A Red Giants' Toy Story

In spite of the spectacular progress accomplished by stellar evolution theory some simple questions remain unanswered. One of these questions is ``Why do stars become Red Giants?''. Here we present a relatively simple analytical answer to this question. We validate our analysis by constructing a quantitative toy-model of a red giant and comparing its predictions to full stellar evolutionar models. We find that the envelope forces the value of $\nabla=d \ln T/d \ln P$ at, and above, the burning shell into a very narrow range of possible values. Together with the fact that the stellar material at the burning shell both provides and transports most of the stellar luminosity, this leads to tight relations between the thermodynamic variables at the burning shell and the mass and radius of the core -- $T_s(M_c,R_s)$, $P_s(M_c,R_s)$ and $\rho_s(M_c,R_s)$. When complemented by typical mass-radius relations of the helium cores, this implies that for all stellar masses the evolution of the core dictates the values of $T_s$, $P_s$ and $\rho_s$. We show that for all stellar masses evolution leads to an increase in the pressure and density contrasts between the shell and the core, forcing a huge expansion of the layers on top of the burning shell. Besides explaining why stars become red giants our analysis also offers a mathematical demonstration of the so-called shell homology relations, and provides simple quantitative answers to some properties of low-mass red giants.