Standard Solar Model - Evolution of The Sun

Evolution of The Sun

Nuclear reactions in the core of the Sun change its composition, by converting hydrogen nuclei into helium nuclei by the proton-proton chain and (to a lesser extent in the Sun than in more massive stars) the CNO cycle. This decreases the mean molecular weight in the core of the Sun, which should lead to a decrease in pressure. This does not happen as instead the core contracts. By the Virial Theorem half of the gravitational potential energy released by this contraction goes towards raising the temperature of the core, and the other half is radiated away. By the ideal gas law this increase in temperature also increases the pressure and restores the balance of hydrostatic equilibrium. The luminosity of the Sun is increased by the temperature rise, increasing the rate of nuclear reactions. The outer layers expand to compensate for the increased temperature and pressure gradients, so the radius also increases.

No star is completely static, but stars stay on the main sequence (burning hydrogen in the core) for long periods. In the case of the Sun, it has been on the main sequence for roughly 4.6 billion years, and will become a red giant in roughly 6.5 billion years for a total main sequence lifetime of roughly 1010 years. Thus the assumption of steady state is a very good approximation. For simplicity, the stellar structure equations are written without explicit time dependence, with the exception of the luminosity gradient equation:

Here L is the luminosity, ε is the nuclear energy generation rate per unit mass and ε_ν is the luminosity due to neutrino emission (see below for the other quantities). The slow evolution of the Sun on the main sequence is then determined by the change in the nuclear species (principally hydrogen being consumed and helium being produced). The rates of the various nuclear reactions are estimated from particle physics experiments at high energies, which are extrapolated back to the lower energies of stellar interiors (the Sun burns hydrogen rather slowly). Historically, errors in the nuclear reaction rates have been one of the biggest sources of error in stellar modelling. Computers are employed to calculate the varying abundances (usually by mass fraction) of the nuclear species. A particular species will have a rate of production and a rate of destruction, so both are needed to calculate its abundance over time, at varying conditions of temperature and density. Since there are many nuclear species, a computerised reaction network is needed to keep track of how all the abundances vary together.

According to the Vogt-Russel theorem, the mass and the composition structure throughout a star uniquely determine its radius, luminosity, and internal structure, as well as its subsequent evolution (though this "theorem" was only intended to apply to the slow, stable phases of stellar evolution and certainly does not apply to the transitions between stages and rapid evolutionary stages). The information about the varying abundances of nuclear species over time, along with the equations of state, is sufficient for a numerical solution by taking sufficiently small time increments and using iteration to find the unique internal structure of the star at each stage.