  The generally accepted representation of $\kappa$-distributions in space
plasma physics allows for two different alternatives, namely assuming either
the temperature or the thermal velocity to be $\kappa$-independent. The present
paper aims to clarify the issue concerning which of the two possible choices
and the related physical interpretation is the correct one. A quantitative
comparison of the consequences of the use of both distributions for specific
physical systems leads to their correct interpretation. It is found that both
alternatives can be realized, but are valid for principally different physical
systems. The investigation demonstrates that, before employing one of the two
alternatives, one should be conscious about the nature of the physical system
one intends to describe, otherwise one would possibly obtain unphysical
results.
