Episodic volcanism of tidally heated satellites with application to Io

Abstract

A simple model of the coupled thermal and orbital evolution of a tidally heated satellite in an orbital resonance is presented and applied specifically to Io. The model quantitatively demonstrates how a feedback mechanism between the orbital and thermal energy of such a satellite can lead to periodic variations in surface heatflow and orbital eccentricity. The convective heatflow and ( k/ Q) of the satellite are parameterized as local power laws of the temperature, where Q is the quality factor and k is the second-degree tidal potential Love number. The time evolution of the model is determined by two nonlinear equations: an equation governing the orbital eccentricity, and a simple heat-balance equation determining the temperature. A linear stability analysis reveals that the time-independent solution is unstable if n > m + p, where n and m are the exponents in the power laws for ( k/ Q) and convective heatflow, respectively, and p is the ratio of the convective cooling time scale to the time scale for equilibration of the eccentricity. Numerical integration of the nonlinear equations reveals behavior in qualitative agreement with this relation. Laboratory data on near-solidus peridotites suggest 20 ≲ n ≲ 30 and parameterized convection schemes suggest m ∼ 10. Since p is of order unity, it follows that tidally heated satellites are probably in the unstable regime if they are operating near the solidus. It is thus probable that Io has no thermal steady state. The model is made more realistic by (1) arresting the reduction of ( k/ Q) at low temperature, and (2) arresting the growth of temperature at the mantle solidus and allowing volcanism to remove the excess heat. When the second modification is included, the unstable regime becomes periodic. In addition, a global k substantially larger than the elastic value is possible for a mostly solid Io because the body may begin to behave viscously when the tidal period is longer than or comparable to the Maxwell time. This requires a solid-state viscosity of ≲4 × 10 15 Pa sec, which may be achievable with a small amount of partial melt. The model can easily be adjusted to pass through Io's current observed heatflow (1–2 W m −2) and eccentricity (∼0.004) for reasonable choices of parameters ( Q/ k) min ∼ 100, ( Q/ k) max ∼ few × 10 3, solidus viscosity ∼10 15–10 17 Pa sec, and Q J within the required dynamical bounds. The periods of high heatflow and acceptable eccentricity typically have durations of ∼20–30 myr, separated in time by ∼80–100 myr. Spatial heterogeneities in Io's thermal structure are likely to make the behavior more complicated. The model predicts that Io's mean motion may be currently increasing, a possibility suggested by recent estimates of n dot 1 from eclipse data. Since Europa's eccentricity mimics that of Io, the model also implies that the tidal stresses in Europa's ice shell may have recently been large enough to produce the observed fracturing. The episodic heating mechanism may be responsible for the resurfacing of Enceladus < 10 9 years ago.