Compressible Liquid Drop Research Articles

Inference of neutron-star properties with unified crust-core equations of state for parameter estimation

Context. Relating different global neutron-star (NS) properties, such as tidal deformability and radius, or mass and radius, requires an equation of state (EoS). Determining the NS EoS is therefore not only the science goal of a variety of observational projects, but it also enters in the analysis process; for example, to predict a NS radius from a measured tidal deformability via gravitational waves (GW) during the inspiral of a binary NS merger. To this aim, it is important to estimate the theoretical uncertainties on the EoS, one of which is the possible bias coming from an inconsistent treatment of the low-density region; that is, the use of a so called non-unified NS crust. Aims. We propose a numerical tool allowing the user to consistently match a nuclear-physics informed crust to an arbitrary high-density EoS describing the core of the star. Methods. We introduce an inversion procedure of the EoS close to saturation density that allows users to extract nuclear-matter parameters and extend the EoS to lower densities in a consistent way. For the treatment of inhomogeneous matter in the crust, a standard approach based on the compressible liquid-drop (CLD) model approach was used in our work. A Bayesian analysis using a parametric agnostic EoS representation in the high-density region is also presented in order to quantify the uncertainties induced by an inconsistent treatment of the crust. Results. We show that the use of a fixed, realistic-but-inconsistent model for the crust causes small but avoidable errors in the estimation of global NS properties and leads to an underestimation of the uncertainties in the inference of NS properties. Conclusions. Our results highlight the importance of employing a consistent EoS in inference schemes. The numerical tool that we developed to reconstruct such a thermodynamically consistent EoS, CUTER, has been tested and validated for use by the astrophysical community.

Unified neutron star equations of state calibrated to nuclear properties

Recently, a dataset of several equations of state (EOSs) for purely nucleonic stellar matter based on a nonlinear relativistic mean-field model prescription and constrained to properties of nuclear matter, state-of-the-art chiral effective-field theory calculations for low-density neutron matter, and astrophysical data were proposed. In this work, 21 unified neutron star EOSs were chosen from that dataset in such a way that a large range of values of the slope of the symmetry energy at saturation is covered. Several quantities are calculated and discussed, such as the proton fraction and the direct Urca behavior, the density dependence of the speed of sound and the trace anomaly, the crust-core transition properties, the compatibility with astrophysical observations, and the neutron matter properties from chiral effective-field theory calculations and pQCD constraints. We construct unified EOSs where the outer crust is given by the BSk22 functional and the inner crust is calculated from a compressible liquid drop approximation. The core is purely nucleonic; made of protons, neutrons, electrons, and muons; under charge neutrality; and in beta -equilibrium conditions. The correlation of the slope of the symmetry energy at saturation with the crust-core transition density and proton fraction is analyzed, and equations that translate these relations are proposed. Moreover, the spectral representation for all the EOSs is given, which is a convenient representation to study quasi-periodic oscillations with realistic EOSs. We show that several of these EOSs have in the center of the most massive neutron star a speed of sound squared on the order of $ 0.5$. Most of the EOSs predict a maximum central density on the order of about six times the nuclear saturation density. Three of the EOSs satisfy all of the constraints imposed. The 21 unified EOSs are available in the zenodo platform.

Symmetry Energy in the Inner Crust Region of Neutron Stars: A Study Around the Neutron Drip Density Point using the Compressed Liquid Drop Model

The properties of neutron star’s inner crust have been investigated using the Compressible Liquid Drop Model, particularly around the neutron drip density region, which is the boundary between the outer and inner crust. Symmetry energy in the equations of state SLy4 and BSk3 was calculated to determine the density data between the outer and inner crusts. The properties of the inner crust can be understood through parameters such as the change in the number of nucleons in the atomic nucleus, the asymmetry parameter in surface energy, and volume energy. It was shown that the choice of the symmetry energy expansion coefficient (L) of order-1 results in a distinct energy range of 9-11 MeV within the neutron drip region. This contrasts significantly with the energy symmetry observed around saturation density, which reaches (30 ± 4) MeV in the reference models used. Furthermore, it was found that symmetry energy affects the neutron composition in the inner crust. As the density increases, neutron numbers rise, while proton counts exhibit relative stability within the range of 40 to 50 for each atomic nucleus. Importantly, we observe a marked decrease in proton fraction at the onset of the neutron drip region, where ???? ≅10 MeV, suggesting electron capture processes transforming protons into neutrons. This phenomenon contributes to the presence of neutron and free neutron gas layers within the inner crust.

Strong magnetic fields and pasta phases reexamined

In this work, we compute the structure and composition of the inner crust of a neutron star in the presence of a strong magnetic field, such as it can be found in magnetars. To determine the geometry and characteristics of the crust inhomogeneities, we consider the compressible liquid drop model, where surface and Coulomb terms are included in the variational equations, and we compare our results with previous calculations based on more approximate treatments. For the equation of state (EoS), we consider two non-linear relativistic mean-field models with different slopes of the symmetry energy, and we show that the extension of the inhomogeneous region inside the star core due to the magnetic field strongly depends on the behavior of the symmetry energy in the crustal EoS. Finally, we argue that the extended spinodal instability observed in previous calculations can be related to the presence of small amplitude density fluctuations in the magnetar outer core, rather than to a thicker solid crust. The compressible liquid drop model formalism, while in overall agreement with the previous calculations, leads to a systematic suppression of the metastable solutions, thus allowing a more precise estimation of the crust-core transition density and pressure, and therefore a better estimation of the crustal radius.

The proto-neutron star inner crust in the liquid phase

Context. The crust of a neutron star is known to melt at a temperature that increases with increasing matter density, up to about 1010 K. At such high temperatures and beyond, the crustal ions are put into collective motion and the associated entropy contribution can affect both the thermodynamic properties and the composition of matter. Aims. We studied the importance of this effect in different thermodynamic conditions relevant to the inner crust of the proto-neutron star, both at beta equilibrium and in the fixed-proton-fraction regime. Methods. To this aim, we solved the hydrodynamic equations for an ion moving in an incompressible, irrotational, and non-viscous fluid, with different boundary conditions, thus leading to different prescriptions for the ion effective mass. We then employed a compressible liquid-drop approach in the one-component plasma approximation, including the renormalisation of the ion mass to account for the influence of the surrounding medium. Results. We show that the cluster size is determined by the competition between the ion centre-of-mass motion and the interface properties, namely the Coulomb, surface, and curvature energies. In particular, including the translational free energy in the minimisation procedure can significantly reduce the optimal number of nucleons in the clusters and lead to an early dissolution of clusters in dense beta-equilibrated matter. On the other hand, we find that the impact of translational motion is reduced in scenarios where the proton fraction is assumed constant and is almost negligible on the inner-crust equation of state. Conclusions. Our results show that the translational degrees of freedom affect the equilibrium composition of beta-equilibrated matter and the density and pressure of the crust-core transition in a non-negligible way, highlighting the importance of its inclusion when modelling the finite-temperature inner crust of the (proto-)neutron star.

Magnetized neutron star crust within effective relativistic mean-field model

Even though the crystallize nature of the neutron star crust plays a pivotal role in describing various fascinating astrophysical observations, its microscopic structure is not fully understood in the presence of a colossal magnetic field. In the present work, we study the crustal properties of a neutron star within an effective relativistic mean field framework in the presence of magnetic field strength $\sim 10^{17}$G. We calculate the equilibrium composition of the outer crust by minimizing the Gibbs free energy using the most recent atomic mass evaluations. The magnetic field significantly affects the equation of state (EoS) and the properties of the outer crust, such as neutron drip density, pressure, and melting temperature. For the inner crust, we use the compressible liquid drop model for the first time to study the crustal properties in a magnetic environment. The inner crust properties, such as mass and charge number distribution, isospin asymmetry, cluster density, etc., show typical quantum oscillations (De Haas-van Alphen effect) sensitive to the magnetic field's strength. The density-dependent symmetry energy influences the magnetic inner crust like the field-free case. We study the probable modifications in the pasta structures and it is observed that their mass and thickness changes by $\sim 10-15 \%$ depending upon the magnetic field strength. The fundamental torsional oscillation mode frequency is investigated for the magnetized crust in the context of quasiperiodic oscillations (QPO) in soft gamma repeaters. The magnetic field strengths considered in this work influences only the EoS of outer and shallow regions of the inner crust, which results in no significant change in global neutron star properties. However, the outer crust mass and its moment of inertia increase considerably with increase in magnetic field strength.

Neutron star inner crust: reduction of shear modulus by nuclei finite size effect

ABSTRACT The elasticity of neutron star crust is important for adequate interpretation of observations. To describe elastic properties one should rely on theoretical models. The most widely used is Coulomb crystal model (system of point-like charges on neutralizing uniform background), in some works it is corrected for electron screening. These models neglect finite size of nuclei. This approximation is well justified except for the innermost crustal layers, where nuclei size becomes comparable with the inter-nuclear spacing. Still, even in those dense layers it seems reasonable to apply the Coulomb crystal result, if one assumes that nuclei are spherically symmetric: Coulomb interaction between them should be the same as interaction between point-like charges. This argument is indeed correct; however, as we point here, shear of crustal lattice generates (microscopic) quadrupole electrostatic potential in a vicinity of lattice cites, which induces deformation on the nuclei. We analyse this problem analytically within compressible liquid drop model. In particular, for ground state crust composition the effective shear modulus is reduced for a factor of $1-u^{5/3}/(2+3\, u-4\, u^{1/3})$, where u is the ratio of the nuclei volume to the volume of the cell. This result is universal, i.e. it does not depend on the applied nucleon interaction model within applied approach. For the innermost layers of inner crust u ∼ 0.2 leading to reduction of the shear modulus by $\sim 25{{\ \rm per\ cent}}$, which can be important for correct interpretation of quasi-periodic oscillations in the tails of magnetar flares.

Pasta Phases in Neutron Star Mantle: Extended Thomas–Fermi vs. Compressible Liquid Drop Approaches

Nuclear pasta phases in the neutron stars mantle can affect the mechanical and transport properties of superdense matter, thus playing an important role in the dynamics and evolution of neutron stars. In this paper, we compare results obtained by the Extended Thomas–Fermi (ETF) method with the compressible liquid drop model (CLDM), based on the thermodynamically consistent description of the surface properties calculated for the two-phase plane interface and the same energy-density functional (for numerical illustration, we applied the Skyrme-type functional SLy4). Our ETF calculations found that pasta phases in cylindrical form cover a significant crustal region (both normal and inverse phases, aka spaghetti and bucatini are presented). Meanwhile, within the applied CLDM framework, which includes the thermodynamically required effect of neutron adsorption on the cluster’s surface but neglects curvature corrections, only the spaghetti phase was found to be energetically favorable in the small density range prior to crust–core transition. On the other hand, the recent CLDM of Dinh Thi et al., 2021, which, on the contrary, accounts for curvature term but neglects neutron adsorption, predicts pasta phase onset in better agreement with the ETF. This fact highlights the importance of the curvature effects and allows counting on the potential validity of the CLDMs as a convenient, transparent and accurate tool for investigation of the pasta-phase properties.

From neutron skins and neutron matter to the neutron star crust

We present the first Bayesian inference of neutron star crust properties to incorporate neutron skin data, including the recent PREX measurement of the neutron skin of 208Pb, combined with recent chiral effective field theory predictions of pure neutron matter with statistical errors. Using a compressible liquid drop model with an extended Skyrme energy-density functional, we obtain the most stringent constraints to date on the transition pressure Pcc=0.33−0.07+0.07 MeV fm−3 and chemical potential μcc=12.6−1.9+1,8 MeV (which control the mass, moment of inertia and thickness of a neutron star crust), the proton fractions that bracket the pasta phases yp=0.115−0.017+0.016 and ycc=0.041−0.006+0.007, as well as the relative mass and moment of inertia ΔMp/ΔMc≈ΔIp/ΔIc=0.54−0.09+0.05 and thickness ΔRp/ΔRc=0.129−0.030+0.019 of the layers of non-spherical nuclei (nuclear pasta) in the crust.

Pasta properties of the neutron star within effective relativistic mean-field model

We study the properties of pasta structures and their influence on the neutron star observables employing the effective relativistic mean-field (E-RMF) model. The compressible liquid drop model is used to incorporate the finite size effects, considering the possibility of nonspherical structures in the inner crust. The unified equation of states are constructed for several E-RMF parameters to study various properties such as pasta mass and thickness in the neutron star's crust. The majority of the pasta properties are sensitive to the symmetry energy in the subsaturation density region. Using the results from Monte Carlo simulations, we estimate the shear modulus of the crust in the context of quasiperiodic oscillations from soft gamma-ray repeaters and calculate the frequency of fundamental torsional oscillation mode in the inner crust. Global properties of the neutron star such as mass-radius profile, the moment of inertia, crustal mass, crustal thickness, and fractional crustal moment of inertia are worked out. The results are consistent with various observational and theoretical constraints.

Partially accreted crusts of neutron stars

Context. Neutron stars in low-mass binary systems are subject to accretion when material originating from the companion star accumulates on the surface. In most cases, the justified and common assumption in studying the properties of the neutron star crust is the fully accreted crust approximation. However, observations of some X-ray transient sources indicate that the original crust has not been completely replaced by accreted material, but is partly composed of the compressed original crust. Aims. The crust of an accreting neutron star beyond the fully accreted crust approximation was studied; a two-part (or hybrid) crust made of the original crust that is compressed and of the accreted material crashing onto it was reconstructed as a function of the accretion stage. The differences in the composition and energy sources for the fully accreted and hybrid crusts influence the cooling and transport properties. Methods. A simple semi-empirical formula of a compressible liquid drop was used to compute the equation of state and composition of the hybrid crust. Calculations were based on the single-nucleus model, with a more accurate treatment of the neutron drip point. We compared the nuclear reactions triggered by compression in the original crust and in the accreted matter part of the hybrid crust. We discuss another crust compression astrophysical phenomenon related to spinning neutron stars. Results. The compression of the originally catalyzed outer crust triggers exothermic reactions (electron captures and pycnonuclear fusions) that deposit heat in the crust. The heat sources are cataloged as a function of the compression until the fully accreted crust approximation is reached. The pressure at which neutron drip occurs is a nonmonotonic function of the depth, leading to a temporary neutron drip anomaly. The additional potential source of energy for partially accreted crusts is the occurrence of a density inversion phenomenon between some compressed layers. Conclusions. The original crust of a neutron star cannot be neglected for the initial period of accretion, when the original crust is not fully replaced by the accreted matter. The amount of heat associated with the compression of the original crust is on the same order of magnitude as that from the sources acting in the accreted part of the hybrid crust.

Crustal properties of a neutron star within an effective relativistic mean-field model

We use the effective relativistic mean-field (E-RMF) model to study the crustal properties of the neutron star. The unified equations of state (EoS) are constructed using recently developed E-RMF parameter sets, such as FSUGarnet, IOPB-I, and G3. The outer crust composition is determined using the atomic mass evaluation 2020 data [Huang et al. Chin. Phys. C 45, 030002 (2021) along with the available Hartree-Fock-Bogoliubov mass models [Goriely et al. Phys. Rev. C 88, 024308 (2013) for neutron-rich nuclei. The structure of the inner crust is estimated by performing the compressible liquid drop model calculations using the same E-RMF functional as that for the uniform nuclear matter in the liquid core. Various neutron star properties such as mass-radius ($M\ensuremath{-}R$) relation, the moment of inertia ($I$), the fractional crustal moment of inertia (${I}_{\text{crust}}/I$), mass (${M}_{\text{crust}}$) and thickness (${l}_{\text{crust}}$) of the crust are calculated with three unified EoSs. The crustal properties are found to be sensitive to the density-dependent symmetry energy and slope parameter advocating the importance of the unified treatment of neutron star EoS. The three unified EoSs, IOPB-I-U, FSUGarnet-U, and G3-U, reproduced the observational data obtained with different pulsars, NICER, and glitch activity and are found suitable for further description of the structure of the neutron star.

Non-spherical nucleon clusters in the mantle of a neutron star: CLDM based on Skyrme-type forces

Neutron stars are superdense compact astrophysical objects. The central region of the neuron star (the core) consists of locally homogeneous nuclear matter, while in the outer region (the crust) nucleons are clustered. In the outer crust these nuclear clusters represent neutron-rich atomic nuclei and all nucleons are bound within them. Whereas in the inner crust some neutrons are unbound, but nuclear clusters still keeps generally spherical shape. Here we consider the region between the crust and the core of the star, so-called mantle, where non-spherical nuclear clusters may exist. We apply compressible liquid drop model to calculate the energy density for several shape types of nuclear clusters. It allows us to identify the most energetically favorable configuration as function of baryon number density. Employing four Skyrme-type forces (SLy4 and BSk24, BSk25, BSk26), which are widely used in the neutron star physics, we faced with strong model dependence of the ground state composition. In particular, in agreement with previous works within liquid drop model, mantle is absent for SLy4 (nuclear spheres directly transit into homogeneous nuclear matter; exotic nuclear shapes do not appear).

Prior Probability Distributions of Neutron Star Crust Models

Abstract To make best use of multifaceted astronomical and nuclear data sets, probability distributions of neutron star models that can be used to propagate errors consistently from one domain to another are required. We take steps toward a consistent model for this purpose, highlight where model inconsistencies occur, and assess the resulting model uncertainty. Using two distributions of nuclear symmetry energy parameters—one uniform, the other based on pure neutron matter theory—we prepare two ensembles of neutron star inner crust models. We use an extended Skyrme energy density functional within a compressible liquid drop model (CLDM). We fit the surface parameters of the CLDM to quantum 3D Hartree–Fock calculations of crustal nuclei. All models predict that more than 50% of the crust by mass and 15% of the crust by thickness comprises pasta with medians of around 62% and 30%, respectively. We also present 68% and 95% ranges for the crust composition as a function of density. We examine the relationships between crust–core boundary and pasta transition properties, the thickness of the pasta layers, the symmetry energy at saturation and subsaturation densities, and the neutron skins of 208Pb and 48Ca. We quantify the correlations using the maximal information coefficient, which can effectively characterize nonlinear relationships. Future measurements of neutron skins, information from nuclear masses and giant resonances, and theoretical constraints on PNM will be able to place constraints on the location of the pasta and crust–core boundaries and the amount of pasta in the crust.

Inner crust of a neutron star at the point of crystallization in a multicomponent approach (Corrigendum)

The possible presence of amorphous and heterogeneous phases in the inner crust of a neutron star is expected to reduce the electrical conductivity of the crust, with potentially important consequences on the magneto-thermal evolution of the star. In cooling simulations, the disorder is quantified by an impurity parameter which is often taken as a free parameter. We aim to give a quantitative prediction of the impurity parameter as a function of the density in the crust,performing microscopic calculations including up-to-date microphysics of the crust. A multi-component approach is developed at finite temperature using a compressible liquid drop description of the ions with an improved energy functional based on recent microscopic nuclear models and optimized on extended Thomas-Fermi calculations. Thermodynamic consistency is ensured by adding a rearrangement term and deviations from the linear mixing rule are included in the liquid phase. The impurity parameter is consistently calculated at the crystallization temperature as determined in the one-component plasma approximation for the different functionals. Our calculations show that at the crystallization temperature the composition of the inner crust is dominated by nuclei with charge number around $Z \approx 40$, while the range of the $Z$ distribution varies from about 20 near the neutron drip to about 40 closer to the crust-core transition. This reflects on the behavior of the impurity parameter that monotonically increases with density up to around 40 in the deeper regions of the inner crust. Our study shows that the contribution of impurities is non-negligible, thus potentially having an impact on the transport properties in the neutron-star crust. The obtained values of the impurity parameter represent a lower limit; larger values are expected in the presence of non-spherical geometries and/or fast cooling dynamics.

Surface energy of nuclear matter above and below neutron drip with the Skyrme-type effective interactions

Neutron star crust consists of highly neutron excess nuclei, which are inaccessible for laboratory experiments. In the deepest region of the crust (so-called inner crust, located after the neutron drip) atomic nuclei are immersed into the sea of degenerate unbound neutrons. Study of the crust structure and equation of state in such conditions relies on the theoretical nuclear mass models. In particular, it is convenient to use the compressible liquid drop model, which contains the surface tension term. A thermodynamically consistent description must take into account adsorption of neutrons on the nucleus surface (neutron skin) and dependence of surface tension on matter properties in two-phase equilibrium. We calculate the surface tension of nuclear matter by the extended Thomas-Fermi approach. For this aim, we parametrize the number density profile of the two-phase system by Fermi-Dirac type functions, totally containing 5 parameters, and minimize the thermodynamic potential Ω to obtain equilibrium configuration. We use Skyrme-type nuclear interactions SLy4 and BSk24, fulfilling experimental data of atomic nuclei, observational constraints on the maximal neutron star mass and theoretical calculations of high-density nuclear matter. The results are presented as a function of neutron chemical potential, which is useful for compressible liquid drop models in the inner crust of a neutron star.

Nuclear pasta in hot and dense matter and its influence on the equation of state for astrophysical simulations

We explore the properties of nuclear pasta appearing in supernova matter, i.e., matter at finite temperature with a fixed proton fraction. The pasta phases with a series of geometric shapes are studied using the compressible liquid-drop (CLD) model, where nuclear matter separates into a dense liquid phase of nucleons and a dilute gas phase of nucleons and $\alpha$ particles. The equilibrium conditions for two coexisting phases are derived by minimization of the total free energy including the surface and Coulomb contributions, which are clearly different from the Gibbs conditions for phase equilibrium due to the finite-size effects. Compared to the results considering only spherical nuclei, the inclusion of pasta phases can delay the transition to uniform matter and enlarge the region of nonuniform matter in the phase diagram. The thermodynamic quantities obtained in the present calculation with the CLD model are consistent with those in the realistic equation of state table for astrophysical simulations using the Thomas--Fermi approximation. It is found that the density ranges of various pasta shapes depend on both the temperature $T$ and the proton fraction $Y_p$. Furthermore, the nuclear symmetry energy and its density dependence may play crucial roles in determining the properties of pasta phases. Our results suggest that the pasta phase diagram is most sensitively dependent on the symmetry energy slope $L$ especially in the low-$Y_p$ and high-$T$ region.

Crystallization of the inner crust of a neutron star and the influence of shell effects

Context. In the cooling process of a non-accreting neutron star, the composition and properties of the crust are thought to be fixed at the finite temperature where nuclear reactions fall out of equilibrium. A lower estimate for this temperature is given by the crystallization temperature, which can be as high as ≈7 × 109 K in the inner crust, potentially leading to sizeable differences with respect to the simplifying cold-catalyzed matter hypothesis. Aims. We extend a recent work on the outer crust to the study of the crystallization of the inner crust and the associated composition in the one-component plasma approximation. Methods. The finite temperature variational equations for non-uniform matter in both the liquid and the solid phases are solved using a compressible liquid-drop approach with parameters optimized on four different microscopic models that cover current uncertainties in nuclear modeling. Results. We consider the effect of the different nuclear ingredients with their associated uncertainties separately: the nuclear equation of state, the surface properties in the presence of a uniform gas of dripped neutrons, and the proton shell effects arising from the ion single-particle structure. Our results suggest that the highest source of model dependence comes from the smooth part of the nuclear functional. Conclusions. We show that shell effects play an important role at the lowest densities close to the outer crust, but the most important physical ingredient to be settled for a quantitative prediction of the inner crust properties is the surface tension at extreme isospin values.

Parametrization of the surface energy in the ETF approximation

We perform extended Thomas–Fermi calculations in the Wigner–Seitz cell below and above neutron drip with realistic functionals. The resulting energy density is decomposed as a sum of bulk terms and a surface term, and a compressible liquid drop analytical formula is used to fit the surface tension. The effect of curvature terms and neutron skin is studied in detail. A very good reproduction of the microscopic data is obtained using an expression that depends only on the mass and charge of the cluster, showing that the in-medium modifications of the nuclear energy in the presence of an external neutron gas can be effectively accounted for in the isospin dependence of the surface tension. In this first application, aimed at establishing the fitting protocol, we concentrate on the Sly4 energy functional, but the study can be easily generalized to different functionals, and the resulting parametrizations can be used for direct applications in pasta calculations in neutron star crusts and supernova matter.

Equation of state and composition of the inner crust of an accreting neutron star: multicomponent model

Correct interpretation of X-ray observations of transiently accreting neutron stars requires modeling of nuclear-physical processes in these objects. We consider a chain of nuclear reactions that drives the crust composition in an accreting neutron star and heats up the star. We constructed multicomponent approach with the kinetics of nuclear reactions described in simplified stepwise manner. The redistribution of nucleons between nuclei by emission and capture of neutrons is shown to significantly affect the nuclear reaction chains and the composition of the inner crust. In particular, even if the outer crust has one-component composition, the appearance of free neutrons in the inner crust leads to branching of reaction chains and formation of the multicomponent composition. We apply the compressible liquid drop nuclear model, which includes effects of free neutrons on nuclear energies. It allows us to calculate the composition, the heating profile and the equation of state of matter up to densities ρ ≃ 2 × 1013 g cm−3.