  The acceleration of the solar coronal plasma to supersonic speeds is one of
the most fundamental yet unresolved problem in heliophysics. Despite the
success of Parker's pioneering theory on an isothermal solar corona, the
realistic solar wind is observed to be non-isothermal, and the decay of its
temperature with radial distance usually can be fitted to a polytropic model.
In this work, we use Parker Solar Probe data from the first nine encounters to
estimate the polytropic index of solar wind protons. The estimated polytropic
index varies roughly between 1.25 and 1.5 and depends strongly on solar wind
speed, faster solar wind on average displaying a smaller polytropic index. We
comprehensively analyze the 1D spherically symmetric solar wind model with
polytropic index $\gamma \in [1,5/3]$. We derive a closed algebraic equation
set for transonic stellar flows, i.e. flows that pass the sound point smoothly.
We show that an accelerating wind solution only exists in the parameter space
bounded by $C_0/C_g < 1$ and $(C_0/C_g)^2 > 2(\gamma-1)$ where $C_0$ and $C_g$
are the surface sound speed and one half of the escape velocity of the star,
and no stellar wind exists for $\gamma > 3/2$. With realistic solar coronal
temperatures, the observed solar wind with $\gamma \gtrsim 1.25$ cannot be
explained by the simple polytropic model. We show that mechanisms such as
strong heating in the lower corona that leads to a thick isothermal layer
around the Sun and large-amplitude Alfv\'en wave pressure are necessary to
remove the constraint in $\gamma$ and accelerate the solar wind to high speeds.
