On a matrix method for the study of small perturbations in galaxies

A matrix method is formulated in a Lagrangian representation for the solution of the characteristic value problem governing modes of oscillation and instability in a collisionless stellar system. The underlying perturbation equations govern the Lagrangian displacement of a star in the six-dimensional phase space. This matrix method has a basis in a variational principle. The method is developed in detail for radial oscillations of a spherical system. The basis vectors required for the representation of the Lagrangian displacement in this case are derived from solutions of the Lagrangian perturbation equations for radial perturbations in a homogeneous sphere. The basis vectors are made divergence free in the six-dimensional phase space in accordance with the requirement of Liouville's theorem that the flow of the system in the phase space must be incompressible. The basis vectors are made orthogonal with respect to a properly chosen set of adjoint vectors with the aid of a Gram-Schmidt procedure. Some basis vectors are null vectors in the sense that their inner products with their own adjoint vectors vanish. The characteristic frequencies of the lowest radial modes are calculated in several approximations for members of a family of spherical models which span a wide range of central concentrations. The present formulation of the matrix method can be generalized for nonradial modes in spherical systems and for modes in axisymmetric systems.Comment: 30 pages, 11 figures, submitted to the Monthly Notices of the Royal Astronomical Societ

Characterising the Gravitational Instability in Cooling Accretion Discs

We perform numerical analyses of the structure induced by gravitational instabilities in cooling gaseous accretion discs. For low enough cooling rates a quasi-steady configuration is reached, with the instability saturating at a finite amplitude in a marginally stable disc. We find that the saturation amplitude scales with the inverse square root of the cooling parameter beta = t_cool / t_dyn, which indicates that the heating rate induced by the instability is proportional to the energy density of the induced density waves. We find that at saturation the energy dissipated per dynamical time by weak shocks due is of the order of 20 per cent of the wave energy. From Fourier analysis of the disc structure we find that while the azimuthal wavenumber is roughly constant with radius, the mean radial wavenumber increases with radius, with the dominant mode corresponding to the locally most unstable wavelength. We demonstrate that the density waves excited in relatively low mass discs are always close to co-rotation, deviating from it by approximately 10 per cent. This can be understood in terms of the flow Doppler-shifted phase Mach number -- the pattern speed self-adjusts so that the flow into spiral arms is always sonic. This has profound effects on the degree to which transport through self-gravity can be modelled as a viscous process. Our results thus provide (a) a detailed description of how the self-regulation mechanism is established for low cooling rates, (b) a clarification of the conditions required for describing the transport induced by self-gravity through an effective viscosity, (c) an estimate of the maximum amplitude of the density perturbation before fragmentation occurs, and (d) a simple recipe to estimate the density perturbation in different thermal regimes.Comment: 16 pages, 22 figures. Accepted for publication in MNRAS 11 November 200

Human sarcopenia reveals an increase in SOCS-3 and myostatin and a reduced efficiency of Akt phosphorylation

Age-related skeletal muscle sarcopenia is linked with increases in falls, fractures, and death and therefore has important socioeconomic consequences. The molecular mechanisms controlling age-related muscle loss in humans are not well understood, but are likely to involve multiple signaling pathways. This study investigated the regulation of several genes and proteins involved in the activation of key signaling pathways promoting muscle hypertrophy, including GH/STAT5, IGF-1/Akt/GSK-3&beta;/4E-BP1, and muscle atrophy, including TNF&alpha;/SOCS-3 and Akt/FKHR/atrogene, in muscle biopsies from 13 young (20 &plusmn; 0.2 years) and 16 older (70 &plusmn; 0.3 years) males. In the older males compared to the young subjects, muscle fiber cross-sectional area was reduced by 40&ndash;45% in the type II muscle fibers. TNF&alpha; and SOCS-3 were increased by 2.8 and 1.5 fold, respectively. Growth hormone receptor protein (GHR) and IGF-1 mRNA were decreased by 45%. Total Akt, but not phosphorylated Akt, was increased by 2.5 fold, which corresponded to a 30% reduction in the efficiency of Akt phosphorylation in the older subjects. Phosphorylated and total GSK-3&beta; were increased by 1.5 and 1.8 fold, respectively, while 4E-BP1 levels were not changed. Nuclear FKHR and FKHRL1 were decreased by 73 and 50%, respectively, with no changes in their atrophy target genes, atrogin-1 and MuRF1. Myostatin mRNA and protein levels were significantly elevated by 2 and 1.4 fold. Human sarcopenia may be linked to a reduction in the activity or sensitivity of anabolic signaling proteins such as GHR, IGF-1, and Akt. TNF&alpha;, SOCS-3, and myostatin are potential candidates influencing this anabolic perturbation.<br /