1064–74.Google Scholar
Rogers, F.J., Swenson, F.J., and Iglesias, C. A. 1996. OPAL Equation-of-State Tables for Astrophysical Applications. ApJ, 456(Jan.), 902.Google Scholar
Roxburgh, I. and Vorontsov, S. 2003. Diagnostics of the Internal Structure of Stars using the Differential Response Technique. Ap&SS, 284, 187–91.Google Scholar
Roxburgh, I. W., Audard, N., Basu, S., Christensen-Dalsgaard, J., and Vorontsov, S. V. 1998. Proc. IAU Symp. 181: Sounding Solar and Stellar Interiors, (poster vol.). eds. J., Provost, F.-X., Schmider, 245.
Schou, J., Kosovichev, A.G., Goode, P.R., and Dziembowski, W. A. 1997. Determination of the Sun's seismic radius from the SOHO Michelson Doppler Imager. ApJ, 489(Nov.), L197.Google Scholar
Schou, J., Christensen-Dalsgaard, J., Howe, R., Larsen, R.M., Thompson, M.J., and Toomre, J. 1998. Slow poles and shearing flows from heliospheric observations with mdi and gong spanning a year. Structure and Dynamics of the Interior of the Sun and Sun-like Stars, 418, 845.Google Scholar
Sekii, T. 1997. Internal solar rotation. Proc IA U Symp. 181, Sounding Solar and Stellar Interiors. Pages 189–202 of: J., Provost and F.-X., Schmider (ed.). Dordrecht, Holland: Kluwer.
Serenelli, A.M., Basu, S., Ferguson, J.W., and Asplund, M. 2009. New solar composition: the problem with solar models revisited. ApJ, 705(Nov.), L123–L127.Google Scholar
Thompson, M. J. 1993. Seismic investigation of the Sun's internal structure and rotation. GONG 1992. Seismic Investigation of the Sun and Stars. ASPCS, 42(Jan.), 141.Google Scholar
Ulrich, R. K. 1970. The five-minute oscillations on the solar surface. ApJ, 162(Dec.), 993.Google Scholar
Unno, W., Osaki, Y., Ando, H., Saio, H., and Shibahashi, H. 1989. Nonradial oscillations of stars, 2nd ed. Tokyo, Japan: University of Tokyo Press.

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×