Titan's Dynamic Love Number Implies Stably-Stratified Ocean

Abstract

The dynamic quadrupole Love number of Titan measured by \Cassini is $k_\mathrm{2,obs}=0.616\pm 0.067$, strongly indicating a global subsurface ocean. However, the theoretical Love number due to equilibrium tides is at most $k_\mathrm{2,eq}^\mathrm{max}\approx 0.48$ in the absence of an ice shell on top of the ocean. In reality, there is an outer ice shell of thickness $ 100\,\mathrm{km}$, reducing the equilibrium-tide Love number to $k_\mathrm{2,eq}\approx 0.42$. Therefore, other types of tidal response, like dynamic tides, may be also present in Titan. We propose that the ocean is stably stratified. As a result, there exist standing ocean waves (gravity modes) with eigen-frequencies close to the tidal frequency. Such a gravity mode (g-mode) is resonantly excited. It bends the outer ice shell radially and thus enhances the dynamic Love number by $k_\mathrm{2,g}$. In order for $k_\mathrm{2,g}$ to account for the discrepancy between $k_\mathrm{2,eq}$ and $k_\mathrm{2,obs}$, the Brunt-Vaisala frequency in the ocean is required to be $3.3\times 10^{-4}\,\mathrm{rad\, s^{-1}}$. It is compatible with the volatile-rich model for Titan that was proposed to explain the methane-rich atmosphere. The three components of the tidal potential with azimuthal degrees, $m=-2,0,2$, correspond to the three components of the quadrupole Love number, $k_\mathrm{2,-2}$, $k_\mathrm{2,0}$ and $k_\mathrm{2,2}$. They can excite retrograde, axisymmetric and prograde g-modes equally in the absence of rotation. However, Coriolis force induced by Titan's rotation breaks the symmetry among these modes. Most likely, only one of the Love-number components is significantly enhanced by a g-mode, while the other two are still attributed to equilibrium tides. This prediction is testable by observation. If confirmed, the smaller components of the Love number can be used to constrain the thickness of the outer ice shell.