  The Earth's magnetosphere represents a natural plasma laboratory that allows
us to study the behavior of particle distribution functions in the absence of
Coulomb collisions, typically described by the Kappa distributions. We have
investigated the properties of these functions for ions and electrons in
different magnetospheric regions, thereby making it possible to reveal the
$\kappa$-parameters for a wide range of plasma beta ($\beta$) values (from
$10^{-3}$ to $10^{2}$). This was done using simultaneous ion and electron
measurements from the five Time History of Events and Macroscale Interactions
during Substorms (THEMIS) spacecraft spanning the years 2008 to 2018. It was
found that for a fixed plasma $\beta$, the $\kappa$-index and core energy
($E_c$) of the distribution can be modeled by the power-law
$\kappa=AE_c^\gamma$ for both species, and the relation between $\beta$,
$\kappa$, and $E_c$ is much more complex than earlier reported: both $A$ and
$\gamma$ exhibit systematic dependencies with $\beta$. Our results indicate
that $\beta \sim 0.1-0.3$ is a range where the plasma is more dynamic since it
is influenced by both the magnetic field and temperature fluctuations, which
suggests that the transition between magnetically dominated plasmas to
kinetically dominated plasmas occurs at these values of $\beta$. For $\beta > 1
$, both $A$ and $\gamma$ take nearly constant values, a feature that is
especially notable for the electrons and might be related to their
demagnetization. The relation between $\beta$, $\kappa$, and $E_c$ that we
present is an important result that can be used by theoretical models in the
future.
