# Research Projects in Astrophysics

## Planet Formation, Astrophysical Discs and Planetary Rings, Disc-Planet Interactions

### Dynamics of Accretion Discs

Discs of gaseous or solid material in Keplerian orbital motion around a central massive body occur in astronomical systems ranging from Saturn's rings to accretion discs around black holes at the centres of galaxies. Despite differences in scale and constitution, these systems share many dynamical properties. Planets form in discs of dusty gas around young stars and their observed orbital characteristics are the result of their dynamical interaction with the disc, any other planets, the central star and any external influences. Of special interest are extrasolar planetary systems involving resonant configurations, highly eccentric orbits, or planets in extreme proximity to the star.

The right plot illustrates the complex magnetic field structure in local midplane/vertical sections of a turbulent protoplanetary disc.

### Disc-Planet Interactions

Solid material embedded in a gas disc is subject to radial migration. Small particles (up to a few 100 km in size) are subject to aerodynamic drag, causing them to migrate inward on time scales that can be alarmingly fast (100 yr for m-sized objects).

Earth-like planets are massive enough to excite tidal sound waves in the disc, one inside its orbit and one outside. These are of unequal strength in general, mainly due to the non-linear nature of Keplerian rotation (the rotation profile is asymmetric with respect to the orbit of the planet). This leads to a torque on the planet, that usually corresponds to inward migration called Type I migration. For Superearth planets (and also possibly the cores of giant planets) the time scale for inward migration can be as short as 10^{5} years. This has to be compared to a disc life time of 10^{7} years, implying that essentially there should be no planets at large (1-10 AU) radii! This is a serious problem, and possible solutions include MHD turbulence and non-linear effects on the corotation torque, especially in non-barotropic discs.

Massive planets, comparable to Jupiter, excite non-linear waves in the disc, leading to the formation of an annular gap (see picture). The migration of the planet slows down considerably, and is now called Type II migration. Basically, in this regime the planet just accretes with the gas onto the star, making the migration time scale comparable to the disc life time.

For intermediate masses, comparable to Saturn, rapid Type III migration can occur. This mode of migration is driven by a strong coorbital flow, and can in principle be directed either inward or outward. This is a very challenging problem, numerically, because it relies on the flow in the direct vicinity of the planet which is difficult to treat in a realistic way.

### Planet formation in binary systems

Making planets is difficult in general, and it is even more difficult in systems that have an additional strong perturber in the form of a companion star. Nevertheless, a planet has been found at 2 AU around a star that has a companion of 0.5 Msun at only 20 AU (orbital distance of Uranus!). Moreover, this massive companion in on a very eccentric orbit. The strong gravitational perturbations due to this second star puts planets-to-be in all sorts of trouble. One fundamental problem arises when you try to grow a population of km-sized planetesimals into larger bodies that will eventually accrete gas to form a giant planet. The perturbations of the secondary star excites the eccentricities of the planetesimals. This leads to very high encounter velocities (up to 1 km/s), which makes collisions between planetesimals very violent and destructive. The presence of a gas disc to damp the eccentricities of the planetesimals does not help. Since the strength of gas drag is size-dependent, planetesimals of different size will still be on very different orbits. This so-called differential orbital phasing leads again to very violent collisions between bodies of unequal size. It is difficult to see how to overcome this problem and proceed with building bigger bodies.

### Possible PhD Projects

Areas of current interest in which PhD projects might be formulated include:

- Tidal interactions between planets and stars, involving the excitation and dissipation of low-frequency internal waves
- Dynamics of general Keplerian discs, which may be warped or eccentric
- Eccentricity and inclination in planet-disc systems, coupled by secular and resonant interactions
- The nonlinear magnetorotational dynamo in accretion discs
- Kinetic theory and nonlinear wave dynamics in planetary rings
- Jet launching and magnetic flux evolution in accretion discs

Please contact Henrik Latter, Gordon Ogilvie or John Papaloizou for more information.

## Solar, Stellar and Planetary Magnetism

### Dynamo theory:

Magnetic fields in large astrophysical bodies such as the Sun, Earth and stars are produced by dynamo action due to fluid motions in their interiors or atmospheres. Observations show that the resulting fields are highly complex in space and time. We are interested in developing theoretical models that can explain the observations and also aid the fundamental understanding of the nature of the dynamo process.

The picture shows a mean-field solar dynamo simulation with contours of toroidal magnetic field as a function of latitude and time. Like the large-scale magnetic field with the Sun, the magnetic field in this simulation is dipolar and migrates equatorwards during each cycle.

Recent observations from various ground-based and space instruments have revealed fascinating details of the highly complex magnetic field structures in the solar photosphere. These are produced by the mutual interaction of magnetic fields and convection. We investigate these structures by a combination of numerical simulation and theoretical modelling.

The picture shows a snapshot of a 3d simulation of compressible magnetoconvection. Left: Magnetic energy density. Right: Temperature distribution. This solution illustrates the phenomenon of flux separation - vigourous convection in a flux-free region in the centre of the box is surrounded by weak (magnetically-dominated) small-scale convection.

Please contact Michael Proctor for more information.

## Pattern Formation and Nonlinear Dynamics

Studies of order and disorder in the natural world have a long history. Indeed, a diverse collection of physical, chemical and biological systems naturally organise themselves into states with degrees of symmetry, i.e. patterns. A realisation has emerged over the past forty years that the same classes of qualitative behaviour appear repeatedly, and universal mathematical models have been developed to understand each characteristic situation, see, for example, the review by Cross and Hohenberg, Rev. Mod. Phys. 65 (1993). These mathematical models provide a unifying viewpoint and have, in turn, stimulated further research in the relevant experimental disciplines.

One of the longest-studied examples of a pattern forming instability is the Rayleigh-Benard problem of thermal convection in a planar fluid layer; this system has come to be regarded as archetypal. Pattern formation occurs in many other contexts, including liquid crystals in externally imposed electric fields, nonlinear optics, directional solidification, Faraday waves in vertically vibrated layers of fluids or granular media, chemical reactions (such as the Belousov-Zhabotinsky and CIMA systems) and surface catalysis.

The mathematical methods involved in analysing these problems make use of the symmetric nature of the problem. Mathematically (using equivariant bifurcation theory) it is often possible to build up a classification of possible regular patterns; experimental systems select particular patterns from this classification. Instabilities of regular patterns to modulational, long-wavelength, disturbances are also well understood. Recent attention has focused on instabilities near 'mode interaction' points where an initial spatially homogeneous state is simultaneously unstable to instabilities with different wavelengths.

Reduced models (especially ODEs) for pattern-forming systems are often a rich source of interesting nonlinear dynamical behaviour in their own right. Ongoing research into these theoretic systems is concentrating on the dynamics near heteroclinic cycles and networks.

Please contact Michael Proctor for more information.