A few of my interests are:
Please see my publications page or ADS for papers and work in progress.Enceladus
In 2005, the Cassini spacecraft flew by Enceladus and observed active plumes originating from the south polar region (the "tiger stripes"). The plumes are spewing out enough water to form the E ring of Saturn! This activity is surprising for a tiny body like Enceladus, only 252 km in radius, that was predicted to be geologically dead.Estimates of the heat flow associated with the solar polar region have continued to be revised upwards since the first flyby. In 2006, the heat flow was given as 5.8 ± 1.9 GW (Spencer et al. 2006). More recent estimates are 13.6 ± 1.4 GW, from observations in March 2008, and 17.5 +2.1 or -1.9 GW, from observations in October 2008 (Spencer et. al 2009).
The observed heat flow and geyser activity on Enceladus are inconsistent with conventional estimates of tidal heating. In Meyer and Wisdom (2007), we showed that the equilibrium tidal heating in the current 2:1 Enceladus-Dione mean motion resonance is only 1.1(18,000/QS) GW, an order of magnitude too small to match the observations. This calculation relied only on conservation of energy and angular momentum, not on any unknown physical parameters of Enceladus such as rigidity. This result implies that either Enceladus is not in equilibrium (orbitally or thermally), the heat source is not tidal, or the Q of Saturn is somehow less than the theoretical minimum of 18,000.
In Meyer and Wisdom (2008a), we constructed a resonance model to study the evolution of Enceladus and other Saturnian satellites through various mean motion resonances in order to look for non-equilibrium dynamical behavior. We found no current non-equilibrium behavior and in fact, the orbital dynamics indicated that Enceladus is at or very near its equilibrium eccentricity.
We then looked for non-equilibrium thermal behavior (Meyer and Wisdom 2008b) using the only published thermal oscillation model (Ojakangas and Stevenson 1986). This model was designed to explain Io's anomalous large heat flow, but when applied to Enceladus does not produce oscillations or any heating above the equilibrium value.
This leaves an unspecified nontidal heat source for Enceladus or a Q of Saturn smaller than 18,000. The constraint that the Q of Saturn is less than 18,000 comes from not allowing Mimas' semimajor axis to begin interior to the synchronous radius of Saturn at the beginning of the solar system. If Mimas' semimajor axis was inside the synchronous value, Mimas would have evolved inward, not outward, as a result of tides and could not possibly be at its current semimajor axis. For an average Q value of 18,000, Mimas would evolve from just outside the synchronous radius to its current location in the age of the solar system. Possible ways of getting around this constraint are to have time or frequency dependence of Saturn's Q. Then either today's Q is different than the average value of 18,000 or the Q at Mimas' orbital frequency is different than the one that applies to Enceladus. Calculating the Q of giant planets is tricky business (i.e. Ogilvie and Lin 2005 and Wu 2005). Time and frequency dependence are surely both important.
In short, Enceladus is still a unsolved mystery.
The Evolution of the Moon
The Moon is the best studied body in our solar system, besides the Earth, yet is still not completely understood. The Moon is the most dramatic example of tidal evolution in the solar system. It is believed that the Moon formed as the result of a giant impact with the Earth at a semimajor axis of about 3 Earth radii and then tides swept the Moon out to its current orbit with a semimajor axis of about 60 Earth radii. The dynamics of the Moon as it evolves outward are, well, dynamic. Two of the many outstanding problems associated with the Moon are its shape and its magnetism.The Shape of the Moon
The shape of the Moon is not the shape that would be in hydrostatic equilibrium with the current lunar orbit, as was first noticed by Laplace in 1799. In fact, the shape is not a hydrostatic shape for any circular orbit. This led Ian Garrick-Bethell and collaborators (2006) to propose that the shape of the Moon could be explained by a high orbital eccentricity for the Moon in the past - either the Moon froze in its shape in synchronous rotation at a semimajor axis of 22.9 Earth radii with an orbital eccentricity of 0.49, or it froze in its shape in one of two 3:2 spin-orbit resonance configurations.
In Meyer, Elkins-Tanton, and Wisdom (2010,2011), we investigated whether the Moon could have passed through the synchronous shape solution of Garrick-Bethell et al. (2006) and whether the Moon could have recorded the shape from this time. The early Moon was likely covered by a magma ocean that gradually solidified. As the Moon solidified, the heavy crystals sank to the bottom of the magma ocean and any crystals lighter than the liquid, such as plagioclase, rose to the top. The plagioclase formed a lid at the surface of the Moon that both insulated the magma ocean, prolonging the freezing process, and enhanced the tidal heating on the early Moon. The tidal heating on the Moon was already large because the Moon was much closer to the Earth and would be enhanced further by a thin lid flexing atop a liquid layer.
We constructed a coupled thermal-orbital model to model the interplay between the freezing magma ocean, tides, and the evolving orbit. We found that we could only match the synchronous shape solution of Garrick-Bethell et al. (2006) by stretching the dissipation in the Earth to unphysical values. However, we also used a simple minimum energy argument adapted from Goldreich and Mitchell (2009) to show that the Moon would not record a shape from this epoch anyway.
The argument goes like this: the Moon instantaneously has the shape dictated by the shape solution at some high eccentricity orbit. Then the eccentricity changes to a low value (in our model, this happens very rapidly) and the shape is no longer in hydrostatic equilibrium. The Moon then has a choice to make - it can deform to match the new hydrostatic shape, storing elastic energy in the lithosphere as a result, or it can keep its shape and store gravitational potential energy instead. We can assume that the Moon will choose the lowest energy configuration, so we can just compute the magnitudes of the elastic energy and the gravitational potential energy for each of these choices and compare them. We find that for any rigidity assumed for the plagioclase lid, the Moon will either break or deform, losing its shape. We conclude that the Moon will not be able to record its odd, non-hydrostatic shape until after the magma ocean is completely frozen.
Magnetism on the Moon
Paleomagnetic measurements of lunar rocks show magnetic remanence most easily explained by a long-lived early lunar dynamo (Garrick-Bethell 2009). Dwyer and Stevenson (2005) argued that the only plausible driving force for an early lunar dynamo is mechanical stirring of the liquid core due to the relative motion between the core and mantle. Further work by Dwyer and collaborators has found that stirring could drive a dynamo powerful enough to explain the observed paleomagnetism for up to a billion years. However, this driving mechanism is only an option if the core of the Moon does not precess along with the mantle.
If the core spin axis and mantle spin axis precess together, there will be no stirring due to mutual motion and thus no dynamo. Goldreich (1967) showed that a liquid lunar core of low viscosity would not precess with the mantle today; the spin axis of the lunar core is nearly normal to the ecliptic. Therefore, today a dynamo is not excluded, if only the stirring was large enough.
However, for the Earth, the core precesses with the mantle because of the inertial coupling mechanism (Poincare 1910, Toomre 1966). The core spin axis and mantle spin axis are nearly parallel and regress with the same period. So precession-driven stirring could not produce a dynamo on the Earth - good thing we have strong convection in the core to drive the dynamo instead. The inertial coupling mechanism relies on torques due to differential pressures along the core-mantle boundary to align the two spin axes.
Goldreich (1967) showed that the inertial coupling mechanism fails for the Moon, arguing that the ellipticity of the core-mantle boundary is smaller than required to cause the core to precess with the mantle. In Meyer and Wisdom (2011), we point out that early on, when the Moon was closer to the Earth, the ellipticity of the core-mantle boundary was larger - large enough to allow inertial coupling to operate and preclude a lunar dynamo. We found that the core and mantle spin axes would be locked together before Moon's orbit evolved out to 26.0-29.0 Earth radii.
Notably, this means that the core and mantle spin axes were free to move separately at the time of the Cassini transition (34 Earth radii). During this transition, the spin axis of the mantle reoriented by more than 75 degrees. Our work indicates that the core would not have to follow along, meaning that the Cassini transition would be a large stirring event for the fluid core. The Cassini transition may be our best bet for sparking a lunar dynamo.
Tidal Theory and the Problem of Q
Traditionally, solar system studies of tides have relied on the constant time lag model, which is the only tidal model with published analytic (not numerical) expressions for tidal heating and orbital decay at high eccentricity (Wisdom 2008). In this model, tides are raised by an imaginary perturber displaced by a constant time lag along the orbit from the actual perturber. If the orbit expands due to the effect of tides, the time lag is constant and so the phase lag is forced to decrease along with the orbital frequency.The constant time lag model predicts that tidal dissipation is linearly proportional to the orbital frequency for small phase lags. If we define the tidal quality factor Q so that 1/Q is the tangent of the phase lag, then the constant time lag model predicts that Q is inversely proportional to the orbital frequency. Unfortunately, this frequency-dependence is not in agreement with the few measured values.
For instance, the measured Q of the Moon is roughly frequency-independent over the limited frequency range studied. Williams and Boggs (2009, p. 101) report a Q of approximately 30 for a forcing frequency of one month and a Q of about 35 for one year. Terrestrial geophysicists have studied the Earth over a broader frequency range and have found more complicated frequency dependences, such as described by the Andrade model or Burgers model. Planetary scientists do not understand the frequency dependence of Q, but as far as we do, the constant time lag model seems like a poor choice. The widespread use of the constant time lag model is due to its mathematical tractability, not to any physical evidence in favor of it.
Models of stellar tides take a different approach. Stellar tides are generally described as the excitation of various modes of the star. The theory of the equilibrium tide was pioneered by Zahn (1975), who computed the viscous dissipation resulting from the velocity field in turbulent convective zones in the star. If dissipation is dominated by the equilibrium tide, the fundamental and acoustic modes of the star are the modes excited by the lower frequency tidal forcing. All dissipation occurs in turbulent regions.
The theory of the dynamical tide describes the response of the star to modes of higher frequency than the fundamental modes. For example, Terquem et al. (1998) and Barker and Ogilvie (2010) examine the excitation of g-modes (gravity modes) and resonances between the tidal forcing and the normal modes of the star. g-modes have a restoring force due to buoyancy. Inertial modes have a restoring force from the Coriolis force and are important contributors to stellar dissipation. Recently, Penev and Sasselov (2011) have re-examined the equilibrium tide and constrained the tidal quality factor that applies for extrasolar planets.
In Meyer and Wisdom (2011), we describe a new formulation of solid body tides that models tidal displacements as a sum of excited elastic modes, analogous to the modeling of stellar tides as excited vibrational modes. The assumptions about the tidal frequency dependence enter near the end of the calculation in a modular and mathematically clean manner. This modularity will allow us to easily compare tidal dissipation and its effects for different rheologies in future work. Here we describe the theory and derive general expressions for wobble damping, tidal heating, tidal despinning, and rate of change of semimajor axis and eccentricity for a system with a zero-obliquity perturber in an eccentric, noninclined orbit. We then specify a Kelvin-Voigt rheology, which corresponds to the constant time lag model, and verify our model with the classic results.