Hydrostatic equilibrium

About: Hydrostatic equilibrium is a research topic. Over the lifetime, 2451 publications have been published within this topic receiving 62172 citations.

Radiation Pressure Acceleration by X-rays in Active Galactic Nuclei

Abstract: We present calculations of the dynamics of highly ionized gas clouds that are confined by external pressure, and are ionized by AGN continuum. We focus on the gas that is seen in absorption in the X-ray spectrum of many AGN and show that such gas can reach hydrostatic equilibrium under various conditions. The principal conclusion is that the clouds can be accelerated to high velocities by the central X-ray source. The dynamical problem can be reduced to the calculation of a single parameter, the average force multiplier, . The typical value of is ~10 suggesting that radiation pressure acceleration by X-rays is efficient for L/L_Edd>0.1. The terminal velocity scales with the escape velocity at the base of the flow and can exceed it by a large factor. The typical velocity for a HIG flow that originates at R=1e17 cm in a source with L_x=1e44 erg/s is ~1000 km/s, i.e. similar to the velocities observed in several X-ray and UV absorption systems. Highly ionized AGN clouds are driven mainly by bound-free absorption and bound-bound processes are less important unless the lines are significantly broadened or the column density is very small. Pressure laws that result in constant or outward decreasing ionization parameters are most effective in accelerating the flow.

Radiation Pressure Supported AGN Tori with Hard X-Ray and Stellar Heating

Abstract: The dynamics and structure of toroidal obscuration around AGN remain uncertain and controversial. In this paper we extend earlier work on the dynamical role of infrared radiation pressure by adding the effects of two kinds of distributed heating: Compton-heating due to hard X-rays from the nucleus and local starlight heating. We find numerical solutions to the axisymmetric hydrostatic equilibrium, energy balance, and photon diffusion equations including these effects. Within the regime of typical parameters, the two different sources of additional heating have very similar effects: the density profile within the torus becomes shallower both radially and vertically, but for plausible heating rates, there is only minor change (relative to the source-free case) in the distribution of column density with solid angle. The most interesting consequence of distributed heating is that it selects out a relatively narrow range of parameters permitting an equilibrium, particularly $(L/L_E)/\tau_T$. We discuss the implications of both the narrowness of the permitted range and its approximate coincidence with the range inferred from observations.

Hydrostatic equilibrium in fluid interfaces

Abstract: The statistical mechanical expression for the pressure tensor is used to determine the density distribution in a fluid interface in hydrostatic equilibrium. The two principal assumptions in the model are, firstly, that an expansion of the pair distribution function in the interfacial gradient converges so rapidly that only the zero and first order terms are necessary to determine the density profile, and, secondly, that the radial distribution function for particles in an isochore plane can be obtained from the continuation of the bulk radial distribution function into interfacial densities. The resulting monotonic density functions agree well with otherwise obtained functions. It is tested whether the equation for hydrostatic equilibrium, in the present formulation gives, as it should do, the same interfacial density profiles as the first integrodifferential equation in the Born–Green–Yvon–Bogolyubov hierarchy.

Mechanical stability of shallow magma chambers

Abstract: The mechanical stability of the wall of a magma chamber depends on the value of the stresses likely to be developed in the immediate vicinity of this boundary in relation to the conditions for failure to occur. Various structural and physical factors contributing to these stresses are systematically analyzed. Models of volcanic systems having an axial symmetry around a vertical axis with an isolated reservoir, filled with magma under pressure, in a homogeneous elastic medium are investigated. Spheroidal magma chambers of different aspect ratios (0.7–2) and approximately the same volume (13.5–14 km3), with tops at 3.5–4.5 km depth, are considered. Stress distributions including the effect of gravity, free surface of the Earth, and remote stresses increasing with depth are calculated by using a numerical finite element method. The development of tensile tangential stresses greater than 10 MPa in the elements adjacent to the wall of the chamber is assumed to be a sufficient condition for its instability. The standard state of stress assumed for the crust (i.e., the boundary conditions imposed at large horizontal distance from the chamber) is varied in a continuous range having as lower limit a state of uniaxial strain in the vertical direction and as upper limit a state of hydrostatic stress. Calculations are first performed by making the assumption that the magma pressure P acting at the top of a chamber equals the lithostatic pressure of the overlying rocks. Alternatively, the upper limit of the values of P (critical pressure Pc) for which the initial shape of a chamber may be stable, according to the assumed necessary conditions for stability, is determined. For boundary conditions corresponding to uniaxial strain and Poisson's ratio v = 0.25, Pc turns out to be approximately half of the lithostatic pressure of the overlying rocks. A simple criterion is proposed to estimate if a chamber may evolve toward a new stable shape when P > Pc. This is highly improbable if P equals the lithostatic pressure of the overburden. Larger values of Poisson's ratio (0.30–0.35) favor stability, yielding critical pressures exceeding the lithostatic pressure. Stability is extremely sensitive to the standard state of stress assumed for the crust. Boundary conditions progressively approaching a hydrostatic state of remote stresses yield increasingly higher critical pressures. It turns out that the density contrast between magma and host rocks is not crucial for stability (for Δρ/ρ < 11%). For the rather large spheroidal chambers investigated, the shape is not a critical factor either. The effect of topographical details (e.g., a volcanic edifice, 1.5 km high and 6 km in radius) is practically irrelevant.