The solar surface sports filamentary structures called coronal loops. These are stream of hot plasma. They are characterised by several parameters, including the pressure at their base $ p_0 $, their length $ L $, peak temperature $ T_{\max} $ and heating rate $ E_H $. These loops persist for many dynamical times, so there has to be equilibrium between the heating, cooling and thermal conduction. Assuming Spitzer thermal conduction we get

$ E_H \propto \nabla \left(T^{5/2} \nabla \left(T\right) \right) \propto \frac{T^{7/2}}{L^2} $

Similarly, balancing the heating and cooling, assuming a phenomenological cooling rate $ \Lambda \propto T^{-1/2} $ and using the ideal gas law $ n \propto p/T $ we get

$ E_H \propto \Lambda n^2 \propto T_{\max}^{-1/2} \frac{p_0^2}{T_{\max}^2} \Rightarrow T_{\max} \propto \left(p_0 L\right)^{1/3} $