Аннотация:Formation of kappa distribution functions and their relaxation to Maxwellian distributions are the main feature of astrophysical and space collisionless plasmas. In this work, we use the magnetosphere of the Earth as a giant plasma laboratory to study the properties of ion kappa distribution functions. Four years of measurements, performed by the multi-satellite Time History of Events and Macroscale Interactions during Substorms (THEMIS) mission during quiet geomagnetic conditions, at geocentric distances from three Earth radii (RE) to the magnetopause at daytime (of the order of 10RE), and up to 20RE at night time are used for the analyses. We find a dependence of the k parameter on the core energy E0 of a single kappa distribution inside the magnetospheric ring current and in the plasma sheet, for different values of the plasma parameter (the ratio between the plasma and magnetic pressures). We show that k increases with E0 for all values of plasma parameter, which supports earlier results obtained for the magnetospheres of the Earth, Jupiter, and Saturn, but using lower statistics. However, contrary to previous results, our studies show that the relation between k and E0 is nonlinear, and most probably is a power law with a nearly constant index. The results obtained are relevant to solve the problem of thermalization of kappa distributions.