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We establish, in the framework of the theory of nested figures, the expressions for the gravitational moments $J_{2n}$ of a systems made of ${\cal L}$ homogeneous layers separated by spheroidal surfaces and in relative rotational motion. We then discuss how to solve the inverse problem, which consists in finding the equilibrium configurations (i.e. internal structures) that reproduce ``exactly'' a set of observables, namely the equatorial radius, the total mass, the shape and the first gravitational moments. Two coefficients $J_{2n}$ being constrained per surface, ${\cal L}=1+\frac{n}{2}$ layers ($n$ even) are required to fix $J_2$ to $J_{2n}$. As shown, this problem already suffers from a severe degeneracy, inherent in the fact that two spheroidal surfaces in the system confocal with each other leave unchanged all the moments. The complexity, which increases with the number of layers involved, can be reduced by considering the rotation rate of each layer. Jupiter is used as a test-bed to illustrate the method, concretely for ${\cal L}=2,3$ and $4$. For this planet, the number of possible internal structures is infinite for ${\cal L} > 2$. Intermediate layers can have smaller or larger oblateness, and can rotate slower or faster than the surroundings. Configurations with large and massive cores are always present. Low-mass cores (of the order a few Earth masses) are predicted for ${\cal L} \ge 4$. The results are in good agreement with the numerical solutions obtained from the Self-Consistent-Field method., Comment: Accepted for publication in MNRAS, 24 pages

Irradiation Assisted Stress Corrosion Cracking (IASCC) is a material degradation phenomenon affecting austenitic stainless steels used in nuclear Pressurized Water Reactors (PWR), leading to the initiation and propagation of intergranular cracks. Such phenomenon belongs to the broader class of InterGranular Stress Corrosion Cracking (IGSCC). A micromechanical analysis of IGSCC of an irradiated austenitic stainless steel is performed in this study to assess local cracking conditions. A 304L proton irradiated sample tested in PWR environment and showing intergranular cracking is investigated. Serial sectioning, Electron BackScatter Diffraction (EBSD) and a two-step misalignment procedure are performed to reconstruct the 3D microstructure over an extended volume, to assess statistically cracking criteria. A methodology is also developed to compute Grain Boundary (GB) normal orientations based on the EBSD measurements. The statistical analysis shows that cracking occurs preferentially for GB normals aligned with the mechanical loading axis, but also for low values of the Luster-Morris slip transmission parameter. Micromechanical simulations based on the reconstructed 3D microstructure, FFT-based solver and crystal plasticity constitutive equations modified to account for slip transmission at grain boundaries are finally performed. These simulations rationalize the correlation obtained experimentally into a single stress-based criterion. The actual strengths and weaknesses of such micromechanical approach are discussed.

The mechanical response of isotropic elastoplastic materials containing random distributions of initially spherical voids is investigated computationally based on Fast Fourier Transform simulations. Numerical limit-analysis simulations at constant stress triaxiality allow to determine the yield surfaces, leading in particular to the determination of a Representative Volume Element size for the onset of coalescence / inhomogeneous yielding. Moreover, two different coalescence regimes are observed that differ by the presence of shearing. The yield surfaces are found to be consistent with the combination of two models proposed in the literature, a GTN-type model calibrated for homogeneous yielding of random porous materials and an inhomogeneous yielding model accounting for both coalescence with or without shear. Finite strain simulations performed for different hardening moduli and stress triaxialities under axisymmetric loading conditions confirm the existence of a RVE up to the onset of inhomogeneous yielding. Coalescence strains are found to be significantly smaller for random porous materials than for periodic distribution of voids. A homogenized model is finally proposed that reproduces quantitatively the behavior of isotropic elastoplastic materials containing random distributions of voids under finite strains.

We perform a bivariate Taylor expansion of the axisymmetric Green function in order to determine the exterior potential of a static thin toroidal shell having a circular section, as given by the Laplace equation. This expansion, performed at the centre of the section, consists in an infinite series in the powers of the minor-to-major radius ratio $e$ of the shell. It is appropriate for a solid, homogeneous torus, as well as for inhomogeneous bodies (the case of a core stratification is considered). We show that the leading term is identical to the potential of a loop having the same main radius and the same mass | this "similarity" is shown to hold in the ${\cal O}(e^2)$ order. The series converges very well, especially close to the surface of the toroid where the average relative precision is $\sim 10^{-3}$ for $e\! = \!0.1$ at order zero, and as low as a few $10^{-6}$ at second order. The Laplace equation is satisfied {\em exactly} in every order, so no extra density is induced by truncation. The gravitational acceleration, important in dynamical studies, is reproduced with the same accuracy. The technique also applies to the magnetic potential and field generated by azimuthal currents as met in terrestrial and astrophysical plasmas., Comment: 15 pages, 12 figures, accepted for publication in MNRAS

We used new ALMA $^{13}$CO and C$^{18}$O(3-2) observations obtained at high angular resolution ($\sim$0.2") together with previous CO(3-2) and (6-5) ALMA data and continuum maps at 1.3 and 0.8 mm in order to determine the gas properties (temperature, density, and kinematics) in the cavity and to a lesser extent in the outer disk of GG Tau A, the prototype of a young triple T Tauri star that is surrounded by a massive and extended Keplerian outer disk. By deprojecting, we studied the radial and azimuthal gas distribution and its kinematics. We also applied a new method to improve the deconvolution of the CO data and in particular better quantify the emission from gas inside the cavity. We perform local and nonlocal thermodynamic equilibrium studies in order to determine the excitation conditions and relevant physical parameters inside the ring and in the central cavity. Residual emission after removing a smooth-disk model indicates unresolved structures at our angular resolution, probably in the form of irregular rings or spirals. The outer disk is cold, with a temperature $<20$ K beyond 250 au that drops quickly (r$^{-1}$). The kinematics of the gas inside the cavity reveals infall motions at about 10% of the Keplerian speed. We derive the amount of gas in the cavity, and find that the brightest clumps, which contain about 10% of this mass, have kinetic temperatures 40$-$80 K, CO column densities of a few 10$^{17}$ cm$^{-2}$, and H$_2$ densities around 10$^7$ cm$^{-3}$. Although the gas in the cavity is only a small fraction of the disk mass, the mass accretion rate throughout the cavity is comparable to or higher than the stellar accretion rate. It is accordingly sufficient to sustain the circumstellar disks on a long timescale., Comment: 18 pages, 14 figures

An experimental methodology is developed to evaluate size effects in nanovoids deformation under macroscopic uniaxial stress loading conditions. Quantitative evaluation of voids deformation as a function of voids size shows both a crystallographic effect, albeit small compared to the scatter, and no evidence of size effects for voids diameter larger than 10 nm, while a slight effect is present for smaller voids. Critical assessment of the data in light of theoretical models indicates that these results may be compatible with the presence of a hardened layer at the void/matrix interface, which is illustrated through finite element simulations accounting for surface tension.

Size effects have been predicted at the micro- or nano-scale for porous ductile materials from Molecular Dynamics, Discrete Dislocation Dynamics and Continuum Mechanics numerical simulations, as a consequence of Geometrically Necessary Dislocations or due to the presence of a void matrix interface. As voids size decreases, higher stresses are needed to deform the material, for a given porosity. However, the majority of the homogenized models for porous materials used in ductile fracture modeling are size-independent, even though micrometric or nanometric voids are commonly observed in structural materials. Based on yield criteria proposed in the literature for nanoporous materials, a size-dependent homogenized model for porous materials is proposed for axisymmetric loading conditions, including void growth and coalescence as well as void shape effects. Numerical implementation of the constitutive equations is detailed. The homogenized model is validated through comparisons to porous unit cells finite element simulations that consider interfacial stresses, consistently with the model used for the derivation of the yield criteria, aiming at modeling an additional hardening at the void matrix interface. Potential improvements of the model are finally discussed with respect to the theoretical derivation of refined yield criteria and evolution laws.

The equilibrium of incompressible spheroid-ring systems in rigid rotation is investigated by numerical means for a unity density contrast. A great diversity of binary configurations is obtained, with no limit neither in the mass ratio, nor in the orbital separation. We found only detached binaries, meaning that the end-point of the $\epsilon_2$-sequence is the single binary state in strict contact, easily prone to mass-exchange. The solutions show a remarkable confinement in the rotation frequency-angular momentum diagram, with a total absence of equilibrium for $\Omega^2/ \pi G \rho \gtrsim 0.21$. A short band of degeneracy is present next to the one-ring sequence. We unveil a continuum of bifurcations all along the ascending side of the Maclaurin sequence for eccentricities of the ellipsoid less than $\approx 0.612$ and which involves a gradually expanding, initially massless loop., Comment: Accepted for publication in MNRAS

We have investigated the toroidal analog of ellipsoidal shells of matter, which are of great significance in Astrophysics. The exact formula for the gravitational potential $\Psi(R,Z)$ of a shell with a circular section at the pole of toroidal coordinates is first established. It depends on the mass of the shell, its main radius and axis-ratio $e$ (i.e. core-to-main radius ratio), and involves the product of the complete elliptic integrals of the first and second kinds. Next, we show that successive partial derivatives $\partial^{n +m} \Psi/\partial_{R^n} \partial_{Z^m}$ are also accessible by analytical means at that singular point, thereby enabling the expansion of the interior potential as a bivariate series. Then, we have generated approximations at orders $0$, $1$, $2$ and $3$, corresponding to increasing accuracy. Numerical experiments confirm the great reliability of the approach, in particular for small-to-moderate axis ratios ($e^2 \lesssim 0.1$ typically). In contrast with the ellipsoidal case (Newton's theorem), the potential is not uniform inside the shell cavity as a consequence of the curvature. We explain how to construct the interior potential of toroidal shells with a thick edge (i.e. tubes), and how a core stratification can be accounted for. This is a new step towards the full description of the gravitating potential and forces of tori and rings. Applications also concern electrically-charged systems, and thus go beyond the context of gravitation., Comment: Accepted for publication in MNRAS

Voids can be observed at various scales in ductile materials, frequently of sizes lower than the grain size, leading to porous single crystals materials. Two local deformation modes of porous ductile (single crystals) materials have been identified and referred to as void growth and void coalescence, the latter being characterized by strong interactions between neighboring voids. A simple semi-analytical coalescence criterion for porous single crystals with periodic arrangement of voids is proposed using effective isotropic yield stresses associated with a criterion derived for isotropic materials. An extension of the coalescence criterion is also proposed to account for shear with respect to the coalescence plane. Effective yield stresses are defined using Taylor theory of single crystal deformation, and rely ultimately on the computation of average Taylor factors. Arbitrary sets of slip systems can be considered. The coalescence criterion is assessed through comparisons to numerical limit analysis results performed using a Fast-Fourier-Transform based solver. A good agreement is observed between the proposed criterion and numerical results for various configurations including different sets of slip systems (Face-Centered-Cubic, Hexagonal-Close-Packed), crystal orientations, void shapes and loading conditions. The competition between void growth and void coalescence is described for specific conditions, emphasizing the strong influence of both crystal orientation and void lattice, as well as their interactions.

From a general point of view, InterGranular Stress-Corrosion Cracking (IGSCC) results from the interplay between mechanical loading and grain boundaries opening. The former leads to intergranular stresses in polycrystalline aggregates, the latter being either stress-accelerated or stress-induced. This work aims at obtaining intergranular normal stress distributions in uncracked polycrystalline aggregates, which is considered as a key milestone towards IGSCC initiation predictive modelling. Based on the finite element method, numerical simulations have been performed on Voronoi polycrystalline aggregates considering a wide variety of material constitutive equations: crystal elasticity (cubic and hexagonal symmetries) with different anisotropy ratios and crystal plasticity for different sets of slip systems under the assumption of uniform critical resolved shear stress: Face-Centered Cubic (FCC), Body-Centered Cubic (BCC) and Hexagonal Close Packed (HCP) with or without hardening, and for both uniaxial and equibiaxial macroscopic loading conditions. In the elastic regime, a correlation between standard deviations of intergranular normal stress distributions and a universal elastic anisotropy index proposed recently is found and explained through a simple model. For macroscopic strain larger than the yield strain, the evolution of standard deviations with strain is rationalized by accounting only for the macroscopic elastic strain and the standard deviation of Taylor factor. These numerical results associated with physically-based simple models allow to estimate easily intergranular normal stress concentrations, constituting a tool for classifying polycrystalline aggregates according to their potential susceptibility to IGSCC.

A decrease of fracture toughness of irradiated materials is usually observed, as reported for austenitic stainless steels in Light Water Reactors (LWRs) or copper alloys for fusion applications. For a wide range of applications (e.g. structural steels irradiated at low homologous temperature), void growth and coalescence fracture mechanism has been shown to be still predominant. As a consequence, a comprehensive study of the effects of irradiation-induced hardening mechanisms on void growth and coalescence in irradiated materials is required. The effects of irradiation on ductile fracture mechanisms - void growth to coalescence - are assessed in this study based on model experiments. Pure copper thin tensile samples have been irradiated with protons up to 0.01 dpa. Micron-scale holes drilled through the thickness of these samples subjected to uniaxial loading conditions allow a detailed description of void growth and coalescence. In this study, experimental data show that physical mechanisms of micron-scale void growth and coalescence are similar between the unirradiated and irradiated copper. However, an acceleration of void growth is observed in the later case, resulting in earlier coalescence, which is consistent with the decrease of fracture toughness reported in irradiated materials. These results are qualitatively reproduced with numerical simulations accounting for irradiation macroscopic hardening and decrease of strain-hardening capability.

We investigate the impact of an external pressure on the structure of self-gravitating polytropes for axially symmetric ellipsoids and rings. The confinement of the fluid by photons is accounted for through a boundary condition on the enthalpy $H$. Equilibrium configurations are determined numerically from a generalised "Self-Consistent-Field"-method. The new algorithm incorporates an intra-loop re-scaling operator ${\cal R}(H)$, which is essential for both convergence and getting self-normalised solutions. The main control parameter is the external-to-core enthalpy ratio. In the case of uniform rotation rate and uniform surrounding pressure, we compute the mass, the volume, the rotation rate and the maximum enthalpy. This is repeated for a few polytropic indices $n$. For a given axis ratio, over-pressurization globally increases all output quantities, and this is more pronounced for large $n$. Density profiles are flatter than in the absence of an external pressure. When the control parameter asymptotically tends to unity, the fluid converges toward the incompressible solution, whatever the index, but becomes geometrically singular. Equilibrium sequences, obtained by varying the axis ratio, are built. States of critical rotation are greatly exceeded or even disappear. The same trends are observed with differential rotation. Finally, the typical response to a photon point source is presented. Strong irradiation favours sharp edges. Applications concern star forming regions and matter orbiting young stars and black holes., Comment: 16 pages

Ductile fracture through void growth and coalescence depends significantly on the plastic anisotropy of the material and on void size, as shown by experiments and/or numerical simulations through several studies. Macroscopic (homogenized) yield criteria aiming at modeling nanoporous materials have been proposed only for the growth regime, i.e. non-interacting voids. The aim of this study is thus to provide a yield criterion for nanoporous materials relevant for the coalescence regime, i.e. when plastic flow is localized between voids. Through homogenization and limit analysis, and accounting for interface stresses at the void-matrix interface, analytical coalescence criterion is derived under the following conditions: axisymmetric loading, orthotropic material obeying Hill's plasticity, cylindrical voids in cylindrical unit-cell. Incidentally, an orthotropic extension of the existing isotropic modeling of interface stresses through limit analysis is described and used. The proposed coalescence criterion is then extended to account for combined tension and shear loading conditions. Numerical limit analyses have been performed under specific conditions / materials parameters to get supposedly exact (up to numerical errors) results of coalescence stress. A good agreement between the analytical coalescence criteria derived in this study and numerical results is found for elongated spheroidal voids, making them usable to predict the onset of void coalescence in ductile fracture modeling of nanoporous materials.

We compute the structure of a self-gravitating torus with polytropic equation-of-state (EOS) rotating in an imposed centrifugal potential. The Poisson-solver is based on isotropic multigrid with optimal covering factor (fluid section-to-grid area ratio). We work at $2$nd-order in the grid resolution for both finite difference and quadrature schemes. For soft EOS (i.e. polytropic index $n \ge 1$), the underlying $2$nd-order is naturally recovered for Boundary Values (BVs) and any other integrated quantity sensitive to the mass density (mass, angular momentum, volume, Virial Parameter, etc.), i.e. errors vary with the number $N$ of nodes per direction as $\sim 1/N^2$. This is, however, not observed for purely geometrical quantities (surface area, meridional section area, volume), unless a subgrid approach is considered (i.e. boundary detection). Equilibrium sequences are also much better described, especially close to critical rotation. Yet another technical effort is required for hard EOS ($n < 1$), due to infinite mass density gradients at the fluid surface. We fix the problem by using kernel splitting. Finally, we propose an accelerated version of the SCF-algorithm based on a node-by-node pre-conditionning of the mass density at each step. The computing time is reduced by a factor $2$ typically, regardless of the polytropic index. There is a priori no obstacle to applying these results and techniques to ellipsoidal configurations and even to $3$D-configurations., Comment: accepted for publication; 18 pages; 26 figures

Coalescence of voids by internal necking is in most cases the last microscopic event related to ductile fracture and corresponds to a localized plastic flow between adjacent voids. Macroscopic load associated to the onset of coalescence is classically estimated based on limit analysis. However, a rigorous upper-bound mathematical expression for the limitload required for flat voids coalescence that remains finite for penny-shaped voids/cracks is still unavailable. Therefore, based on limit analysis, theoretical upper-bound estimates - both integral expression and closed-form formula - are obtained for the limit-load of cylindrical flat voids in cylindrical unit-cell subjected to boundary conditions allowing the assessment of coalescence, for axisymmetric stress state. These estimates, leading to finite limit-loads for pennyshaped cracks, are shown to be in very good agreement with numerical limit analysis, for both cylindrical and spheroidal voids. Approximate formula is also given for coalescence under combined tension and shear loading. These coalescence criteria can thus be used to predict onset of coalescence of voids by internal necking in ductile fracture modelling.

In order to predict InterGranular Stress Corrosion Cracking (IGSCC) of post-irradiated austenitic stainless steel in Light Water Reactor (LWR) environment, reliable predictions of intergranular stresses are required. Finite elements simulations have been performed on realistic polycrystalline aggregate with a recently proposed physically-based crystal plasticity constitutive equations validated for neutron-irradiated austenitic stainless steel. Intergranular normal stress probability density functions are found with respect to plastic strain and irradiation level, for uniaxial loading conditions. In addition, plastic slip activity jumps at grain boundaries are also presented. Intergranular normal stress distributions describe, from a statistical point of view, the potential increase of intergranular stress with respect to the macroscopic stress due to grain-grain interactions. The distributions are shown to be well described by a master curve once rescaled by the macroscopic stress, in the range of irradiation level and strain considered in this study. The upper tail of this master curve is shown to be insensitive to free surface effect, which is relevant for IGSCC

We study experimentally the distribution of local stresses in a granular material confined inside a vertical cylinder. We use an image correlation technique to measure the displacement field of the container induced by the forces exerted by the grains on the inner wall. We describe an optimization procedure based on the linear theory of elastic shells to deduce the distribution of these forces from the measured displacement field. They correspond to the stress field of the granular material close to the container's inner wall. We first confirm the validity of Janssen's description for various experiments, including the influence of the bead diameter and the effect of an additional mass on top of the granular column. We then apply this method to determine the stress field during the gravity-driven discharge of a silo through an aperture.

We introduce a novel technique for grabbing water with a flexible solid. This new passive pipetting mechanism was inspired by floating flowers and relies purely on the coupling of the elasticity of thin plates and the hydrodynamic forces at the liquid interface. Developing a theoretical model has enabled us to design petal-shaped objects with maximum grabbing capacity.

Blisters are commonly observed when an adhesive sheet is carelessly deposited on a plate. Although such blisters are usually not desired for practical applications, we show through model experiments on angular blisters how material properties can be deduced from height profile measurements. In particular the typical curvature of the crests is found proportional to an elasto-capillary length that compares the bending stiffness of the sheet with adhesion energy. In addition, the radius of the tip allows to estimate the product of this length with the thickness of the sheet. The relevance of these results to realistic random configurations is finally confirmed.

We present a combined theoretical and experimental study of the buckling of a thin film wrapped around a sphere under the action of capillary forces. A rigid sphere is coated with a wetting liquid, and then wrapped by a thin film into an initially cylindrical shape. The equilibrium of this cylindrical shape is governed by the antagonistic effects of elasticity and capillarity: elasticity tends to keep the film developable while capillarity tends to curve it in both directions so as to maximize the area of contact with the sphere. In the experiments, the contact area between the film and the sphere has cylindrical symmetry when the sphere radius is small, but destabilises to a non-symmetric, wrinkled configuration when the radius is larger than a critical value. We combine the Donnell equations for near-cylindrical shells to include a unilateral constraint with the impenetrable sphere, and the capillary forces acting along a moving edge. A non-linear solution describing the axisymmetric configuration of the film is derived. A linear stability analysis is then presented, which successfully captures the wrinkling instability, the symmetry of the unstable mode, the instability threshold and the critical wavelength. The motion of the free boundary at the edge of the region of contact, which has an effect on the instability, is treated without any approximation.

We study the peculiar wrinkling pattern of an elastic plate stamped into a spherical mold. We show that the wavelength of the wrinkles decreases with their amplitude, but reaches a maximum when the amplitude is of the order of the thickness of the plate. The force required for compressing the wrinkled plate presents a maximum independent of the thickness. A model is derived and verified experimentally for a simple one-dimensional case. This model is extended to the initial situation through an effective Young modulus representing the mechanical behavior of wrinkled state. The theoretical predictions are shown to be in good agreement with the experiments. This approach provides a complement to the "tension field theory" developed for wrinkles with unconstrained amplitude.

By using various properties of the complete elliptic integrals, we have derived an alternative expression for the gravitational potential of axially symmetric bodies, which is free of singular kernel in contrast with the classical form. This is mainly a radial integral of the local surface density weighted by a regular "mean Green function" which depends explicitly on the body's vertical thickness. Rigorously, this result stands for a wide variety of configurations, as soon as the density structure is vertically homogeneous. Nevertheless, the sensitivity to vertical stratification | the Gaussian profile has been considered | appears weak provided that the surface density is conserved. For bodies with small aspect ratio (i.e. geometrically thin discs), a first-order Taylor expansion furnishes an excellent approximation for this mean Green function, the absolute error being of the fourth order in the aspect ratio. This formula is therefore well suited to studying the structure of self-gravitating discs and rings in the spirit of the "standard model of thin discs" where the vertical structure is often ignored, but it remains accurate for discs and tori of finite thickness. This approximation which perfectly saves the properties of Newton's law everywhere (in particular at large separations), is also very useful for dynamical studies where the body is just a source of gravity acting on external test particles., Comment: Accepted for publication in MNRAS, 11 pages

We study the adhesion of an elastic sheet on a rigid spherical substrate. Gauss'Theorema Egregium shows that this operation necessarily generates metric distortions (i.e. stretching) as well as bending. As a result, a large variety of contact patterns ranging from simple disks to complex branched shapes are observed as a function of both geometrical and material properties. We describe these different morphologies as a function of two non-dimensional parameters comparing respectively bending and stretching energies to adhesion. A complete configuration diagram is finally proposed.

We present the exact calculus of the gravitational potential and acceleration along the symmetry axis of a plane, homogeneous, polar cell as a function of mean radius a, radial extension e, and opening angle f. Accurate approximations are derived in the limit of high numerical resolution at the geometrical mean of the inner and outer radii (a key-position in current FFT-based Poisson solvers). Our results are the full extension of the approximate formula given in the textbook of Binney & Tremaine to all resolutions. We also clarify definitely the question about the existence (or not) of self-forces in polar cells. We find that there is always a self-force at radius except if the shape factor a.f/e reaches ~ 3.531, asymptotically. Such cells are therefore well suited to build a polar mesh for high resolution simulations of self-gravitating media in two dimensions. A by-product of this study is a newly discovered indefinite integral involving complete elliptic integral of the first kind over modulus., Comment: 4 pages, 4 figures, A&A accepted

The first-order ordinary differential equation (ODE) that describes the mid-plane gravitational potential in flat finite size discs in which the surface density is a power-law function of the radius R with exponent s (Hur\'e & Hersant 2007) is solved exactly in terms of infinite series. The formal solution of the ODE is derived and then converted into a series representation by expanding the elliptic integral of the first kind over its modulus before analytical integration. Inside the disc, the gravitational potential consists of three terms: a power law of radius R with index 1+s, and two infinite series of the variables R and 1/R. The convergence of the series can be accelerated, enabling the construction of reliable approximations. At the lowest-order, the potential inside large astrophysical discs (s ~ -1.5 +/- 1) is described by a very simple formula whose accuracy (a few percent typically) is easily increased by considering successive orders through a recurrence. A basic algorithm is given. Applications concern all theoretical models and numerical simulations where the influence of disc gravity must be checked and/or reliably taken into account., Comment: Accepted for publication in A&A, 13 pages, 8 figures

We report a simple method to generate potential/surface density pairs in flat axially symmetric finite size disks. Potential/surface density pairs consist of a ``homogeneous'' pair (a closed form expression) corresponding to a uniform disk, and a ``residual'' pair. This residual component is converted into an infinite series of integrals over the radial extent of the disk. For a certain class of surface density distributions (like power laws of the radius), this series is fully analytical. The extraction of the homogeneous pair is equivalent to a convergence acceleration technique, in a matematical sense. In the case of power law distributions, the convergence rate of the residual series is shown to be cubic inside the source. As a consequence, very accurate potential values are obtained by low order truncation of the series. At zero order, relative errors on potential values do not exceed a few percent typically, and scale with the order N of truncation as 1/N**3. This method is superior to the classical multipole expansion whose very slow convergence is often critical for most practical applications., Comment: Accepted for publication in Astronomy & Astrophysics 7 pages, 8 figures, F90-code available at http://www.obs.u-bordeaux1.fr/radio/JMHure/intro2applawd.html

We show that the gravitational potential in the plane of an axisymmetrical flat disk where the surface density varies as a power of the radius obeys an inhomogeneous first-order Ordinary Differential Equation (ODE) solvable by standard techniques. The potential being exactly known at the origin for any power index (and at infinity as well), the search for solutions consists of a Two-point Boundary Value Problem (TBVP) with Dirichlet conditions. The computating time is then linear with the number of grid points, instead of quadratic from direct summation methods. Complex mass distributions which can be decomposed into a mixture of power law surface density profiles are easily accessible through the superposition principle. This ODE definitively takes the place of the untractable bidimensional Poisson equation for planar calculations. It opens new horizons to investigate various aspects related to self-gravity in astrophysical disks (force calculations, stability analysis, etc.)., Comment: 5 pages, 1 figure, accepted for publication in A&A

Structure of a quasi-stationary stellar cluster is modelled assuming that it is embedded in the gravitational field of a super-massive black hole. Gradual orbital decay of stellar trajectories is caused by the dissipative interaction with an accretion disc. Gravitational field of the disc is constructed and its effect on the cluster structure is taken into account as an axially symmetric perturbation. Attention is focused on a circumnuclear region (r<10^4 gravitational radii) where the effects of the central black hole and the disc dominate over the influence of an outer galaxy. It is shown how the stellar system becomes gradually flattened towards the disc plane. For certain combinations of the model parameters, a toroidal structure is formed by a fraction of stars. Growing anisotropy of stellar velocities as well as their segregation occur. The mass function of the inner cluster is modified and it progressively departs from the asymptotic form assumed in the outer cluster. A new stationary distribution can be characterized in terms of velocity dispersion of the stellar sample in the central region of the modified cluster., Comment: Accepted for publication in MNRAS; 12 pages, 10 figures

We investigate the analogy between circumstellar disks and the Taylor-Couette flow. Using the Reynolds similarity principle, the analogy results in a number of parameter-free predictions about stability of the disks, and their turbulent transport properties, provided the disk structure is available. We discuss how the latter can be deduced from interferometric observations of circumstellar material. We use the resulting disk structure to compute the molecular transport coefficients, including the effect of ionization by the central object. The resulting control parameter indicates that the disk is well into the turbulent regime. The analogy is also used to compute the effective accretion rate, as a function of the disk characteristic parameters (orbiting velocity, temperature and density). These values are in very good agreement with experimental, parameter-free predictions derived from the analogy. The turbulent viscosity is also computed and found to correspond to an $\alpha$-parameter $2\times 10^{-4}<\alpha<2\times 10^{-2}$. Predictions regarding fluctuations are also checked: luminosity fluctuations in disks do obey the same universal distribution as energy fluctuations observed in a laboratory turbulent flow. Radial velocity dispersion in the outer part of the disk is predicted to be of the order of 0.1 km/s, in agreement with available observations. All these issues provide a proof of the turbulent character of the circumstellar disks, as well as a parameter-free theoretical estimate of effective accretion rates., Comment: Accepted for publication in Astronomy and Astrophysics, 13 pages

Fluid discs and tori around black holes are discussed within different approaches and with the emphasis on the role of disc gravity. First reviewed are the prospects of investigating the gravitational field of a black hole--disc system by analytical solutions of stationary, axially symmetric Einstein's equations. Then, more detailed considerations are focused to middle and outer parts of extended disc-like configurations where relativistic effects are small and the Newtonian description is adequate. Within general relativity, only a static case has been analysed in detail. Results are often very inspiring, however, simplifying assumptions must be imposed: ad hoc profiles of the disc density are commonly assumed and the effects of frame-dragging and completely lacking. Astrophysical discs (e.g. accretion discs in active galactic nuclei) typically extend far beyond the relativistic domain and are fairly diluted. However, self-gravity is still essential for their structure and evolution, as well as for their radiation emission and the impact on the environment around. For example, a nuclear star cluster in a galactic centre may bear various imprints of mutual star--disc interactions, which can be recognised in observational properties, such as the relation between the central mass and stellar velocity dispersion., Comment: Accepted for publication in CQG; high-resolution figures will be available from http://www.iop.org/EJ/journal/CQG

There is apparently a widespread belief that the gravitational field (and subsequently the rotation curve) ``inside'' razor-thin, axially symmetric disks can not be determined accurately from elliptic integrals because of the singular kernel in the Poisson integral. Here, we report a simple and powerful method to achieve this task numerically using the technique of ``density splitting''., Comment: 8 pages, Postscript, 1 figure, accepted in ApJ

Self-gravitating accretion discs have only been studied in a few nearby objects using maser spots at the parsec-scale. We find a new spectral window for observing the self-gravitating accretion disc in super-Eddington accreting Active Galactic Nuclei (AGNs). This window is determined by calculating the outermost radius (rsg) of a non self-gravitating disc and the corresponding emission wavelength (lsg) as a function of various disc parameters. We find that lsg reaches ~ 4000 AA for alpha=0.1, when Mdot > 70 (M_BH / 10^7 M_Sun)^-1 L_Edd/c^2 (where alpha, Mdot, M_BH and L_Edd are, respectively, the viscosity parameter, gas accretion rate onto the central black hole (BH), the BH mass and the Eddington luminosity). Moreover, lsg is as small as 1500 AA for alpha = 0.001, which is the smallest alpha case in this study. Therefore, the window for observing the self-gravitating part of an AGN accretion disc is from 2 micron to lsg. Incidentally, rsg can be less than the photon trapping radius for Mdot >~ 10^3.3 L_Edd/c^2. Namely, a self-gravitating, optically-thick, advection-dominated accretion disc is expected to appear in the extremely high accretion rate regime. Next, we demonstrate that the Mid-Infrared to X-ray spectrum of a bright, well-studied Narrow-Line Seyfert 1 galaxy, Ton S 180, is indeed well fitted by the spectrum arising from the following three components: an inner slim disc (with a corona), an outer, self-gravitating non-Keplerian disc and a dusty torus. The total mass, BH mass plus the entire disc mass, is found to be about (1.4 - 8.0) M_BH. If the surface density varies with radius r in proportion to r^-0.6, the total mass is consistent with the central mass estimated by H_beta and [O III] widths., Comment: 8 pages (including 5 figures), Latex, Accepted for publication in Astronomy and Astrophysics