  We present explicit, analytical heating formulas that predict the heating
rates of spherical, nonneutral plasmas stored in a Paul trap as a function of
cloud size $S$, particle number $N$, and Paul-trap control parameter $q$ in the
low-temperature regime close to the cloud $\rightarrow$ crystal phase
transition. We find excellent agreement between our analytical heating formulas
and detailed, time-dependent molecular-dynamics simulations of the trapped
plasmas. We also present the results of our numerical solutions of a
temperature-dependent mean-field equation, which are consistent with our
numerical simulations and our analytical results. This is the first time that
analytical heating formulas are presented that predict heating rates with
reasonable accuracy, uniformly for all $S$, $N$, and $q$.
