  Inspired by investigations of Bose-Einstein condensates (BECs) produced in
the Cold Atom Laboratory (CAL) aboard the International Space Station, we
present a study of thermodynamic properties of shell-shaped BECs. Within the
context of a spherically symmetric `bubble trap' potential, we study the
evolution of the system from small filled spheres to hollow, large, thin shells
via the tuning of trap parameters. We analyze the bubble trap spectrum and
states, and track the distinct changes in spectra between radial and angular
modes across the evolution. This separation of the excitation spectrum provides
a basis for quantifying dimensional cross-over to quasi-2D physics at a given
temperature. Using the spectral data, for a range of trap parameters, we
compute the critical temperature for a fixed number of particles to form a BEC.
For a set of initial temperatures, we also evaluate the change in temperature
that would occur in adiabatic expansion from small filled sphere to large thin
shell were the trap to be dynamically tuned. We show that the system cools
during this expansion but that the decrease in critical temperature occurs more
rapidly, thus resulting in depletion of any initial condensate. We contrast our
spectral methods with standard semiclassical treatments, which we find must be
used with caution in the thin-shell limit. With regards to interactions, using
energetic considerations and corroborated through Bogoliubov treatments, we
demonstrate that they would be less important for thin shells due to reduced
density but vortex physics would become more predominant. Finally, we apply our
treatments to traps that realistically model CAL experiments and borrow from
the thermodynamic insights found in the idealized bubble case during adiabatic
expansion.
