RTV Scaling Laws

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

$E_{H}\propto \Lambda n^{2}\propto T_{\max }^{-1/2}{\frac {p_{0}^{2}}{T_{\max }^{2}}}\Rightarrow T_{\max }\propto \left(p_{0}L\right)^{1/3}$