Long Period Non-Synchronous Rotation of Io

when we refer to long period non-synchronous rotation, we are referring the the perturbation on a mostly-synchronous rotation rate, so “long period” refers to the extra rotation taking place, and we will use the acronym “NSR”. Searches for imaging evidence of NSR have been unsuccessful, indicating that NSR periods are of order 10 yr or more [3]. Theoretical arguments show that only a small amount of permanent (“frozen in”) topography is required to overcome the torques on the tidal bulge and ensure exactly synchronous rotation; [2] estimate a 10 km tall, 100 km radius mountain would synchronize Io, but in reality Io has many volcanoes which will partially cancel each other out, so it is not clear whether Io’s topography alone is enough to synchronize it. On the other hand, the lack of a pronounced apex-antapex asymmetry in crater distributions has been used to argue that NSR does happen [4]. Despite the lack of clear evidence of NSR up to ∼kyr timescales, materials will behave more viscously at longer timescales, so it is worth exploring on what timescale (if any) bodies like Io will experience NSR. Io Background: We are interested in exploring the potential for NSR on Io for two reasons. First, Io is at least partially molten, making it likely that it has a fairly low mantle viscosity compared to other satellites. Second, observations of Io’s volcanoes [5] have found them offset from where they are expected to form in canonical solid body tidal heating models [see discussion in 5]. NSR could explain this by constantly shifting volcanoes away from where they are forming. Previous searches have not found any deviation from synchronous rotation and constrain any Io NSR to have a period longer than about 4 kyr [3]. Methods: Broadly, figuring out how quickly a body rotates (above synchronous) requires figuring out its very low frequency tidal response. We explore this in two steps: First, we analytically solve equations for tidal evolution to determine how the low frequency tidal response determines the rate of NSR. Then, we use numerical codes to determine that tidal response for a suite of potential Io structures and rheologies. Tidal Calculations: We begin with a general formula for the orbitally averaged tidal torque on a body [6] as a function its response to tides (k2) and how dissipative it is (Q), which are both functions of frequency: 〈T 〉orb = − 3Gm0R 5 2Ca6 ∞ ∑