Gordon I. Ogilvie

Tidal Dissipation in Rotating Giant Planets

Many extrasolar planets orbit sufficiently close to their host stars that significant tidal interactions can be expected, resulting in an evolution of the spin and orbital properties of the planets. The accompanying dissipation of energy can also be an important source of heat, leading to the inflation of short-period planets and even mass loss through Roche lobe overflow. Tides may therefore play an important role in determining the observed distributions of mass, orbital period, and eccentricity of the extrasolar planets. In addition, tidal interactions between gaseous giant planets in the solar system and their moons are thought to be responsible for the orbital migration of the satellites, leading to their capture into resonant configurations. Traditionally, the efficiency of tidal dissipation is simply parameterized by a quality factor Q, which depends, in principle, in an unknown way on the frequency and amplitude of the tidal forcing. In this paper we treat the underlying fluid dynamical problem with the aim of determining the efficiency of tidal dissipation in gaseous giant planets such as Jupiter, Saturn, or the short-period extrasolar planets. Efficient convection enforces a nearly adiabatic stratification in these bodies, which may or may not contain solid cores. With some modifications, our approach can also be applied to low-mass stars with extended convective envelopes. In cases of interest, the tidal forcing frequencies are typically comparable to the spin frequency of the planet but are small compared to its dynamical frequency. We therefore study the linearized response of a slowly and possibly differentially rotating planet to low-frequency tidal forcing. Convective regions of the planet support inertial waves, which possess a dense or continuous frequency spectrum in the absence of viscosity, while any radiative regions support generalized Hough waves. We formulate the relevant equations for studying the excitation of these disturbances and present a set of illustrative numerical calculations of the tidal dissipation rate. We argue that inertial waves provide a natural avenue for efficient tidal dissipation in most cases of interest. In the presence of a solid core, the excited disturbance tends to be localized on a web of rays rather than resembling a smooth eigenfunction. The resulting value of Q depends, in principle, in a highly erratic way on the forcing frequency, but we provide analytical and numerical evidence that the frequency-averaged dissipation rate may be asymptotically independent of the viscosity in the limit of small Ekman number. For a smaller viscosity, the tidal disturbance has a finer spatial structure and individual resonances are more pronounced. In short-period extrasolar planets, tidal dissipation via inertial waves becomes somewhat less efficient once they are spun down to a synchronous state. However, if the stellar irradiation of the planet leads to the formation of a radiative outer layer that supports generalized Hough modes, the tidal dissipation rate can be enhanced, albeit with significant uncertainty, through the excitation and damping of these waves. The dissipative mechanisms that we describe offer a promising explanation of the historical evolution and current state of the Galilean satellites, as well as the observed circularization of the orbits of short-period extrasolar planets.

Three-dimensional calculations of high- and low-mass planets embedded in protoplanetary discs

We analyse the non-linear, three-dimensional response of a gaseous, viscous protoplanetary disc to the presence of a planet of mass ranging from 1 Earth mass (1 M⊕) to 1 Jupiter mass (1 MJ) by using the ZEUS hydrodynamics code. We determine the gas f ow pattern, and the accretion and migration rates of the planet. The planet is assumed to be in a f xed circular orbit about the central star. It is also assumed to be able to accrete gas without expansion on the scale of its Roche radius. Only planets with masses Mp 0.1 MJ produce significan perturbations in the surface density of the disc. The fl w within the Roche lobe of the planet is fullythree-dimensional.GasstreamsgenerallyentertheRochelobeclosetothediscmid-plane, but produce much weaker shocks than the streams in two-dimensional models. The streams supply material to a circumplanetary disc that rotates in the same sense as the orbit of the planet. Much of the mass supply to the circumplanetary disc comes from non-coplanar fl w. The accretion rate peaks with a planet mass of approximately 0.1 MJ and is highly efficient occurring at the local viscous rate. The migration time-scales for planets of mass less than 0.1 MJ, based on torques from disc material outside the Roche lobes of the planets, are in excellent agreement with the linear theory of type I (non-gap) migration for three-dimensional discs.ThetransitionfromtypeItotypeII(gap)migrationissmooth,withchangesinmigration times of about a factor of 2. Starting with a core which can undergo runaway growth, a planet can gain up to a few MJ with little migration. Planets with fina masses of the order of 10 MJ would undergo large migration, which makes formation and survival difficult

TIDAL DISSIPATION IN ROTATING SOLAR-TYPE STARS

We calculate the excitation and dissipation of low-frequency tidal oscillations in uniformly rotating solar-type stars. For tidal frequencies smaller than twice the spin frequency, inertial waves are excited in the convective envelope and are dissipated by turbulent viscosity. Enhanced dissipation occurs over the entire frequency range rather than in a series of very narrow resonant peaks and is relatively insensitive to the effective viscosity. Hough waves are excited at the base of the convective zone and propagate into the radiative interior. We calculate the associated dissipation rate under the assumption that they do not reflect coherently from the center of the star. Tidal dissipation in a model based on the present Sun is significantly enhanced through the inclusion of the Coriolis force but may still fall short of that required to explain the circularization of close binary stars. However, the dependence of the results on the spin frequency, tidal frequency, and stellar model indicate that a more detailed evolutionary study including inertial and Hough waves is required. We also discuss the case of higher tidal frequencies appropriate to stars with very close planetary companions. The survival of even the closest hot Jupiters can be plausibly explained provided that the Hough waves they generate are not damped at the center of the star. We argue that this is the case because the tide excited by a hot Jupiter in the present Sun would marginally fail to achieve nonlinearity. As conditions at the center of the star evolve, nonlinearity may set in at a critical age, resulting in a relatively rapid inspiral of the hot Jupiter.

Precessing warped accretion discs in X-ray binaries

We study the radiation-driven warping of accretion discs in the context of X-ray binaries. The latest evolutionary equations are adopted, which extend the classical alpha theory to time-dependent thin discs with non-linear warps. We also develop accurate, analytical expressions for the tidal torque and the radiation torque, including self-shadowing. We investigate the possible non-linear dynamics of the system within the framework of bifurcation theory. First, we re-examine the stability of an initially flat disc to the Pringle instability. Then we compute directly the branches of non-linear solutions representing steadily precessing discs. Finally, we determine the stability of the non-linear solutions. Each problem involves only ordinary differential equations, allowing a rapid, accurate and well-resolved solution. We find that radiation-driven warping is probably not a common occurrence in low-mass X-ray binaries. We also find that stable, steadily precessing discs exist for a narrow range of parameters close to the stability limit. This could explain why so few systems show clear, repeatable ‘superorbital’ variations. The best examples of such systems, Her X-1, SS 433 and LMC X-4, all lie close to the stability limit for a reasonable choice of parameters. Systems far from the stability limit, including Cyg X-2, Cen X-3 and SMC X-1, probably experience quasi-periodic or chaotic variability as first noticed recently by Wijers and Pringle. We show that radiation-driven warping provides a coherent and persuasive framework but that it does not provide a generic explanation for the long-term variabilities in all X-ray binaries.

Aligning spinning black holes and accretion discs

We consider the alignment torque between a spinning black hole and an accretion disc whose angular momenta are misaligned. This situation must hold initially in almost all gas accretion events on to supermassive black holes, and may occur in binaries where the black hole receives a natal supernova kick. We show that the torque always acts to align the hole’s spin with the total angular momentum without changing its magnitude. The torque acts dissipatively on the disc, reducing its angular momentum, and aligning it with the hole if and only if the angle θ between the angular momenta J d of the disc and J h of the hole satisfy the inequality cos θ> −J d/2J h .I fthis condition fails, which requires both θ> π/2 and J d < 2J h, the disc counteraligns. Ke yw ords: accretion, accretion discs ‐ black hole physics.

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.

Observational implications of precessing protostellar discs and jets

We consider the dynamics of a protostellar disc in a binary system where the disc is misaligned with the orbital plane of the binary, with the aim of determining the observational consequences for such systems. The disc wobbles with a period approximately equal to half the orbital period of the binary and precesses on a longer time-scale. We determine the characteristic time-scale for realignment of the disc with the orbital plane as a result of dissipation. If the dissipation is determined by a simple isotropic viscosity then we find, in line with previous studies, that the alignment time-scale is of the order of the viscous evolution time-scale. However, for typical protostellar disc parameters, if the disc tilt exceeds the opening angle of the disc, then tidally induced shearing within the disc is transonic. In general, hydrodynamic instabilities associated with the internally driven shear result in extra dissipation that is expected to drastically reduce the alignment time-scale. For large disc tilts the alignment time-scale is then comparable with the precession time-scale, while for smaller tilt angles δ, the alignment time-scale varies as (sin δ)−1. We discuss the consequences of the wobbling, precession and rapid realignment for observations of protostellar jets and the implications for binary star formation mechanisms.

The non-linear fluid dynamics of a warped accretion disc

The dynamics of a viscous accretion disc subject to a slowly varying warp of large amplitude is considered. Attention is restricted to discs in which self-gravitation is negligible, and to the generic case in which the resonant wave propagation found in inviscid Keplerian discs does not occur. The equations of fluid dynamics are derived in a coordinate system that follows the principal warping motion of the disc. They are reduced using asymptotic methods for thin discs, and solved to extract the equation governing the warp. In general, this is a wave equation of parabolic type with non-linear dispersion and diffusion, which describes fully non-linear bending waves. This method generalizes the linear theory of Papaloizou & Pringle to allow for an arbitrary rotation law, and extends it into the non-linear domain, where it connects with a generalized version of the theory of Pringle. The astrophysical implications of this analysis are discussed briefly.

Viscous effects on the interaction between the coplanar decretion disc and the neutron star in Be/X-ray binaries

We study the viscous effects on the interaction between the coplanar Be-star disc and the neutron star in Be/X-ray binaries, using a three-dimensional, smoothed particle hydrodynamics code. For simplicity, we assume the Be disc to be isothermal at the temperature of half the stellar effective temperature. In order to mimic the gas ejection process from the Be star, we inject particles with the Keplerian rotation velocity at a radius just outside the star. Both the Be star and the neutron star are treated as point masses. We find that the Be-star disc is effectively truncated if the Shakura‐Sunyaev viscosity parameter αSS � 1, which confirms the previous semi-analytical result. In the truncated disc, the material decreted from the Be star accumulates, so that the disc becomes denser more rapidly than if around an isolated Be star. The resonant truncation of the Be disc results in a significant reduction of the amount of gas captured by the neutron star and a strong dependence of the mass-capture rate on the orbital phase. We also find that an eccentric mode is excited in the Be disc through direct driving as a result of a one-armed bar potential of the binary. The strength of the mode becomes greater in the case of a smaller viscosity. In a high-resolution simulation with αSS = 0.1, the eccentric mode is found to precess in a prograde sense. The mass-capture rate by the neutron star modulates as the mode precesses.

Tidal Dissipation in Stars and Giant Planets

Astrophysical fluid bodies that orbit close to one another induce tidal distortions and flows that are subject to dissipative processes. The spin and orbital motions undergo a coupled evolution over astronomical timescales, which is relevant for many types of binary star, short-period extrasolar planetary systems, and the satellites of the giant planets in the Solar System. I review the principal mechanisms that have been discussed for tidal dissipation in stars and giant planets in both linear and nonlinear regimes. I also compare the expectations based on theoretical models with recent observational findings.