General Relativistic Stars : Polytropic Equations of State

In this paper, the gravitational field equations for static spherically symmetric perfect fluid models with a polytropic equation of state, $p=k\rho^{1+1/n}$, are recast into two complementary 3-dimensional {\it regular} systems of ordinary differential equations on compact state spaces. The systems are analyzed numerically and qualitatively, using the theory of dynamical systems. Certain key solutions are shown to form building blocks which, to a large extent, determine the remaining solution structure. In one formulation, there exists a monotone function that forces the general relativistic solutions towards a part of the boundary of the state space that corresponds to the low pressure limit. The solutions on this boundary describe Newtonian models and thus the relationship to the Newtonian solution space is clearly displayed. It is numerically demonstrated that general relativistic models have finite radii when the polytropic index $n$ satisfies $0\leq n \lesssim 3.339$ and infinite radii when $n\geq 5$. When $3.339\lesssim n<5$, there exists a 1-parameter set of models with finite radii and a finite number, depending on $n$, with infinite radii.Comment: 31 pages, 10 figure

Spectral Representations of Neutron-Star Equations of State

Methods are developed for constructing spectral representations of cold (barotropic) neutron-star equations of state. These representations are faithful in the sense that every physical equation of state has a representation of this type, and conversely every such representation satisfies the minimal thermodynamic stability criteria required of any physical equation of state. These spectral representations are also efficient, in the sense that only a few spectral coefficients are generally required to represent neutron-star equations of state quiet accurately. This accuracy and efficiency is illustrated by constructing spectral fits to a large collection of "realistic" neutron-star equations of state.Comment: 10 pages, 8 figures, 2 tables; updated to accepted Phys. Rev. D versio

Nonlinear integral equations for finite volume excited state energies of the O(3) and O(4) nonlinear sigma-models

We propose nonlinear integral equations for the finite volume one-particle energies in the O(3) and O(4) nonlinear sigma-models. The equations are written in terms of a finite number of components and are therefore easier to solve numerically than the infinite component excited state TBA equations proposed earlier. Results of numerical calculations based on the nonlinear integral equations and the excited state TBA equations agree within numerical precision.Comment: numerical results adde

Injection-suction control for Navier-Stokes equations with slippage

We consider a velocity tracking problem for the Navier-Stokes equations in a 2D-bounded domain. The control acts on the boundary through a injection-suction device and the flow is allowed to slip against the surface wall. We study the well-posedness of the state equations, linearized state equations and adjoint equations. In addition, we show the existence of an optimal solution and establish the first order optimality condition.Comment: 23 page

Perfect fluids and generic spacelike singularities

We present the conformally 1+3 Hubble-normalized field equations together with the general total source equations, and then specialize to a source that consists of perfect fluids with general barotropic equations of state. Motivating, formulating, and assuming certain conjectures, we derive results about how the properties of fluids (equations of state, momenta, angular momenta) and generic spacelike singularities affect each other.Comment: Considerable changes have been made in presentation and arguments, resulting in sharper conclusion

Residual equilibrium schemes for time dependent partial differential equations

Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In this paper, inspired by micro-macro decomposition methods for kinetic equations, we present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent partial differential equations including nonlinear diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed.Comment: 23 pages, 12 figure

Nucleon self-energies for supernova equations of state

Nucleon self-energies and interaction potentials in supernova (SN) matter, which are known to have an important effect on nucleosynthesis conditions in SN ejecta are investigated. Corresponding weak charged-current interaction rates with unbound nucleons that are consistent with existing SN equations of state (EOSs) are specified. The nucleon self-energies are made available online as electronic tables. The discussion is mostly restricted to relativistic mean-field models. In the first part of the article, the generic properties of this class of models at finite temperature and asymmetry are studied. It is found that the quadratic expansion of the EOS in terms of asymmetry works reasonably well at finite temperatures and deviations originate mostly from the kinetic part. The interaction part of the symmetry energy is found to be almost temperature independent. At low densities, the account of realistic nucleon masses requires the introduction of a linear term in the expansion. Finally, it is shown that the important neutron-to-proton potential difference is given approximately by the asymmetry of the system and the interaction part of the zero-temperature symmetry energy. The results of different interactions are then compared with constraints from nuclear experiments and thereby the possible range of the potential difference is limited. In the second part, for a certain class of SN EOS models, the formation of nuclei is considered. Only moderate modifications are found for the self-energies of unbound nucleons that enter the weak charged-current interaction rates. This is because in the present approach the binding energies of bound states do not contribute to the single-particle energies of unbound nucleons.Comment: 25 pages, 12 figures, v3: editorial corrections, matches published versio

Determination of the Equation of State of Dense Matter

Nuclear collisions can compress nuclear matter to densities achieved within neutron stars and within core-collapse supernovae. These dense states of matter exist momentarily before expanding. We analyzed the flow of matter to extract pressures in excess of 10^34 pascals, the highest recorded under laboratory-controlled conditions. Using these analyses, we rule out strongly repulsive nuclear equations of state from relativistic mean field theory and weakly repulsive equations of state with phase transitions at densities less than three times that of stable nuclei, but not equations of state softened at higher densities because of a transformation to quark matter.Comment: 26 pages, 6 figures; final versio