  Motivated by the recent discovery of subsurface oceans on planetary moons and
the interest they have generated, we explore convective flows in shallow
spherical shells of dimensionless gap width $\varepsilon^2\ll 1$ in the rapid
rotation limit $\mathrm{E}\ll1$, where $\mathrm{E}$ is the Ekman number. We
employ direct numerical simulations (DNS) of the Boussinesq equations to
compute the local heat flux $\mathrm{Nu}(\lambda)$ as a function of the
latitude $\lambda$ and use the results to characterize the trapping of
convection at low latitudes, around the equator. We show that these results are
quantitatively reproduced by an asymptotically exact nonhydrostatic equatorial
$\beta$-plane convection model at a much more modest computational cost than
DNS. We identify the trapping parameter $\beta=\varepsilon \mathrm{E}^{-1}$ as
the key parameter that controls the vigor and latitudinal extent of convection
for moderate thermal forcing when $\mathrm{E}\sim\varepsilon$ and
$\varepsilon\downarrow 0$. This model provides a new theoretical paradigm for
nonlinear investigations.
