Magma Ocean Waves and Thermal Variability on Lava Worlds

TL;DR

This paper develops a fluid-dynamical framework to study lava worlds with dayside magma oceans subjected to eccentricity tides, solving the Laplace tidal equations for creeping-flow lava and coupling the surface tidal dissipation to deep mantle convection. By decomposing the tidal response into hemispheric eigenfunctions and summing over forcing frequencies, it demonstrates that modest eccentricities can sustain deep magma oceans and drive complex, aperiodic heat patterns across the dayside, including drifting hotspots. The analysis identifies thermal equilibria where tidal heating is balanced by convective cooling, predicting ocean depths ranging from hundreds of kilometers to near-core scales for Earth-like and super-Earth planets, and shows that tidal energy can significantly reshape surface temperatures and light curves on timescales shorter than orbital periods. The findings imply observable, time-variable thermal emission and hotspot motion for lava worlds, and highlight the role of exterior companions in maintaining eccentricity, with broad implications for interpreting phase curves and informing future observational campaigns.

Abstract

Lava worlds are rocky planets with dayside skins made molten by stellar irradiation. Tidal heating on these shortest-period planets is more than skin deep. We show how orbital eccentricities of just a few percent (within current observed bounds and maintained secularly by exterior companions) can create deep magma oceans. ``Lava tidal waves'' slosh across these oceans; we compute the multi-modal response of the ocean to tidal forcing, subject to a coastline at the day-night terminator and a parameterized viscous drag. Wave interference produces a dayside heat map that is spatially irregular and highly time-variable; hotspots can wander both east and west of the substellar point, and thermal light curves can vary and spike aperiodically, from orbit to orbit and within an orbit. Heat deposited by tides is removed in steady state by a combination of fluid, mushy, and solid-state convection in the mantle. For Earth-sized planets with sub-day periods, the entire mantle may be tidally liquified.

Figures (9)

• Figure 1: Secular forcing of a short-period rocky planet's eccentricity by an outer companion. Colored contours plot solutions to Eq. (\ref{['forced_ecc_general']}) for the forced eccentricity $e_{\rm p}$, as a function of the outer companion's eccentricity $e_{\rm c}$ and the inner-to-outer semimajor axis ratio $a_{\rm p}/a_{\rm c}$. These contours assume the inner planet is a super-Earth of mass $M_{\rm p}=8M_{\oplus}$ on a 3-day orbit, while the outer planet has a mass $M_{\rm c}$ of either $10M_\oplus$ (upper panel) or $1M_{\rm J}$ (lower panel). The gray area at top right marks the region where inner and outer orbits cross and our (octopole-level) secular theory is invalid. Overlaid on the contours are data for known outer companions ($M_{\rm c}\leq 30M_\oplus$ in the upper panel, $M_{\rm c}>30M_\oplus$ in the lower panel) to known rocky planets with $P_{\rm orb}\leq 3~$days. These observational data in combination with the theoretical contours suggest that a subset of short-period rocky planets may have $e_{\rm p}> 0.01$ and are significantly heated by eccentricity tides. For details, including calculations for how long $e_{\rm p}$ and $e_{\rm c}$ can be maintained against tidal dissipation, see Appendix \ref{['Section_maintaining_eccentricity']}.
• Figure 2: Eigenfunctions describing tidal lava flows on a hemispherical dayside magma ocean. The nightside is masked in brown. Left half shows the scalar potential functions $\phi_{r}$, computed by taking the real part of Eq. (\ref{['phi_r_eq']}) and plotted for $n=0,...,4$ and $m=0,...,n$. Right half shows the stream functions $\psi_r$ computed by taking the real part of Eq. (\ref{['psi_r_eq']}) and plotted for $n=1,...,4$ and $m=1,...,n$. Bright and dark colors correspond to negative and positive values of the eigenfunctions, respectively. Left streamlines follow the tidal irrotational flow $\grad\phi_r$, and right streamlines the tidal non-divergent flow $\grad\psi_r\cross\hat{r}$.
• Figure 3: Time-and-space averaged heat generated by tidal motion of a magma ocean on our fiducial short-period planet (modeled after 55 Cancri e), assuming an orbital eccentricity of $e_{\rm p}=0.05$ and two values for the Rayleigh drag frequency $\sigma_{\rm R}$, compared to the average stellar insolation. The tidal power peaks and exceeds the stellar insolation for intermediate ocean thicknesses $H$ that roughly satisfy Eq. (\ref{['H_peak']}).
• Figure 4: The effect of lava tidal flows on the thermal emission of our fiducial short-period rocky planet. The surface equilibrium temperature $T_{\rm s}$ is plotted vs. time accounting for either stellar insolation alone (black curve), or insolation plus tidal heating within the magma ocean (red curve). Orbital eccentricity $e_{\rm p}$ varies between different panels (note different y-axis scales). These results presume $H/R_{\rm p} = 1\%$ and $\sigma_{\rm R} = 10^{-3}~{\rm s}^{-1}$, which according to Fig. \ref{['Fig_PT_over_Pins']} implies $\overline{\mathcal{P}}_{\rm T}/\overline{\mathcal{P}}_{\rm ins} \simeq 5$ (tidal heating dominates) for $e_{\rm p} = 0.05$. Whether $\overline{\mathcal{P}}_{\rm T}/\overline{\mathcal{P}}_{\rm ins} > 1$ or $<1$ in thermal equilibrium depends on $e_{\rm p}$ and other model details (see section \ref{['Section_thermal_equilibria']} for the equilibrium theory).
• Figure 8: Sample interior structures for different ($R_{\rm p}, M_{\rm p}$). Solid curves plot the thickness $H$ of the fluid magma ocean as a function of potential temperature $T_{\rm p}$, while dashed curves plot the thickness of the mushy layer, both relative to the full thickness of the mantle. We adopt an Earth-like mantle volume fraction of $84\%$; the corresponding mantle radius is $0.46R_{\rm p}$. For calculation details, see Appendix \ref{['Appendix_mantle_phase_diagram']}.