### 3.1 Basic energetics of coronal expansion

The majority of the models have been concerned with the fast solar wind which, at least during solar minimum, appears to be the basic or equilibrium mode of flow. Its properties can be reproduced by using (single- or multi-) fluid models involving waves. Such studies show that electrons may remain hot because of their high heat conduction. Although protons (and other ions) can be accelerated by magnetohydrodynamic wave pressure, it is necessary that they are heated preferentially in the corona. This can be concluded from a simple consideration of the energetics of a polytropic model of coronal expansion, in which the sum of the specific enthalpy, binding gravitational energy and kinetic energy is conserved (Bernoulli’s equation). This conservation law takes the simple form:
where is Boltzmann’s constant, Newton’s gravitational constant, the proton mass, the Sun’s mass, its radius, the terminal wind speed, and the coronal temperature. The escape speed from the solar surface is , giving 618 km s–1.

The above constraint requires a coronal energy per proton-electron pair of about 5 keV, to release the fast wind from the Sun’s gravitational potential well and attain its high asymptotic speed. If , there is no critical point, so that if the corona appears gravitationally bound. To obtain a fast flow according to Equation (2), an average temperature of about is needed for and . However, isothermal models, for which , require an infinite amount of internal energy, since formally their enthalpy diverges. Anyway, the key issue of coronal heating is not even addressed in a polytropic model. However, one has to deal with the thermodynamics of the weakly collisional and turbulent corona and wind, a complex problem which requires the kinetic approach. Apparently, in the single-fluid description the coronal temperature profile entirely determines the coronal expansion and solar wind outflow, which is a natural consequence of a hot corona. Here we do not want to address fluid-modelling issues, but refer the reader to the book chapter by Marsch et al. (2003) and further references therein.