Kappa distribution

A Kappa distribution is a heavy-tailed distribution used to model particle velocities with a suprathermal population. One isotropic three-dimensional convention is $f_{\kappa }(\mathbf{v})=A_{\kappa }[1+v^2/(\kappa {\theta }^2){]}^{-(\kappa +1)}$, where $A_{\kappa }$ normalizes the velocity density and $\theta$ is a speed parameter.

For fixed $\theta$, the limit $\kappa \to \infty$ is proportional to $e^{-v^2/{\theta }^2}$. Finite $\kappa$ gives a power-law tail instead. The kinetic second moment requires $\kappa >3/2$: multiplying the large-speed density by $v^4$ gives a radial integral proportional to $\int v^{2-2\kappa }dv$. A fitted speed parameter must not be identified with temperature without checking the convention.^[ref]