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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., Comment: 26 pages, 10 figures. Accepted for publication in Physical Review Fluids