Fluid In Hydrostatic Equilibrium Research Articles

Constraints on axions from neutron star in HESS J1731-347

To constrain the allowed range for the axion decay constant fa or, equivalently, for the axion mass ma, we consider the cooling of a neutron star with strong proton superfluidity and normal (non-superfluid) neutrons inside its core and without strong magnetic field, by analogy with the observed supernova remnant in HESS J1731-347. For this specific case, we demonstrate that after the thermal relaxation is over, the hydrostatic structure of the neutron star can be well described with the aid of solution of Einstein field equations, applied to a sphere of fluid in hydrostatic equilibrium, derived by Tolman. The internal temperature of the neutron star is calculated assuming that the cooling occurs dominantly due to production of neutrino pairs and axions in the nn-bremsstrahlung. To impose a constraint to the axion decay constant the fact is used that the currently observed neutron star surface temperature does not deviate from the neutrino cooling scenario. For the KSVZ-axion model we find that fa>1. 9× 108 GeV, while for the DFSZ-axion model we obtain fa>4. 7× 109 GeV.

Haumea’s Shape, Composition, and Internal Structure

We have calculated the figure of equilibrium of a rapidly rotating, differentiated body to determine the shape, structure, and composition of the dwarf planet Haumea. Previous studies of Haumea's light curve have suggested Haumea is a uniform triaxial ellipsoid consistent with a Jacobi ellipsoid with axes $\approx 960 \times 774 \times 513$ km, and bulk density $\approx 2600 \, {\rm kg} \, {\rm m}^{-3}$. In contrast, observations of a recent stellar occultation by Haumea indicate its axes are $\approx 1161 \times 852 \times 523$ km and its bulk density $\approx 1885 \, {\rm kg} \, {\rm m}^{-3}$; these results suggest that Haumea cannot be a fluid in hydrostatic equilibrium and must be partially supported by interparticle forces. We have written a code to reconcile these contradictory results and to determine if Haumea is in fact a fluid in hydrostatic equilibrium. The code calculates the equilibrium shape, density, and ice crust thickness of a differentiated Haumea after imposing (semi-) axes lengths $a$ and $b$. We find Haumea is consistent with a differentiated triaxial ellipsoid fluid in hydrostatic equilibrium with axes of best fit $a$ = 1050 km, $b$ = 840 km, and $c$ = 537 km. This solution for Haumea has $\rho_{\rm avg} = 2018 \, {\rm kg} \, {\rm m}^{-3}$, $\rho_{\rm core} = 2680 \, {\rm kg} \, {\rm m}^{-3}$, and core axes $a_{\rm c} = 883$ km, $b_{\rm c} = 723$ km, and $c_{\rm c}=470$ km, which equates to an ice mantle comprising $\sim 17\%$ of Haumea's volume and ranging from 67 to 167 km in thickness. The thick ice crust we infer allows for Haumea's collisional family to represent only a small fraction of Haumea's pre-collisional ice crust. For a wide range of parameters, the core density we calculate for Haumea suggests that today the core is composed of hydrated silicates and likely underwent serpentinization in the past.

Field theory of the correlation function of mass density fluctuations for self-gravitating systems

The mass density distribution of Newtonian self-gravitating systems is studied analytically in the field theoretical method. Modeling the system as a fluid in hydrostatic equilibrium, we apply Schwinger’s functional derivative on the average of the field equation of mass density, and obtain the field equation of 2-point correlation function ξ(r) of the mass density fluctuation, which includes the next order of nonlinearity beyond the Gaussian approximation. The 3-point correlation occurs hierarchically in the equation, and is cut off by the Groth-Peebles ansatz, making it closed. We perform renormalization and write the equation with three nonlinear coefficients. The equation tells us that ξ depends on the point mass m and the Jeans wavelength scale λ0, which are different for galaxies and clusters. Applying this to large scale structures, it predicts that the profile of ξcc for clusters is similar to ξgg for galaxies but with a higher amplitude, and that the correlation length increases with the mean separation between clusters, i.e., a scaling behavior r0 ≃ 0.4d. The solution yields the galaxy correlation ξgg(r) ≃ (r0/r)1.7 valid only in a range 1 < r < 10 h−1 Mpc. At larger scales the solution ξgg deviates below the power law and goes to zero around ∼50 h−1 Mpc, just as the observations show. We also derive the field equation of the 3-point correlation function in the Gaussian approximation and its analytical solution, for which the Groth-Peebles ansatz with Q = 1 holds.

Galactic Constraints on Fermionic Dark Matter

In order to explain Galactic structures, a self-gravitating system composed of massive fermions in spherical symmetry is considered. The finite mass distribution of such a component is obtained after solving the Einstein equation for a thermal and semi-degenerate fermionic gas, described by a perfect fluid in hydrostatic equilibrium and exposed to cutoff effects (e.g. evaporation). Within this more general approach a family of density profiles arises, which explains dark matter halo constraints of the Galaxy and provides at the same time an alternative to the central black hole scenario in Sgr A*. This analysis narrows the allowed particle mass to mc2 = 48−345 keV.

Does space-time torsion determine the minimum mass of gravitating particles?

We derive upper and lower limits for the mass–radius ratio of spin-fluid spheres in Einstein–Cartan theory, with matter satisfying a linear barotropic equation of state, and in the presence of a cosmological constant. Adopting a spherically symmetric interior geometry, we obtain the generalized continuity and Tolman–Oppenheimer–Volkoff equations for a Weyssenhoff spin fluid in hydrostatic equilibrium, expressed in terms of the effective mass, density and pressure, all of which contain additional contributions from the spin. The generalized Buchdahl inequality, which remains valid at any point in the interior, is obtained, and general theoretical limits for the maximum and minimum mass–radius ratios are derived. As an application of our results we obtain gravitational red shift bounds for compact spin-fluid objects, which may (in principle) be used for observational tests of Einstein–Cartan theory in an astrophysical context. We also briefly consider applications of the torsion-induced minimum mass to the spin-generalized strong gravity model for baryons/mesons, and show that the existence of quantum spin imposes a lower bound for spinning particles, which almost exactly reproduces the electron mass.

A new equilibrium torus solution and GRMHD initial conditions

General relativistic magnetohydrodynamic (GRMHD) simulations are providing influential models for black hole spin measurements, gamma ray bursts, and supermassive black hole feedback. Many of these simulations use the same initial condition: a rotating torus of fluid in hydrostatic equilibrium. A persistent concern is that simulation results sometimes depend on arbitrary features of the initial torus. For example, the Bernoulli parameter (which is related to outflows), appears to be controlled by the Bernoulli parameter of the initial torus. In this paper, we give a new equilibrium torus solution and describe two applications for the future. First, it can be used as a more physical initial condition for GRMHD simulations than earlier torus solutions. Second, it can be used in conjunction with earlier torus solutions to isolate the simulation results that depend on initial conditions. We assume axisymmetry, an ideal gas equation of state, constant entropy, and ignore self-gravity. We fix an angular momentum distribution and solve the relativistic Euler equations in the Kerr metric. The Bernoulli parameter, rotation rate, and geometrical thickness of the torus can be adjusted independently. Our torus tends to be more bound and have a larger radial extent than earlier torus solutions. While this paper was in preparation, several GRMHD simulations appeared based on our equilibrium torus. We believe it will continue to provide a more realistic starting point for future simulations.

Self-force as probe of internal structure

The self-force acting on a (scalar or electric) charge held in place outside a massive body contains information about the body’s composition, and can therefore be used as a probe of internal structure. We explore this theme by computing the (scalar or electromagnetic) self-force when the body is a spherical ball of perfect fluid in hydrostatic equilibrium, under the assumption that its rest-mass density and pressure are related by a polytropic equation of state. The body is strongly self-gravitating, and all computations are performed in exact general relativity. The dependence on internal structure is best revealed by expanding the self-force in powers of r−10, with r0 denoting the radial position of the charge outside the body. To the leading order, the self-force scales as r−30 and depends only on the square of the charge and the body’s mass; the leading self-force is universal. The dependence on internal structure is seen at the next order, r−50, through a structure factor that depends on the equation of state. We compute this structure factor for relativistic polytropes, and show that for a fixed mass, it increases linearly with the body’s radius in the case of the scalar self-force, and quadratically with the body’s radius in the case of the electromagnetic self-force. In both cases we find that for a fixed mass and radius, the self-force is smaller if the body is more centrally dense, and larger if the mass density is more uniformly distributed.

Dynamical analysis of continental overflow

Recent field and laboratory studies indicate that the viscosity of the middle and lower continental crust may be 10 18 Pa s or less in regions of high heat flow. Such strikingly low viscosities must strongly facilitate gravitational collapse of overthickened crust (thicker than would be possible without lateral tectonic support). In such areas, the ductile zone is so weak that lateral flow in it can relax all lateral pressure gradients in a geologically short time. The ductile or “fluid” zone is, for geological purposes, a fluid in hydrostatic equilibrium, floating a brittle cap of upper crust above it. A continent built in this way can collapse by overflow onto adjacent ocean basins if the continental elevation exceeds a critical threshold related to the mean geothermal gradient. This model permits relatively simple identification of the terms of the energy budget of gravitational collapse. Here I show that the budget is likely to be dominated by the balance between liberated gravitational energy (a source) and normal fault friction (a sink). Examination of this budget provides a stability criterion for gravitational collapse by continental overflow onto ocean basins: that crustal elevation above the ocean basin beyond an amount between a third and a half of the brittle cap thickness should drive a propagating overflow front.

Tropical waves

On a flat equatorial beta plane the equations of fluid motion and continuity have low‐frequency wavelike solutions that are separable into vertical and horizontal modes. The horizontal modal structure is the same in the case of a compressible fluid in hydrostatic equilibrium (the usual meteorological case) and in the case of an incompressible fluid under the Boussinesq approximation (the usual oceanographic case), but the vertical modal structure, normally expressed in terms of an equivalent depth, is different in the two cases. The theory of low‐frequency tropical waves (excluding tides) is used to highlight the similarities and the disparities in the oceanographic and meteorological approaches to them. This is followed by a brief review of the observational evidence for the existence of these low‐frequency waves in the oceans, atmosphere, and ionosphere of the tropics.

Laminar natural convection about an isothermally heated sphere at small Grashof number

The flow induced by gravity about a very small heated isothermal sphere introduced into a fluid in hydrostatic equilibrium is studied. The natural-convection flow is taken to be steady and laminar. The conditions under which the Boussinesq model is a good approximation to the full conservation laws are described. For a concentric finite cold outer sphere with radius, in ratio to the heated sphere radius, roughly less than the Grashof number to the minus one-half power, a recirculating flow occurs; fluid rises near the inner sphere and falls near the outer sphere. For a small heated sphere in an unbounded medium an ordinary perturbation expansion essentially in the Grashof number leads to unbounded velocities far from the sphere; this singularity is the natural-convection analogue of the Whitehead paradox arising in three-dimensional low-Reynolds-number forced-convection flows. Inner-and-outer matched asymptotic expansions reveal the importance of convective transport away from the sphere, although diffusive transport is dominant near the sphere. Approximate solution is given to the nonlinear outer equations, first by seeking a similarity solution (in paraboloidal co-ordinates) for a point heat source valid far from the point source, and then by linearization in the manner of Oseen. The Oseen solution is matched to the inner diffusive solution. Both outer solutions describe a paraboloidal wake above the sphere within which the enthalpy decays slowly relative to the rapid decay outside the wake. The updraft above the sphere is reduced from unbounded growth with distance from the sphere to constant magnitude by restoration of the convective accelerations. Finally, the role of vertical stratification of the ambient density in eventually stagnating updrafts predicted on the basis of a constant-density atmosphere is discussed.