Adrian J. Barker

On the tidal evolution of Hot Jupiters on inclined orbits

Tidal friction is thought to be important in determining the long-term spin-orbit evolution of short-period extrasolar planetary systems. Using a simple model of the orbit-averaged effects of tidal friction, we study the evolution of close-in planets on inclined orbits, due to tides. We analyse the effects of the inclusion of stellar magnetic braking by performing a phase-plane analysis of a simplified system of equations, including the braking torque. The inclusion of magnetic braking is found to be important, and its neglect can result in a very different system history. We then present the results of numerical integrations of the tidal evolution equations, where we find that it is essential to consider coupled evolution of the orbital and rotational elements, including dissipation in both the star and planet, to accurately model the evolution. The main result of our integrations is that for typical Hot Jupiters, tidal friction aligns the stellar spin with the orbit on a similar time as it causes the orbit to decay. This tells us that if a planet is observed to be aligned, then it probably formed coplanar. This reinforces the importance of Rossiter–McLaughlin effect observations in determining the degree of spin-orbit alignment in transiting systems. We apply these results to the only observed system with a spin-orbit misalignment, XO-3, and constrain the efficiency of tidal dissipation (i.e. the modified tidal quality factors Q′) in both the star and the planet in this system. Using a model in which inertial waves are excited by tidal forcing in the outer convective envelope and dissipated by turbulent viscosity, we calculate Q′ for a range of F-star models, and find it to vary considerably within this class of stars. This means that using a single Q′, and assuming that it applies to all stars, is probably incorrect. In addition, we propose an explanation for the survival of two of the planets on the tightest orbits, WASP-12 b and OGLE-TR-56 b, in terms of weak dissipation in the star, as a result of their internal structures and slow rotation periods.

Non-linear evolution of tidally forced inertial waves in rotating fluid bodies

We perform one of the first studies into the nonlinear evolution of tidally excited inertial waves in a uniformly rotating fluid body, exploring a simplified model of the fluid envelope of a planet (or the convective envelope of a solar-type star) subject to the gravitational tidal perturbations of an orbiting companion. Our model contains a perfectly rigid spherical core, which is surrounded by an envelope of incompressible uniform density fluid. The corresponding linear problem was studied in previous papers which this work extends into the nonlinear regime, at moderate Ekman numbers (the ratio of viscous to Coriolis accelerations). By performing high-resolution numerical simulations, using a combination of pseudo-spectral and spectral element methods, we investigate the effects of nonlinearities, which lead to time-dependence of the flow and the corresponding dissipation rate. Angular momentum is deposited non-uniformly, leading to the generation of significant differential rotation in the initially uniformly rotating fluid, i.e. the body does not evolve towards synchronism as a simple solid body rotator. This differential rotation modifies the properties of tidally excited inertial waves, changes the dissipative properties of the flow, and eventually becomes unstable to a secondary shear instability provided that the Ekman number is sufficiently small. Our main result is that the inclusion of nonlinearities eventually modifies the flow and the resulting dissipation from what linear calculations would predict, which has important implications for tidal dissipation in fluid bodies. We finally discuss some limitations of our simplified model, and propose avenues for future research to better understand the tidal evolution of rotating planets and stars. ; Comment: 17 pages, 17 figures, accepted for publication in MNRAS

Three-dimensional simulations of internal wave breaking and the fate of planets around solar-type stars

We study the fate of internal gravity waves approaching the centre of an initially non-rotating solar-type star, by performing three-dimensional numerical simulations using a Boussinesq-type model. These waves are excited at the top of the radiation zone by the tidal forcing of a short-period planet on a circular, coplanar orbit. This extends previous work done in two dimensions by Barker & Ogilvie. We first derive a linear wave solution, which is not exact in three dimensions; however, the reflection of ingoing waves from the centre is close to perfect for moderate amplitude waves. Waves with sufficient amplitude to cause isentropic overturning break, and deposit their angular momentum near the centre. This forms a critical layer, at which the angular velocity of the flow matches the orbital angular frequency of the planet. This efficiently absorbs ingoing waves, and spins up the star from the inside out, while the planet spirals into the star. We also perform numerical integrations to determine the linearized adiabatic tidal response throughout the star, in a wide range of solar-type stellar models with masses in the range 0.5 ≤m★/M⊙≤ 1.1, throughout their main-sequence lifetimes. The aim is to study the influence of the launching region for these waves at the top of the radiation zone in more detail, and to determine the accuracy of a semi-analytic approximation for the tidal torque on the star, which was derived under the assumption that all ingoing wave angular momentum is absorbed in a critical layer. The main conclusion of this work is that this non-linear mechanism of tidal dissipation could provide an explanation for the survival of all short-period extrasolar planets observed around FGK stars, while it predicts the destruction of more massive planets. This work provides further support for the model outlined in a previous paper by Barker & Ogilvie, and makes predictions that will be tested by ongoing observational studies, such as WASP and Kepler.

On the vertical-shear instability in astrophysical discs

We explore the linear stability of astrophysical discs exhibiting vertical shear, which arises when there is a radial variation in the temperature or entropy. Such discs are subject to a ‘vertical-shear instability’, which recent non-linear simulations have shown to drive hydrodynamic activity in the MRI-stable regions of protoplanetary discs. We first revisit locally isothermal discs using the quasi-global reduced model derived by Nelson et al. This analysis is then extended to global axisymmetric perturbations in a cylindrical domain. We also derive and study a reduced model describing discs with power-law radial entropy profiles (‘locally polytropic discs’), which are somewhat more realistic in that they possess physical (as opposed to numerical) surfaces. The fastest growing modes have very short wavelengths and are localized at the disc surfaces (if present), where the vertical shear is maximal. An additional class of modestly growing vertically global body modes is excited, corresponding to destabilized classical inertial waves (‘r modes’). We discuss the properties of both types of modes, and stress that those that grow fastest occur on the shortest available length-scales (determined either by the numerical grid or the physical viscous length). This ill-posedness makes simulations of the instability difficult to interpret. We end with some brief speculation on the non-linear saturation and resulting angular momentum transport.

Non-linear evolution of the elliptical instability in the presence of weak magnetic fields

We investigate whether the elliptical instability is important for tidal dissipation in gaseous planets and stars. In a companion paper, we found that the conventional elliptical instability results in insufficient dissipation because it produces long-lived vortices that then quench further instability. Here, we study whether the addition of a magnetic field prevents those vortices from forming, and hence leads to enhanced dissipation. We present results from magnetohydrodynamic simulations that evolve the elliptical instability in a local patch of a rotating planet or star, in the presence of a weak magnetic field. We find that magnetic fields do indeed prevent vortices from forming, and hence greatly enhance the steady-state dissipation rate. In addition, the resulting turbulence acts as a small-scale dynamo, amplifying the initially weak field. The inferred tidal dissipation is potentially important at short orbital periods. For example, it can circularize hot Jupiters with orbital periods shorter than 2.5 d and synchronize their spins with their orbits out to 6 d. However, it appears unable to account for the hot Jupiters that appear to have been circularized out to 6–10 d orbital periods. It also cannot account for the inferred circularization of many close binary stars.

Non-linear evolution of the tidal elliptical instability in gaseous planets and stars

Tidally distorted rotating stars and gaseous planets are subject to a well-known linear fluid instability – the elliptical instability. It has been proposed that this instability might drive enough energy dissipation to solve the long-standing problem of the origin of tidal dissipation in stars and planets. But the non-linear outcome of the elliptical instability has yet to be investigated in the parameter regime of interest, and the resulting turbulent energy dissipation has not yet been quantified. We do so by performing three-dimensional hydrodynamical simulations of a small patch of a tidally deformed fluid planet or star subject to the elliptical instability. We show that when the tidal deformation is weak, the non-linear outcome of the instability leads to the formation of long-lived columnar vortices aligned with the axis of rotation. These vortices shut off the elliptical instability, and the net result is insufficient energy dissipation to account for tidal dissipation. However, further work is required to account for effects neglected here, including magnetic fields, turbulent convection and realistic boundary conditions.

Theory and simulations of rotating convection

We study thermal convection in a rotating fluid in order to better understand the properties of convection zones in rotating stars and planets. We first derive mixing-length theory for rapidly-rotating convection, arriving at the results of Stevenson (1979) via simple physical arguments. The theory predicts the properties of convection as a function of the imposed heat flux and rotation rate, independent of microscopic diffusivities. In particular, it predicts the mean temperature gradient; the rms velocity and temperature fluctuations; and the size of the eddies that dominate heat transport. We test all of these predictions with high resolution three-dimensional hydrodynamical simulations of Boussinesq convection in a Cartesian box. The results agree remarkably well with the theory across more than two orders of magnitude in rotation rate. For example, the temperature gradient is predicted to scale as the rotation rate to the 4/5th power at fixed flux, and the simulations yield

Hydrodynamic instability in eccentric astrophysical discs

0.75\pm 0.06

Non-linear tides in a homogeneous rotating planet or star: global simulations of the elliptical instability

. We conclude that the mixing length theory is a solid foundation for understanding the properties of convection zones in rotating stars and planets.

The effect of ambipolar resistivity on the formation of dense cores

Eccentric Keplerian discs are believed to be unstable to three-dimensional hydrodynamical instabilities driven by the time-dependence of fluid properties around an orbit. These instabilities could lead to small-scale turbulence, and ultimately modify the global disc properties. We use a local model of an eccentric disc, derived in a companion paper, to compute the non-linear vertical (‘breathing mode’) oscillations of the disc. We then analyse their linear stability to locally axisymmetric disturbances for any disc eccentricity and eccentricity gradient using a numerical Floquet method. In the limit of small departures from a circular reference orbit, the instability of an isothermal disc is explained analytically. We also study analytically the small-scale instability of an eccentric neutrally stratified polytropic disc with any polytropic index using a Wentzel–Kramers–Brillouin (WKB) approximation. We find that eccentric discs are generically unstable to the parametric excitation of small-scale inertial waves. The non-linear evolution of these instabilities should be studied in numerical simulations, where we expect them to lead to a decay of the disc eccentricity and eccentricity gradient as well as to induce additional transport and mixing. Our results highlight that it is essential to consider the three-dimensional structure of eccentric discs, and their resulting vertical oscillatory flows, in order to correctly capture their evolution.