We construct C2 representations of the background quantities that characterize the interior of the Sun and its atmosphere starting from the data-points of the standard solar model S. This model is further extended considering an isothermal atmosphere, that we refer to as model AtmoI. It is not trivial to build the C2 representations of the parameters from a discrete set of values, in particular in the transition region between the end of model S and the atmosphere. This technical work is needed as a crucial building block to study theoretically and numerically the propagation of waves in the Sun, using the equations of solar oscillations (also referred to as Galbrun's equation in aeroacoustics). The constructed models are available at http://phaidra.univie.ac.at/o:1097638.