Equilibria and the protomodel of the Sun’s atmosphere by Karl Schwarzschild in hindsight

The concepts of radiative and adiabatic equilibria, introduced by Karl Schwarzschild in his seminal paper Ueber das Gleichgewicht der Sonnenatmosphäre published in January 1906, are the founding blocks of the theory of radiative transfer, stellar structure, and solar physics. Careful reading of the paper and its later English translation reveals small formal inaccuracies and ambiguities but with no consequences whatsoever for the final outcomes and conclusions. This paper offers their adjustments with respective derivations using contemporary formalism and sets Schwarzschild’s paper in context with a historical and modern perspective. Particular attention is paid to Schwarzschild’s largely forgotten limb-darkening formula for adiabatic equilibrium. The paper also reproduces Schwarzschild’s radiative equilibrium protomodel of the Sun’s atmosphere in graphical form and compares it with modern models presented in some of the most cited papers in stellar and solar physics.