State Of Hydrostatic Equilibrium Research Articles

DETERMINING ALL GAS PROPERTIES IN GALAXY CLUSTERS FROM THE DARK MATTER DISTRIBUTION ALONE

We demonstrate that all properties of the hot X-ray emitting gas in galaxy clusters are completely determined by the underlying dark matter (DM) structure. Apart from the standard conditions of spherical symmetry and hydrostatic equilibrium for the gas, our proof is based on the Jeans equation for the DM and two simple relations which have recently emerged from numerical simulations: the equality of the gas and DM temperatures, and the almost linear relation between the DM velocity anisotropy profile and its density slope. For DM distributions described by the NFW or the Sersic profiles, the resulting gas density profile, the gas-to-total-mass ratio profile, and the entropy profile are all in good agreement with X-ray observations. All these profiles are derived using zero free parameters. Our result allows us to predict the X-ray luminosity profile of a cluster in terms of its DM content alone. As a consequence, a new strategy becomes available to constrain the DM morphology in galaxy clusters from X-ray observations. Our results can also be used as a practical tool for creating initial conditions for realistic cosmological structures to be used in numerical simulations.

Reconstruction of Gas Temperature and Density Profiles of the Galaxy Cluster RX J1347.5–1145

We use observations of Sunyaev-Zel'dovich effect and X-ray surface brightness to reconstruct the radial profiles of gas temperature and density under the assumption of a spherically symmetric distribution of the gas. The method of reconstruction, first raised by Silk & White, depends directly on the observations of the Sunyaev-Zel'dovich effect and the X-ray surface brightness, without involving additional assumptions such as the equation of state of the gas or the conditions of hydrostatic equilibrium. We applied this method to the cluster RX J1347.5-1145, which has both the Sunyaev-Zel'dovich effect and X-ray observations with relative high precision. It is shown that it will be an effective method to obtain the gas distribution in galaxy clusters. Statistical errors of the derived temperature and density profiles of gas were estimated according to the observational uncertainties.

Hydrostatic equilibrium of causally consistent and dynamically stable neutron star models

We show that the mass-radius $(M-R)$ relation corresponding to the stiffest equation of state (EOS) does not provide the necessary and sufficient condition of dynamical stability for the equilibrium configurations, since such configurations can not satisfy the `compatibility criterion'. In this connection, we construct sequences composed of core-envelope models such that, like the stiffest EOS, each member of these sequences satisfy the extreme case of causality condition, $v = c = 1$, at the centre. We, thereafter, show that the $M-R$ relation corresponding to the said core-envelope model sequences can provide the necessary and sufficient condition of dynamical stability only when the `compatibility criterion' for these sequences is `appropriately' satisfied. However, the fulfillment of `compatibility criterion' can remain satisfied even when the $M-R$ relation does not provide the necessary and sufficient condition of dynamical stability for the equilibrium configurations. In continuation to the results of previous study, these results explicitly show that the `compatibility criterion' {\em independently} provides, in general, the {\em necessary} and {\em sufficient} condition of hydrostatic equilibrium for any regular sequence. Beside its fundamental feature, this study can also explain simultaneously, both (the higher as well as lower) values of the glitch healing parameter observed for the Crab and the Vela-like pulsars respectively, on the basis of starquake model of glitch generation.

QUANTUM CONDENSATES IN EXTREME GRAVITY: IMPLICATIONS FOR COLD STARS AND DARK MATTER

Stable end-point stars currently fall into two distinct classes — white dwarfs and neutron stars — differing enormously in central density and radial size. No stable cold dead stars are thought to span the intervening densities or have masses beyond ~2–3 solar masses. I show, however, that the general-relativistic condition of hydrostatic equilibrium augmented by the equation of state of a neutron condensate at 0 K generates stable sequences of cold stars that span the density gap and can have masses well beyond prevailing limits. The radial sizes and mass limit of each sequence are determined by the mass and scattering length of the composite bosons. Solutions for hypothetical bosons of ultrasmall mass and large scattering length yield huge self-gravitating systems of low density, resembling galactic dark matter halos.

Designing Carbon Nanotube Membranes for Efficient Water Desalination

The transport of water and ions through membranes formed from carbon nanotubes ranging in diameter from 6 to 11 A is studied using molecular dynamics simulations under hydrostatic pressure and equilibrium conditions. Membranes incorporating carbon nanotubes are found to be promising candidates for water desalination using reverse osmosis, and the size and uniformity of tubes that is required to achieve a desired salt rejection is determined. By calculating the potential of mean force for ion and water translocation, we show that ions face a large energy barrier and will not pass through the narrower tubes studied ((5,5) and (6,6) "armchair" type tubes) but can pass through the wider (7,7) and (8,8) nanotubes. Water, however, faces no such impediment due to the formation of stable hydrogen bonds and crosses all of the tubes studied at very large rates. By measuring this conduction rate under a hydrostatic pressure difference, we show that membranes incorporating carbon nanotubes can, in principle, achieve a high degree of desalination at flow rates far in excess of existing membranes.

Rayleigh–Taylor shock waves

Beginning from a state of hydrostatic equilibrium, in which a heavy gas rests atop a light gas in a constant gravitational field, Rayleigh–Taylor instability at the interface will launch a shock wave into the upper fluid. We have performed a series of large-eddy simulations which suggest that the rising bubbles of light fluid act like pistons, compressing the heavy fluid ahead of the fronts and generating shocklets. These shocklets coalesce in multidimensional fashion into a strong normal shock, which increases in strength as it propagates upwards. The simulations demonstrate that the shock Mach number increases faster in three dimensions than it does in two dimensions. The generation of shocks via Rayleigh–Taylor instability could play an important role in type Ia supernovae.

An Evolving Entropy Floor in the Intracluster Gas?

Nongravitational processes, such as feedback from galaxies and their active nuclei, are believed to have injected excess entropy into the intracluster gas, and therefore to have modified the density profiles in galaxy clusters during their formation. Here we study a simple model for this so-called preheating scenario, and ask (1) whether it can simultaneously explain both global X-ray scaling relations and number counts of galaxy clusters, and (2) whether the amount of entropy required evolves with redshift. We adopt a baseline entropy profile that fits recent hydrodynamic simulations, modify the hydrostatic equilibrium condition for the gas by including ≈20% nonthermal pressure support, and add an entropy floor K0 that is allowed to vary with redshift. We find that the observed luminosity-temperature (L − T) relations of low-redshift ( z = 0.05) HIFLUGCS clusters and high-redshift ( z = 0.80) WARPS clusters are best simultaneously reproduced with an entropy floor that evolves from ≈200 h−1/3 keV cm 2 at z ≈ 0.8 to 300 h−1/3 keV cm 2 at z < 0.05. This evolution may take place predominantly at low redshift (z 0.2). If we restrict our analysis to the subset of bright (kT 3 keV) clusters, we find that the evolving entropy floor can mimic a self-similar evolution in the L − T scaling relation. This degeneracy with self-similar evolution is, however, lifted when 0.5 keV kT 3 keV clusters are included. Using the cosmological parameters from the WMAP 3 yr data, but treating σ8 as a free parameter, our model can reproduce the number counts of the X-ray galaxy clusters in the 158 deg2 ROSAT PSPC survey, with a best-fit value of σ8 = 0.80 ± 0.05.

Equation of state for macromolecules of variable flexibility in good solvents: A comparison of techniques for Monte Carlo simulations of lattice models

The osmotic equation of state for the athermal bond fluctuation model on the simple cubic lattice is obtained from extensive Monte Carlo simulations. For short macromolecules (chain length N=20 ) we study the influence of various choices for the chain stiffness on the equation of state. Three techniques are applied and compared in order to critically assess their efficiency and accuracy: the "repulsive wall" method, the thermodynamic integration method (which rests on the feasibility of simulations in the grand canonical ensemble), and the recently advocated sedimentation equilibrium method, which records the density profile in an external (e.g., gravitationlike) field and infers, via a local density approximation, the equation of state from the hydrostatic equilibrium condition. We confirm the conclusion that the latter technique is far more efficient than the repulsive wall method, but we find that the thermodynamic integration method is similarly efficient as the sedimentation equilibrium method. For very stiff chains the onset of nematic order enforces the formation of an isotropic-nematic interface in the sedimentation equilibrium method leading to strong rounding effects and deviations from the true equation of state in the transition regime.

The structure of the accretion disk in NGC 4258

AbstractA wealth of new information about the structure of the maser disk in NGC 4258 has been obtained from a series of 18 VLBA observations spanning three years, as well as from 32 additional epochs of spectral monitoring data from 1994 to the present, acquired with the VLA, Effelsberg, and GBT. The warp of the disk has been defined precisely. The thickness of the maser disk has been measured to be 12 micro-arcseconds (FWHM), which is slightly smaller than previously quoted upper limits. Under the assumption that the masers trace the true vertical distribution of material in the disk, from the condition of hydrostatic equilibrium the sound speed is 1.5 km s−1, corresponding to a thermal temperature of 600K. The accelerations of the high velocity maser components have been accurately measured for many features on both the blue and red side of the spectrum. The azimuthal offsets of these masers from the midline (the line through the disk in the plane of the sky) and derived projected offsets from the midline based on the warp model correspond well with the measured offsets. This result suggests that the masers are well described as discrete clumps of masing gas, which accurately trace the Keplerian motion of the disk. However, we have continued to search for evidence of apparent motions caused by “phase effects.” This work provides the foundation for refining the estimate of the distance to NGC 4258 through measurements of feature acceleration and proper motion. The refined estimate of this distance is expected to be announced in the near future.

Structure and evolution of magnetized clusters: entropy profiles,S–TandLX–Trelations

We study the impact of an intracluster magnetic field on the main structural properties of clusters and groups of galaxies: the radial density and entropy profiles, the entropy – temperature relation and the X-ray luminosity – temperature relation for groups and clusters of galaxies. To this aim, we develop a description of the intra-cluster gas based on the Hydrostatic Equilibrium condition and on the Magnetic Virial Theorem in the presence of a radial distribution of the magnetic field $B(r) = B_* (\rho_{\rm g}(r))^{\alpha}$, with $\alpha \approx 0.9$, as the one indicated by observations and numerical simulation. Our analysis shows that such a description is able to provide, at once, a possible explanation of three problematic aspects of the cluster structure: i) the flattening of the entropy profile in the cluster center; ii) the flatness of the $S - T$ relation; iii) the increasing steepening of the $L_{\rm X}-T$ relation from the cluster scale towards the group scales. The available entropy and X-ray luminosity data indicate that an increase of the magnetic field $B_* \sim T^{0.5 \pm 0.1}$ is required to reproduce at the same time both the $S - T$ and the $L_{\rm X}-T$ relations. It follows that a consistent description of the magnetized ICM can provide a simple explanation of several (or of all) of these still open problems, and thus weakens the need for the inclusion of other non-gravitational effects which have been proposed so far for the explanation of some of these features. This (initial, but not conclusive) analysis can be regarded as a starting point for a more refined analytical exploration of the physics of the magnetized intra-cluster medium, and it provides testable predictions that can be proven or disproven with the next coming sensitive observations of groups and clusters in the X-ray band and in the radio frequency band.

NECESSARY AND SUFFICIENT CONDITION FOR HYDROSTATIC EQUILIBRIUM IN GENERAL RELATIVITY

We present explicit examples to show that the "compatibility criterion" (recently obtained by us toward providing equilibrium configurations compatible with the structure of general relativity) which states that for a given value of σ[≡ (P0/E0) ≡ the ratio of central pressure to central energy-density], the compactness ratio u[≡ (M/R), where M is the total mass and R is the radius of the configuration] of any static configuration cannot exceed the compactness ratio, uh, of the homogeneous density sphere (i.e., u ≤ uh) is capable of providing a necessary and sufficient condition for any regular configuration to be compatible with the state of hydrostatic equilibrium. This conclusion is drawn on the basis of the finding that the M–R relation gives the necessary and sufficient condition for dynamical stability of equilibrium configurations only when the compatibility criterion for these configurations is appropriately satisfied. In this regard, we construct an appropriate sequence composed of core-envelope models on the basis of compatibility criterion such that each member of this sequence satisfies the extreme case of causality condition v = c = 1 at the center. The maximum stable value of u ≃ 0.3389 (which occurs for the model corresponding to the maximum value of mass in the mass–radius relation) and the corresponding central value of the local adiabatic index, (Γ1)0 ≃ 2.5911, of this model are found fully consistent with those of the corresponding absolute values, u max ≤ 0.3406 and (Γ1)0 ≤ 2.5946, which impose strong constraints on these parameters of such models. In addition to this example, we also study dynamical stability of pure adiabatic polytropic configurations on the basis of variational method for the choice of the "trial function," ξ = reν/4, as well as the mass–central density relation, since the compatibility criterion is appropriately satisfied for these models. The results of this example provide additional proof in favor of the statement regarding compatibility criterion mentioned above. Together with other results, this study also confirms the previous claim that just the choice of the "trial function," ξ = reν/4, is capable of providing the necessary and sufficient condition for dynamical stability of a mass on the basis of variational method. Obviously, the upper bound on the compactness ratio of neutron stars, u ≅ 0.3389, which belongs to two-density model studied here, turns out to be much stronger than the corresponding "absolute" upper bound mentioned in the literature.

Improved estimation of soil water retention characteristics from hydrostatic column experiments

The soil water retention characteristic, θ(h), is required for modeling and predicting water and solute transport in unsaturated porous media. Commonly, θ(h) is determined by relating pressure heads, h, to mean water contents, θ, that are measured in column experiments under hydrostatic equilibrium conditions and fitting a parametric retention function to these data pairs. Implicit to this method is the assumption that the mean water content of the column is equivalent to a point measurement in the column center. Dependent on the nonlinearity of the vertical water content distribution, θ(z), in the column, this assumption may be invalid and introduces a systematic error. A sensitivity analysis shows that the magnitude of the error caused by neglecting the nonlinearity of the water content distribution may reach several percent if coarse materials with low air entry values and tall soil columns are investigated. Furthermore, neglecting θ(z) yields a smoothed retention characteristic and thus may lead to wrong conclusions about the most appropriate parametric model for the water retention characteristic. If the hydraulic conductivity function K(h) is predicted from such an incorrect retention function, it can differ greatly from the true function. In this paper, we propose to consider the measured water content of a soil column explicitly as an integral of the equilibrium water content distribution with depth. We show that this eliminates systematic parameter estimation errors and leads to improved estimates of the soil water retention function.

Collimation of Astrophysical Jets: The Role of the Accretion Disk Magnetic Field Distribution

We have applied axisymmetric MHD simulations to investigate the impact of the accretion disk magnetic flux profile on the jet collimation. Using the ZEUS-3D code modified for magnetic diffusivity, our simulations evolve from an initial hydrostatic equilibrium state in a force-free magnetic field. Considering a power law for the disk magnetic field profile Bp ~ r^{-mu} and for the disk wind density profile rho ~ r^{-mu_rho} we performed a systematic study over a wide parameter range mu and mu_rho. We find a degree of (ratio of mass flow rates in axial and lateral direction) decreasing for steeper disk magnetic field profiles (increasing mu). Varying the total magnetic flux doesn't change the degree of jet substantially, it only affects the time scale of outflow evolution and the terminal jet speed. As our major result we find a general relation between the degree with the disk wind magnetization power law exponent. Outflows with high degree resulting from a flat disk magnetic field profile tend to be unsteady, producing axially propagating knots as discussed earlier. Depending slightly on the inflow density profile this unsteady behavior sets in for mu < 0.4. We also performed simulations of jet formation with artificially enhanced decay of the toroidal magnetic field in order to investigate the idea of a purely poloidal collimation discussed in the literature. These outflows remain weakly collimated and propagate with lower velocity. Thanks to our large numerical grid size (7x14 AU for protostars), we may apply our results to recently observed hints of jet rotation (DG Tau) indicating a relatively flat disk magnetic field profile, mu ~ 0.5. In general, our results are applicable to both stellar and extragalactic sources of MHD jets.

Exact Solutions of Einstein's Field Equations

We examine various well known exact solutions available in the literature to investigate the recent criterion obtained in Negi and Durgapal [Gravitation and Cosmology 7, 37 (2001)] which should be fulfilled by any static and spherically symmetric solution in the state of hydrostatic equilibrium. It is seen that this criterion is fulfilled only by (i) the regular solutions having a vanishing surface density together with pressure, and (ii) the singular solutions corresponding to a non-vanishing density at the surface of the configuration. On the other hand, the regular solutions corresponding to a non-vanishing surface density do not fulfill this criterion. Based upon this investigation, we point out that the exterior Schwarzschild solution itself provides necessary conditions for the types of the density distributions to be considered inside the mass, in order to obtain exact solutions or equations of state compatible with the state of hydrostatic equilibrium in general relativity. The regular solutions with finite centre and non-zero surface densities which do not fulfill the criterion given by Negi and Durgapal (2001), in fact, cannot meet the requirement of the‘actual mass’, set up by exterior Schwarzschild solution. The only regular solution which could be possible in this regard is represented by uniform (homogeneous) density distribution. This criterion provides a necessary and sufficient condition for any static and spherical configuration (including core-envelope models) to be compatible with the structure of general relativity [that is, the state of hydrostatic equilibrium in general relativity]. Thus, it may find application to construct the appropriate core-envelope models of stellar objects like neutron stars and may be used to test various equations of state for dense nuclear matter and the models of relativistic star clusters with arbitrary large central redshifts.

Phase Separation of a Model Binary Polymer Solution in an External Field

The phase separation of a simple binary mixture of incompatible linear polymers in solution is investigated using an extension of the sedimentation equilibrium method, whereby the osmotic pressure of the mixture is extracted from the density profiles of the inhomogeneous mixture in a gravitational field. In Monte Carlo simulations the field can be tuned to induce significant inhomogeneity, while keeping the density profiles sufficiently smooth for the macroscopic condition of hydrostatic equilibrium to remain applicable. The method is applied here for a simplified model of ideal but mutually avoiding polymers, which readily phase separate at relatively low densities. The Monte Carlo data are interpreted with the help of an approximate bulk phase diagram calculated from a simple, second-order virial coefficient theory. By derivation of effective potentials between polymer centers of mass, the binary mixture of polymers is coarse-grained to a "soft colloid" picture reminiscent of the Widom-Rowlinson model for incompatible atomic mixtures. This approach significantly speeds up the simulations and accurately reproduces the behavior of the full monomer resolved model.

Emplacement mechanism of the Great Whin and Midland Valley dolerite sills

The Great Whin and Midland Valley dolerite sill complexes appear to have been emplaced laterally from the walls of feeder dykes. The overall thickness of each sill increases with depth, as would be expected if the magma finally reached a state of hydrostatic equilibrium. Variations in the thickness of the sills with estimated intrusion depth imply that the head of magma was about 100 m below the contemporary ground surface in the areas of the present-day outcrops at the end of the intrusive episodes. Before hydrostatic equilibrium was established, the magma pressure would probably have been somewhat greater, so it is likely that intrusion of the sills was accompanied by the extrusion of flood basalts. Step-and-stair transgressions of the bedding are commonly found within the sills, mostly stepping downwards in the direction of bedding dip. The reason for this directionality is that the weight of sediments floating on an intruding sill has a downdip component that applies a tensile stress to intersecting fractures below the sill when magma is moving downdip, and to intersecting fractures above the sill when magma is moving updip.

Dynamical Friction and Cooling Flows in Galaxy Clusters

We investigate a model of galaxy clusters in which the hot intracluster gas is efficiently heated by dynamical friction (DF) of galaxies. We allow for both subsonic and supersonic motions of galaxies and use the gravitational drag formula in a gaseous medium presented by Ostriker (1999). The energy lost by the galaxies is either redistributed locally or into a Gaussian centered on the galaxy. We find that the condition of hydrostatic equilibrium and strict energy balance yields a trivial isothermal solution T_iso, independent of radius, or rising temperature distributions provided T_iso/gamma < T < T_iso, where gamma is the adiabatic index of the gas. The isothermal temperature corresponds to the usual scaling relation between the gas temperatures and the velocity dispersions of galaxies. However the minimal temperature associated with the rising solutions is ~ 0.5 T_vir, larger than that inferred from observations, the radial distribution of galaxy masses notwithstanding. Heating by supersonically moving galaxies cannot suppress thermal instability, although it can lengthen the growth time up to the level comparable to the ages of clusters when Mach number of galaxies is less than about two. We show using numerical hydrodynamic simulations that DF-induced heating is generally unable to produce stable equilibrium cores by evolving arbitrary non-equilibrium clusters, although it can lengthen the cooling time. We conclude that DF-induced heating alone is an unlikely solution to the cooling flow problem, although it can still be an important heat supplier, considerably delaying cooling catastrophe. We discuss other potential consequences of DF of galaxies in galaxy clusters.

Basic properties of anisotropic stellar wind expansion in the fluid approach

[1] Whereas in an isotropic temperature atmosphere both the hydrostatic equation and the momentum equation give the same conditions for hydrostatic equilibrium, in the anisotropic case the situation is ambiguous. It is found that for an anisotropic temperature the hydrostatic equilibrium conditions have to be deduced from the momentum equation. The condition for hydrostatic equilibrium is that both the parallel and the perpendicular temperatures, to the radial direction, decrease more rapidly than 1/r in the general case and that the perpendicular temperature decreases more rapidly than 1/r when the parallel temperature is constant. The momentum equation displays a transonic solution when at least one of the temperatures, parallel or perpendicular to the radial direction, decreases less rapidly than 1/r in the general case and when the perpendicular temperature decreases less rapidly than 1/r when the parallel temperature is constant. In an anisotropic atmosphere the parallel thermal velocity is the critical velocity. The properties of the transonic expansion in an isothermal and anisotropic atmosphere are studied. The initial velocity, the critical distance position, the terminal velocity, and the density profile are significantly different from the isotropic case. These properties are opposite with respect to the value of the anisotropy T⊥/T∥ > 1.0 and T⊥/T∥ 2.0. The extension of this study to multimoment anisotropic models can be useful for the interpretation of particle simulation of rarefied stellar atmosphere expansion.

HYDROSTATIC EQUILIBRIUM OF INSULAR, STATIC, SPHERICALLY SYMMETRIC, PERFECT FLUID SOLUTIONS IN GENERAL RELATIVITY

An analysis of insular solutions of Einstein's field equations for static, spherically symmetric, source mass, on the basis of exterior Schwarzschild solution is presented. Following the analysis, we demonstrate that the regular solutions governed by a self-bound (that is, the surface density does not vanish together with pressure) equation of state (EOS) or density variation cannot exist in the state of hydrostatic equilibrium, because the source mass which belongs to them, does not represent the "actual mass" appears in the exterior Schwarzschild solution. The only configuration which could exist in this regard is governed by the homogeneous density distribution (i.e. the interior Schwarzschild solution). Other structures which naturally fulfill the requirement of the source mass, set up by exterior Schwarzschild solution (and, therefore, can exist in hydrostatic equilibrium) are either governed by gravitationally-bound regular solutions (i.e. the surface density also vanishes together with pressure), or self-bound singular solutions (i.e. the pressure and density both become infinity at the centre).

A theory of the equilibrium figure and gravitational field of the Galilean satellite Io: The second approximation

We construct a theory of the equilibrium figure and gravitational field of the Galilean satellite Io to within terms of the second order in the small parameter α. We show that to describe all effects of the second approximation, the equation for the figure of the satellite must contain not only the components of the second spherical function, but also the components of the third and fourth spherical functions. The contribution of the third spherical function is determined by the Love number of the third order h3, whose model value is 1.6582. Measurements of the third-order gravitational moments could reveal the extent to which the hydrostatic equilibrium conditions are satisfied for Io. These conditions are J3=C32=0 and C31/C33=−6. We have calculated the corrections of the second order of smallness to the gravitational moments J2 and C22. We conclude that when modeling the internal structure of Io, it is better to use the observed value of k2 than the moment of inertia derived from k2. The corrections to the lengths of the semiaxes of the equilibrium figure of Io are all positive and equal to ∼64.5, ∼26, and ∼14 m for the a, b, and c axes, respectively. Our theory allows the parameters of the figure and the fourth-order gravitational moments that differ from zero to be calculated. For the homogeneous model, their values are:\(s_4 = \frac{{885}}{{224}}\alpha ^2 ,s_{42} = - \frac{{75}}{{224}}\alpha ^2 ,s_{44} = \frac{{15}}{{896}}\alpha ^2 ,J_4 = - \frac{{885}}{{224}}\alpha ^2 ,C_{42} = \frac{{75}}{{224}}\alpha ^2 ,C_{44} = \frac{{15}}{{896}}\alpha ^2 \).