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Unlike in Chapter 5, this project aims at finding a real mass density distribution of a hydrogen star of given mass. For that purpose an equilibrium condition for the gravitational and pressure-induced forces acting on a mass element is utilised. Using the integral form of Gauss’s law and the equation of state, we establish an integro-differential equation describing the mass density distribution. To numerically solve the integro-differential equation, we adapt the Adams–Bashforth method and implement a linear extrapolation based on known data points. This approach involves modelling the star as a gas under pressure using an exponential form for the equation of state, which helps in avoiding gravitational collapse. The equation of state is derived based on density functional theory data. We also discuss the constraints of this model and the significance of the parameters within it. The chapter concludes by suggesting potential numerical experiments to examine the influence of these parameters and their physical interpretation. This analysis aims to provide a more comprehensive understanding of stellar structure and the behaviour of mass density distribution within stars.
We complete the basic equations of cosmology by introducing the second Friedmann and continuity equations. We next introduce the equation of state parameter, and describe the evolution of the universe in the simplest cosmological models: matter-only, radiation-only, and lambda-only. We introduce the concept of a cosmological horizon, and explain how to calculate the age of the universe in a given model. Along the way, we establish the fiducial cosmological model -- a set of cosmological parameters that we will use for all our results in the remainder of the book. We end by introducing two observationally accessible distance measures -- luminosity distance and angular-diameter distance.
Equilibrium distribution functions are determined for fermions (e.g. electrons) and bosons (e.g. photons). The Saha–Boltzmann equation, the Maxwellian distribution, and relativistic Maxwell–Juttner distribution are derived. The relativistic equation of state for a distribution where particle velocities approach the speed of light is examined.
Stable neutron stars exist with masses approximately between one and two solar masses, and radii of approximately 10 to 11 km. The structure is determined primarily by a balance between gravitation and the repulsion between adjacent neutrons. The configuration depends on the equation of state of the neutron fluid. The rotation of the strong dipolar magnetic field generates a magnetosphere of charged particles, which co-rotates with the star.
Thanks to a rapid progress of high-power lasers since the birth of laser by T. H. Maiman in 1960, intense lasers have been developed mainly for studying the scientific feasibility of laser fusion. Inertial confinement fusion with an intense laser has attracted attention as a new future energy source after two oil crises in the 1970s and 1980s. From the beginning, the most challenging physics is known to be the hydrodynamic instability to realize the spherical implosion to achieve more than 1000 times the solid density. Many studies have been performed theoretically and experimentally on the hydrodynamic instability and resultant turbulent mixing of compressible fluids. During such activities in the laboratory, the explosion of supernova SN1987A was observed in the sky on 23 February 1987. The X-ray satellites have revealed that the hydrodynamic instability is a key issue to understand the physics of supernova explosion. After collaboration between laser plasma researchers and astrophysicists, the laboratory astrophysics with intense lasers was proposed and promoted around the end of the 1990s. The original subject was mainly related to hydrodynamic instabilities. However, after two decades of laboratory astrophysics research, we can now find a diversity of research topics. It has been demonstrated theoretically and experimentally that a variety of nonlinear physics of collisionless plasmas can be studied in laser ablation plasmas in the last decade. In the present paper, we shed light on the recent 10 topics studied intensively in laboratory experiments. A brief review is given by citing recent papers. Then, modeling cosmic-ray acceleration with lasers is reviewed in a following session as a special topic to be the future main topic in laboratory astrophysics research.
In this work we present experimental results on the behavior of diamond at megabar pressure. The experiment was performed using the PHELIX facility at GSI in Germany to launch a planar shock into solid multi-layered diamond samples. The target design allows shock velocity in diamond and in two metal layers to be measured as well as the free surface velocity after shock breakout. As diagnostics, we used two velocity interferometry systems for any reflector (VISARs). Our measurements show that for the pressures obtained in diamond (between 3 and 9 Mbar), the propagation of the shock induces a reflecting state of the material. Finally, the experimental results are compared with hydrodynamical simulations in which we used different equations of state, showing compatibility with dedicated SESAME tables for diamond.
Gravitational waves from coalescing neutron stars encode information about nuclear matter at extreme densities, inaccessible by laboratory experiments. The late inspiral is influenced by the presence of tides, which depend on the neutron star equation of state. Neutron star mergers are expected to often produce rapidly rotating remnant neutron stars that emit gravitational waves. These will provide clues to the extremely hot post-merger environment. This signature of nuclear matter in gravitational waves contains most information in the 2–4 kHz frequency band, which is outside of the most sensitive band of current detectors. We present the design concept and science case for a Neutron Star Extreme Matter Observatory (NEMO): a gravitational-wave interferometer optimised to study nuclear physics with merging neutron stars. The concept uses high-circulating laser power, quantum squeezing, and a detector topology specifically designed to achieve the high-frequency sensitivity necessary to probe nuclear matter using gravitational waves. Above 1 kHz, the proposed strain sensitivity is comparable to full third-generation detectors at a fraction of the cost. Such sensitivity changes expected event rates for detection of post-merger remnants from approximately one per few decades with two A+ detectors to a few per year and potentially allow for the first gravitational-wave observations of supernovae, isolated neutron stars, and other exotica.
The thermodynamics potentials for describing matter at nonzero temperatures and densities or chemical potentials are summarized. Emphasis is put on the thermodynamically correct description within the canonical and grand canonical ensemble for dense matter. The notion of chemical equilibrium is introduced for several conserved quantities and used to describe matter in β-equilibrium where charge and baryon number are conserved. The limit for nonrelativistic and relativistic particles is worked out in detail. The concept of an equation of state is introduced and applied to free Fermi gases. The pressure integral is solved analytically and the nonrelativistic and relativistic limits for the equation of state are delineated. Finally, the properties of polytropes are discussed and connected to the limiting cases of the equation of state of a free Fermi gas.
This self-contained introduction to compact star physics explains important concepts from areas such as general relativity, thermodynamics, statistical mechanics, and nuclear physics. Containing many tested exercises, and written by an international expert in the research field, the book provides important insights on the basic concepts of compact stars, discusses white dwarfs, neutron stars, quark stars and exotic compact stars. Included are sections on astrophysical observations of compact stars, and present and future terrestrial experiments related to compact stars physics, as the study of exotic nuclei and relativistic heavy-ion collisions. Major developments in the field such as the discovery of massive neutron stars, and a discussion of the recent gravitational wave measurement of a neutron star merger are also presented. This book is ideal for graduate students and researchers working on the physics of compact stars, general relativity and nuclear physics.
A detailed description of the properties of dense matter in extreme conditions, as those within Neutron Star cores, is still an open problem, whose solution is hampered by both the lack of empirical data, and by the difficulties in developing a suitable theoretical framework for the microscopic nuclear dynamics in such regimes.
We report here the results of a study aimed at inferring the properties of the repulsive three-nucleon interaction, driving the stiffness of the equation of state at high densities, by performing bayesian inference on current and future astrophysical observations.
The nuclear equation-of-state (EOS) describing newly formed proto-neutron stars (PNSs) in core-collapse supernovae (CCSNe) is yet uncertain, and varying its prescription affects multimessenger signatures in CCSN simulations. Focusing on the gravitational wave (GW) signal, we demonstrate the effect of varying parameter values in the EOS. We conclude that an especially important parameter is the effective mass of nucleons which affect thermal properties and subsequently the PNS compactness, regulating the GW signal in both amplitude and frequency. By radially decomposing the GW emission, we show where in the PNS the GWs originate from.
On August 17, 2017, the LIGO/VIRGO collaboration detected the first gravitational wave signal coming from the merger of two neutron stars. This groundbreaking discovery, referred to as GW170817, revealed to us how heavy elements, such as gold and platinum, are synthesized through a mechanism known as rapid neutron capture (r-process). In order to fully understand these signals, we need to simulate the resulting accretion disk around a black hole, and its outflows. This task requires efficient computing codes that include general relativity magnetohydrodynamics (GRMHD), neutrino physics, and a model for matter at high densities. We present the implementation of a tabulated equation of state that takes care of matter at high densities and a neutrino leakage scheme that considers the impact of neutrinos into HARM3D, a GRMHD parallelized code. We also apply the tools to a magnetized torus.
During the late stages of a neutron star binary inspiral finite-size effects come into play, with the tidal deformability of the supranuclear density matter leaving an imprint on the gravitational-wave signal. As demonstrated in the case of GW170817—the first direct detection of gravitational waves from a neutron star binary—this can lead to strong constraints on the neutron star equation of state. As detectors become more sensitive, effects which may have a smaller influence on the neutron star tidal deformability need to be taken into consideration. Dynamical effects, such as oscillation mode resonances triggered by the orbital motion, have been shown to contribute to the tidal deformability, especially close to the neutron star coalesence. We calculate the contribution of the various stellar oscillation modes to the tidal deformability and demonstrate the (anticipated) dominance of the fundamental mode, showing what the impact of the matter composition is on the tidal deformability.
The conclusion of Zaffiro et al. (2019; Constraints on the Equations of State of stiff anisotropic minerals: rutile, and the implications for rutile elastic barometry. Mineralogical Magazine, 83, 339–347) that the Mie–Grüneisen–Debye (MGD) Equation of State (EoS) cannot fit the available data for rutile is shown to be incorrect, even though rutile exhibits significant anisotropic thermal pressure which invalidates the quasi-harmonic approximation used as the basis for the MGD EoS. The refined parameters for the MGD EoS of rutile are: KTR0= 205.05(25) GPa, $K_{TR0}^{\prime} $ = 7.2(5), θD = 399(20) K, γ0= 1.40(2) and q = 1.5(7). This EoS predicts volumes, bulk moduli and volume thermal expansion coefficients for rutile at metamorphic conditions that are statistically indistinguishable from those predicted by the ‘isothermal’ type of EoS reported previously.
Compact objects are of great interest in astrophysical research. There are active research interests in understanding better various aspects of formation and evolution of these objects. In this paper we addressed some problems related to the compact objects mass limit. We employed Einstein field equations (EFEs) to derive the equation of state (EoS). With the assumption of high densities and low temperature of compact sources, the derived equation of state is reduced to polytropic kind. Studying the polytropic equations we obtained similar physical implications, in agreement to previous works. Using the latest version of Mathematica-11 in our numerical analysis, we also obtained similar results except slight differences in accuracy.
As internal energy is a function of entropy, volume and number of moles, its differential is given by the Gibbs relation, and temperature, pressure and chemical potentials are defined as conjugate variables. Extensivity implies the Euler relation. The Gibbs-Duhem relation will find applications later, in the analysis of phase transitions. Legendre transformations are introduced, leading to the definition of the thermodynamic potentials: free energy, enthalpy and Gibbs free energy. When a system is coupled to a thermal reservoir or heat bath, its equlibrium is characterised by a minimum of the free energy; when it is a work reservoir, the enthalpy is minimum, and when it is a work and heat reservoir, the Gibbs free energy is minimum. Maxwell relations establish relationships between quantities that would not immediately be associated. The cyclic chain rule links together the derivatives of one property function with respect to two others. It is conveniently applied to analyse the Joule expansion and Joule-Thomson effect.
We adopt two- and three-body nuclear forces derived at the next-to-next-to-leading-order in the framework of effective chiral perturbation theory to calculate the equation of state of β-stable neutron star matter using the Brueckner–Hartree–Fock many-body approach. We use the recent optimized chiral two-body nuclear interaction at next-to-next-to-leading-order derived by Ekström et al. and two different parametrizations of the three-body next-to-next-to-leading-order interaction: the first one is fixed to reproduce the saturation point of symmetric nuclear matter while the second one is fixed to reproduce the binding energies of light atomic nuclei. We show that in the second case the properties of nuclear matter are not well determined whereas in the first case various empirical nuclear matter properties around the saturation density are well reproduced. We finally calculate various neutron star properties and in particular the mass-radius and mass-central density relations. We find that the adopted interactions based on a fully microscopic framework, are able to provide an equation of state which is consistent with the present data of measured neutron star masses.
The equations of state of dense hydrous magnesium silicates (DHMS), determined from high-pressure single-crystal X-ray diffraction are reviewed, including hydroxylchondrodite, hydroxylclinohumite, phase A, phase B (anhydrous and hydrous), superhydrous phase B and phase E. The phases along the forsterite–brucite join, Mg2SiO4–Mg(OH)2, display near (increasing) linearity in compressibility with respect to water content and increasing bulk moduli (KT) with density. Such trends allow prediction of the as yet unknown bulk moduli of phases such as OH-Mg norbergite. The addition of water also reduces the bulk modulus of the B-phases and the anisotropy observed in axial compression. The alternating layers of octahedra and octahedra + tetrahedra completely control compression of the B phases, with the stacking direction becoming more compressible with addition of water. The enigmatic Phase E has the highest KT' yet measured for a hydrous silicate and one of the lowest KT. In contrast with other DHMS, Phase E is only slightly anisotropic in axial compression and we attribute this to the role of the intralayer cations in the structure. The degree of hydration and the vacancy concentration appear to be coupled in Phase E.
The equation of state of synthetic deuterated burtite, CaSn(OD)6, has been determined to 7.25 GPa at 298 K by synchrotron X-ray powder diffraction. Fitting to a third-order Birch-Murnaghan equation of state gives K0 = 44.7(9) GPa and K0′ = 5.3(4). A second-order fit gives K0 = 47.4(4) GPa. Within experimental error the two fits are indistinguishable over the pressure range studied. The decrease in the a parameter with pressure is smooth and no phase transitions were observed. Burtite is much more compressible (by a factor of three or four) than CaSnO3 and CdSnO3 perovskites, indicating that the absence of a cavity cation has a major effect upon the compressibility of the octahedral framework. Burtite is also markedly more compressible than the closely-related mineral stottite FeGe(OH)6 (K0 = 78 GPa). Their different compressibilities correlate with the relative compressibilities of stannate and germanate perovskites. Although different octahedral compressions are likely to be the primary reason for the different compressibilities of burtite and stottite, we also consider the possible secondary role of hydrogen-bonding topology in affecting the compressibilities of protonated octahedral frameworks. Burtite and stottite have different hydrogen-bonding topologies due to their different octahedral-tilt system. Burtite, space group Pn and tilt system a+a+a+, has a hydrogen-bonded network of linked four-membered rings of O-H…O linkages, whereas stottite, space group P42/n and tilt system a+a+c−, has <100> O-H…O crankshafts and isolated four-membered rings. These different hydrogen-bonded configurations lead to different bracing of the empty cavity sites by the O-H…O linkages and very different hydrogen-bonding connectivities in these two minerals that may also enhance the difference between their compressibilities.
An investigation of the relative stabilities and equations of state of possible Fe3S polymorphs was conducted using first-principles pseudopotential calculations. These calculations were based on density functional theory and performed using ultrasoft Vanderbilt pseudopotentials within the generalized gradient approximation. In accord with experiment, we found that the tetragonal Fe3P-type polymorph is the only stable phase along the 0 K isotherm as a function of pressure. Fe3S exhibits permanent magnetism at ambient conditions (Fei et al., 2000), but magnetism is suppressed by pressure and temperature, and therefore non-magnetic data are appropriate ones to use for modelling planetary interiors. For this reason, and because the Fe3P-type polymorph of Fe3S contains 32 atoms per unit cell it was impractical to incorporate magnetic properties into the simulations of this phase, we studied the behaviour of the non-magnetic phase. We obtained values of 250 GPa for the bulk modulus, K0, and 4.61 for its first derivative withrespect to pressure, K0′, by fitting a 3rd order Birch-Murnaghan equation of state to the calculated internal energy as a function of volume for the non-magnetic Fe3P-type Fe3S. This suggests that a pressure far greater than that expected in the Martian interior would be needed to achieve a density comparable to that of the Martian core. We therefore conclude that it is unlikely that the core of Mars contains significant amounts of solid Fe3S.