The dynamical interactions that occur in newly formed planetary systems may reflect the conditions occurring in the protoplanetary disk out of which they formed. With this in mind, we explore the attainment and maintenance of orbital resonances by migrating planets in the terrestrial mass range. Migration time scales varying between ~106 yr and ~103 yr are considered. In the former case, for which the migration time is comparable to the lifetime of the protoplanetary gas disk, a 2:1 resonance may be formed. In the latter, relatively rapid migration regime commensurabilities of high degree such as 8:7 or 11:10 may be formed. However, in any one large-scale migration several different commensurabilities may be formed sequentially, each being associated with significant orbital evolution. We also use a simple analytic theory to develop conditions for first order commensurabilities to be formed. These depend on the degree of the commensurability, the imposed migration and circularization rates, and the planet mass ratios. These conditions are found to be consistent with the results of our simulations.

The magnetorotational instability (MRI) is the most likely source of MHD turbulence in accretion disks. Recently, it has been realized that microscopic diffusion coefficients (viscosity and resistivity) are important in determining the saturated state of the turbulence and thereby the rate of angular momentum transport. In this paper, we use a set of numerical simulations performed with a variety of numerical methods to investigate the dependance of α, the rate of angular momentum transport, on these coefficients. We show that α is an increasing function of the magnetic Prandtl number Pm, the ratio of viscosity over resistivity. In the absence of a mean field, we also find that MRI–induced MHD turbulence decays at low Pm.

Many multi-planet systems have been discovered in recent years. Some of them are in mean-motion resonances (MMR). Planet formation theory was successful in explaining the formation of 2:1, 3:1 and other low resonances as a result of convergent migration. However, higher order resonances require high initial orbital eccentricities in order to be formed by this process and these are in general unexpected in a dissipative disk. We present a way of generating large initial eccentricities using additional planets. This precedure allows us to form high order MMRs and predict new planets using a genetic N-body code.

The aim of this talk is to present the most recent advances in establishing plausible planetary system architectures determined by the gravitational tidal interactions between the planets and the disc in which they are embedded during the early epoch of planetary system formation. We concentrate on a very well defined and intensively studied process of the disc-planet interaction leading to the planet migration. We focus on the dynamics of the systems in which low-mass planets are present. Particular attention is devoted to investigation of the role of resonant configurations. Our studies, apart from being complementary to the fast progress occurring just now in observing the whole variety of planetary systems and uncovering their structure and origin, can also constitute a valuable contribution in support of the missions planned to enhance the number of detected multiple systems.

Non-linear ideal MHD equilibria in axisymmetric systems are examined, both in diffuse and free boundary cases. Attention is restricted to the situation in present low-β tokamak experiments, in which there are no current reversals. Both general qualitative results on the uniqueness and bifurcation of solutions are provided and exact solutions of several problems in a circular cylinder are given, exhibiting bifurcation phenomena. In particular sufficient conditions for bifurcation behaviour are conjectured, for both diffuse and free boundary equilibria.

We review processes leading to the orbital migration of bodies ranging from dust particles up to protoplanets in the earth mass range in protoplanetary disks. The objects most at risk of being lost from the disk have dimensions of the order of metres. To retain these it may be necessary to invoke either strong turbulence or trapping due to special flow features such as vortices. Migration also becomes important for protoplanets with mass exceeding $0.1M_{\oplus}$. In this case it can also lead to the attainment of commensurabilities for pairs of protoplanets. Such pairs could be left behind after disk dispersal. We review some recent work on the attainment of commensurabilities for protoplanets in the earth mass range.

We present evolutionary calculations which show the tumbling instability found in spherical stellar systems by Allen et al. (1992) also occurs in some initially oblate rotating systems.

We consider the linear stability of spherical stellar systems by solving the Vlasov and Poisson equations which yield a matrix eigenvalue problem to determine the growth rate. We consider this for purely growing modes in the limit of vanishing growth rate. We show that a large class of anisotropic models are unstable and derive growth rates for the particular example of generalized polytropic models. We present a simple method for testing the stability of general anisotropic models. Our anlysis shows that instability occurs even when the degree of anisotropy is very slight.

This talk summarizes recent work that has been done on cataclysmic variables at Oxford and Cambridge. Full details should be sought elsewhere, either from the authors concerned or from the references given below.

In previous work (Papaloizou and Pringle, 1978), we investigated analytically the low frequency spectrum of a rotating pulsating star, taking the effects of rotation fully into account. In that case we found that not only are the g-modes substantially modified by the rotation, but also that a new set of modes appear (the r modes) which are similar to Rossby waves in the Earth’s atmosphere and oceans.

Massive main sequence stars are vibrationally unstable to small perturbations excited in the region of nuclear energy generation. This instability may, however, be limited at finite amplitude. Only the fundamental radial mode is unstable with an e-folding time of 105 to 106 pulsation periods; in contrast the overtones have damping times short compared with the growth time of the fundamental. Non-linear effects couple the linear modes. At very low amplitude the energy transfer time between modes is long compared with both growth and damping times and, as the overtones decay, a pure fundamental should result. However, as the amplitude increases, the transfer time becomes less than the growth and damping times. Energy can then be transferred from the fundamental to the overtones and damping can lead to a limitation of amplitude.