to the density ratio $\rho_d/\rho_{\rm out}=780$, while the inner and outer kinematic viscosities are $\nu_d = 10^{-5}$ m$^2$/s and $\nu_{\rm out}=1.5\times 10^{-5}$ m$^2$/s, leading to the viscosity ratio of $\nu_d/\nu_{\rm out}=0.66$. The viscosity of both fluids is sufficiently low that the stability diagram approaches that of the Mathieu equation. (One of the conclusions of [@Ali1] is that the Mathieu equation describes the inviscid Faraday instability even in a spherical geometry.) More specifically, $\nu_d/({R}^2\omega)$ is between $3\times 10^{-4}$ and $7\times 10^{-3}$ for the capillary cases and between $6\times 10^{-4}$ and $9\times 10^{-4}$ for the gravitational cases. Usually, Faraday waves are subharmonic, i.e. their period is twice the forcing period $T$. However, in this low-viscosity regime, we can easily excite harmonic waves as well [@K1996], whose period is the same as the forcing period. For the capillary cases, we set the surface tension to be $\sigma=0.02$ kg/s$^2$, the constant radial acceleration to be $g=0$, and the drop radius to be $R=0.06$ m, while for the gravitational cases, we set the constant radial acceleration to be $g=-1$ m/s$^2$, the surface tension to be $\sigma=0$, and the drop radius to be $R=0.05$ m. We will use these parameters to non-dimensionalize the forcing frequencies and 