For me, and for many astrophysicists of my generation, Chandrasekhar’s book An Introduction to the Study of Stellar Structure was very important. I could not have done my PhD (1962–1965) without it. Much more recently (1998) I realized that I could not have written my lecture course on thermodynamics and statistical mechanics without much of it, particularly the first chapter. I shall present anecdotal evidence that the influence of his discussion on the second law of thermodynamics has been important not just for astrophysics but for a much wider range of physics.

Chandrasekhar’s discussion of polytropes was masterly. Even today polytropes play an important role as an aid for understanding stellar structure. I believe that to the list of analytic solutions of the polytrope only one more has to be added: a curious $n = 5$ model of Srivastava (1962).

Stellar structure is nowadays a very computationally intensive subject. I shall illustrate this with a couple of topics from my experience with Djehuty, a supercomputer code for modelling stars in 3D. Nevertheless it remains true, I believe, that analytical mathematical entities like polytropes are fundamental as aids for understanding what the computers churn out.

How close are we to seeing a book with the title `The Last Word on the Study of Stellar Structure’? Not very, although much has been learned in 70 years. I shall discuss a few of the aspects of stellar evolution that are problematic today.

I shall discuss a couple of aspects where I believe analysis of `piecewise polytropic’ structures sheds light on the question `Why do stars become red giants?’