  We develop the kappa-tail fitting technique, which analyzes observations of
power-law tails of distributions and energy-flux spectra and connects them to
theoretical modeling of kappa distributions, to determine the thermodynamics of
the examined space plasma. In particular, we (i) construct the associated
mathematical formulation, (ii) prove its decisive lead for determining whether
the observed power-law is associated with kappa distributions; and (iii)
provide a validation of the technique using pseudo-observations of typical
input plasma parameters. Then, we apply this technique to a case-study by
determining the thermodynamics of solar energetic particle (SEP) protons, for a
SEP event observed on April 17, 2021, by the PSP/ISOIS instrument suite onboard
PSP. The results show SEP temperatures and densities of the order of $\sim 1$
MeV and $ \sim 5 \cdot 10^{-7} $ cm$^{-3}$, respectively.
