Professor
Ohio University, Department of Physics and Astronomy
Born and raised in idyllic Mysore, India, Madappa Prakash earned his Bachelor's and Master's degrees at the University of Mysore
and his Ph. D. degree in physics from the University of Bombay (1979), while he was employed as a scientific officer at the Bhabha
Atomic Research Center, Bombay (1974-1981). His post-doctoral studies were conducted at the Niels Bohr Institute, Copenhagen,
Denmark (1979-81) and the State University of New York at Stony Brook, Stony Brook, New York (1982-86), where he taught and engaged
in research as a research professor and principal research scientist untill 2005. Prakash joined Ohio University in 2005 as a full
professor, in part to be the ``glue'' between the different research areas pursued by the Institute of Nuclear and Particle Physics
and the Astrophysics Institute under the sponsorship of Ohio University's Research Priority Program: ``Structure of the Universe:
From Quarks to Super-Clusters''.
Prakash has a broad range of interests that include nuclear and
particle physics, the physics of high energy heavy-ion collisions, and
astrophysics. Beginning with the nuclear fission process in his
Ph.D. thesis, his research has covered many of the myriad phenomena
that occur in nuclei and in nuclear matter under extreme conditions of
density, temperature, and magnetic fields. His work has illuminated
the role of the equation of state of dense matter in nuclear
collisions from low to very high energies and the structure of neutron
stars. Investigations of strangeness-bearing matter led him to
discover new pathways to form stellar mass black holes from metastable
neutron stars. His work includes the evolution of neutron stars from
their birth to old age, neutrino interactions and propagation in dense
astrophysical systems, and the theoretical interpretations of the
growing number of observations of neutron stars. He develops equations
of state, neutrino opacities, and transport characteristics in dense
matter for studies of supernovae and binary mergers involving neutron
stars and black holes.
In 2001, Prakash was elected a Fellow of the American Physical
Society (Division of Nuclear Physics) `` For fundamental research into
the properties of hot and dense matter, providing a basis for
understanding relativistic heavy ion collisions and the structure and
composition of neutron stars''.
Prakash loves teaching. ``Classrooms are fun, especially as there are
so many young people who want to learn. Students, with their
questions, answers and doubts, make me learn more and more. It is a
two-way street.'' Devising new projects that suit the interests and
proclivities of students, be they high school students or
undergraduate or graduate students, is a challenge he loves to
embrace. Whether with students or other professionals, getting along
on a very basic level of mutual liking and respect is very important
to Prakash.
When not involved in science, Prakash's obsessions include inventing
match-stick games (personal peculiarity), cricket (what a game!),
reading (what a pleasure!), gardening (what a joy!), hiking (nature's
fault for being so!) and films (why isn't there more time? oh, well!).
Prakash's favorite quote outside of science: ``We must become the
change we want to see'', by Mohandas Karamchand Gandhi.
Ohio University, Department of Physics and Astronomy
SUNNY at Stony Brook, Department of Physics
SUNNY at Stony Brook, Department of Physics
SUNY at Stony Brook, Department of Physics
SUNY at Stony Brook, Department of Physics
The Niels Bohr Institute, Copenhagen, Denmark
Bhabha Atomic Research Centre, Bombay, India
Ph.D. in Physics
University of Bombay, India
Master of Science
University of Mysore, India
Bachelor of Science in Physics
University of Mysore, India
The major areas of Prakash′s research are: (1) the interface between nuclear theory and nuclear, neutrino and gravitational astrophysics, and (2) the extreme energy density physics encountered in the collisions of highly energetic nuclei as at the relativistic heavy-ion colliders. Questions addressed in connection with astrophysical phenomena include: (i) How do new theoretical calculations of the equation of state and neutrino processes in dense matter impact numerical simulations of supernovae, proto-neutron stars and cooling neutron stars? (ii) How do astronomical observations of supernovae, neutron stars (including pulsars) and black holes delineate dense matter properties, such as its symmetry energy, specific heat and compressibility? Do exotic phases that contain hyperons, Bose condensates or deconfined quark matter exist in observable dense matter? Do they have distinct signatures? (iii) How do nuclear experiments, such as those involving rare isotope accelerators, heavy ion collisions, and parity-violating electron scattering reactions, restrict the parameters of nuclear equation of state models? (iv) How do gravitational wave detections elucidate the properties of dense baryonic matter? In the area of relativistic heavy-ion collisions (RHIC and LHC), Prakash addresses issues related with (i) How do fundamental interactions between quarks and hadrons determine their thermal and transport properties? How are these properties manisfested in observables of heavy-ion collisions? Prakash works in concert with the Joint Institute for Nuclear Astrophysics -- Center for the Evolution of the Elements, or JINA-CEE, of which the INPP is an affiliate member. His role in this collaboration is to provide microphysical inputs (to large-scale computer simulations of astrophysical phenomena) on the equation of state of and neutrino reaction rates in the dense matter encountered in core collapse supernovae, young and old neutron stars, and, binary mergers involving neutron stars and black holes.
Equation of state of stellar (supernova and neutron star) matter
Neutron stars: Composition, structure and evolution
Rotational properties
Short and long term cooling of neutron stars
Multi-wavelength observations of neutron stars
Superfluidity in dense matter
Inferences from experimental constraints
Neutrino interactions in hot and dense matter
Neutrino signals from stars and supernovae
Relic neutrinos and cosmology
X-ray and gamma-ray bursters
Binary star mergers - Gravity wave detections
Intense magnetic fields in the universe: Applications to neutron stars, supernovae, and binary mergers
QCD at high temperature and high density
QCD sum rules
Nuclear processes in relation to QCD
Transparency in nuclear exclusive processes
Effective field theories of hadrons
Many-body theories of hadronic matter
Possible pion, kaon and quark-hadron phase transitions
Experimental consequences and signatures
Inferences from nuclear collisions
Nuclear properties in their ground & excited states
Nuclei far off stability
Nuclei under extreme conditions of rotation and entropy
Large amplitude collective phenomena
Theoretical framework to describe RHIC
Consequences of deconfinement and/or chiral phase transitions
Statistical mechanics of hadrons, quarks and gluons
Equation of state and transport properties
Spacetime simulations: Hydrodynamics and sequential scattering
Diagnostics of dynamics: Hadrons and electromagnetic probes
We present a thermodynamically consistent method by which equations of state based on nonrelativistic potential models can be modified so that they respect causality at high densities, both at zero and finite temperature (entropy). We illustrate the application of the method by using the high-density phase parametrization of the well-known Akmal–Pandharipande–Ravenhall model in its pure neutron matter configuration as an example. We also show that, for models with only contact interactions, the adiabatic speed of sound is independent of the temperature in the limit of very large temperature. This feature is approximately valid for models with finite-range interactions as well, insofar as the temperature dependence they introduce to the Landau effective mass is weak. In addition, our study reveals that in first-principle nonrelativistic models of hot and dense matter, contributions from higher-than-two-body interactions must be screened at high density to preserve causality.
The formalism of next-to-leading order Fermi Liquid Theory is employed to calculate the thermal properties of symmetric nuclear and pure neutron matter in a relativistic many-body theory beyond the mean field level which includes two-loop effects. For all thermal variables, the semi-analytical next-to-leading order corrections reproduce results of the exact numerical calculations for entropies per baryon up to 2. This corresponds to excellent agreement down to sub-nuclear densities for temperatures up to 20 MeV. In addition to providing physical insights, a rapid evaluation of the equation of state in the homogeneous phase of hot and dense matter is achieved through the use of the zero-temperature Landau effective mass function and its derivatives.
Analytical formulas for next-to-leading order temperature corrections to the thermal state variables of interacting nucleons in bulk matter are derived in the degenerate limit. The formalism developed is applicable to a wide class of non-relativistic and relativistic models of hot and dense matter currently used in nuclear physics and astrophysics (supernovae, proto-neutron stars and neutron star mergers) as well as in condensed matter physics. We consider the general case of arbitrary dimensionality of momentum space and an arbitrary degree of relativity (for relativistic mean-field theoretical models). For non-relativistic zero-range interactions, knowledge of the Landau effective mass suffices to compute next-to-leading order effects, but in the case of finite-range interactions, momentum derivatives of the Landau effective mass function up to second order are required. Numerical computations are performed to compare results from our analytical formulas with the exact results for zero- and finite-range potential and relativistic mean-field theoretical models. In all cases, inclusion of next-to-leading order temperature effects substantially extends the ranges of partial degeneracy for which the analytical treatment remains valid.
It is found that for level schemes obtained from a folded Yukawa potential, the Strutinsky smearing procedure for the evaluation of the shell correction to the total potential energy of nuclei does not lead to a unique value for nuclear shapes near and beyond the outer fission barrier deformations and consequently introduces uncertainties in the relative fission barrier heights.
The exact penetrability for an inverted biharmonic oscillator potential is calculated and compared with that obtained from the WKB method.
An earlier penetrability calculation through a triple-humped barrier for ^{234}Th nucleus by Sharma and Leboeuf failed to reproduce the right order of magnitude for penetrability. It is shown here that one can obtain the correct penetrability by assuming that Back et al. actually determined the parameters of the second saddle, third minimum, and the third saddle for ^{234}Th from an analysis of their (t, pf) fission data. The height of the first barrier is assumed to be that given by microscopic calculations for this nucleus. NUCLEAR REACTIONS, FISSION Fission penetrability calculated through triple-humped potential barrier in ^{234}Th, quantitative resolution of thorium anomaly along the lines suggested by Möller and Nix.
Exact penetrability calculations through a double-humped barrier, a portion of which is taken to be biharmonic in nature, have been performed. It is shown that a family of fission barrier shapes exist which give roughly the same penetrability as that through a double-humped barrier made up of harmonic wells and barriers. This leads to an uncertainty in the position of the barrier extrema as determined from an analysis of the experimental data on fission probability and angular distributions.
Nuclear matter overlap integrals between two nuclei are calculated for density distributions generated by folding a short range function over arbitrarily shaped sharp surfaces of the two nuclei. Explicit formulas are derived for two choices of the folding function, viz., a Gaussian and a Yukawa function. An approximation scheme is given to facilitate a quick evaluation of the overlap integrals and is found to give results which are in good agreement with the exact results over a wide range of masses and spheroidal deformations of the two nuclei.
NUCLEAR REACTIONS, Heavy-ion reactions, fission. Calculation of overlap integrals between arbitrarily shaped diffuse surface nuclei, Yukawa and Gaussian folding functions, approximation scheme for spheroidal nuclei, comparison of exact and approximate results.
Effect of friction on the post-scission motion of fission fragments is studied. It is shown that, with values of the frictional constant in the neighbourhood of those obtained from heavy-ion reaction data, no significant pre-scission kinetic energy can exist.
It is shown that no definite conclusions can be drawn regarding the magnitude of energy dissipation in fission during the descent from the saddle point, by a comparison of the calculated deformation and interaction energies of fission fragments near scission, with the experimental fragment excitation and kinetic energies. This is contrary to the conclusions drawn by Schultheis and Schultheis on the basis of a similar analysis. The reasons for this discrepancy are also discussed.
NUCLEAR REACTIONS, FISSION Bounds on energy dissipation, comparison of calculated scission energy and experimental post-scission energies, calculated results for ^{252}Cf (sf).
We derive analytic expressions for the Wigner transform (WT) of the one-body density matrix of a system of independent fermions in an N-dimensional harmonic oscillator (HO) potential. Considerable simplification is obtained for closed-shell configurations (all degenerate states occupied), when the Wigner transform (WT) depends only on the energy. Illustrative examples for one- and three-dimensional cases are discussed.
We present results of microscopic calculations of the drift and diffusion coefficients in deep inelastic heavy ion collisions. Assuming the two nuclei to interact weakly, we make use of first-order perturbation theory (Golden Rule) for nucleon transfer and take into account the Pauli blocking effect. The nucleon transfer probability from one nucleus to the other is obtained using single particle wave functions. The results obtained are consistent with empirical estimates.
The nucleon exchange process in deep-inelastic collisions is investigated by studying the one-way current constructed using the Wigner transform of the time-dependent Hartree-Fock one-body density matrix. Detailed calculations for head-on ^{16}O + ^{16}O collisions at three different bombarding energies have been performed. A combination of two simple models for the contributions from relative motion and tunneling is shown to predict one-way currents close to the time-dependent Hartree-Fock results.
Using smoothed single-particle occupation numbers, the contribution of shell and surface effects to the static Wigner phase-space distribution function for non-interacting nucleons in an external, spherical, harmonic oscillator potential well is determined. The smoothing procedure illustrates clearly the strong quantum oscillations of the distributions function around the constant value for an infinite Fermi gas, and allows the separation of the contributions arising from shell effects connected with the structure of the single-particle level spectrum near the Fermi surface, and surface effects associated with single reflections of nucleons from the surface of the potential well. The temperature dependence of these effects in the distribution function is also investigated and a thermodynamical definition of the shell contribution to the kinetic energy density is suggested.
Prompt emission of nucleons is observed in time-dependent Hartree-Fock calculations of ^{16}O + ^{93}Nb collisions. These nucleons are emitted in the forward direction in about 2×10^{-22}s after the contact of the ions, with a mean velocity somewhat less than the beam velocity. At E(lab)=204 MeV, the estimated total nucleon cross section is ∼230 mb. For central collisions a lower energy threshold of 130 MeV is established. Force dependence of the results is studied in detail.
NUCLEAR REACTIONS ^{16}O(^{93}Nb,x) in time-dependent Hartree-Fock approximation. Energy and angular momentum dependence and effect of nuclear interaction on preequilibrium emission of particles.
We study the flow of matter, momentum and energy in low-energy heavy-ion collisions. This is done by using the Wigner phase space distribution function to calculate quantum mechanical analogs of the classical distributions of the observables. We apply the Wigner transformation to time-dependent Hartree-Fock calculations of ^{16}O + ^{16}O and ^{40}Ca + ^{40}Ca reactions. Both static and dynamic features of the distribution functions are demonstrated. We find a significant energy flow perpendicular to the beam direction. The one-way matter current obtained from time-dependent Hartree-Fock calculations is reproduced well by a combination of two simplified models for the relative motion and tunneling.
The influence of momentum and frequency dependence of the nuclear mean field ∑ on the level density parameter a is studied. By introducing a local quasiparticle effective mass we determine the volume, surface and curvature coefficients of a, expressed as a = a_{V}A + a_{S}A^{2/3} + a_{c}A^{1/3}. It is found that the magnitude of a_{C} is significant due to the very nature of the surface. A purely momentum dependent field ∑ determined from the nonlocality of the optical potential gives a level density close to a self-consistent Thomas-Fermi description with Skyrme III forces. An explicit frequency dependence of ∑ mainly influences a_{S}, enhancing it considerably for medium and heavy nuclei.
Analytical formulas for next-to-leading order temperature corrections to the thermal state variables of interacting nucleons in bulk matter are derived in the degenerate limit. The formalism developed is applicable to a wide class of non-relativistic and relativistic models of hot and dense matter currently used in nuclear physics and astrophysics (supernovae, proto-neutron stars and neutron star mergers) as well as in condensed matter physics. We consider the general case of arbitrary dimensionality of momentum space and an arbitrary degree of relativity (for relativistic mean-field theoretical models). For non-relativistic zero-range interactions, knowledge of the Landau effective mass suffices to compute next-to-leading order effects, but in the case of finite-range interactions, momentum derivatives of the Landau effective mass function up to second order are required. Numerical computations are performed to compare results from our analytical formulas with the exact results for zero- and finite-range potential and relativistic mean-field theoretical models. In all cases, inclusion of next-to-leading order temperature effects substantially extends the ranges of partial degeneracy for which the analytical treatment remains valid.
Current studies of heavy-ion reactions make extensive use of fission-fragment angular distributions to prove the limits of statistical equilibrium. The "standard" theory was derived for a rigid-rotor, transition-state nucleus. Here a more complete theory is developed by inclusion of the dependence of transition-state shape on orientation of the symmetry axis. It is concluded that even this improved application of the liquid-drop model cannot give an adequate prediction of the Z^{2}/A dependence of fission-fragment anisotropies.
A unified approach is used to describe quasielastic and fusion processes in heavy-ion reactions at near barrier energies. The spin distributions for fusion are used in conjunction with transition-state theory to calculate fission fragment angular distributions. Results for the reaction ^{16}O + ^{208}Pb show that when transfer contributions are small, satisfactory agreement with data is obtained. The role of fragment anisotropy as a useful probe of entrance channel effects at energies close to the barrier is pointed out.
We use the extended Thomas-Fermi approximation and Skyrme-type interactions to describe the energy density of a semi-infinite slab of neutron-rich nuclear matter at zero temperature. We allow for the existence of a drip phase at low proton fractions in addition to the more dense nuclear phase. We determine various bulk properties of both phases when the system is in equilibrium. We extend the usual definition of the surface energy to apply to the case where drip is present. Assuming the density profile has the form of a Fermi function to a power, we perform a constrained variational calculation to determine the parameters of the density profile. The surface and curvature energies are calculated for proton fractions ranging from 0.5 (symmetric nuclear matter) to 0 (pure neutron matter) for typical Skyrme-type interactions. We find significantly different asymmetry dependences for different interactions. For proton fractions close to 0.5, our results are in close agreement with the predictions of the droplet model. We also present results of calculations for fission barrier properties and phase transitions between nuclei and bubbles to highlight the role of surface and curvature energies in the neutron-rich regime.
The asymmetry dependence of the incompressibility at fixed density, the density at fixed pressure, and the isobaric incompressibility are studied for neutron-rich nuclear matter at arbitrary densities. Simple expressions valid to all orders in density and to second order in asymmetry are derived involving the density derivatives of the Landau parameters of the symmetric system. Illustrative calculations are performed with different equations of state.
Pion production cross sections in heavy ion-induced reactions at energies close to the absolute threshold are calculated using a generalized version of the compound nucleus model of Aichelin and Bertsch. The calculated results are consistent with the data for laboratory energies per nucleon in excess of 50 MeV. For lower beam energies and for high pion kinetic energies, the model tends to underpredict the data. This indicates the presence of a nonstatistical component in the pion production process.
An equation of state for cold nuclear matter for the region of densities ρ_{nm}-4ρ_{nm}, where ρ_{nm} is empirical nuclear-matter density, is constructed. We begin from the detailed calculation of Day and Wiringa for the two-body interactions; these give a saturation density of ∼2ρ_{nm}. This density is brought down to ρ_{nm} by the addition of relativistic corrections. Additional binding is obtained from three-body forces. A reasonable picture is obtained with the Day-Wiringa compression modulus for the two-body calculation, but the picture can be further improved by choosing this to be smaller. Our equation of state is similar to that of Friedman and Pandharipande in the region of nuclear matter density ρ_{nm}, but, due to higher-order terms in the loop correction, is substantially softer at high density. Basically what happens is that the many-body effects saturate with increasing density, leaving only the two-body interactions. With this equation of state, prompt supernova explosions are very powerful when the compression modulus of neutron-rich matter (twice as many neutrons as protons) is ∼150 MeV, which corresponds to K_{nm}> ∼ 190 MeV for symmetric nuclear matter. Analysis shows that hot nuclear matter formed in heavy ion collisions demands a very stiff equation of state. We understand this as arising from the strong velocity dependence in the real part of the optical model potential which follows chiefly from the Lorentz character of the interactions, the vector mean field growing with increasing density and the scalar one decreasing. This gives a substantial repulsive contribution to the energy per particle and produces a stiff effective equation of state for several hundred MeV heavy-ion collisions. With increasing degree of equilibration the magnitude of the repulsive energy decreases since equilibration decreases the effective momentum. Given the strong velocity dependence in the interaction, the hot equation of state can be reconciled with the cool one.
The stress-energy tensor and entropy are calculated for non-interacting quarks and gluons confined to a slab of thickness L at finite temperature T. Slab propagators for quarks and gluons are constructed. For T = 200 MeV corrections to the Stefan-Boltzmann results are less than 10% if L ⩾ 1.5 fm. If L is less than this, corrections become substantial, for example, pressures exhibit marked directional asymmetries. Implications for the calculation of the evolution of quark-gluon plasmas expected in heavy-ion collisions are discussed.
We examine the role of many-body effects provided by the chiral sigma model in the equation of state of symmetric nuclear matter and neutron-rich matter. Using the parameters determined by fitting the known equilibrium properties of symmetric and slightly neutron-rich matter, the equation of state with arbitrary neutron excess is calculated. The role of the nuclear symmetry energy on the structure of beta stable neutron stars is discussed.
We extend the relativistic Brueckner-Hartree-Fock approach for nuclear matter to dense neutron matter. Special attention is devoted to contributions arising from tensor interactions due to the exchange of π-and ϱ-mesons and the dependence upon neutron-proton asymmetry. The role of the symmetry energy in studies of beta-stable neutron stars is also discussed.
Incorporation of effective masses into negative energy states (nucleon loop corrections) gives rise to repulsive many-body forces, as has been known for some time. Rather than renormalizing away the three- and four-body terms, we introduce medium corrections into the effective σ-exchange, which roughly cancel the nucleon loop terms for densities ϱ ∼ ϱ_{nm}, where ϱ_{nm} is nuclear matter density. Going to higher densities, the repulsive contributions tend to saturate whereas the attractive ones keep on growing in magnitude. The latter is achieved through use of a density-dependent effective mass for the σ-particle, m_{σ} = mσ(ϱ), such that m_{σ}(ϱ) decreases with increasing density. Such a behavior is seen e.g. in the Nambu-Jona-Lasinio model. It is argued that a smooth transition to chiral restoration implies a similar behavior. The resulting nuclear equation of state is, because of the self-consistency in the problem, immensely insensitive to changes in the mass or coupling constant of the σ-particle.
Drawing upon the experience of a previous calculation, we show that it is possible to make simple qualitative estimates of the transverse momenta generated in a Boltzmann-Uehling-Uhlenbeck calculation where the mean field is obtained from a Gogny force and a Skyrme force.
Photon production cross sections in intermediate energy heavy-ion reactions are calculated using detailed balance. The model gives substantial photon yields and reproduces qualitatively the experimental observations. A possible influence of the Δ-isobar degree of freedom on the high energy photon yield is pointed out.
We consider the possibility that the suppression of the J/ψ signal observed in oxygen-uranium collisions at CERN is due to interactions with a hot hadron gas. Assuming a longitudinally expanding meson gas, we obtain an overall suppression compatible with the CERN data. This suppression, however, varies only slowly with the J/ψ transverse momentum. We also discuss the possibility of J/ψ regeneration from the more abundantly produced χ and η_{c} states, an effect that might become important at RHIC energies.
For near central collisions of Nb on Nb at a laboratory energy of 650 MeV per projectile nucleon we calculate inclusive cross sections as a function of the azimuthal angle where this angle is measured from the reaction plane. The azimuthal dependence is strongly influenced by the nuclear equation of state and is a useful quantity to measure.
A simple model which includes the effects of primary degradation is shown to be able to reproduce FERMILAB data for transverse energy production in pA collisions of 800 GeV protons on all targets from protons to ^{208}Pb.
The relationship between the maximum neutron-star mass and observable parameters of the equation of state is explored. In particular, the roles of the nuclear incompressibility and the symmetry energy are considered. It is concluded that, for realistic symmetry energies, the compression modulus cannot, by itself, be severely limited by observed neutron-star masses. Several directions for further study are suggested.
Charmonium bound states immersed in a coherent chromo-electric field are easily ripped apart for field strengths comparable to the QCD string tension. Estimates based on flux tube models suggest that field strengths of such magnitude may be achieved in heavy-ion collisions at ultrarelativistic energies. Our results suggest that charmonium suppression would not discriminate between a coherent and a thermalized post collision state in relativistic heavy-ion collisions.
A new kind of first-order phase transition at finite baryon density is shown to occur using an effective lagrangian which simulates the parity doubling of the nucleon found recently in lattice QCD. Within the mean field approximation, we obtain a critical density of about five times the saturation density of nuclear matter. The implications of this phase transition to neutron star structure are discussed.
We compare three test-particle methods currently used in numerical simulations of Boltzmann-type equations for the analysis of intermediate-energy heavy-ion collisions with an exact solution of the Krook-Wu model. These methods are the full ensemble, parallel ensemble, and hybrid techniques. We find that collisional relaxation is sensitive to the method of simulation used. The full ensemble approach is found to agree with the exact results of the Krook-Wu model. The parallel ensemble procedure provides a reasonable approximation to the analytical relaxation rate for a wide range of systems, while the hybrid method overestimates the relaxation rate. We further compare transverse flow data from the first two of these methods in a cascade simulation of heavy-ion collisions, and find reasonable agreement provided the two-body cross section is not enhanced by a large factor over its free space value. This has implications for quantitative comparisons of calculations to experimental data.
We analyze the quark mass dependence of the chiral parameters calculated from an SU(2) Nambu-Jona-Lasinio model in the context of chiral perturbation theory. We find that in the mean-field approximation the model is unable to reproduce, within the experimental bounds, two parameters of the low-energy SU(2) effective lagrangian of Gasser and Leutwyler. Our results suggest that the model is not suited to give quantitative predictions concerning quark mass dependences. We also comment on the strange quark mass dependence of the chiral parameters in the SU(3) version of the model.
We first discuss calculations in the Nambu-Jona-Lasinio model which show that the mass of the σ-meson, m_{σ}, decreases with increasing density at the same rate as nucleon effective mass m_{N}^{*}. We next discuss linear response theory in which Δ-particle, nucleon-hole insertions are put into pion lines; then masses of all the mesons seem to decrease with increasing density. To the extent that all of the masses scale in the same way, the common scale can be taken out of the hamiltonian, giving a new hamiltonian which involves only vacuum masses. The nucleon effective mass is calculated in the Dirac-Brueckner Hartree-Fock formalism, with inclusion of nucleon one-loop terms. The latter tend to keep m_{N}^{*}/m_{N} large ~0.9. Finally, a full DBHF calculation is carried out, including the nucleon one-loop term and density dependent meson masses. With inclusion of the latter, conventional mechanisms in nuclear physics for saturation disappear and a density dependent two loop term ΔU_{σ}(ρ) must be introduced. We motivate its introduction by empirical as well as formal arguments. In the discussion it is pointed out that the scaled meson masses can be considered, for a convenient density, as renormalization points.
We investigate the generation of transverse momentum in high-energy heavy-ion collisions and its relation to the nuclear equation of state. We find that streamer chamber data can be fitted by Boltzmann-Uehling-Uhlenbeck calculations with a momentum-dependent potential that closely models realistic nuclear matter interactions and yields an equation of state with K=215 MeV.
We discuss fits of the Landau-Milekhin model to the transverse momentum spectra measured in 200 GeV/c nucleus-nucleus collisions. It is observed that the data are fit by a range of anticorrelated values of the breakup temperature and the average transverse hydrodynamic velocity. These fits indicate that a better understanding of transverse flow in ultrarelativistic nuclear collisions is required to uniquely determine the breakup temperature of the system.
It has recently been claimed that a submillisecond pulsar has been observed in the remnant of SN 1987A. If the pulsations are due to rotation of a neutron star at this frequency, the equation of state (EOS) must be severely limited. In general, soft equations of state allow rapid enough rotational frequencies, but the softness is limited by the observed neutron star mass of 1.44 solar mass in the binary pulsar PSR 1913 + 16. In order to simultaneously satisfy these two constraints, that, in the vicinity of ordinary nuclear density, the pressure must vary relatively slowly with density, but at higher densities, the EOS must become relatively stiff and approach the causality limit. Specifically, in the absence of phase transitions above nuclear density and for cases in which the nuclear symmetry energy is an increasing function of density, the compression modulus of symmetric nuclear matter at the saturation density 0.16/cu fm must be less than about 160 MeV. If either of these situations occurs, higher values for the compression modulus may be possible, but, in any case, the EOS must still stiffen to the causal limit at high density. Examples of phase transitions that might permit rapid enough rotation are those due to pion or kaon condensates or to parity-doublet matter occurring around a few times the nuclear matter density.
We calculate the maximum keplerian frequencies of stars containing strange matter that are in uniform rotation. We consider both self-bound stars and stars containing quark matter cores but normal surfaces. For both cases studied, use of perturbative QCD results for the equation of state of quark matter implies a limit on the maximum keplerian frequency of ∼1×10^{4}s^{-1}.
We calculate the speed of sound in a pion gas including interactions between pions using the results of chiral perturbation theory as well as a relativistic virial expansion with experimental phase shifts. Our result is that the speed of sound is considerably less than that for the corresponding ideal, relativistic gas. The ramifications of this result in the description of heavy-ion collisions at ultrarelativistic energies are noted.
Models that are successful in hadronic spectroscopy, viz., the nonrelativistic quark model, bag models, and QCD sum rules, are used to shed light on the Okamoto-Nolen-Schiffer anomaly in nuclear physics.
The QCD sum-rule approach for a nuclear medium is developed. The medium dependence of the neutron-proton mass difference calculated from this approach gives effects in nuclei which have direct relevance for the resolution of the Okamoto-Nolen-Schiffer anomaly.
We show that the direct URCA process can occur in neutron stars if the proton concentration exceeds some critical value in the range 11–15%. The proton concentration, which is determined by the poorly known symmetry energy of matter above nuclear density, exceeds the critical value in many current calculations. If it occurs, the direct URCA process enhances neutrino emission and neutron star cooling rates by a large factor compared to any process considered previously.
We study a simple quantum-mechanical model that illustrates various conceptual questions associated with color transparency.
Nuclear matter at high density requires a relativistic description. Recent two-loop calculations in local quantum field theory models at finite density yield disastrously large contributions. We argue that the composite structure of nucleons ought to soften these contributions. This argument is supported by a model calculation in which we find very strong reductions in the Lamb shift and vacuum fluctuation contributions. At nuclear matter densities and beyond, the total-loop result is perturbatively small in comparison to the one-loop calculation.
Direct Urca processes with hyperons and/or nucleon isobars can occur in dense matter as long as the concentration of Lambda hyperons exceeds a critical value that is less than 3 percent and is typically about 0.1 percent. The neutrino luminosities from the hyperon Urca processes are about 5-100 times less than the typical luminosity from the nucleon direct Urca process, if the latter process is not forbidden, but they are larger than those expected from other sources. These direct Urca processes provide avenues for rapid cooling of neutron stars which invoke neither exotic states nor the large proton fraction (of order 0.11-0.15) required for the nucleon direct Urca process.
We study hadronic matter at temperatures close to the pion mass. In the dilute-gas stage, the relativistic virial expansion is used with empirical phase shifts to investigate the thermodynamic properties of an interacting gas of pions, kaons and nucleons. The interplay between the attractive and repulsive interactions which governs the size of the interacting contributions is discussed. In the dense gas stage characterized by the presence of many massive resonances, schematic models that include the effects of repulsive interactions are analysed. Limits on the size of repulsive interactions are established by causality constraints. The causal behaviour of some of the conventional excluded-volume approaches is examined. Since our work is also aimed at studying those equations of state which may be used in dynamical simulations of heavy-ion collisions, phase diagrams are constructed including the thermodynamics of the quark-gluon phase.
The correlation between the compression modulus and the skewness coefficient of the nuclear matter ground state implied by the breathing mode data is compared with predictions of the relativistic Hartree approximation (RHA). Retaining explicit dependence on the renormalization scale μ, employed in the RHA, this correlation, together with other considerations, suggests a value for μ/M ⋍ 1.2.
We analyze the ′t Hooft equation for two-dimensional QCD in the limit of large number of colors using the semiclassical approximation. We show that the correct meson energies and wave functions are reproduced both in the light and heavy quark limits.
We analyze the quark number susceptibility at high temperature in three-dimensional QCD where the susceptibility carries nonperturbative temperature effects at next-to-leading order. Using arguments of dimensional reduction, we show that the Coulomb bound state description used recently to describe the DeTar correlators is consistent with the free quark susceptibility, and offers a nonperturbative framework for discussing next-to-leading-order effects. The relevance of our results to four-dimensional QCD at high temperature is discussed.
We study various non-equilibrium properties of a mixture comprised of pions, kaons and nucleons. The concentration of the constituents is chosen to mimic those expected in the mid-rapidity interval of CERN and RHIC experiments. The heavier kaons and nucleons are found to equilibrate more slowly than pions. These results shed new light on the influence of collective flow effects on the transverse momentum distributions of kaons and nucleons versus those of pions in ultrarelativistic nuclear collisions.
Recently proposed neutrino emission processes in high-density matter result in the relatively rapid cooling of a neutron star's interior followed by a precipitous drop in the surface temperature. We show that the time interval between the formation of the neutron star and the drop in the surface temperature is primarily determined by the structure of the neutron star and is relatively insensitive to the rapid cooling mechanism itself. Thus, observations of thermal emissions from neutron stars have the potential for constraining the neutron star's structure and the underlying equation of state of dense matter.
We have examined the properties of neutron-rich matter and finite nuclei in the modified relativistic Hartree approximation for several values of the renormalization parameter, μ, around the standard choice of μ equal to the nucleon mass M. Observed neutron-star masses do not effectively constrain the value of μ. However, for finite nuclei the value μ/M = 0.79, suggested by nuclear matter data, provides a good account of the bulk properties with a sigma mass of about 600 MeV. This value of μ/M renders the effective three- and four-body scalar self-couplings to be zero at 60% of equilibrium nuclear matter density. We have also found that the matter part of the exchange diagram has little impact on the bulk properties of neutron stars.
We investigate the possibility of kaon condensation in the dense interior of neutron stars through the s-wave interaction of kaons with nucleons. We include nucleon-nucleon interactions by using simple parametrizations of realistic forces, and include electrons and muons in β-equilibrium. The equation of state above the condensate threshold is derived in the mean field approximation. The conditions under which kaon condensed cores undergo a transition to quark matter containing strange quarks are also established. The critical density for kaon condensation lies in the range (2.3–5.0)ϱ_{0} where ϱ_{0} = 0.16 fm^{-3} is the equilibrium density of nuclear matter. The critical density depends largely on the value of the strangeness content of the proton, the size of which is controversial. For too large a value of the strangeness content, matter with a kaon condensate is not sufficiently stiff to support the lower limit of 1.44M_{⊙} for a neutron star. Kaon condensation dramatically increases the proton abundance of matter and even allows positrons to exist inside the core. We also consider the case when neutrinos are trapped in the matter, a situation that applies to newly-formed neutron star matter that is less than about 10 s old. Neutrino trapping shifts both kaon condensation and the quark matter transition to higher densities than in the case of cold, catalyzed matter. A newly-formed neutron star is expected to have a rather low central density, the density rising only after mass accretion onto the star ends after a few seconds. Thus, it is likely that if kaon condensation and/or the quark-hadron phase transition occur, they do so only during or after the mass accretion and neutrino trapping stages. We suggest that neutrino observations from a galactic supernova may provide direct evidence for or against a condensate and/or a quark-hadron transition.
We explore the influence of three-particle interactions, in either the initial or final state, on the collision rate in a high temperature plasma, and on the rate of quark and anti-quark pair (flavor) production. When the interactions are taken to be screened at the Debye wave numberqd≈T, three-particle interactions contribute significantly to the collision rate, but only marginally enhance flavor production over that from two-particle interactions. The magnitudes of the rates are, however, sensitive to the infra-red thresholds, which emphasizes the need for a reliable analysis of this issue. Our results also highlight the importance of treating many-particle processes adequately in the space-time evolution of quarks and gluons produced in ultrarelativistic heavy-ion collisions. We thank members of the Theoretical Physics Institute and the School of Physics and Astronomy at the University of Minnesota for their kind hospitality. Special thanks are due to J.I. Kapusta for stimulating discussions. The stay of P. L. at the University of Minnesota was supported by the U.S. Department of Energy under grant No. DOE/DE-FG02-87ER-40328; travel expenses were borne by the grant MŠMŠ SR 01/35. Research support for M. P. by the U.S. Department of Energy under grant No. DE-FG02-88ER-40388 is acknowledged. The paper was written in its final form at the Institute of Theoretical Physics, Santa Barbara, during the research program Strong Interactions at Finite Temperatures. The authors express gratitude for the warm hospitality extended there and acknowledge the support of the National Science Foundation under Grant No. PHY89-04035.
Based on the Kaplan-Nelson Lagrangian, we investigate kaon condensation in dense neutron star matter allowing for the explicit presence of hyperons. Using various models we find that the condensate threshold is sensitive to the behavior of the scalar density; the more rapidly it increases with baryon density, the lower is the threshold for condensation. The presence of hyperons, particularly the Σ^{−}, shifts the threshold for K^{−} condensation to a higher density. In the mean field approach, with hyperons, the condensate amplitude grows sufficiently rapidly that the nucleon effective mass vanishes at a finite density and a satisfactory treatment of the thermodynamics cannot be achieved. Thus, calculations of kaon-baryon interactions beyond the mean field level appear to be necessary.
We investigate the role of trapped neutrinos on the composition of protoneutron stars with quark-hadron phase transitions. Trapped neutrinos shift the transition to higher baryon densities and also reduce the extent of a mixed phase in comparison to neutrino-free matter. Thus, a mixed phase of baryons and quarks is more likely to occur after a neutrino diffusion time scale of several seconds. In contrast with stars containing only nucleons and leptons, stars with quarks have larger maximum masses when neutrinos are trapped than for neutrino-free stars. Hence, black hole formation is likely delayed for the neutrino diffusion time scale when quarks are present.
We consider the rate at which energy is emitted by neutrinos from the dense interior of neutron stars containing a Bose condensate of pions or kaons. The rates obtained are larger, by a factor of 2, than those found earlier, and are consistent with those found for the direct Urca processes.
We examine the presence of strangeness-bearing components, hyperons and kaons, in dense neutron star matter. Calculations are performed using relativistic mean field models, in which both the baryon-baryon and kaon-baryon interactions are mediated by meson exchange. Results of kaon condensation are found to be qualitatively similar to previous work with chiral models, if compatibility of the kaon optical potentials is required. The presence of strangeness, be it in the form of hyperons or kaons, implies a reduction in the maximum mass and a relatively large number of protons, sufficient to allow rapid cooling to take place. The need to improve upon the poorly known couplings of the strange particles, which determine the composition and structure of neutron stars, is stressed. We also discuss generic problems with effective masses in mean field theories.
We have examined the contribution of the filled negative energy sea of hyperons to the energy/particle in nuclear matter at the one and two loop levels. While this has the potential to be significant, we find a strong cancellation between the one and two loop contributions for our chosen parameters so that hyperon effects can be justifiably neglected.
Immediately after they are born, neutron stars are characterized by an entropy per baryon of order unity and by the presence of trapped neutrinos. If the only hadrons in the star are nucleons, these effects slightly reduce the maximum mass relative to cold, catalyzed matter. However, if negatively charged particles in the form of hyperons, a kaon condensate, or quarks are also present, these effects result in an increase in the maximum mass of ∼0.2M_{⊙}∼0.2M_{⊙} compared to that of a cold, neutrino-free star. This could lead to the delayed formation of a black hole; such a scenario is consistent with our present knowledge of SN1987A.
We study relativistic S+Au collisions at 200 A GeV/c using a hydrodynamical approach. We test various equations of state (EOSs), which are used to describe the strongly interacting matter at densities attainable in the CERN-SPS heavy ion experiments. For each EOS, suitable initial conditions can be determined to reproduce the experimental hadron spectra; this emphasizes the ambiguity between the initial conditions and the EOS in such an approach. Simultaneously, we calculate the resulting thermal photon and dielectron spectra, and compare with experiments. If one allows the excitation of resonance states with increasing temperature, the electro-magnetic signals from scenarios with and without phase transition are very similar and are not resolvable within the current experimental resolution. With regard to the CERES dilepton data, none of the EOSs considered, in conjunction with the standard leading order dilepton rates, succeed in reproducing the observed excess of dileptons below the rho peak. Our work, however, suggests that an improved measurement of the photon and dilepton spectra has the potential to strongly constrain the EOS.
We calculate neutrino cross sections from neutral-current reactions in the dense matter encountered in the evolution of a newly born neutron star. Effects of composition and of strong interactions in the deleptonization and cooling phases of the evolution are studied. The influence of the possible presence of strangeness-rich hyperons on the neutrino scattering cross sections is explored. Due to the large vector couplings of the ∑^{-} and Ξ^{-}, |C_{V}| ∼ 2, these particles, if present in protoneutron star matter, give significant contributions to neutrino scattering. In the deleptonization phase, the presence of strangeness leads to large neutrino energies, which results in large enhancements in the cross sections compared to those in matter with nucleons only. In the cooling phase, in which matter is nearly neutrino-free, the response of the Σ- hyperons to thermal neutrinos is the most significant. Neutrinos couple relatively weakly to the Λ hyperons and, hence, their contributions are significant only at high density.
We study the charged and neutral current weak interaction rates relevant for the determination of neutrino opacities in dense matter found in supernovae and neutron stars. We establish an efficient formalism for calculating differential cross sections and mean free paths for interacting, asymmetric nuclear matter at arbitrary degeneracy. The formalism is valid for both charged and neutral current reactions. Strong interaction corrections are incorporated through the in-medium single particle energies at the relevant density and temperature. The effects of strong interactions on the weak interaction rates are investigated using both potential and effective field-theoretical models of matter. We investigate the relative importance of charged and neutral currents for different astrophysical situations, and also examine the influence of strangeness-bearing hyperons. Our findings show that the mean free paths are significantly altered by the effects of strong interactions and the multi-component nature of dense matter. The opacities are then discussed in the context of the evolution of the core of a protoneutron star.
We study the thermal and chemical evolution during the Kelvin-Helmholtz phase of the birth of a neutron star, employing neutrino opacities that are consistently calculated with the underlying equation of state (EOS). Expressions for the diffusion coefficients appropriate for general relativistic neutrino transport in the equilibrium diffusion approximation are derived. The diffusion coefficients are evaluated using a field-theoretical finite-temperature EOS that includes the possible presence of hyperons. The variation of the diffusion coefficients is studied as a function of EOS and compositional parameters. We present results from numerical simulations of proto–neutron star cooling for internal stellar properties as well as emitted neutrino energies and luminosities. We discuss the influence of the initial stellar model, the total mass, the underlying EOS, and the addition of hyperons on the evolution of the proto–neutron star and on the expected signal in terrestrial detectors. We find that the differences in predicted luminosities and emitted neutrino energies do not depend much upon the details of the initial models or the underlying high-density EOS for early times (t<10 s), provided that opacities are calculated consistently with the EOS. The same holds true for models that allow for the presence of hyperons, except when the initial mass is significantly larger than the maximum mass for cold, catalyzed matter. For times larger than about 10 s, and prior to the occurrence of neutrino transparency, the neutrino luminosities decay exponentially with a time constant that is sensitive to the high-density properties of matter. We also find the average emitted neutrino energy increases during the first 5 s of evolution and then decreases nearly linearly with time. In general, increasing the proto–neutron star mass increases the average energy and the luminosity of neutrinos, as well as the overall evolutionary timescale. The influence of hyperons or variations in the dense matter EOS is increasingly important at later times. Metastable stars, those with hyperons that are unstable to collapse upon deleptonization, have relatively long evolution times, which increase the nearer the mass is to the maximum mass supportable by a cold, deleptonized star.
An extensive study of the effects of correlations on both charged and neutral current weak interaction rates in dense matter is performed. Both strong and electromagnetic correlations are considered.The propagation of particle-hole interactions in the medium plays an important role in determining the neutrino mean free paths. The effects due to Pauli-Blocking and density, spin, and isospin correlations in the medium significantly reduce the neutrino cross sections. Due to the lack of experimental information at high density, these correlations are necessarily model dependent. For example, spin correlations in nonrelativistic models are found to lead to larger suppressions of neutrino cross sections compared to those of relativistic models. This is due to the tendency of the nonrelativistic models to develop spin instabilities. Notwithstanding the above caveats, and the differences between nonrelativistic and relativistic approaches such as the spin- and isospin-dependent interactions and the nucleon effective masses, suppressions of order 2--3, relative to the case in which correlations are ignored, are obtained. Neutrino interactions in dense matter are especially important for supernova and early neutron star evolution calculations. The effects of correlations for protoneutron star evolution are calculated. Large effects on the internal thermodynamic properties of protoneutron stars, such as the temperature, are found. These translate into significant early enhancements in the emitted neutrino energies and fluxes, especially after a few seconds. At late times, beyond about 10 seconds, the emitted neutrino fluxes decrease more rapidly compared to simulations without the effects of correlations, due to the more rapid onset of neutrino transparency in the protoneutron star
Using a hydrodynamic approach to describe Pb+Au collisions at 158 A GeV/c, we analyze e^{+}e^{-} yields from matter containing baryons in addition to mesons. We employ e^{+}e^{-} production rates from two independent calculations, which differ both in their input physics and in their absolute magnitudes, especially in the mass range where significant enhancements over expected backgrounds exist in the CERES data. Although the presence of baryons leads to significant enhancement of e^{+}e^{-} emission relative to that from mesons-only matter, the calculated results fall below the data in the range 400 < M _{e+e-}/MeV < 600, by a factor of 2–3. Since the calculated e^{+}e^{-} spectra are relatively insensitive to the equation of state for initial conditions that fit the observed hadronic spectra, either in-medium modifications of the e^{+}e^{-} sources more significant than so far considered, or the presence of hitherto unidentified additional sources of e^{+}e^{-} is indicated.
In a dilute system of instantons and antiinstantons, the U_{A}(1) and scale anomalies are shown to be directly related to the bulk susceptibility and compressibility of the system. Using 1 ∕ Nc (where Nc is the number of colors) as a book-keeping argument, mesonic, baryonic and gluonic correlators are worked out in p-space and Fourier transformed to x-space for a comparison with recently simulated correlators. The results are in overall agreement with simulations and lattice calculations, for distances up to 1.5 fm, despite the fact that some channels lack the necessary physical singularities. We analyze various space-like form factors of the nucleon and show that they are amenable to constituent quark form factors to leading order in 1∕Nc. Issues related to the lack of confinement in the model and its consequence on the various correlation functions and form factors are also discussed.
We study the effects of very strong magnetic fields on the equation of state (EOS) in multicomponent, interacting matter by developing a covariant description for the inclusion of the anomalous magnetic moments of nucleons. For the description of neutron star matter, we employ a field-theoretical approach, which permits the study of several models that differ in their behavior at high density. Effects of Landau quantization in ultrastrong magnetic fields (B>10^{14} G) lead to a reduction in the electron chemical potential and a substantial increase in the proton fraction. We find the generic result for B>10^{18} G that the softening of the EOS caused by Landau quantization is overwhelmed by stiffening due to the incorporation of the anomalous magnetic moments of the nucleons. In addition, the neutrons become completely spin polarized. The inclusion of ultrastrong magnetic fields leads to a dramatic increase in the proton fraction, with consequences for the direct Urca process and neutron star cooling. The magnetization of the matter never appears to become very large, as the value of |H/B| never deviates from unity by more than a few percent. Our findings have implications for the structure of neutron stars in the presence of large frozen-in magnetic fields.
A first order phase transition at high baryon density implies that a mixed phase can occupy a significant region of the interior of a neutron star. In this article we investigate the effect of a droplet phase on neutrino transport inside the core. Two specific scenarios of the phase transition are examined, one having a kaon condensate and the other having quark matter in the high density phase. The coherent scattering of neutrinos off the droplets greatly increases the neutrino opacity of the mixed phase. We comment on how the existence of such a phase will affect a supernova neutrino signal.
We study the equation of state (EOS) of kaon-condensed matter including the effects of temperature and trapped neutrinos. It is found that the order of the phase transition to a kaon-condensed phase, and whether or not Gibbs' rules for phase equilibrium can be satisfied in the case of a first order transition, depend sensitively on the choice of the kaon-nucleon interaction. The main effect of finite temperature, for any value of the lepton fraction, is to mute the effects of a first order transition, so that the thermodynamics becomes similar to that of a second order transition. Above a critical temperature, found to be at least 30--60 MeV depending upon the interaction, the first order transition disappears. The phase boundaries in baryon density versus lepton number and baryon density versus temperature planes are delineated. We find that the thermal effects on the maximum gravitational mass of neutron stars are as important as the effects of trapped neutrinos, in contrast to previously studied cases in which the matter contained only nucleons or in which hyperons and/or quark matter were considered. Kaon-condensed EOSs permit the existence of metastable neutron stars, because the maximum mass of an initially hot, lepton-rich protoneutron star is greater than that of a cold, deleptonized neutron star. The large thermal effects imply that a metastable protoneutron star's collapse to a black hole could occur much later than in previously studied cases that allow metastable configurations.
The mixed phase of quarks and hadrons which might exist in the dense matter encountered in the varying conditions of temperature and trapped neutrino fraction in proto-neutron stars is studied. The extent that the mixed phase depends upon the thermodynamical parameters as well as on the stiffness of matter in the hadronic and quark phases is discussed. We show that hadronic equations of state that maximize the quark content of matter at a given density generally minimize the extent of the mixed phase region in a neutron star of a given mass, and that only in extreme cases could a pure quark star result. For both the Nambu Jona-Lasinio and MIT bag quark models, neutrino trapping inhibits the appearance of a mixed phase which leads to possible proto-neutron star metastability. The main difference between the two quark models is the small abundance of strange quarks in the former. We also demonstrate that ∂T/∂n<0 along adiabats in the quark-hadron mixed phase, opposite to what is found for the kaon condensates-hadron mixed phase. This could lead to core temperatures which are significantly lower in stars containing quarks than in those not containing quarks.
Baryon and quark superfluidity in the cooling of neutron stars are investigated. Observations could constrain combinations of the neutron or Lambda-hyperon pairing gaps and the star's mass. However, in a hybrid star with a mixed phase of hadrons and quarks, quark gaps larger than a few tenths of an MeV render quark matter virtually invisible for cooling. If the quark gap is smaller, quark superfluidity could be important, but its effects will be nearly impossible to distinguish from those of other baryonic constituents.
The structure of neutron stars is considered from theoretical and observational perspectives. We demonstrate an important aspect of neutron star structure: the neutron star radius is primarily determined by the behavior of the pressure of matter in the vicinity of nuclear matter equilibrium density. In the event that extreme softening does not occur at these densities, the radius is virtually independent of the mass and is determined by the magnitude of the pressure. For equations of state with extreme softening or those that are self-bound, the radius is more sensitive to the mass. Our results show that in the absence of extreme softening, a measurement of the radius of a neutron star more accurate than about 1 km will usefully constrain the equation of state. We also show that the pressure near nuclear matter density is primarily a function of the density dependence of the nuclear symmetry energy, while the nuclear incompressibility and skewness parameters play secondary roles. In addition, we show that the moment of inertia and the binding energy of neutron stars, for a large class of equations of state, are nearly universal functions of the star's compactness. These features can be understood by considering two analytic, yet realistic, solutions of Einstein's equations, by, respectively, Buchdahl and Tolman. We deduce useful approximations for the fraction of the moment of inertia residing in the crust, which is a function of the stellar compactness and, in addition, the pressure at the core-crust interface.
We present simulations of the evolution of a proto-neutron star in which kaon-condensed matter might exist, including the effects of finite temperature and trapped neutrinos. The phase transition from pure nucleonic matter to the kaon condensate phase is described using Gibbs' rules for phase equilibrium, which permit the existence of a mixed phase. A general property of neutron stars containing kaon condensates, as well as other forms of strangeness, is that the maximum mass for cold, neutrino-free matter can be less than the maximum mass for matter containing trapped neutrinos or that has a finite entropy. A proto-neutron star formed with a baryon mass exceeding that of the maximum mass of cold, neutrino-free matter is therefore metastable, that is, it will collapse to a black hole at some time during the Kelvin-Helmholtz cooling stage. The effects of kaon condensation on metastable stars are dramatic. In these cases, the neutrino signal from a hypothetical galactic supernova (distance ~8.5 kpc) will stop suddenly, generally at a level above the background in the Super-Kamiokande and Sudbury Neutrino Observatory detectors, which have low-energy thresholds and backgrounds. This is in contrast to the case of a stable star, for which the signal exponentially decays, eventually disappearing into the background. We find the lifetimes of kaon-condensed metastable stars to be restricted to the range of 40-70 s and weakly dependent on the proto-neutron star mass, in sharp contrast to the significantly larger mass dependence and range (1-100 s) of hyperon-rich metastable stars. We find that a unique signature for kaon condensation will be difficult to identify. The formation of the kaon condensate is delayed until the final stages of the Kelvin-Helmholtz epoch, when the neutrino luminosity is relatively small. In stable stars, modulations of the neutrino signal caused by the appearance of the condensate will therefore be too small to be clearly distinguished with current detectors, despite the presence of a first-order phase transition in the core. In metastable stars, the sudden cessation in the neutrino signal occurs whether it is caused by kaon condensation, hyperons, or quarks. However, if the lifetime of the metastable star is less than about 30 s, we find that it is not likely to be due to kaon condensation.
We study static neutron stars with poloidal magnetic fields and a simple class of electric current distributions consistent with the requirement of stationarity. For this class of electric current distributions, we find that magnetic fields are too large for static configurations to exist when the magnetic force pushes a sufficient amount of mass off-center that the gravitational force points outward near the origin in the equatorial plane. (In our coordinates an outward gravitational force corresponds to ∂ln g_{tt}/∂r > 0, where t and r are respectively time and radial coordinates and g_{tt} is coefficient of dt^{2} in the line element.) For the equations of state (EOSs) employed in previous work, we obtain configurations of higher mass than had been reported; we also present results with more recent EOSs. For all EOSs studied, we find that the maximum mass among these static configurations with magnetic fields is noticeably larger than the maximum mass attainable by uniform rotation, and that for fixed values of baryon number the maximum mass configurations are all characterized by an off-center density maximum.
Neutrino opacities important in the evolution of a proto-neutron star containing quark matter are studied. The results for pure quark matter are compared with limiting expressions previously derived, and are generalized to the temperatures, neutrino degeneracies and lepton contents encountered in a proto-neutron star's evolution. We find that the appearance of quarks in baryonic matter drastically reduces the neutrino opacity for a given entropy, the reduction being sensitive to the thermodynamic conditions in the mixed quark-hadron phase.
Neutrino fluxes from proto-neutron stars with and without quarks are studied. Observable differences become apparent after 10--20 s of evolution. Sufficiently massive stars containing negatively-charged, strongly interacting, particles collapse to black holes during the first minute of evolution. Since the neutrino flux vanishes when a black hole forms, this is the most obvious signal that quarks (or other types of strange matter) have appeared. The metastability timescales for stars with quarks are intermediate between those containing hyperons and kaon condensates.
For the low energy Standard Model neutrino-matter interactions, we calculate neutrino pair (νν¯) emissivites in superfluid quark matter. Just below the critical temperature, Cooper pairs continuously break and recombine, resulting in the emission of νν¯ pairs with a rate that greatly exceeds the standard quark modified Urca and bremsstrahlung rates. At the same baryon density in baryonic and quark matter, the ratio of baryon to quark νν¯ emissivities lies in the range 2-5 for the densities of interest in the long-term cooling of solar mass compact stars. We also find that in matter containing hyperons, νν¯ emission can occur with hyperons of all species.
We discuss efforts to determine the mass, radius, and surface composition of the nearby compact object RX J185635-3754 from its multi-wavelength spectral energy distribution. We compute non-magnetized model atmospheres and emergent spectra for selected compositions and gravities, and discuss efforts to fit existing and new observational data from ROSAT, EUVE and the HST. The spectral energy distribution matches that expected from a heavy-element dominated atmosphere, but not from a uniform temperature blackbody. Non-magnetic light element atmospheres cannot be simultaneously reconciled with the optical and X-ray data. We extend previous studies, which were limited to one fixed neutron star mass and radius. For uniform temperature models dominated by heavy elements, the redshift z is constrained to be 0.3 ≤ z ≤ 0.4 and the best-fit mass and radius are M approx 0.9 solar masses and R approx 6 km (for a 61 pc distance). These values for M and R together are not permitted for any plausible equation of state, including that of a self-bound strange quark star. A simplified two-temperature model allows masses and radii up to about 50% larger, or a factor of 2 in the case of a black body. The observed luminosity is consistent with the thermal emission of an isolated neutron star no older than about 1 million years, the age inferred from available proper motion and parallax information
Using a chirally invariant effective Lagrangian, we calculate the density and isospin dependences of the in-medium axial coupling, g_{A}*, in spatially uniform matter present in core collapse supernovae and neutron stars. The quenching of g_{A}∗ with density in matter with different proton fractions is found to be similar. However, our results suggest that the quenching of the nucleon's g_{A}* in matter with hyperons is likely to be significantly greater than in matter with nucleons only.
We address the role of fluctuations in strongly interacting matter during the dense stages of a heavy-ion collision through its electromagnetic emission. Fluctuations of isospin charge are considered in a thermal system at rest as well as in a moving hadronic fluid at fixed proper time within a finite bin of pseudorapidity. In the former case, we use general thermodynamic relations to establish a connection between fluctuations and the spacelike screening limit of the retarded photon self-energy, which directly relates to the emissivities of dileptons and photons. Effects of hadronic interactions are highlighted through two illustrative calculations. In the latter case, we show that a finite time scale τ inherent in the evolution of a heavy-ion collision implies that equilibrium fluctuations involve both spacelike and timelike components of the photon self-energy in the system. Our study of nonthermal effects, explored here through a stochastic treatment, shows that an early and large fluctuation in isospin survives only if it is accompanied by a large temperature fluctuation at freeze-out, an unlikely scenario in hadronic phases with large heat capacity. We point out prospects for the future which include (1) a determination of the Debye mass of the system at the dilute freeze-out stage of a heavy-ion collision and (2) a delineation of the role of charge fluctuations during the dense stages of the collision through a study of electromagnetic emissivities.
We investigate the effects of very strong magnetic fields upon the equation of state of dense baryonic matter in which hyperons are present. In the presence of a magnetic field, the equation of state above nuclear density is significantly affected both by Landau quantization and magnetic moment interactions, but only for field strengths B >5×10 18 G. The former tends to soften the EOS and increase proton and lepton abundances, while the latter produces an overall stiffening of the EOS. Each results in a suppression of hyperons relative to the field-free case. The structure of a neutron star is, however, primarily determined by the magnetic field stress. We utilize existing general relativistic magneto-hydrostatic calculations to demonstrate that maximum average fields within a stable neutron are limited to values B ⩽1–3×10 18 G. This is not large enough to significantly influence particle compositions or the matter pressure, unless fluctuations dominate the average field strengths in the interior or configurations with significantly larger field gradients are considered.
We calculate neutrino emissivities from the decay and scattering of Goldstone bosons in the color-flavor-locked (CFL) phase of quarks at high baryon density. Interactions in the CFL phase are described by an effective low-energy theory. For temperatures in the tens of keV range, relevant to the long-term cooling of neutron stars, the emissivities involving Goldstone bosons dominate over those involving quarks, because gaps in the CFL phase are ∼100 MeV while the masses of Goldstone modes are on the order of 10 MeV. For the same reason, the specific heat of the CFL phase is also dominated by the Goldstone modes. Notwithstanding this, both the emissivity and the specific heat from the massive modes remain rather small, because of their extremely small number densities. The values of the emissivity and the specific heat imply that the timescale for the cooling of the CFL core in isolation is ∼1026 y, which makes the CFL phase invisible as the exterior layers of normal matter surrounding the core will continue to cool through significantly more rapid processes. If the CFL phase appears during the evolution of a proto-neutron star, neutrino interactions with Goldstone bosons are expected to be significantly more important since temperatures are high enough (∼20−40 MeV) to admit large number densities of Goldstone modes.
We investigate the consequences of enforcing local color neutrality on the color superconducting phases of quark matter by utilizing the Nambu-Jona-Lasinio model supplemented by diquark and the t'Hooft six-fermion interactions. In neutrino free matter at zero temperature, color neutrality guarantees that the number densities of u, d, and s quarks in the Color-Flavor-Locked (CFL) phase will be equal even with physical current quark masses. Electric charge neutrality follows as a consequence and without the presence of electrons. In contrast, electric charge neutrality in the less symmetric 2-flavor superconducting (2SC) phase with ud pairing requires more electrons than the normal quark phase. The free energy density cost of enforcing color and electric charge neutrality in the CFL phase is lower than that in the 2SC phase, which favors the formation of the CFL phase. With increasing temperature and neutrino content, an unlocking transition occurs from the CFL phase to the 2SC phase with the order of the transition depending on the temperature, the quark and lepton number chemical potentials. The astrophysical implications of this rich structure in the phase diagram, including estimates of the effects from Goldstone bosons in the CFL phase, are discussed.
Invariant mass distributions of the hadronic decay products from resonances formed in relativistic heavy-ion collision (RHIC) experiments are investigated with a view to disentangle the effects of thermal motion and the phase space of decay products from those of intrinsic changes in the structure of resonances at the freeze-out conditions. Analytic results of peak mass shifts for the cases of both equal and unequal mass decay products are derived. The shift is expressed in terms of the peak mass and width of the vacuum or medium-modified spectral functions and temperature. Examples of expected shifts in meson (e.g.,ρ,ω, and σ) and baryon (e.g., Δ) resonances that are helpful to interpret recent RHIC measurements at BNL are provided. Although significant downward mass shifts are caused by widened widths of the ρ meson in medium, a downward shift of at least 50 MeV in its intrinsic mass is required to account for the reported downward shift of 60–70 MeV in the peak of the ρ invariant mass distribution. An observed downward shift from the vacuum peak value of the Δ distinctively signals a significant downward shift in its intrinsic peak mass, since unlike for the ρ meson, phase space functions produce an upward shift for the Δ isobar.
The differential rates and emissivities of neutrino pairs from an equilibrium plasma are calculated for the wide range of density and temperature encountered in astrophysical systems. New analytical expressions are derived for the differential emissivities which yield total emissivities in full agreement with those previously calculated. The photon and plasmon pair production and absorption kernels in the source term of the Boltzmann equation for neutrino transport are provided. The appropriate Legendre coefficients of these kernels, in forms suitable for multi-group flux-limited diffusion schemes are also computed.
Explicit expressions for the differential and total rates and emissivities of neutrino pairs from the photoneutrino process e±+→γe±+ν+¯ν in hot and dense matter are derived. Full information about the emitted neutrinos is retained by evaluating the squared matrix elements for this process which was hitherto bypassed through the use of Lenard’s identity in obtaining the total neutrino emissivities. Accurate numerical results are presented for widely varying conditions of temperature and density. Analytical results helpful in understanding the qualitative behaviors of the rates and emissivities in limiting situations are derived. The corresponding production and absorption kernels in the source term of the Boltzmann equation for neutrino transport are developed. The appropriate Legendre coefficients of these kernels, in forms suitable for multigroup flux-limited diffusion schemes, are also provided.
The photon emissivity from the bremsstrahlung process e−e−→e−e−γ occurring in the electrosphere at the bare surface of a strange quark star is calculated. For surface temperatures T<109K, the photon flux exceeds that of e+e− pairs that are produced via the Schwinger mechanism in the presence of a strong electric field that binds electrons to the surface of the quark star. The average energy of photons emitted from the bremsstrahlung process can be around 0.5 MeV, which is comparable to that in e+e− pair annihilation. The observation of this distinctive photon spectrum would constitute an unmistakable signature of a strange quark star and shed light on color superconductivity at stellar densities.
A new classification of neutron star cooling scenarios, involving either "minimal" cooling or "enhanced" cooling, is proposed. The minimal cooling scenario replaces and extends the so-called standard cooling scenario to include neutrino emission from the Cooper pair breaking and formation process. This emission dominates that due to the modified Urca process for temperatures close to the critical temperature for superfluid pairing. Minimal cooling is distinguished from enhanced cooling by the absence of neutrino emission from any direct Urca process, due either to nucleons or to exotica such as hyperons, Bose condensates, or deconfined quarks. Within the minimal cooling scenario, theoretical cooling models can be considered to be a four parameter family involving the equation of state (including various compositional possibilities) of dense matter, superfluid properties of dense matter, the composition of the neutron star envelope, and the mass of the neutron star. The consequences of minimal cooling are explored through extensive variations of these parameters. The results are compared with the inferred properties of thermally emitting neutron stars in order to ascertain if enhanced cooling occurs in any of them. All stars for which thermal emissions have been clearly detected are at least marginally consistent with the lack of enhanced cooling, given the combined uncertainties in ages and temperatures or luminosities. The two pulsars PSR 0833-45 (Vela) and PSR 1706-44 would require enhanced cooling in case their ages and/or temperatures are on the lower side of their estimated values, whereas the four stars PSR 0656+14, PSR 1055-52, Geminga, and RX J0720.4-3125 may require some source of internal heating in case their age and/or luminosity are on the upper side of their estimated values. The new upper limits on the thermal luminosity of PSR J0205+6449 (in the supernova remnant 3C 58) and RX J0007.0+7302 (in CTA 1) are indicative of the occurrence of some enhanced neutrino emission beyond the minimal scenario.
We demonstrate that the largest measured mass of a neutron star establishes an upper bound to the energy density of observable cold baryonic matter. An equation of state-independent expression satisfied by both normal neutron stars and self-bound quark matter stars is derived for the largest energy density of matter inside stars as a function of their masses. The largest observed mass sets the lowest upper limit to the density. Implications from existing and future neutron star mass measurements are discussed.
Close binary systems of compact stars, due to the emission of gravitational radiation, may evolve into a phase in which the less massive star transfers mass to its companion. We describe mass transfer by using the model of Roche lobe overflow, in which mass is transferred through the first, or innermost, Lagrange point. Under conditions in which gravity is strong, the shapes of the equipotential surfaces and the Roche lobes are modified compared to the Newtonian case. We present calculations of the Roche lobe utilizing the second order post-Newtonian (2PN) approximation in the Arnowitt-Deser-Misner gauge. Heretofore, calculations of the Roche lobe geometry beyond the Newtonian case have not been available. Beginning from the general N-body Lagrangian derived by Damour and Schaffer, we develop the Lagrangian for a test particle in the vicinity of two massive compact objects. As an exact result for the transverse-traceless part of the Lagrangian is not available, we devise an approximation that is valid for regions close to the less massive star. We calculate the Roche lobe volumes, and provide a simple fitting formula for the effective Roche lobe radius analogous to that for the Newtonian case furnished by Eggleton. In contrast to the Newtonian case, in which the effective Roche radius depends only upon the mass ratio q=m1/m2, in the 2PN case the effective Roche lobe radius also depends on the ratio z=2(m1+m2)/a of the total mass and the orbital separation.
Binary mergers involving black holes and neutron stars have been proposed as major sources of gravitational waves, r--process nucleosynthesis, and gamma ray bursters. In addition, they represent an important, and possibly unique, observable that could distinguish between normal and self--bound neutron stars. These two families of stars have distinctly different mass--radius relationships resulting from their equations of state which can be revealed during their mergers if stable mass transfer ensues. We consider two cases of gravitational-radiation induced binary mergers: (i) a black hole and a normal neutron star, and (ii) a black hole and a self-bound strange quark matter star. We extend previous Newtonian analyses to incorporate the pseudo-general relativistic Paczy\'nski-Wiita potential or a potential correct to second--order post-Newtonian order in Arnowitt--Deser--Misner coordinates. These potentials are employed to study both the orbital evolution of the binary and the Roche lobe geometry that determines when and if stable mass transfer between the components is possible. The Roche lobe geometry with pseudo-general relativistic or post-Newtonian potentials has not heretofore been considered. Our analysis shows that differences in the evolution of normal neutron stars and strange quark matter stars are significant and could be detected in gravity waves. Both the amplitude and frequencies of the wave pattern are affected. In addition, details of the equation of state for either normal neutron stars or strange quark stars may be learned. A single merger could reveal one or two individual points of the mass-radius relation, and observations of several mergers could map a significant portion of this relation.
Recently, muon production in electron-proton scattering has been suggested as a possible candidate reaction for the identification of lepton-flavor violation due to physics beyond the standard model. Here we point out that the standard-model processes e−p→μ−p¯νμνe and e−p→e−nμ+νμ can cloud potential beyond-the-standard-model signals in ep collisions. We find that standard-model ep→μX cross sections exceed those from lepton-flavor-violating operators by several orders of magnitude. We also discuss the possibility of using a nuclear target to enhance the ep→μX signal.
The shear viscosity of the crust might have a damping effect on the amplitude of r-modes of rotating neutron stars. This damping has implications for the emission of gravitational waves. We calculate the contribution to the shear viscosity coming from the ions using both semianalytical methods that consider binary collisions and molecular dynamics simulations. We compare these results with the contribution coming from electrons. We study how the shear viscosity depends on density for conditions of interest in neutron star envelopes and outer crusts. In the low-density limit, we find good agreement between results of our molecular dynamics simulations and those of classical semianalytic calculations.
The coefficients of diffusion, thermal conductivity, and shear viscosity are calculated for a system of non-relativistic particles interacting via a delta-shell potential V(r)=-v \delta(r-R) when the average distance between particles is smaller than R. The roles of resonances and long scattering lengths including the unitary limit are examined. Results for ratios of diffusion to viscosity and viscosity to entropy density are presented for varying scattering lengths.
The minimal cooling paradigm for neutron star cooling assumes that enhanced cooling due to neutrino emission from any direct Urca process, due either to nucleons or to exotica such as hyperons, Bose condensates, or deconfined quarks, does not occur. Previous studies showed that the observed temperatures of young, cooling, isolated neutron stars with ages between 102 and 105 yr, with the possible exception of the pulsar in the supernova remnant CTA 1, are consistent with predictions of the minimal cooling paradigm as long as the neutron 3 P 2 pairing gap present in the stellar core is of moderate size. Recently, it has been found that Cooper-pair neutrino emission from the vector channel is suppressed by a large factor, of the order of 10–3, compared to the original estimates that violated vector current conservation. We show that Cooper-pair neutrino emission remains, nevertheless, an efficient cooling mechanism through the axial channel. As a result, the elimination of neutrino emission from Cooper-paired nucleons through the vector channel has only minor effects on the long-term cooling of neutron stars within the minimal cooling paradigm. We further quantify precisely the effect of the size of the neutron 3 P 2 gap and demonstrate that consistency between observations and the minimal cooling paradigm requires that the critical temperature Tc for this gap covers a range of values between T min c lesssim 0.2 × 109 up to T max c gsim 0.5 × 109 in the core of the star. This range of values guarantees that the Cooper-pair neutrino emission is operating efficiently in stars with ages between 103 to 105 yr, leading to the coldest predicted temperatures for young neutron stars. In addition, it is required that young neutron stars have heterogeneous envelope compositions: some must have light-element compositions and others must have heavy-element compositions. Unless these two conditions are fulfilled, about half of the observed young cooling neutron stars are inconsistent with the minimal cooling paradigm and provide evidence for the existence of enhanced cooling.
Gravitational waves from the final stages of inspiraling binary neutron stars are expected to be one of the most important sources for ground-based gravitational wave detectors. The masses of the components are determinable from the orbital and chirp frequencies during the early part of the evolution, and large finite-size (tidal) effects are measurable toward the end of inspiral, but the gravitational wave signal is expected to be very complex at this time. Tidal effects during the early part of the evolution will form a very small correction, but during this phase the signal is relatively clean. The accumulated phase shift due to tidal corrections is characterized by a single quantity related to a star’s tidal Love number. The Love number is sensitive, in particular, to the compactness parameter M/R and the star’s internal structure, and its determination could provide an important constraint to the neutron star radius. We show that Love numbers of self-bound strange quark matter stars are qualitatively different from those of normal neutron stars. Observations of the tidal signature from coalescing compact binaries could therefore provide an important, and possibly unique, way to distinguish self-bound strange quark stars from normal neutron stars. Tidal signatures from self-bound strange quark stars with masses smaller than 1M⊙ are substantially smaller than those of normal stars owing to their smaller radii. Thus tidal signatures of stars less massive than 1M⊙ are probably not detectable with Advanced LIGO. For stars with masses in the range 1–2M⊙, the anticipated efficiency of the proposed Einstein telescope would be required for the detection of tidal signatures.
We propose that the observed cooling of the neutron star in Cassiopeia A is due to enhanced neutrino emission from the recent onset of the breaking and formation of neutron Cooper pairs in the 3P2 channel. We find that the critical temperature for this superfluid transition is ≃0.5×109 K. The observed rapidity of the cooling implies that protons were already in a superconducting state with a larger critical temperature. This is the first direct evidence that superfluidity and superconductivity occur at supranuclear densities within neutron stars. Our prediction that this cooling will continue for several decades at the present rate can be tested by continuous monitoring of this neutron star.
The supernova remnant Cassiopeia A contains the youngest known neutron star which is also the first one for which real time cooling has ever been observed. In order to explain the rapid cooling of this neutron star, we first present the fundamental properties of neutron stars that control their thermal evolution with emphasis on the neutrino emission processes and neutron/proton superfluidity/superconductivity. Equipped with these results, we present a scenario in which the observed cooling of the neutron star in Cassiopeia A is interpreted as being due to the recent onset of neutron superfluidity in the core of the star. The manner in which the earlier occurrence of proton superconductivity determines the observed rapidity of this neutron star's cooling is highlighted. This is the first direct evidence that superfluidity and superconductivity occur at supranuclear densities within neutron stars.
A quantitative comparison between the results of shear viscosities from the Chapman-Enskog and relaxation time methods is performed for selected test cases with specified elastic differential cross sections: (i) the non-relativistic, relativistic and ultra-relativistic hard sphere gas with angle and energy independent differntial cross section, (ii) the Maxwell gas, (iii) chiral pions and (iv) massive pions. Our quantitative results reveal that the extent of agreement (or disagreement) depends very sensitively on the energy dependence of the differential cross sections employed.
The interpretation of the measured elliptic and higher order collective flows in heavy-ion collisions in terms of viscous hydrodynamics depends sensitively on the ratio of shear viscosity to entropy density. Here we perform a quantitative comparison between the results of shear viscosities from the Chapman-Enskog and relaxation time methods for selected test cases with specified elastic differential cross sections: (i) the nonrelativistic, relativistic and ultrarelativistic hard sphere gas with angle and energy independent differential cross section, (ii) the Maxwell gas, (iii) chiral pions, and (iv) massive pions for which the differential elastic cross section is taken from experiments. Our quantitative results (i) reveal that the extent of agreement (or disagreement) depends sensitively on the energy dependence of the differential cross sections employed, and (ii) stress the need to perform quantum molecular dynamical (URQMD) simulations that employ Green-Kubo techniques with similar cross sections to validate the codes employed and to test the accuracy of other methods.
We study the mass-radius curve of hybrid stars, assuming a single first-order phase transition between nuclear and quark matter, with a sharp interface between the quark matter core and nuclear matter mantle. We use a generic parametrization of the quark matter equation of state, which has a constant, i.e. density-independent, speed of sound. We argue that this parametrization provides a framework for comparison and empirical testing of models of quark matter. We obtain the phase diagram of possible forms of the hybrid star mass-radius relation, where the control parameters are the transition pressure, energy density discontinuity, and the quark matter speed of sound. We find that this diagram is sensitive to the quark matter parameters but fairly insensitive to details of the nuclear matter equation of state (EoS). We calculate the maximum hybrid star mass as a function of the parameters of the quark matter EoS, and find that there are reasonable values of those parameters that give rise to hybrid stars with mass above 2M⊙.
Implications of recently well-measured neutron star masses, particularly near and above 2 solar masses, for the equation of state (EOS) of neutron star matter are highlighted. Model-independent upper limits to thermodynamic properties in neutron stars, which only depend on the neutron star maximum mass, established from causality considerations are presented. The need for non-perturbative treatments of quark matter in neutron stars is stressed through studies of self-bound quark matter stars, and of nucleon-quark hybrid stars. The extent to which several well-measured masses and radii of individual neutron stars can establish a model-independent EOS through an inversion of the stellar structure equations is briefly discussed.
Shear viscosity η and entropy density s of a hadronic resonance gas are calculated using the Chapman-Enskog and virial expansion methods using the K-matrix parametrization of hadronic cross sections which preserves the unitarity of the T matrix. In the π−K−N−η mixture considered, a total of 57 resonances up to 2 GeV were included. Comparisons are also made to results with other hadronic cross sections such as the Breit-Wigner (BW) and, where available, experimental phase shift parameterizations. Hadronic interactions forming resonances are shown to decrease the shear viscosity and increase the entropy density leading to a substantial reduction of η/s as the QCD phase transition temperature is approached.
We investigate the thermal properties of the potential model equation of state of Akmal, Pandharipande, and Ravenhall. This equation of state approximates the microscopic model calculations of Akmal and Pandharipande, which feature a neutral pion condensate. We treat the bulk homogeneous phase for isospin asymmetries ranging from symmetric nuclear matter to pure neutron matter and for temperatures and densities relevant for simulations of core-collapse supernovae, protoneutron stars, and neutron star mergers. Numerical results of the state variables are compared with those of a typical Skyrme energy density functional with similar properties at nuclear densities but which differ substantially at supranuclear densities. Analytical formulas, which are applicable to nonrelativistic potential models such as the equations of state we are considering, are derived for all state variables and their thermodynamic derivatives. A highlight of our work is its focus on thermal response functions in the degenerate and nondegenerate situations, which allow checks of the numerical calculations for arbitrary degeneracy. These functions are sensitive to the density-dependent effective masses of neutrons and protons, which determine the thermal properties in all regimes of degeneracy. We develop the “thermal asymmetry free energy” and establish its relation to the more commonly used nuclear symmetry energy. We also explore the role of the pion condensate at supranuclear densities and temperatures. Tables of matter properties as functions of baryon density, composition (i.e., proton fraction), and temperature are being produced which are suitable for use in astrophysical simulations of supernovae and neutron stars.
Gerry Brown has had the most influence on my career in Physics, and my life after graduate studies. This article gives a brief account of some of the many ways in which Gerry shaped my research. Focus is placed on the significant strides on neutron star research made by the group at Stony Brook, which Gerry built from scratch. Selected puzzles about neutron stars that remain to be solved are noted.
We explore the thermal properties of hot and dense matter using a model that reproduces the empirical properties of isospin symmetric and asymmetric bulk nuclear matter, optical-model fits to nucleon-nucleus scattering data, heavy-ion flow data in the energy range 0.5–2 GeV/A, and the largest well-measured neutron star mass of 2M⊙. This model, which incorporates finite range interactions through a Yukawa-type finite range force, is contrasted with a conventional zero range Skyrme model. Both models predict nearly identical zero-temperature properties at all densities and proton fractions, including the neutron star maximum mass, but differ in their predictions for heavy-ion flow data. We contrast their predictions of thermal properties, including their specific heats, and provide analytical formulas for the strongly degenerate and nondegenerate limits. We find significant differences in the results of the two models for quantities that depend on the density derivatives of nucleon effective masses. We show that a constant value for the ratio of the thermal components of pressure and energy density expressed as Γth=1+(Pth/ɛth), often used in simulations of proto-neutron stars and merging compact object binaries, fails to adequately describe results of either nuclear model. The region of greatest discrepancy extends from subsaturation densities to a few times the saturation density of symmetric nuclear matter. Our results suggest alternate approximations for the thermal properties of dense matter that are more realistic.
Analytical formulas for next-to-leading order temperature corrections to the thermal state variables of interacting nucleons in bulk matter are derived in the degenerate limit. The formalism developed is applicable to a wide class of non-relativistic and relativistic models of hot and dense matter currently used in nuclear physics and astrophysics (supernovae, proto-neutron stars and neutron star mergers) as well as in condensed matter physics. We consider the general case of arbitrary dimensionality of momentum space and an arbitrary degree of relativity (for relativistic mean-field theoretical models). For non-relativistic zero-range interactions, knowledge of the Landau effective mass suffices to compute next-to-leading order effects, but in the case of finite-range interactions, momentum derivatives of the Landau effective mass function up to second order are required. Numerical computations are performed to compare results from our analytical formulas with the exact results for zero- and finite-range potential and relativistic mean-field theoretical models. In all cases, inclusion of next-to-leading order temperature effects substantially extends the ranges of partial degeneracy for which the analytical treatment remains valid.
Properties of hot and dense matter are calculated in the framework of quantum hadrodynamics by including contributions from two-loop (TL) diagrams arising from the exchange of isoscalar and isovector mesons between nucleons. Our extension of mean field theory (MFT) employs the same five density-independent coupling strengths which are calibrated using the empirical properties at the equilibrium density of isospin-symmetric matter. Results of calculations from the MFT and TL approximations are compared for conditions of density, temperature, and proton fraction encountered in the study of core-collapse supernovae, young and old neutron stars, and mergers of compact binary stars. The TL results for the equation of state (EOS) of cold pure neutron matter at sub- and near-nuclear densities agree well with those of modern quantum Monte Carlo and effective field-theoretical approaches. Although the high-density EOS in the TL approximation for cold and β-equilibrated neutron-star matter is substantially softer than its MFT counterpart, it is able to support a 2M⊙ neutron star required by recent precise determinations. In addition, radii of 1.4M⊙ stars are smaller by ∼1 km than those obtained in MFT and lie in the range indicated by analysis of astronomical data. In contrast to MFT, the TL results also give a better account of the single-particle or optical potentials extracted from analyses of medium-energy proton-nucleus and heavy-ion experiments. In degenerate conditions, the thermal variables are well reproduced by results of Landau's Fermi-liquid theory in which density-dependent effective masses feature prominently. The ratio of the thermal components of pressure and energy density expressed as Γth=1+(Pth/εth), often used in astrophysical simulations, exhibits a stronger dependence on density than on proton fraction and temperature in both MFT and TL calculations. The prominent peak of Γth at supranuclear density found in MFT is, however, suppressed in TL calculations. This outcome is analogous to results of nonrelativistic models when exchange contributions from finite-range interactions are included in addition to those of contact interactions.
We examine the application of transition-state theory for fission-fragment angular distributions to composite nuclei near the limits of stability. The possible roles of saddle-point and scission-point configurations are explored. For many heavy-ion reactions that involve large angular momenta, the observed anisotropies are between the predictions of the saddle-point and scisson--point models. Empirical correlations are shown between the effective moments of inertia and the spin and Z^{2}A of the compound nucleus. These correlations provide evidence for a class of transition-state nuclei intermediate between saddle- and scission-point configurations. An important indication of these patterns is that the speed of collective deformation toward fission may well be slow enough to allow for statistical equilibrium in the tilting mode even for configurations well beyond the saddle point.
The equilibration of hot hadronic matter is studied in the framework of relativistic kinetic theory. Various non-equilibrium properties of a mixture comprised of pions, kaons and nucleons are calculated in the dilute limit for small deviations from local thermal equilibrium. Interactions between these constituents are specified through the empirical phase shifts. The properties calculated include the relaxation/collision times, momentum and energy persistence ratios in elastic collisions, and transport properties such as the viscosity, the thermal conductivity, and the diffusion and thermal diffusion coefficients. The Chapman-Enskog formalism is extended to extract relaxation times associated with shear and heat flows, and drag and diffusion flows in a mixture. The equilibrium number concentration of the constituents is chosen to mimic those expected in the mid-rapidity interval of CERN and RHIC experiments. In this case, kaons and nucleons are found to equilibrate significantly more slowly than pions. These results shed new light on the influence of collective flow effects on the transverse momentum distributions of kaons and nucleons versus those of pions in ultra-relativistic nuclear collisions.
The thermal evolution of neutron stars has long been regarded as the source of information about the possible physical states of dense matter. Until recently, the general view was that, if matter consisted of nucleons, cooling would be relatively slow, while if matter were in an exotic state (Bose condensates or quark matter), cooling would be faster — so fast, in fact, that thermal emission from the star's surface would be unobservable. Recently, this view has been called into question following the demonstration that ordinary matter can cool by the so-called direct Urca process even more rapidly than matter in an exotic state. The conditions under which very rapid cooling would occur through energy loss via emission of neutrinos from the interior are as follows. The direct Urca process can occur in neutron stars if the proton concentration exceeds some critical value in the range 11–15%. The proton concentration, which is determined by the poorly known symmetry energy of matter above nuclear density, exceeds the critical value in many current calculations. If it occurs, the direct Urca process enhances neutrino emission and neutron star cooling rates by a large factor compared to any process considered previously. Direct Urca processes with hyperons and/or nucleon isobars can also occur in dense matter as long as the concentration of Λ hyperons exceeds a critical value that is less than 3% and is typically about 0.1%. The neutrino luminosities from the hyperon Urca processes are about 5–100 times less than the typical luminosity from the nucleon direct Urca process, but they are larger than those expected from other sources. These new direct Urca processes provide avenues for rapid cooling of neutron stars which invoke neither exotic states nor the large proton fraction required for the nucleon direct Urca process. It is thus likely that all neutron stars will cool rapidly, whether they contain the so-called exotic matter or not. If so, the ramifications are many and are briefly discussed.
We investigate the structure of neutron stars shortly after they are born, when the entropy
per baryon is of order 1 or 2 and neutrinos are trapped on dynamical timescales. We find that
the structure depends more sensitively on the composition of the star than on its entropy, and
that the number of trapped neutrinos play an important role in determining the composition.
Since the structure is chiefly determined by the pressure of the strongly interacting constituents
and the nature of the strong interactions is poorly understood at high density, we consider
several models of dense matter, including matter with strangeness-rich hyperons, a kaon condensate
and quark matter.
In all cases, the thermal effects for an entropy per baryon of order 2 or less are small when
considering the maximum neutron star mass. Neutrino trapping, however, significantly changes
the maximum mass due to the abundance of electrons. When matter is allowed to contain only
nucleons and leptons, trapping decreases the maximum mass by an amount comparable to, but
somewhat larger than, the increase due to finite entropy. When matter is allowed to contain
strongly interacting negatively charged particles, in the form of strange baryons, a kaon
condensate, or quarks, trapping instead results in an increase in the maximum mass, which adds
to the effects of finite entropy. A net increase of order 0.2M_{⊙} occurs.
The presence of negatively-charged particles has two major implications for the neutrino
signature of gravitational collapse supernovae. First, the value of the maximum mass will
decrease during the early evolution of a neutron star as it loses trapped neutrinos, so that
if a black hole forms, it either does so immediately after the bounce (accretion being completed
in a second or two) or it is delayed for a neutrino diffusion timescale of ~ 10 s. The latter
case is most likely if the maximum mass of the hot star with trapped neutrinos is near 1.5M_{⊙}.
In the absence of negatively-charged hadrons, black hole formation would be due to accretion and
therefore is likely to occur only immediately after bounce. Second, the appearance of hadronic
negative charges results in a general softening of the equation of state that may be observable
in the neutrino luminosities and average energies. Further, these additional negative charges
decrease the electron fraction and may be observed in the relative excess of electron neutrinos
compared to other neutrinos.
The equation of state (EOS) of dense matter plays an important role in the supernova phenomenon, the structure of neutron stars, and in the mergers of compact objects (neutron stars and black holes). During the collapse phase of a supernova, the EOS at subnuclear densities controls the collapse rate, the amount of deleptonization and thus the size of the collapsing core and the bounce density. Properties of nuclear matter that are especially crucial are the symmetry energy and the nuclear specific heat. The nuclear incompressibility, and the supernuclear EOS, play supporting roles. In a similar way, although the maximum masses of neutron stars are entirely dependent upon the supernuclear EOS, other important structural aspects are more sensitive to the equation of state at nuclear densities. The radii, moments of inertia, and the relative binding energies of neutron stars are, in particular, sensitive to the behavior of the nuclear symmetry energy. The dependence of the radius of a neutron star on its mass is shown to critically influence the outcome of the compact merger of two neutron stars or a neutron star with a small mass black hole. This latter topic is especially relevant to this volume, since it stems from research prompted by the tutoring of David Schramm a quarter century ago.
Even the elusive neutrinos are trapped in matter, albeit transiently, in several astrophysical circumstances. Their interactions with the ambient matter not only reveal the properties of such exotic matter, but also shed light on the fundamental properties of the neutrinos. The physical sites of interest include the early universe, supernovae, and newly born neutron stars. Detection of neutrinos from these vastly different eras using the new generation of neutrino detectors holds great promise for enhancing our understanding of neutrino-matter interactions and astrophysical phenomena.
Neutron stars are some of the densest manifestations of massive objects in the universe. They are ideal astrophysical laboratories for testing theories of dense matter physics and provide connections among nuclear physics, particle physics, and astrophysics. Neutron stars may exhibit conditions and phenomena not observed elsewhere, such as hyperon-dominated matter, deconfined quark matter, superfluidity and superconductivity with critical temperatures near 1010 kelvin, opaqueness to neutrinos, and magnetic fields in excess of 1013 Gauss. Here, we describe the formation, structure, internal composition, and evolution of neutron stars. Observations that include studies of pulsars in binary systems, thermal emission from isolated neutron stars, glitches from pulsars, and quasi-periodic oscillations from accreting neutron stars provide information about neutron star masses, radii, temperatures, ages, and internal compositions.
The roles of isospin asymmetry in nuclei and neutron stars are investigated using a range of potential and field-theoretical models of nucleonic matter. The parameters of these models are fixed by fitting the properties of homogeneous bulk matter and closed-shell nuclei. We discuss and unravel the causes of correlations among the neutron skin thickness in heavy nuclei, the pressure of beta-equilibrated matter at a density of 0.1fm^{-3}, the derivative of the nuclear symmetry energy at the same density and the radii of moderate mass neutron stars. Constraints on the symmetry properties of nuclear matter from the binding energies of nuclei are examined. The extent to which forthcoming neutron skin measurements will further delimit the symmetry properties is investigated. The impact of symmetry energy constraints for the mass and moment of inertia contained within neutron star crusts and the threshold density for the nucleon direct Urca process, all of which are potentially measurable, is explored. We also comment on the minimum neutron star radius, assuming that only nucleonic matter exists within the star.
We investigate how current and proposed observations of neutron stars can lead to an
understanding of the state of their interiors and the key unknowns: the typical neutron
star radius and the neutron star maximum mass. We consider observations made not only with
photons, ranging from radio waves to X-rays, but also those involving neutrinos and gravity
waves. We detail how precision determinations of structural properties would lead to significant
restrictions on the poorly understood equation of state near and beyond the equilibrium density
of nuclear matter.
To begin, a theoretical analysis of neutron star structure, including general relativistic
limits to mass, compactness, and spin rates is made. A review is the made of recent observations
such as pulsar timing (which leads to mass, spin period, glitch and moment of inertia estimates),
optical and X-ray observations of cooling neutron stars (which lead to estimates of core
temperatures and ages and inferences about the internal composition), and X-ray observations of
accreting and bursting sources (which shed light on both the crustal properties and internal
composition). Next, we discuss neutrino emission from proto-neutron stars and how neutrino
observations of a supernova, from both current and planned detectors, might impact our knowledge
of the interiors, mass and radii of neutron stars. We also explore the question of how
superstrong magnetic fields could affect the equation of state and neutron star structure.
This is followed by a look at binary mergers involving neutron stars and how the detection of
gravity waves could unambiguously distinguish normal neutron stars from self-bound strange
quark matter stars.
This work extends the seminal work of Gottfried on the two-body quantum physics of particles interacting through a delta-shell potential to many-body physics by studying a system of non-relativistic particles when the thermal De-Broglie wavelength of a particle is smaller than the range of the potential and the density is such that average distance between particles is smaller than the range. The ability of the delta-shell potential to reproduce some basic properties of the deuteron are examined. Relations for moments of bound states are derived. The virial expansion is used to calculate the first quantum correction to the ideal gas pressure in the form of the second virial coefficient. Additionally, all thermodynamic functions are calculated up to the first order quantum corrections. For small departures from equilibrium, the net flows of mass, energy and momentum, characterized by the coefficients of diffusion, thermal conductivity and shear viscosity, respectively, are calculated. Properties of the gas are examined for various values of physical parameters including the case of infinite scattering length when the unitary limit is achieved.
Neutron stars have long been regarded as extra-terrestrial laboratories from which we can learn about extreme energy density matter at low temperatures. In this article, I highlight some of the recent advances made in astrophysical observations and related theory. Although the focus is on the much needed information on masses and radii of several individual neutron stars, the need for additional knowledge about the many facets of neutron stars is stressed. The extent to which quark matter can be present in neutron stars is summarized with emphasis on the requirement of non-perturbative treatments. Some longstanding and new questions, answers to which will advance our current status of knowledge, are posed.
Recent developments in the theory of pure neutron matter and experiments concerning the symmetry energy of nuclear matter, coupled with recent measurements of high-mass neutron stars, now allow for relatively tight constraints on the equation of state of dense matter. We review how these constraints are formulated and describe the implications they have for neutron stars and core-collapse supernovae. We also examine thermal properties of dense matter, which are important for supernovae and neutron star mergers, but which cannot be nearly as well constrained at this time by experiment. In addition, we consider the role of the equation of state in medium-energy heavy-ion collisions.
The major areas of Prakash′s research are: (1) the interface between nuclear theory and nuclear, neutrino and gravitational astrophysics, and (2) the extreme energy density physics encountered in the collisions of highly energetic nuclei as at the relativistic heavy-ion colliders. Questions addressed in connection with astrophysical phenomena include: (i) How do new theoretical calculations of the equation of state and neutrino processes in dense matter impact numerical simulations of supernovae, proto-neutron stars and cooling neutron stars? (ii) How do astronomical observations of supernovae, neutron stars (including pulsars) and black holes delineate dense matter properties, such as its symmetry energy, specific heat and compressibility? Do exotic phases that contain hyperons, Bose condensates or deconfined quark matter exist in observable dense matter? Do they have distinct signatures? (iii) How do nuclear experiments, such as those involving rare isotope accelerators, heavy ion collisions, and parity-violating electron scattering reactions, restrict the parameters of nuclear equation of state models? (iv) How do gravitational wave detections elucidate the properties of dense baryonic matter? In the area of relativistic heavy-ion collisions (RHIC and LHC), Prakash addresses issues related with (i) How do fundamental interactions between quarks and hadrons determine their thermal and transport properties? How are these properties manisfested in observables of heavy-ion collisions? Prakash works in concert with the Joint Institute for Nuclear Astrophysics -- Center for the Evolution of the Elements, or JINA-CEE, of which the INPP is an affiliate member. His role in this collaboration is to provide microphysical inputs (to large-scale computer simulations of astrophysical phenomena) on the equation of state of and neutrino reaction rates in the dense matter encountered in core collapse supernovae, young and old neutron stars, and, binary mergers involving neutron stars and black holes.
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The major areas of Prakash′s research are: (1) the interface between nuclear theory and nuclear, neutrino and gravitational astrophysics, and (2) the extreme energy density physics encountered in the collisions of highly energetic nuclei as at the relativistic heavy-ion colliders. Questions addressed in connection with astrophysical phenomena include: (i) How do new theoretical calculations of the equation of state and neutrino processes in dense matter impact numerical simulations of supernovae, proto-neutron stars and cooling neutron stars? (ii) How do astronomical observations of supernovae, neutron stars (including pulsars) and black holes delineate dense matter properties, such as its symmetry energy, specific heat and compressibility? Do exotic phases that contain hyperons, Bose condensates or deconfined quark matter exist in observable dense matter? Do they have distinct signatures? (iii) How do nuclear experiments, such as those involving rare isotope accelerators, heavy ion collisions, and parity-violating electron scattering reactions, restrict the parameters of nuclear equation of state models? (iv) How do gravitational wave detections elucidate the properties of dense baryonic matter? In the area of relativistic heavy-ion collisions (RHIC and LHC), Prakash addresses issues related with (i) How do fundamental interactions between quarks and hadrons determine their thermal and transport properties? How are these properties manisfested in observables of heavy-ion collisions? Prakash works in concert with the Joint Institute for Nuclear Astrophysics -- Center for the Evolution of the Elements, or JINA-CEE, of which the INPP is an affiliate member. His role in this collaboration is to provide microphysical inputs (to large-scale computer simulations of astrophysical phenomena) on the equation of state of and neutrino reaction rates in the dense matter encountered in core collapse supernovae, young and old neutron stars, and, binary mergers involving neutron stars and black holes.
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