The great diversity of extrasolar planetary systems has challenged our understanding of how planets form, and how their orbits evolve as they form. Among the various processes that may account for this diversity, the gravitational interaction between planets and their parent protoplanetary disc plays a prominent role in shaping young planetary systems. Planet-disc forces are large, and the characteristic times for the evolution of planets orbital elements are much shorter than the lifetime of protoplanetary discs. The determination of such forces is challenging, because it involves many physical mechanisms and it requires a detailed knowledge of the disc structure. Yet, the intense research of the past few years, with the exploration of many new avenues, represents a very significant improvement on the state of the discipline. This chapter reviews current understanding of planet-disc interactions, and highlights their role in setting the properties and architecture of observed planetary systems.

Standard models of accretion discs study the transport of mass on a viscous time-scale but do not consider the transport of magnetic flux. The evolution of a large-scale poloidal magnetic field is, however, an important problem because of its role in the launching of jets and winds and in determining the intensity of turbulence. As a consequence, the transport of poloidal magnetic flux should be considered on an equal basis to the transport of mass. In this paper, we develop a formalism to study such a transport of mass and magnetic flux in a thin accretion disc. The governing equations are derived by performing an asymptotic expansion in the limit of a thin disc, in the regime where the magnetic field is dominated by its vertical component. Turbulent viscosity and resistivity are included, with an arbitrary vertical profile that can be adjusted to mimic the vertical structure of the turbulence. At a given radius and time, the rates of transport of mass and magnetic flux are determined by a one-dimensional problem in the vertical direction, in which the radial gradients of various quantities appear as source terms. We solve this problem to obtain the transport rates and the vertical structure of the disc. This paper is then restricted to the idealized case of uniform diffusion coefficients, while a companion paper will study more realistic vertical profiles of these coefficients. We show the advection of weak magnetic fields to be significantly faster than the advection of mass, contrary to what a crude vertical averaging might suggest. This results from the larger radial velocities away from the mid-plane, which barely affect the mass accretion owing to the low density in these regions but do affect the advection of magnetic flux. Possible consequences of this larger accretion velocity include a potentially interesting time dependence with the magnetic flux distribution evolving faster than the mass distribution. If the disc is not too thin, this fast advection may also partially solve the long-standing problem of too efficient diffusion of an inclined magnetic field.

The explosion of core-collapse supernova depends on a sequence of events taking place in less than a second in a region of a few hundred kilometers at the centre of a supergiant star, after the stellar core approaches the Chandrasekhar mass and collapses into a proto-neutron star, and before a shock wave is launched across the stellar envelope. Theoretical efforts to understand stellar death focus on the mechanism which transforms the collapse into an explosion. Progress in understanding this mechanism is reviewed with particular attention to its asymmetric character. We highlight a series of successful studies connecting observations of supernova remnants and pulsars properties to the theory of corecollapse using numerical simulations. The encouraging results from first principles models in axisymmetric simulations is tempered by new puzzles in 3D. The diversity of explosion paths and the dependence on the pre-collapse stellar structure is stressed, as well as the need to gain a better understanding of hydrodynamical and MHD instabilities such as standing accretion shock instability and neutrino-driven convection. The shallow water analogy of shock dynamics is presented as a comparative system where buoyancy effects are absent. This dynamical system can be studied numerically and also experimentally with a water fountain. The potential of this complementary research tool for supernova theory is analysed. We also review its potential for public outreach in science museums.

The release of spin-down energy by a magnetar is a promising scenario to power several classes of extreme explosive transients. However, it lacks a firm basis because magnetar formation still represents a theoretical challenge. Using the first three-dimensional simulations of a convective dynamo based on a protoneutron star interior model, we demonstrate that the required dipolar magnetic field can be consistently generated for sufficiently fast rotation rates. The dynamo instability saturates in the magnetostrophic regime with the magnetic energy exceeding the kinetic energy by a factor of up to 10. Our results are compatible with the observational constraints on galactic magnetar field strength and provide strong theoretical support for millisecond protomagnetar models of gammaray burst and superluminous supernova central engines. Acknowledgments:We thank R. Bollig, M. Bugli, T. Foglizzo, B. Gallet, D. Götz, and A. Reboul-Salze for the discussions and comments. We thank the anonymous referees for useful comments, which improved the quality of the paper. We thank the online CompOSE database (https://compose.obspm.fr). Numerical simulations have been carried out at the CINES on the Occigen supercomputer (DARI projects A0030410317 and A0050410317).