Solar structure and evolution Jørgen Christensen-Dalsgaard Institut for Fysik og Astronomi, Aarhus Universitet, Ny Munkegade, DK-8000 Aarhus C, Denmark email: [email protected] http://astro.phys.au.dk/~jcd Last Modified February 25, 2019 Abstract [Some suitable abstract]. 1 1 Introduction [sec:introduction] The study of stellar properties and stellar evolution plays a central role in astrophysics. Ob- servations of stars determine the chemical composition, age and distance of the varied components of the Milky Way Galaxy and hence form the basis for studies of Galactic evolution. Stellar abun- dances and their evolution, particularly for lithium, are also a crucial component of the study of Big-Bang nucleosynthesis. Understanding of the pulsational properties of Cepheids underlies their use as distance indicators and hence the basic unit of distance measurement in the Universe. The detailed properties of supernovae are important for the study of element nucleosynthesis, while supernovae of Type Ia are crucial for determining the large-scale properties of the Universe, in- cluding the evidence for a dominant component of `dark energy'. In all these cases an accurate understanding, and modelling, of stellar interiors and their evolution is required for reliable results. Modelling stellar evolution depends on a detailed treatment of the physics of stellar interiors. Insofar as the star is regarded as nearly spherically symmetric the basic equations of stellar equilib- rium are relatively straightforward (see Section 2.1), but the detailed properties, often referred to as microphysics, of matter in a star are extremely complex, yet of major importance to the modelling. This includes the thermodynamical properties, as specified by the equation of state, the interaction between matter and radiation described by the opacity, the nuclear processes generating energy and causing the evolution of the element composition, and the diffusion and settling of elements. Equally important are potential hydrodynamical processes caused by various instabilities which may contribute to the transport of energy and material, hence causing partial or full mixing of given regions in a star. It is obvious that sufficiently detailed observations of stellar properties, and comparison with models, may provide a possibility for testing the physics used in the model calculation, hence allowing investigations of physical processes far beyond the conditions that can be reached in a terrestrial laboratory. Amongst stars, the Sun obviously plays a very special role, both to our daily life and as an astrophysical object. Its proximity allows very precise, and probably accurate, determination of its global parameters, as well as extremely detailed investigations of phenomena in the solar atmosphere, compared with other stars. Indications are that the Sun is typical for its mass and age [Check further references and coordinate with Section 7].(e.g., Gustafsson, 1998; Robles et al., 2008);1 also, conditions in the solar interior are relatively benign, providing some hope that reasonably realistic modelling can be carried out. Thus it is an ideal case for investigations of stellar structure and evolution. Interestingly, there still remain very significant discrepancies between the observed properties of the Sun and solar models. An excellent overview of the development of the study of stellar structure and evolution was provided by Tassoul and Tassoul (2004). The application of physics to the understanding of stellar interiors developed from the middle of the 19th century. The first derivation of stellar models based on mechanical equilibrium was carried out by Lane (1870).2 Further development of the theory of such models, summarized in an extensive bibliographical note by Chandrasekhar (1939), was carried out by Ritter, Lord Kelvin and others, culminating in the monograph by Emden (1907). These models were based on the condition of hydrostatic equilibrium, combined with a simplified, so-called polytropic, equation of state. Major advances came with the application of the theory of radiative transfer, and quantum-mechanical calculations of atomic absorption coefficients, to the energy transport in stellar interiors. This allowed theoretical estimates to be made of the relation between the stellar mass and luminosity, even without detailed knowledge about the stellar 1However, subtle and potentially very interesting differences have recently been found between the solar surface composition and the composition of similar stars; see Section 7, in particular Fig. 64. 2It is interesting to note that the lengthy title of that paper explicitly refers to the use of `the laws of gases as known to terrestrial experiments'; the application of terrestrial physics to the modelling of stars of course remains a key aspect to the study of stellar interiors. 2 energy sources (for a masterly discussion of these developments, see Eddington 1926). Further investigations of the properties of stellar opacity led to the conclusion that stellar matter was dominated by hydrogen (Str¨omgren, 1932, 1933), in agreement with the detailed determination of the composition of the solar photosphere by Russell (1929), as well as with the analysis of a broad range of stars by Uns¨old (1931). Although stellar modelling had proceeded up to this point without any definite information about the sources of stellar energy, this issue was evidently of very great interest. As early as 1920 Eddington (1920) and others noted that the fusion of hydrogen into helium might produce the required energy, over the solar lifetime, but a mechanism making the fusion possible, given the strong Coulomb repulsion between the nuclei, was lacking. This mechanism was provided by Gamow's development of the treatment of quantum-mechanical barrier penetration between reacting nuclei, resulting in the identification of the dominant reactions in hydrogen fusion through the PP chains and the CNO cycle (cf. Section 2.3.3) by von Weizs¨acker (1937, 1938), Bethe and Chritchfield (1938) and Bethe (1939). With this, the major ingredients required for the modelling of the solar interior and evolution had been established. An important aspect of solar structure is the presence of an outer convection zone. Following the introduction by Schwarzschild (1906) of the criterion for convective instability in stellar atmo- spheres, Uns¨old (1930) noted that such instability would be expected in the lower photosphere of the Sun. As a very important result, Biermann (1932) noted that the temperature gradient result- ing from the consequent convective energy transport would in general be close to adiabatic; as a result, the structure of the convection zone depends little on the details of the convective energy transport. Also, he found that the resulting convective region in the Sun extended to very sub- stantial depths, reaching a temperature of 107 K. Further calculations by, for example, Biermann (1942) and Rudkjøbing (1942), taking into account more detailed models of the solar atmosphere, generally confirmed these results. In an interesting short paper Str¨omgren (1950) summarized these early results. He noted that the presence of 7Li in the solar atmosphere clearly showed that convective mixing could extend at most to a temperature of 3:5 × 107 K,3 beyond which lithium would be destroyed by nuclear reactions. He also pointed out that a revision of determinations of the composition of the solar atmosphere, relative to the one assumed by Biermann, had reduced the heavy-element abundance and that this would reduce the temperature at the base of the con- vection zone to the acceptable value of 2:5 × 106 K. Although these models are highly simplified, the use of the lithium abundance as a constraint on the extent of convective mixing, and the effect of a composition adjustment on the convection-zone depth, remain highly relevant, as discussed below. Specific computations of solar models must satisfy the known observational constraints for the Sun, namely that solar radius and luminosity be reached at solar age, for a 1M⊙ model. As discussed by Schwarzschild et al. (1957) this can be achieved by adjusting the composition and the characteristics of the convection zone. They noted that no independent determination of the initial hydrogen and helium abundances X0 and Y0 was possible and consequently determined models for specified initial values of the hydrogen abundance. The convection zone was assumed to have an adiabatic stratification and to consist of fully ionized material, such that it was characterized by 5=3 the adiabatic constant K in the relation p = Kρ between pressure p and density ρ. Given X0 the values of Y0 and K were then determined to obtain a model with the correct luminosity and radius. Although since substantially refined, this remains the basic principle for the calibration of solar models (see Section 2.6). A detailed discussion of the calibration of the properties of the convection zone was provided by Gough and Weiss (1976). Given the calibration, the observed mass, radius and luminosity clearly provide no test of the validity of the solar model. An important potential for testing solar models became evident with the realization (Fowler, 1958) that nuclear reactions in the solar core produce huge numbers of neutrinos which in principle may be measured, given a suitable detector (Davis, 1964; Bahcall, 3A modern value to this limit is 2:5 × 106 K; see Section 5.3. 3 1964). The first results of a large-scale experiment (Davis et al., 1968) surprisingly showed an upper limit to the neutrino flux substantially below the predictions of the then current solar models. Further experiments using a variety of techniques, and additional computations, did not eliminate this discrepancy, the predictions being higher by a factor 2{3 than the experiment, until the beginning of the present millennium. An independent way of testing solar models, with potentially much higher selectivity, became available with the detection of solar oscillations (see Christensen-Dalsgaard, 2004, for further details on the history of the field). Oscillations with periods near 5 minutes were discovered by Leighton et al.