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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

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