A unified formalism for the core mass-luminosity relations of shell-burning stars

The luminosity evolution of stars with highly condensed cores surrounded by nuclear-burning shell(s) is analytically investigated with the aid of homology relations. With respect to earlier works using a similar approach (e.g. Refsdal & Weigert 1970; Kippenhahn 1981), the major improvement is that we derive all the basic dependences (i.e. on core mass, core radius, and chemical composition) in a completely generalised fashion, then accounting for a large range of possible physical properties characterising the burning shell(s). Parameterised formulas for the luminosity are given as a function of the (i) relative contribution of the gas to the total pressure (gas plus radiation), (ii) opacity source, and (iii) dominant nuclear reaction rates. In this way, the same formalism can be applied to shell-burning stars of various metallicities and in different evolutionary phases. In particular, we present some applications concerning the luminosity evolution of RGB and AGB stars with different chemical compositions, including the case of initial zero metallicity. It turns out that homology predictions provide a good approximation to the results of stellar model calculations. Therefore, the proposed formalism is useful to understand the possible differences in the luminosity evolution of shell-burning stars within a unified interpretative framework, and can be as well adopted to improve the analytical description of stellar properties in synthetic models.