On the 'coarse-grained' evolution of collisionless stellar systems

We have suggested in a previous article that the coarse-grained evolution of a collisionless stellar system could be viewed as a diffusion process in velocity space compensated by an appropriate friction. Using a quasi-linear theory, we calculate the diffusion coefficient associated with this evolution. This provides a new self-consistent relaxation equation for f , the locally averaged distribution function. This equation bears some resemblance to the conventional Fokker–Planck equation of collisional systems but the friction term is non-linear in f (accounting for degeneracy effects) and the relaxation time is much smaller (in agreement with the concept of 'violent relaxation'). Under the condition that the diffusion current vanishes identically at equilibrium, we recover Lynden-Bell's distribution function; but if we allow stars to escape from the system at a constant rate, we can derive a truncated model which coincides with Lynden-Bell's solution in the core but provides a depletion of high-energy stars in the halo. This distribution function has a finite mass and is the generalization of the Michie–King model to the case of (possibly degenerate) collisionless stellar systems.     

Fossil fuels and other energy resources in the 21st century

We study the changing volumes and costs of the energy resources available in the coming century as a result of exploration, technical progress, and consumption. The most reliable and well-documented information involves fossil fuels, which account for nearly 80 percent of the energy mix today. Known and presumed resources are fairly well ranked by cost, and we can estimate future developments. We present two consumption scenarios: the more contrasted reference case and the ecologically driven case presented by the World Energy Council at its last congress (1992). Our scenarios take into account the twofold goal set forth at the Rio summit: to achieve sustainable development and to minimize the climatic effects of pollutant emissions.We show that the most cost-effective mineral resources, including uranium, will run out only gradually in the next century according to these hypotheses. As a result, marginal cost will grow relatively slowly. However, we presuppose a tough policy on curbing consumption and using renewable sources of energy.We point out the need for better cost ranking of energy saving and energy switch possibilities. Additionally, the ecological impact and cost of corrective measures will have to be evaluated. We believe that the medium- and long-term difficulties will stem chiefly from the contradiction between low energy prices and the commitments required. The danger of geopolitical tension will remain a serious concern.     

Degenerate equilibrium states of collisionless stellar systems

We study spherically symmetrical equilibrium states of collisionless stellar systems confined to a spherical box. These equilibrium states correspond to the statistics introduced by Lynden-Bell in his theory of 'violent relaxation', and are described by a Fermi–Dirac distribution function. We compute the corresponding equilibrium diagram and show that a global entropy maximum exists for any accessible control parameter. This equilibrium state shows a pronounced separation between a degenerate core and a halo. We therefore check that degeneracy is able to stop the gravitational collapse (of a collisionless system), and we propose a simple model for the 'core–halo' structure. We also discuss the relevance of our study for real galaxies or other astrophysical systems such as massive neutrinos.