The current state of solar modeling

Science

Volumn 272, Issue 5266, 1996, Pages 1286-1292

  Christensen Dalsgaard, J ^a   Dappen W ^a   Ajukov, S V ^a   Anderson, E R ^a   Antia, H M ^a   Basu, S ^a   Baturin, V A ^a   Berthomieu, G ^a   Chaboyer, B ^a   Chitre, S M ^a   Cox, A N ^a   Demarque, P ^a   Donatowicz, J ^a   Dziembowski, W A ^a   Gabriel, M ^a   Gough, D O ^a   Guenther, D B ^a   Guzik, J A ^a   Harvey, J W ^a   Hill, F ^a   Houdek, G ^a   Iglesias, C A ^a   Kosovichev, A G ^a   Leibacher, J W ^a   Morel, P ^a   Proffitt, C R ^a   Provost, J ^a   Reiter, J ^a   Rhodes Jr , E J ^a   Rogers, F J ^a   Roxburgh, I W ^a   Thompson, M J ^a   Ulrich, R K ^a   more..

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

GLOBAL OSCILLATION NETWORK GROUP; HELIOSEISMOLOGY; STELLAR STRUCTURE;

SUN;

EID: 0029750056     PISSN: 00368075     EISSN: None     Source Type: Journal    
DOI: 10.1126/science.272.5266.1286     Document Type: Article

References (124)

1. M. J. Thompson et al, Science 272, 1300 (1996).
2. Models of the evolution of solar internal rotation assume loss of angular momentum from the convection zone through the solar wind, the coupling to the interior taking place through various types of instability [A. S. Endal and S. Sofia, Astrophys. J 243, 625 (1981); M. H. Pinsonneault, S. D. Kawaler, S. Sofia, P. Demarque, ibid. 338, 424 (1989)]. These models have been useful in interpreting rotational velocities and lithium abundances in stellar clusters and in field stars near the main sequence. When applied to the sun, they fail to reproduce the near-uniform rotation of the present solar interior, as inferred from hellioseismology (1), indicating that other physical mechanisms must contribute to the solar internal spin-down.
3. Models of the evolution of solar internal rotation assume loss of angular momentum from the convection zone through the solar wind, the coupling to the interior taking place through various types of instability [A. S. Endal and S. Sofia, Astrophys. J 243, 625 (1981); M. H. Pinsonneault, S. D. Kawaler, S. Sofia, P. Demarque, ibid. 338, 424 (1989)]. These models have been useful in interpreting rotational velocities and lithium abundances in stellar clusters and in field stars near the main sequence. When applied to the sun, they fail to reproduce the near-uniform rotation of the present solar interior, as inferred from hellioseismology (1), indicating that other physical mechanisms must contribute to the solar internal spin-down.
4. The notion of "standard" models is an evolving concept; in a loose sense, such models should incorporate the physics that is generally recognized as relevant and for which an adequate theoretical descrip-tion exists.
5. Simplified stellar models can be based on the condition of hydrostatic equilibrium and the assumption that pressure is related to density alone [for example, J. H. Lane, Am. J. Sci. Arts 2nd Ser. 50, 57 (1870); R. Emden, Gaskugeln (Teubner, Leipzig, 1907)].
6. Simplified stellar models can be based on the condition of hydrostatic equilibrium and the assumption that pressure is related to density alone [for example, J. H. Lane, Am. J. Sci. Arts 2nd Ser. 50, 57 (1870); R. Emden, Gaskugeln (Teubner, Leipzig, 1907)].
7. A substantial breakthrough in the physics of stellar interiors came with the treatment of radiative energy transport. On the basis of a relation between opacity and temperature gradient, the condition of hydrostatic equilibrium, and the equation of state, one can estimate the luminosity of a star without regard to the sources of energy. This procedure led A. S. Eddington [The Internal Constitution of the Stars, sect. 169 (Cambridge Univ. Press, Cambridge, 1926)] to consider the possibility that the sun consists primarily of hydrogen, contrary to the then prevailing view.
8. K. Schwarzschild, Nachr. Königl. Ges. Wiss. Göttingen 195, 41 (1906).
9. The depth of the solar convection zone has been determined from helioseismic inversion to be (28.7 ± 0.3)% of the solar radius [J. Christensen-Dalsgaard, D. O. Gough, M. J. Thompson, Astrophys. J. 378, 413 (1991); A. G. Kosovichev and A. V. Fedorova, Astron. Zh. 68, 1015 (1991)].
10. The depth of the solar convection zone has been determined from helioseismic inversion to be (28.7 ± 0.3)% of the solar radius [J. Christensen-Dalsgaard, D. O. Gough, M. J. Thompson, Astrophys. J. 378, 413 (1991); A. G. Kosovichev and A. V. Fedorova, Astron. Zh. 68, 1015 (1991)].
11. Here the density is so low that convective velocities reaching a considerable fraction of the sound speed are required to sustain the flux of energy.
12. L. Biermann [Z Astrophys. 5, 117 (1932)] and T. G. Cowling [Mon. Not. R Astron. Soc. 96, 42 (1935)] were among the first to use mixing-length treatments to describe the transport of energy by convection. For a refined version, now commonly used, see E. Böhm-Vitense, Z. Astrophys. 46, 108 (1958); see (58) for an alternative, more recent formulation. Also, hydrodynamical simulations are now able to guide solar modelers [for example, R. F. Stein and Å Nordlund, Astrophys. J. 342, L95 (1989); N. Brummell, F. Cattaneo, J. Toomre, Science 269, 1370 (1995); Y.-C. Kim, P. A. Fox, S. Sofia, P. Demarque, Astrophys. J. 442, 422 (1995)].
13. L. Biermann [Z Astrophys. 5, 117 (1932)] and T. G. Cowling [Mon. Not. R Astron. Soc. 96, 42 (1935)] were among the first to use mixing-length treatments to describe the transport of energy by convection. For a refined version, now commonly used, see E. Böhm-Vitense, Z. Astrophys. 46, 108 (1958); see (58) for an alternative, more recent formulation. Also, hydrodynamical simulations are now able to guide solar modelers [for example, R. F. Stein and Å Nordlund, Astrophys. J. 342, L95 (1989); N. Brummell, F. Cattaneo, J. Toomre, Science 269, 1370 (1995); Y.-C. Kim, P. A. Fox, S. Sofia, P. Demarque, Astrophys. J. 442, 422 (1995)].
14. L. Biermann [Z Astrophys. 5, 117 (1932)] and T. G. Cowling [Mon. Not. R Astron. Soc. 96, 42 (1935)] were among the first to use mixing-length treatments to describe the transport of energy by convection. For a refined version, now commonly used, see E. Böhm-Vitense, Z. Astrophys. 46, 108 (1958); see (58) for an alternative, more recent formulation. Also, hydrodynamical simulations are now able to guide solar modelers [for example, R. F. Stein and Å Nordlund, Astrophys. J. 342, L95 (1989); N. Brummell, F. Cattaneo, J. Toomre, Science 269, 1370 (1995); Y.-C. Kim, P. A. Fox, S. Sofia, P. Demarque, Astrophys. J. 442, 422 (1995)].
15. L. Biermann [Z Astrophys. 5, 117 (1932)] and T. G. Cowling [Mon. Not. R Astron. Soc. 96, 42 (1935)] were among the first to use mixing-length treatments to describe the transport of energy by convection. For a refined version, now commonly used, see E. Böhm-Vitense, Z. Astrophys. 46, 108 (1958); see (58) for an alternative, more recent formulation. Also, hydrodynamical simulations are now able to guide solar modelers [for example, R. F. Stein and Å Nordlund, Astrophys. J. 342, L95 (1989); N. Brummell, F. Cattaneo, J. Toomre, Science 269, 1370 (1995); Y.-C. Kim, P. A. Fox, S. Sofia, P. Demarque, Astrophys. J. 442, 422 (1995)].
16. L. Biermann [Z Astrophys. 5, 117 (1932)] and T. G. Cowling [Mon. Not. R Astron. Soc. 96, 42 (1935)] were among the first to use mixing-length treatments to describe the transport of energy by convection. For a refined version, now commonly used, see E. Böhm-Vitense, Z. Astrophys. 46, 108 (1958); see (58) for an alternative, more recent formulation. Also, hydrodynamical simulations are now able to guide solar modelers [for example, R. F. Stein and Å Nordlund, Astrophys. J. 342, L95 (1989); N. Brummell, F. Cattaneo, J. Toomre, Science 269, 1370 (1995); Y.-C. Kim, P. A. Fox, S. Sofia, P. Demarque, Astrophys. J. 442, 422 (1995)].
17. L. Biermann [Z Astrophys. 5, 117 (1932)] and T. G. Cowling [Mon. Not. R Astron. Soc. 96, 42 (1935)] were among the first to use mixing-length treatments to describe the transport of energy by convection. For a refined version, now commonly used, see E. Böhm-Vitense, Z. Astrophys. 46, 108 (1958); see (58) for an alternative, more recent formulation. Also, hydrodynamical simulations are now able to guide solar modelers [for example, R. F. Stein and Å Nordlund, Astrophys. J. 342, L95 (1989); N. Brummell, F. Cattaneo, J. Toomre, Science 269, 1370 (1995); Y.-C. Kim, P. A. Fox, S. Sofia, P. Demarque, Astrophys. J. 442, 422 (1995)].
18. Early discussions of hydrogen fusion include C. F. von Weizsäcker, Phys. Z. 38, 176 (1937); G. Gamow, Phys. Rev. 53, 595 (1938); and H. A. Bethe, ibid. 55, 434 (1939). The details of the processes dominating energy generation in the sun, in the socalled pp chains, were elucidated by J. B. Oke [J. R. Astron. Soc. Can. 44, 135 (1950)]. Changes in the gravitational and internal energy, resulting from the fusion-induced changes of solar structure, also make a small contribution to the luminosity.
19. Early discussions of hydrogen fusion include C. F. von Weizsäcker, Phys. Z. 38, 176 (1937); G. Gamow, Phys. Rev. 53, 595 (1938); and H. A. Bethe, ibid. 55, 434 (1939). The details of the processes dominating energy generation in the sun, in the socalled pp chains, were elucidated by J. B. Oke [J. R. Astron. Soc. Can. 44, 135 (1950)]. Changes in the gravitational and internal energy, resulting from the fusion-induced changes of solar structure, also make a small contribution to the luminosity.
20. Early discussions of hydrogen fusion include C. F. von Weizsäcker, Phys. Z. 38, 176 (1937); G. Gamow, Phys. Rev. 53, 595 (1938); and H. A. Bethe, ibid. 55, 434 (1939). The details of the processes dominating energy generation in the sun, in the socalled pp chains, were elucidated by J. B. Oke [J. R. Astron. Soc. Can. 44, 135 (1950)]. Changes in the gravitational and internal energy, resulting from the fusion-induced changes of solar structure, also make a small contribution to the luminosity.
21. Early discussions of hydrogen fusion include C. F. von Weizsäcker, Phys. Z. 38, 176 (1937); G. Gamow, Phys. Rev. 53, 595 (1938); and H. A. Bethe, ibid. 55, 434 (1939). The details of the processes dominating energy generation in the sun, in the socalled pp chains, were elucidated by J. B. Oke [J. R. Astron. Soc. Can. 44, 135 (1950)]. Changes in the gravitational and internal energy, resulting from the fusion-induced changes of solar structure, also make a small contribution to the luminosity.
22. -1, respectively. For further details on the neutrino problem, see J. N. Bahcall, Neutrino Astrophysics (Cambridge Univ. Press, Cambridge, 1989); H. Dzitko, S. Turck-Chiėze, P. Delbourgo-Salvador, C. Lagrange, Astrophys. J. 447, 428 (1995); and (59).
23. -1, respectively. For further details on the neutrino problem, see J. N. Bahcall, Neutrino Astrophysics (Cambridge Univ. Press, Cambridge, 1989); H. Dzitko, S. Turck-Chiėze, P. Delbourgo-Salvador, C. Lagrange, Astrophys. J. 447, 428 (1995); and (59).
24. Early evolution calculations were carried out by C. B. Haselgrove and F. Hoyle [Mon. Not. R. Astron. Soc. 116, 515 (1956)] and M. Schwarzschild, R. Howard, and R. Härm [Astrophys. J. 125, 233 (1957)].
25. Early evolution calculations were carried out by C. B. Haselgrove and F. Hoyle [Mon. Not. R. Astron. Soc. 116, 515 (1956)] and M. Schwarzschild, R. Howard, and R. Härm [Astrophys. J. 125, 233 (1957)].
26. Although diffusion and settling were discussed by Eddington (5) and are unavoidable consequences of the normal assumptions of stellar modeling, their inclusion in detailed solar models is comparatively recent. See, for example, P. D. Noerdlinger, Astron. Astrophys. 57, 407 (1977); P. Demarque and D. B. Guenther, Advances in Helio- and Asteroseismology, J. Christensen-Dalsgaard and S. Frandsen, Eds. (IAU Symp. 123, Reidel, Dordrecht, 1988), p. 91; and (60); C. R. Proffitt and G. Michaud, Astrophys. J. 380, 238 (1991). The microscopic diffusion and settling rates were discussed by, for example, Michaud and Proffitt (61) and A. A. Thoul, J. N. Bahcall, and A. Loeb [Astrophys. J. 421, 828 (1994)].
27. Although diffusion and settling were discussed by Eddington (5) and are unavoidable consequences of the normal assumptions of stellar modeling, their inclusion in detailed solar models is comparatively recent. See, for example, P. D. Noerdlinger, Astron. Astrophys. 57, 407 (1977); P. Demarque and D. B. Guenther, Advances in Helio- and Asteroseismology, J. Christensen-Dalsgaard and S. Frandsen, Eds. (IAU Symp. 123, Reidel, Dordrecht, 1988), p. 91; and (60); C. R. Proffitt and G. Michaud, Astrophys. J. 380, 238 (1991). The microscopic diffusion and settling rates were discussed by, for example, Michaud and Proffitt (61) and A. A. Thoul, J. N. Bahcall, and A. Loeb [Astrophys. J. 421, 828 (1994)].
28. Although diffusion and settling were discussed by Eddington (5) and are unavoidable consequences of the normal assumptions of stellar modeling, their inclusion in detailed solar models is comparatively recent. See, for example, P. D. Noerdlinger, Astron. Astrophys. 57, 407 (1977); P. Demarque and D. B. Guenther, Advances in Helio- and Asteroseismology, J. Christensen-Dalsgaard and S. Frandsen, Eds. (IAU Symp. 123, Reidel, Dordrecht, 1988), p. 91; and (60); C. R. Proffitt and G. Michaud, Astrophys. J. 380, 238 (1991). The microscopic diffusion and settling rates were discussed by, for example, Michaud and Proffitt (61) and A. A. Thoul, J. N. Bahcall, and A. Loeb [Astrophys. J. 421, 828 (1994)].
29. Although diffusion and settling were discussed by Eddington (5) and are unavoidable consequences of the normal assumptions of stellar modeling, their inclusion in detailed solar models is comparatively recent. See, for example, P. D. Noerdlinger, Astron. Astrophys. 57, 407 (1977); P. Demarque and D. B. Guenther, Advances in Helio- and Asteroseismology, J. Christensen-Dalsgaard and S. Frandsen, Eds. (IAU Symp. 123, Reidel, Dordrecht, 1988), p. 91; and (60); C. R. Proffitt and G. Michaud, Astrophys. J. 380, 238 (1991). The microscopic diffusion and settling rates were discussed by, for example, Michaud and Proffitt (61) and A. A. Thoul, J. N. Bahcall, and A. Loeb [Astrophys. J. 421, 828 (1994)].
30. The mass of the sun is known from planetary motion, and the radius of the visible surface can be measured directly. The solar luminosity is inferred from spacebased irradiance measurements (R. C. Wilson and H. S. Hudson, Nature 332, 810 (1988)], assuming isotropic radiation and averaging over the solar-cycle variation of about 0.1 % [for stellar evidence, see S. Baliunas and R. Jastrow, Nature 348, 520 (1990)].
31. The mass of the sun is known from planetary motion, and the radius of the visible surface can be measured directly. The solar luminosity is inferred from spacebased irradiance measurements (R. C. Wilson and H. S. Hudson, Nature 332, 810 (1988)], assuming isotropic radiation and averaging over the solar-cycle variation of about 0.1 % [for stellar evidence, see S. Baliunas and R. Jastrow, Nature 348, 520 (1990)].
32. The abundances of elements other than helium can in general be determined with considerable precision [E. Anders and N. Grevesse, Geochim. Cosmochim. Acta 53, 197 (1989); N. Grevesse and A. Noels, in Origin and Evolution of the Elements, N. Prantzos, E. Vangioni, M. Cassé, Eds. (Cambridge Univ. Press, Cambridge, 1993), p. 15). Although helium was first detected in the solar spectrum, the formation of the helium lines is so complex that observation gives no precise information about its abundance.
33. The abundances of elements other than helium can in general be determined with considerable precision [E. Anders and N. Grevesse, Geochim. Cosmochim. Acta 53, 197 (1989); N. Grevesse and A. Noels, in Origin and Evolution of the Elements, N. Prantzos, E. Vangioni, M. Cassé, Eds. (Cambridge Univ. Press, Cambridge, 1993), p. 15). Although helium was first detected in the solar spectrum, the formation of the helium lines is so complex that observation gives no precise information about its abundance.
34. 6 years after this time.
35. P. Demarque and J. R, Percy, Astrophys. J. 140, 541 (1964).
36. Rotationally induced mixing was considered by B. Cnaboyer, P. Demarque, D. B. Guenther, and M. H. Pinsonneault [Astrophys. J. 446, 435 (1995)] and by O. Richard, S. Vauclair, C. Charbonnel, and W. A. Dziembowski [Astron. Astrophys., in press]. Simple models of penetration beneath the convection zone predict a region of nearly adiabatic stratification, followed by a sharp transition to radiative transport [J. H. M. M. Schmitt, R. Rosner, H. U. Bohn, Astrophys. J. 282, 316 (1984); J.-P. Zahn, Astron. Astrophys. 252, 179 (1991)]. The extent of penetration in such simple models has been constrained to a small fraction of a pressure scale height by helioseismic analyses [S. Basu, H. M. Antia, D. Narasimha, Mon. Not. R. Astron. Soc. 267, 209 (1994), M. J. P. F. G. Monteiro, J. Christensen-Dalsgaard, M. J. Thompson, Astron. Astrophys. 283, 247 (1994); I. W. Roxburgh and S. V. Vorontsov, Mon. Not. R. Astron. Soc. 268, 880 (1994)]. Beyond the region of penetration, which is certainly fully mixed, additional mixing might be caused by gravity waves induced by the penetration.
37. Rotationally induced mixing was considered by B. Cnaboyer, P. Demarque, D. B. Guenther, and M. H. Pinsonneault [Astrophys. J. 446, 435 (1995)] and by O. Richard, S. Vauclair, C. Charbonnel, and W. A. Dziembowski [Astron. Astrophys., in press]. Simple models of penetration beneath the convection zone predict a region of nearly adiabatic stratification, followed by a sharp transition to radiative transport [J. H. M. M. Schmitt, R. Rosner, H. U. Bohn, Astrophys. J. 282, 316 (1984); J.-P. Zahn, Astron. Astrophys. 252, 179 (1991)]. The extent of penetration in such simple models has been constrained to a small fraction of a pressure scale height by helioseismic analyses [S. Basu, H. M. Antia, D. Narasimha, Mon. Not. R. Astron. Soc. 267, 209 (1994), M. J. P. F. G. Monteiro, J. Christensen-Dalsgaard, M. J. Thompson, Astron. Astrophys. 283, 247 (1994); I. W. Roxburgh and S. V. Vorontsov, Mon. Not. R. Astron. Soc. 268, 880 (1994)]. Beyond the region of penetration, which is certainly fully mixed, additional mixing might be caused by gravity waves induced by the penetration.
38. Rotationally induced mixing was considered by B. Cnaboyer, P. Demarque, D. B. Guenther, and M. H. Pinsonneault [Astrophys. J. 446, 435 (1995)] and by O. Richard, S. Vauclair, C. Charbonnel, and W. A. Dziembowski [Astron. Astrophys., in press]. Simple models of penetration beneath the convection zone predict a region of nearly adiabatic stratification, followed by a sharp transition to radiative transport [J. H. M. M. Schmitt, R. Rosner, H. U. Bohn, Astrophys. J. 282, 316 (1984); J.-P. Zahn, Astron. Astrophys. 252, 179 (1991)]. The extent of penetration in such simple models has been constrained to a small fraction of a pressure scale height by helioseismic analyses [S. Basu, H. M. Antia, D. Narasimha, Mon. Not. R. Astron. Soc. 267, 209 (1994), M. J. P. F. G. Monteiro, J. Christensen-Dalsgaard, M. J. Thompson, Astron. Astrophys. 283, 247 (1994); I. W. Roxburgh and S. V. Vorontsov, Mon. Not. R. Astron. Soc. 268, 880 (1994)]. Beyond the region of penetration, which is certainly fully mixed, additional mixing might be caused by gravity waves induced by the penetration.
39. Rotationally induced mixing was considered by B. Cnaboyer, P. Demarque, D. B. Guenther, and M. H. Pinsonneault [Astrophys. J. 446, 435 (1995)] and by O. Richard, S. Vauclair, C. Charbonnel, and W. A. Dziembowski [Astron. Astrophys., in press]. Simple models of penetration beneath the convection zone predict a region of nearly adiabatic stratification, followed by a sharp transition to radiative transport [J. H. M. M. Schmitt, R. Rosner, H. U. Bohn, Astrophys. J. 282, 316 (1984); J.-P. Zahn, Astron. Astrophys. 252, 179 (1991)]. The extent of penetration in such simple models has been constrained to a small fraction of a pressure scale height by helioseismic analyses [S. Basu, H. M. Antia, D. Narasimha, Mon. Not. R. Astron. Soc. 267, 209 (1994), M. J. P. F. G. Monteiro, J. Christensen-Dalsgaard, M. J. Thompson, Astron. Astrophys. 283, 247 (1994); I. W. Roxburgh and S. V. Vorontsov, Mon. Not. R. Astron. Soc. 268, 880 (1994)]. Beyond the region of penetration, which is certainly fully mixed, additional mixing might be caused by gravity waves induced by the penetration.
40. Rotationally induced mixing was considered by B. Cnaboyer, P. Demarque, D. B. Guenther, and M. H. Pinsonneault [Astrophys. J. 446, 435 (1995)] and by O. Richard, S. Vauclair, C. Charbonnel, and W. A. Dziembowski [Astron. Astrophys., in press]. Simple models of penetration beneath the convection zone predict a region of nearly adiabatic stratification, followed by a sharp transition to radiative transport [J. H. M. M. Schmitt, R. Rosner, H. U. Bohn, Astrophys. J. 282, 316 (1984); J.-P. Zahn, Astron. Astrophys. 252, 179 (1991)]. The extent of penetration in such simple models has been constrained to a small fraction of a pressure scale height by helioseismic analyses [S. Basu, H. M. Antia, D. Narasimha, Mon. Not. R. Astron. Soc. 267, 209 (1994), M. J. P. F. G. Monteiro, J. Christensen-Dalsgaard, M. J. Thompson, Astron. Astrophys. 283, 247 (1994); I. W. Roxburgh and S. V. Vorontsov, Mon. Not. R. Astron. Soc. 268, 880 (1994)]. Beyond the region of penetration, which is certainly fully mixed, additional mixing might be caused by gravity waves induced by the penetration.
41. Rotationally induced mixing was considered by B. Cnaboyer, P. Demarque, D. B. Guenther, and M. H. Pinsonneault [Astrophys. J. 446, 435 (1995)] and by O. Richard, S. Vauclair, C. Charbonnel, and W. A. Dziembowski [Astron. Astrophys., in press]. Simple models of penetration beneath the convection zone predict a region of nearly adiabatic stratification, followed by a sharp transition to radiative transport [J. H. M. M. Schmitt, R. Rosner, H. U. Bohn, Astrophys. J. 282, 316 (1984); J.-P. Zahn, Astron. Astrophys. 252, 179 (1991)]. The extent of penetration in such simple models has been constrained to a small fraction of a pressure scale height by helioseismic analyses [S. Basu, H. M. Antia, D. Narasimha, Mon. Not. R. Astron. Soc. 267, 209 (1994), M. J. P. F. G. Monteiro, J. Christensen-Dalsgaard, M. J. Thompson, Astron. Astrophys. 283, 247 (1994); I. W. Roxburgh and S. V. Vorontsov, Mon. Not. R. Astron. Soc. 268, 880 (1994)]. Beyond the region of penetration, which is certainly fully mixed, additional mixing might be caused by gravity waves induced by the penetration.
42. Rotationally induced mixing was considered by B. Cnaboyer, P. Demarque, D. B. Guenther, and M. H. Pinsonneault [Astrophys. J. 446, 435 (1995)] and by O. Richard, S. Vauclair, C. Charbonnel, and W. A. Dziembowski [Astron. Astrophys., in press]. Simple models of penetration beneath the convection zone predict a region of nearly adiabatic stratification, followed by a sharp transition to radiative transport [J. H. M. M. Schmitt, R. Rosner, H. U. Bohn, Astrophys. J. 282, 316 (1984); J.-P. Zahn, Astron. Astrophys. 252, 179 (1991)]. The extent of penetration in such simple models has been constrained to a small fraction of a pressure scale height by helioseismic analyses [S. Basu, H. M. Antia, D. Narasimha, Mon. Not. R. Astron. Soc. 267, 209 (1994), M. J. P. F. G. Monteiro, J. Christensen-Dalsgaard, M. J. Thompson, Astron. Astrophys. 283, 247 (1994); I. W. Roxburgh and S. V. Vorontsov, Mon. Not. R. Astron. Soc. 268, 880 (1994)]. Beyond the region of penetration, which is certainly fully mixed, additional mixing might be caused by gravity waves induced by the penetration.
43. turb in the uppermost parts of the convection zone. Significant uncertainties also arise from the treatment of the convective flux (9).
44. ⊙ during the early parts of the main-sequence phase has been suggested to explain the solar surface lithium depletion [A. I. Boothroyd, I.-J. Sackmann, W. A. Fowler, Astrophys. J. 377, 318 (1991); J. A. Guzik and A. N. Cox, ibid. 448, 905 (1995).
45. ⊙ during the early parts of the main-sequence phase has been suggested to explain the solar surface lithium depletion [A. I. Boothroyd, I.-J. Sackmann, W. A. Fowler, Astrophys. J. 377, 318 (1991); J. A. Guzik and A. N. Cox, ibid. 448, 905 (1995).
46. B. Chaboyer, P. Demarque, M. H. Pinsonneault, Astrophys. J. 441, 865 (1995), and references therein. For a review of the solar lithium problem, see G. Michaud and P. Charbonneau, Space Sci. Rev. 57, 1 (1990).
47. B. Chaboyer, P. Demarque, M. H. Pinsonneault, Astrophys. J. 441, 865 (1995), and references therein. For a review of the solar lithium problem, see G. Michaud and P. Charbonneau, Space Sci. Rev. 57, 1 (1990).
48. ⊙ is 0.0245, and the age of the present sun was assumed to be 4.6 billion years. The numerical error in the calculation should be substantially smaller than the physical effects investigated. The present code has been tested against several others, using simplified but precisely defined physics [M. Gabriel, in Challenges to Theories of the Structure of Moderate-Mass Stars, vol. 388 of Lecture Notes in Physics, D. O. Gough and J. Toomre, Eds. (Springer, Heidelberg, 1991), p. 51; J. Christensen-Dalsgaard and J. Reiter, in GONG '94: Helio- and Astero-seismology from Earth and Space, R. K. Ulrich, E. J. Rhodes Jr., W. Däppen, Eds. (ASP Conf. Ser. 76, Astronomical Society of the Pacific, San Francisco, 1995), p. 136]. These and other tests indicate that the numerical error in the models is below 0.005%.
49. ⊙ is 0.0245, and the age of the present sun was assumed to be 4.6 billion years. The numerical error in the calculation should be substantially smaller than the physical effects investigated. The present code has been tested against several others, using simplified but precisely defined physics [M. Gabriel, in Challenges to Theories of the Structure of Moderate-Mass Stars, vol. 388 of Lecture Notes in Physics, D. O. Gough and J. Toomre, Eds. (Springer, Heidelberg, 1991), p. 51; J. Christensen-Dalsgaard and J. Reiter, in GONG '94: Helio- and Astero-seismology from Earth and Space, R. K. Ulrich, E. J. Rhodes Jr., W. Däppen, Eds. (ASP Conf. Ser. 76, Astronomical Society of the Pacific, San Francisco, 1995), p. 136]. These and other tests indicate that the numerical error in the models is below 0.005%.
50. D. O. Gough, J. W. Leibacher, P. H. Scherrer, J. Toomre, Science 272, 1281 (1996).
51. D. O. Gough et al., ibid., p. 1296.
52. Nonadiabatic frequency calculations were carried out by, for example, J. Christensen-Dalsgaard and S. Frandsen [Sol. Phys. 82, 165 (1983)]; Cox, Guzik, and Kidman (60); N. J. Balmforth [Mon. Not. R. Astron. Soc. 255, 632 (1992)]; and D. B. Guenther [Astrophys. J. 422, 400 (1994)], Turbulent pressure and convection introduce additional uncertainties (19).
53. Nonadiabatic frequency calculations were carried out by, for example, J. Christensen-Dalsgaard and S. Frandsen [Sol. Phys. 82, 165 (1983)]; Cox, Guzik, and Kidman (60); N. J. Balmforth [Mon. Not. R. Astron. Soc. 255, 632 (1992)]; and D. B. Guenther [Astrophys. J. 422, 400 (1994)], Turbulent pressure and convection introduce additional uncertainties (19).
54. Nonadiabatic frequency calculations were carried out by, for example, J. Christensen-Dalsgaard and S. Frandsen [Sol. Phys. 82, 165 (1983)]; Cox, Guzik, and Kidman (60); N. J. Balmforth [Mon. Not. R. Astron. Soc. 255, 632 (1992)]; and D. B. Guenther [Astrophys. J. 422, 400 (1994)], Turbulent pressure and convection introduce additional uncertainties (19).
55. J. Christensen-Dalsgaard and D. J. Mullan, Mon. Not. P. Astron. Soc. 270, 921 (1994).
56. -1 [J. Christensen-Dalsgaard and G. Berthomieu, in Solar Interior and Atmosphere, A. N. Cox, W. C. Livingston, M. Matthews, Eds. (Space Science Ser., Univ. of Arizona Press, Tucson, 1991), p. 401].
57. For modes of degree beyond the range covered by GONG, the l-dependence of the near-surface effects becomes significant. D. O. Gough and S. V. Vorontsov, Mon. Not. R. Astron. Soc. 273, 573 (1995); H. M. Antia, ibid. 274, 499 (1995).
58. For modes of degree beyond the range covered by GONG, the l-dependence of the near-surface effects becomes significant. D. O. Gough and S. V. Vorontsov, Mon. Not. R. Astron. Soc. 273, 573 (1995); H. M. Antia, ibid. 274, 499 (1995).
59. The plasma is still ideal as long as the underlying microphysics does not contain dynamical interactions. The "particles," however, can be classical or quantum, material or photonic.
60. In the solar interior, the ratio between Coulomb and kinetic energy is generally small (typically ≃ 0.1), although it reaches -0.4 in the second helium ionization zone.
61. Such equations of state are based on a model for the free energy of the plasma, incorporating approximate statistical-mechanical models. The ionization equilibria are obtained by searching for the minimum of the free energy under the stoichiometric constraints of the reactions considered. The procedure guarantees thermodynamic consistency.
62. Here the only nonideal effect is a simple prescription to ensure full ionization in stellar cores. P. P. Eggleton, J. Faulkner, B. P. Flannery, Astron. Astrophys. 23, 325 (1973).
63. D. G. Hummer and D. M. Mihalas, Astrophys. J. 331, 794 (1988); D. M. Mihalas, W. Däppen, D. G. Hummer, ibid., p. 815; W. Däppen, D. M. Mihalas, D. G. Hummer, B. W. Mihalas, ibid. 332, 261 (1988).
64. D. G. Hummer and D. M. Mihalas, Astrophys. J. 331, 794 (1988); D. M. Mihalas, W. Däppen, D. G. Hummer, ibid., p. 815; W. Däppen, D. M. Mihalas, D. G. Hummer, B. W. Mihalas, ibid. 332, 261 (1988).
65. D. G. Hummer and D. M. Mihalas, Astrophys. J. 331, 794 (1988); D. M. Mihalas, W. Däppen, D. G. Hummer, ibid., p. 815; W. Däppen, D. M. Mihalas, D. G. Hummer, B. W. Mihalas, ibid. 332, 261 (1988).
66. See (62); F. J. Rogers and C. A. Iglesias, Astrophys. J. 401, 361 (1992). As part of the OPAL project, a physical-picture equation of state suitable for stellar models was developed [F. J. Rogers, Astrophys. J. 310, 723 (1986), and references therein; _, F. J. Swenson, C. A. Iglesias,ibid. 456, 902 (1996)]. For general literature on the physical picture, see W. Ebeling, A. Förster, V. E. Fortov, V. K. Gryaznov, A. Ya. Polishchuk, Thermodynamic Properties of Hot Dense Plasmas (Teubner, Stuttgart, 1991).
67. See (62); F. J. Rogers and C. A. Iglesias, Astrophys. J. 401, 361 (1992). As part of the OPAL project, a physical-picture equation of state suitable for stellar models was developed [F. J. Rogers, Astrophys. J. 310, 723 (1986), and references therein; _, F. J. Swenson, C. A. Iglesias,ibid. 456, 902 (1996)]. For general literature on the physical picture, see W. Ebeling, A. Förster, V. E. Fortov, V. K. Gryaznov, A. Ya. Polishchuk, Thermodynamic Properties of Hot Dense Plasmas (Teubner, Stuttgart, 1991).
68. See (62); F. J. Rogers and C. A. Iglesias, Astrophys. J. 401, 361 (1992). As part of the OPAL project, a physical-picture equation of state suitable for stellar models was developed [F. J. Rogers, Astrophys. J. 310, 723 (1986), and references therein; _, F. J. Swenson, C. A. Iglesias,ibid. 456, 902 (1996)]. For general literature on the physical picture, see W. Ebeling, A. Förster, V. E. Fortov, V. K. Gryaznov, A. Ya. Polishchuk, Thermodynamic Properties of Hot Dense Plasmas (Teubner, Stuttgart, 1991).
69. See (62); F. J. Rogers and C. A. Iglesias, Astrophys. J. 401, 361 (1992). As part of the OPAL project, a physical-picture equation of state suitable for stellar models was developed [F. J. Rogers, Astrophys. J. 310, 723 (1986), and references therein; _, F. J. Swenson, C. A. Iglesias,ibid. 456, 902 (1996)]. For general literature on the physical picture, see W. Ebeling, A. Förster, V. E. Fortov, V. K. Gryaznov, A. Ya. Polishchuk, Thermodynamic Properties of Hot Dense Plasmas (Teubner, Stuttgart, 1991).
70. The physical picture has the power to avoid divergent internal partition functions of bound systems (atoms and ions), a problem that has always plagued the chemical picture.
71. J. Christensen-Dalsgaard and W. Däppen, Astron. Astrophys. Rev. 4, 267 (1992).
72. + ions, and helium may be virtually fully ionized in the solar core.
73. The effect would be even more prominent in the hydrogen ionization zone; however, there the uncertainty in the treatment of convection precludes practical applications.
74. For the helium-abundance determination, see S. V. Vorontsov, V. A. Baturin, A. A. Pamyatnykh, Nature 349, 49 (1991); A. G. Kosovichev et al., Mon. Not. R Astron. Soc. 259, 536 (1992); H. M. Antia and S. Basu, Astrophys. J. 426, 801 (1994); A. G. Kosovichev, in GONG '94: Helio- and Astero-seismology from Earth and Space, R. K. Ulrich, E. J. Rhodes Jr., W. Däppen, Eds. (ASP Conf. Ser. 76, Astronomical Society of the Pacific, San Francisco, 1995), p. 89; (42).
75. For the helium-abundance determination, see S. V. Vorontsov, V. A. Baturin, A. A. Pamyatnykh, Nature 349, 49 (1991); A. G. Kosovichev et al., Mon. Not. R Astron. Soc. 259, 536 (1992); H. M. Antia and S. Basu, Astrophys. J. 426, 801 (1994); A. G. Kosovichev, in GONG '94: Helio- and Astero-seismology from Earth and Space, R. K. Ulrich, E. J. Rhodes Jr., W. Däppen, Eds. (ASP Conf. Ser. 76, Astronomical Society of the Pacific, San Francisco, 1995), p. 89; (42).
76. For the helium-abundance determination, see S. V. Vorontsov, V. A. Baturin, A. A. Pamyatnykh, Nature 349, 49 (1991); A. G. Kosovichev et al., Mon. Not. R Astron. Soc. 259, 536 (1992); H. M. Antia and S. Basu, Astrophys. J. 426, 801 (1994); A. G. Kosovichev, in GONG '94: Helio- and Astero-seismology from Earth and Space, R. K. Ulrich, E. J. Rhodes Jr., W. Däppen, Eds. (ASP Conf. Ser. 76, Astronomical Society of the Pacific, San Francisco, 1995), p. 89; (42).
77. For the helium-abundance determination, see S. V. Vorontsov, V. A. Baturin, A. A. Pamyatnykh, Nature 349, 49 (1991); A. G. Kosovichev et al., Mon. Not. R Astron. Soc. 259, 536 (1992); H. M. Antia and S. Basu, Astrophys. J. 426, 801 (1994); A. G. Kosovichev, in GONG '94: Helio- and Astero-seismology from Earth and Space, R. K. Ulrich, E. J. Rhodes Jr., W. Däppen, Eds. (ASP Conf. Ser. 76, Astronomical Society of the Pacific, San Francisco, 1995), p. 89; (42).
78. J. Christensen-Dalsgaard, W. Däppen, Y. Lebreton, Nature 336, 634 (1988); V. A. Baturin, W. Däppen, X. Wang, F. Yang, in Proceedings of the 32nd Liege International Astrophysical Colloquium, A. Noels, M. Gabriel, N. Grevesse, P. Démarque, Eds. (Université de Liège, Liège, Belgium, 1996), p. 33.
79. J. Christensen-Dalsgaard, W. Däppen, Y. Lebreton, Nature 336, 634 (1988); V. A. Baturin, W. Däppen, X. Wang, F. Yang, in Proceedings of the 32nd Liege International Astrophysical Colloquium, A. Noels, M. Gabriel, N. Grevesse, P. Démarque, Eds. (Université de Liège, Liège, Belgium, 1996), p. 33.
80. F. Hill et al., Science 272, 1292 (1996).
81. F. Pérez Hernández and J. Christensen-Dalsgaard, Mon. Not. P. Astron. Soc. 269, 475 (1994).
82. The reason MHD fits better than EFF plus Coulomb correction is not clear. Higher order Coulomb terms bring no advantage over the description involving the leading term only [S. V. Vorontsov, V. A. Baturin, A. A. Pamyatnykh. Mon. Not. R. Astron. Soc. 257, 32 (1992)]. Models computed with the OPAL equation of state appear to provide a better approximation to solar conditions than do those using the MHD formulation [A. G. Kosovichev, Adv. Space Res. 15 (no. 7), 95 (1995); J. Christensen-Dalsgaard. Nucl. Phys. B Suppl., M. Fatas, Ed., in press].
83. The reason MHD fits better than EFF plus Coulomb correction is not clear. Higher order Coulomb terms bring no advantage over the description involving the leading term only [S. V. Vorontsov, V. A. Baturin, A. A. Pamyatnykh. Mon. Not. R. Astron. Soc. 257, 32 (1992)]. Models computed with the OPAL equation of state appear to provide a better approximation to solar conditions than do those using the MHD formulation [A. G. Kosovichev, Adv. Space Res. 15 (no. 7), 95 (1995); J. Christensen-Dalsgaard. Nucl. Phys. B Suppl., M. Fatas, Ed., in press].
84. The reason MHD fits better than EFF plus Coulomb correction is not clear. Higher order Coulomb terms bring no advantage over the description involving the leading term only [S. V. Vorontsov, V. A. Baturin, A. A. Pamyatnykh. Mon. Not. R. Astron. Soc. 257, 32 (1992)]. Models computed with the OPAL equation of state appear to provide a better approximation to solar conditions than do those using the MHD formulation [A. G. Kosovichev, Adv. Space Res. 15 (no. 7), 95 (1995); J. Christensen-Dalsgaard. Nucl. Phys. B Suppl., M. Fatas, Ed., in press].
85. A. N. Cox, J. N. Stewart, D. D. Eilers, Astrophys. J. Suppl. Ser. 11,1 (1965); A. N. Cox and J. N. Stewart, ibid., p. 22; ibid. 19, 243 (1970); ibid., p. 261; A. N. Cox and J. E. Tabor, ibid. 31, 271 (1976).
86. A. N. Cox, J. N. Stewart, D. D. Eilers, Astrophys. J. Suppl. Ser. 11,1 (1965); A. N. Cox and J. N. Stewart, ibid., p. 22; ibid. 19, 243 (1970); ibid., p. 261; A. N. Cox and J. E. Tabor, ibid. 31, 271 (1976).
87. A. N. Cox, J. N. Stewart, D. D. Eilers, Astrophys. J. Suppl. Ser. 11,1 (1965); A. N. Cox and J. N. Stewart, ibid., p. 22; ibid. 19, 243 (1970); ibid., p. 261; A. N. Cox and J. E. Tabor, ibid. 31, 271 (1976).
88. A. N. Cox, J. N. Stewart, D. D. Eilers, Astrophys. J. Suppl. Ser. 11,1 (1965); A. N. Cox and J. N. Stewart, ibid., p. 22; ibid. 19, 243 (1970); ibid., p. 261; A. N. Cox and J. E. Tabor, ibid. 31, 271 (1976).
89. A. N. Cox, J. N. Stewart, D. D. Eilers, Astrophys. J. Suppl. Ser. 11,1 (1965); A. N. Cox and J. N. Stewart, ibid., p. 22; ibid. 19, 243 (1970); ibid., p. 261; A. N. Cox and J. E. Tabor, ibid. 31, 271 (1976).
90. K. Fricke, R. S. Stobie, P. A. Strittmatter, Mon. Not. R. Astron. Soc. 154, 23 (1971); W. B. Stellingwerf, Astron. J. 83, 1184 (1978); J. O. Petersen, Astron. Astrophys. 34, 309 (1974); N. R. Simon, Astrophys. J. 260, L87 (1982).
91. K. Fricke, R. S. Stobie, P. A. Strittmatter, Mon. Not. R. Astron. Soc. 154, 23 (1971); W. B. Stellingwerf, Astron. J. 83, 1184 (1978); J. O. Petersen, Astron. Astrophys. 34, 309 (1974); N. R. Simon, Astrophys. J. 260, L87 (1982).
92. K. Fricke, R. S. Stobie, P. A. Strittmatter, Mon. Not. R. Astron. Soc. 154, 23 (1971); W. B. Stellingwerf, Astron. J. 83, 1184 (1978); J. O. Petersen, Astron. Astrophys. 34, 309 (1974); N. R. Simon, Astrophys. J. 260, L87 (1982).
93. K. Fricke, R. S. Stobie, P. A. Strittmatter, Mon. Not. R. Astron. Soc. 154, 23 (1971); W. B. Stellingwerf, Astron. J. 83, 1184 (1978); J. O. Petersen, Astron. Astrophys. 34, 309 (1974); N. R. Simon, Astrophys. J. 260, L87 (1982).
94. More complete calculations produced an increase by factors of up to 100 in the opacity of pure iron [(62); F. J. Rogers and C. A. Iglesias, Astrophys. J. Supp/. Ser. 79, 507 (1992); C. A. Iglesias, F. J. Rogers, B. G. Wilson, Astrophys. J, 397, 717 (1992)] at temperatures where transitions originating in the M shell of iron dominate the absorption. This huge increase in iron opacity is enough to increase the total opacity by a factor of 2 to 3.
95. More complete calculations produced an increase by factors of up to 100 in the opacity of pure iron [(62); F. J. Rogers and C. A. Iglesias, Astrophys. J. Supp/. Ser. 79, 507 (1992); C. A. Iglesias, F. J. Rogers, B. G. Wilson, Astrophys. J, 397, 717 (1992)] at temperatures where transitions originating in the M shell of iron dominate the absorption. This huge increase in iron opacity is enough to increase the total opacity by a factor of 2 to 3.
96. The Opacity Project [M. J. Seaton, Y. Yan, D. Mihalas, A. K. Pradhan, Mon. Not. R. Astron. Soc. 266, 805 (1994); M. J. Seaton, Ed., The Opacity Project (Institute of Physics Publ., London, 1995), vol. 1, and references therein] is mainly an atomic physics effort; plasma effects on occupation numbers are of secondary interest.
97. The Opacity Project [M. J. Seaton, Y. Yan, D. Mihalas, A. K. Pradhan, Mon. Not. R. Astron. Soc. 266, 805 (1994); M. J. Seaton, Ed., The Opacity Project (Institute of Physics Publ., London, 1995), vol. 1, and references therein] is mainly an atomic physics effort; plasma effects on occupation numbers are of secondary interest.
98. For example, pulsationally inferred masses (P. Moskalik, J. R. Buchler, A. Marom, Astrophys. J. 385, 685 (1992); S, M. Kanburand N. R, Simon, ibid. 420, 880 (1994); J. Christensen-Dalsgaard and J. O. Petersen, Astron. Astrophys. 299, L17 (1995)].
99. For example, pulsationally inferred masses (P. Moskalik, J. R. Buchler, A. Marom, Astrophys. J. 385, 685 (1992); S, M. Kanburand N. R, Simon, ibid. 420, 880 (1994); J. Christensen-Dalsgaard and J. O. Petersen, Astron. Astrophys. 299, L17 (1995)].
100. For example, pulsationally inferred masses (P. Moskalik, J. R. Buchler, A. Marom, Astrophys. J. 385, 685 (1992); S, M. Kanburand N. R, Simon, ibid. 420, 880 (1994); J. Christensen-Dalsgaard and J. O. Petersen, Astron. Astrophys. 299, L17 (1995)].
101. See, for example, J. Christensen-Dalsgaard, T. L. Duvall, D. O. Gough, J. W. Harvey, E. J. Rhodes Jr., Nature 315, 378 (1985); S. G. Korzennik and R. K. Ulrich, Astropnys. J. 339, 1144 (1989).
102. See, for example, J. Christensen-Dalsgaard, T. L. Duvall, D. O. Gough, J. W. Harvey, E. J. Rhodes Jr., Nature 315, 378 (1985); S. G. Korzennik and R. K. Ulrich, Astropnys. J. 339, 1144 (1989).
103. Much of the improvement arose because the enhanced opacity increased the depth of the conveclion zone to match more closely an earlier helioseismic determination (7). However, calculations using the new OPAL opacities and the physical-picture equation of state were able to bring the calculated value much closer to the observation [D. B. Guenther, P. Demarque, Y.-C. Kim, M. H. Pinsonneault, Astrophys. J. 387, 372 (1992); J. N. Bahcall and M. H. Pinsonneault, Rev. Mod. Phys. 64, 885 (1992); J. A. Guzik and A. N. Cox, Astrophys. J. 411, 394 (1993)]. The inclusion of element settling in the models largely explains the residual difference.
104. Much of the improvement arose because the enhanced opacity increased the depth of the conveclion zone to match more closely an earlier helioseismic determination (7). However, calculations using the new OPAL opacities and the physical-picture equation of state were able to bring the calculated value much closer to the observation [D. B. Guenther, P. Demarque, Y.-C. Kim, M. H. Pinsonneault, Astrophys. J. 387, 372 (1992); J. N. Bahcall and M. H. Pinsonneault, Rev. Mod. Phys. 64, 885 (1992); J. A. Guzik and A. N. Cox, Astrophys. J. 411, 394 (1993)]. The inclusion of element settling in the models largely explains the residual difference.
105. Much of the improvement arose because the enhanced opacity increased the depth of the conveclion zone to match more closely an earlier helioseismic determination (7). However, calculations using the new OPAL opacities and the physical-picture equation of state were able to bring the calculated value much closer to the observation [D. B. Guenther, P. Demarque, Y.-C. Kim, M. H. Pinsonneault, Astrophys. J. 387, 372 (1992); J. N. Bahcall and M. H. Pinsonneault, Rev. Mod. Phys. 64, 885 (1992); J. A. Guzik and A. N. Cox, Astrophys. J. 411, 394 (1993)]. The inclusion of element settling in the models largely explains the residual difference.
106. Simple scaling arguments (5) show that the luminosity is inversely proportional to opacity and increases as a high power of the mean molecular weight μ. Thus, to compensate for the increase in opacity, an increase in μ, and hence in Y. is required [V. A. Baturin and S. V. Ajukov, Astron. Rep. 37, 489 (1995)].
107. V. N. Tsytovich, R. Bingham, U, de Angelis, A. Forlani, M. R. Occorsio, Astropart. Phys., in press.
108. Such improvement in helioseismic inversion was first noted by J. Christensen-Dalsgaard, C. R. Proffitt, and M. J. Thompson [Astrophys. J. 403, L75 (1993)].
109. A. G. Kosovichev, in The Sun and Beyond: 2e Rencontres du Vietnam, L. M. Celnikier, Ed. (Editions Frontières, Gif sur Yvette, France, in press). For a recent review, see R. S. Raghavan, Science 267, 45 (1995).
110. A. G. Kosovichev, in The Sun and Beyond: 2e Rencontres du Vietnam, L. M. Celnikier, Ed. (Editions Frontières, Gif sur Yvette, France, in press). For a recent review, see R. S. Raghavan, Science 267, 45 (1995).
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112. J. N. Bahcall and H. A. Bethe, Phys. Rev. Lett. 65, 2233 (1990); J. N. Bahcall, Phys. Lett. B 33, 276 (1994).
113. 37Cl neutrino detections; however, the internal opacity would have to be substantially reduced in such models.
114. T. M. Brown, Annu. Rev. Astron. Astrophys. 32, 37 (1994).
115. V. Canuto and I. Mazzitelli, Astrophys. J. 370, 295 (1991).
116. J. N. Bahcall and M. H. Pinsonneault, Rev. Mod. Phys. 67, 781 (1995).
117. A. N. Cox, J. A. Guzik, R. B. Kidman, Astrophys. J. 342, 1187 (1989).
118. G. Michaud and C. R. Proffitt, in Inside the Stars. A. Baglin and W. W Weiss, Eds. (IAU Colloq. 137, ASP Conf. Ser. 40, Astronomical Society of the Pacific, San Francisco, 1993), p. 246.
119. C. A. Iglesias and F. J. Rogers, Astrophys. J. 371, L73 (1991).
120. D. O. Gough and J. Toomre, Annu. Rev. Astron. Astrophys. 29, 627 (1991).
121. S. Basu and H. M. Antia, J. Astrophys. Astron. 15, 143 (1994). Convection was treated using the Canuto and Mazzitelli formulation (58).
122. A. G. Kosovichev, in Proceedings of the Fourth SOHO Workshop: Helioseismology, J. T. Hoeksema, V. Domingo, B. Fleck, B. Battrick, Eds. (ESASP-376, European Space Research and Technology Centre, Noordwijk, Netherlands, 1995), vol. 1, p. 165.
123. C. S. Rosenthal et al., ibid., vol. 2, p. 459.
124. We thank R. Trampedach for assistance with the graphics. The work reported here was supported in part by the Danish National Research Foundation through the establishment of the Theoretical Astrophysics Center and in part by grant AST-9315112 from National Science Foundation (NSF). The GONG is managed by the National Solar Observatory, a division of the National Optical Astronomy Observatories, operated by Association of Universities for Research in Astronomy under a cooperative agreement with NSF. The data were acquired by instruments operated by the Big Bear Solar Observatory, High Altitude Observatory, Learmonth Solar Observatory, Udaipur Solar Observatory, Instituto de Astrofisica de Canarias, and Cerro Toblo Interamerican Observatory.