Dependence of normalized particle energy ε(v) (normalized by ) on particle speed v for five different potential exponents n with a fixed shape parameter α = 1.33. For small speed, the particle energy follows a parabolic law with particle speed (ε(v) ∝ v² Eq. (30)). While for large speed, the particle energy follows a linear scaling with particle speed (ε(v) ∝ v in Eq. (31)) that is different from gas molecules. Particle energy from N-body simulation is plotted as the dash line. Simulation matches an effective exponent n = −1.2 for virial theorem (not −1 due to the mass cascade and nonzero halo surface energy; Xu 2021b).