CompOSE reference manual

Abstract

CompOSE (CompStar Online Supernovae Equations of State) is an online repository of equations of state (EoS) for use in nuclear physics and astrophysics, e.g., in the description of compact stars or the simulation of core-collapse supernovae and neutron-star mergers, see https://compose.obspm.fr. The main services, offered via the website, are: a collection of data tables in a flexible and easily extendable data format for different EoS types and related physical quantities with extensive documentation and referencing; software for download to extract and to interpolate these data and to calculate additional quantities; webtools to generate EoS tables that are customized to the needs of the users and to illustrate dependencies of various EoS quantities in graphical form. This manual is an update of previous versions that are available on the CompOSE website, at arXiv:1307.5715 [astro-ph.SR], and that was originally published in the journal “Physics of Particles and Nuclei” with doi:10.1134/S1063779615040061. It contains a detailed description of the service, containing a general introduction as well as instructions for potential contributors and for users. Short versions of the manual for EoS users and providers will also be available as separate publications.

Graphical Abstract

1 Preliminary Remark

This CompOSE reference manual, version 3.01, is an updated and extended version of previous manuals that appeared as a preprint arXiv:1307.5715 [astro-ph.SR] on arXiv.org. and as a refereed article in the journal “Physics of Particles and Nuclei” [1]. The content of this publication was adapted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Physics of Particles and Nuclei, Vol. 46, No. 4, 633–664 (“CompOSE CompStar online supernova equations of state harmonising the concert of nuclear physics and astrophysics compose.obspm.fr”, S. Typel, M. Oertel, and T. Klähn), Copyright Pleiades Publishing, Ltd. (2015). Previous versions are available on the CompOSE website compose.obspm.fr.

2 Part I. Introduction

3 What CompOSE can do and what not

The online service CompOSE provides information and data tables for different equations of state (EoS) ready for further use in astrophysical applications, nuclear physics and beyond. See reviews [2,3,4,5] for general introductions.

In general, EoS are available from more-or-less sophisticated theoretical models or parametric descriptions using different representations of the data. In some cases, e.g., for the calculation of properties of cold neutrons stars, simple piecewise polytropes, speed-of-sound parameterizations or other functional forms with a dependence on a single variable like the baryon density are used. This EoS class allows a fast computation of the required quantities and a simple change of the parameters is often possible. But these approaches often lack details on the underlying physics and additional information on the type and composition of matter. For EoS with a dependence on more than one variable, such representations are more difficult to develop, in particular when thermodynamic consistency should be satisfied. On the other hand, results from most theoretical models are usually given only on a grid of certain variables. Data from these approaches are collected in tables of various forms that cannot provide simple functional representations. In this case, a change of model parameters is not easily accomplished. CompOSE focuses on this tabular EoS approach and provides a unified format for disseminating the data. Nevertheless, the incorporation of EoS data using functional representations into CompOSE is under discussion.

CompOSE has three major purposes:

• CompOSE is a repository of EoS tables in a common format for direct usage with information on a large number of thermodynamic properties, on the chemical composition of dense matter and, if available, on microphysical quantities of the constituents.

• CompOSE allows to interpolate the provided tables using different schemes to obtain the relevant quantities, selected by the user, for grids that are tailored to specific applications.

• CompOSE can provide information on additional thermodynamic quantities, which are not stored in the original data tables, and on further quantities, which characterize an EoS, such as nuclear matter parameters and compact star properties.

The format of the files, as well as the calculational mesh, is mainly determined according to the needs of scientific groups performing extensive numerical simulations of astrophysical objects. We try to provide the tables in a large parameter space to cover most applications.

We cannot offer an online service for all features of CompOSE, e.g. to run all codes online. This is mostly due to limitations in storage and computation times, but also gives better control on avoiding unphysical input parameters. However, we offer several computational tools that allow the user to extract the data from the tables that are relevant for her/him. These tools can be downloaded from the CompOSE website

https://compose.obspm.fr .

Appendix E describes how to cite CompOSE and the EoS tables it contains.

CompOSE is designed in a modular way, thus allowing to extend the service over time. More and more models and parameter sets will be provided in time. It is foreseen that additional features will be added in the future, too.

4 How to read this document

While reading this document please always keep in mind: this document was written by physicists for physicists. It is divided into three major parts.

The first one is relevant both to providers and users of equations of state, as it serves as a basis for the discussions in the following parts of the manual and the web site. Both, contributors and users, should first of all have a look at the introductory chapter 3 where we will discuss general conventions and the notation used throughout CompOSE. In addition, we give definitions and details on the system of units that is used within CompOSE and on physical constants that should be used in order to standardise the generation of new equations of state.

The second part concerns those persons who wish to contribute to an extension of the CompOSE data base by the active development of an EoS. CompOSE allows them to make their favourite EoS available for a broad range of astrophysical and nuclear physics applications. In part II, detailed instructions, minimal requirements and recommendations are specified for the preparation of EoS tables that can be incorporated in future versions of the CompOSE data base. If you plan to contribute your EoS, you should contact the CompOSE core team (develop.compose@obspm.fr), see Appendix D. A summary of possible future extensions of the CompOSE data base are summarized in chapter 5.

The third part concerns the users of EoS data, that are provided by CompOSE, who want to test various equations of state in their simulations of core-collapse supernovae, neutron-star mergers, and other scenarios, e.g., the calculation of mass-radius relations of cold neutron stars. The different families of provided EoS data thereby distinguish different preferred applications. In general, they can safely skip the second part and go directly to the third one. This part gives a brief introduction on nuclear matter properties relevant for the construction of an EoS, as well as a classification of different types of EoS models in chapter 6. The models can be distinguished either by using different techniques to treat the many-body system of strongly interacting particles or by assuming a different particle composition. The main aim is to give the relevant information for the interpretation of the data sheets, provided with each available EoS table on the web site. You will find remarks on the range of applicability of the various models, i.e. the range of parameters where the code is tested and/or the made approximations are still valid. Characteristic parameters of each model will be specified. For more detailed information about the physics behind each model, we refer to the original references.

In addition, the third part of the manual explains in chapter 7 how to proceed in order to download an EoS table and the computational tools. The latter allow the generation of tables with a mesh different from the original one via interpolation, in order to adapt the table to the need of the user. In addition, several thermodynamic quantities can be calculated which are not contained in the original tables. The use of the online service and web interface is described in chapter 7 too.

Table 1 Recommended values for physical constants from the 2018 CODATA evaluation and the compilation of the Particle Data Group [7]

4.1 Recent changes

Version 1.00 of this manual was published in [1] and is still available on the CompOSE web site. Version 2.00 contains some major changes, corrections and additions, in particular concerning the structure of the manual and the use of the provided software for handling data. The main changes from version 1.00 to 2.00 in the manual are:

• the numerical values of physical constants, recommended for the preparation of EoS tables, in Table 1 were updated;

• the discussion of constraints on nuclear matter properties in Sect. 6.1 has been modified;

• Sections 6.2 (Overview of models) and 6.3 (Phase transitions) have been removed since a comprehensive review [2] is available;

• the title ‘User registration’ of Sect. 7.1 has been changed to ‘Access to CompOSE’ with an adjusted description of the online service;

• the description of how to use the compose program in Sect. 7.5.2 was adapted to the new version and the description of the files eos.parameters (with a change of the structure) and eos.quantities was moved to Appendix B;

• Section 7.4.3 was moved to Appendix C and the corresponding tables with the (sub-)routines and functions were updated;

• Tables 10, 13, and 17 were amended and a new Table 12 was introduced;

• new Appendices B and C were introduced;

• Appendix D and the Acknowledgements were updated;

• the bibliography has been updated.

The main changes to the compose program are

• most vectors and matrices are allocated dynamically with the appropriate size for the particular EoS used for generating a user-defined EoS table;

• the program compose runs by default in a standard version that guides the user through the selection of output data and the tabulation scheme;

• the input files eos.parameters (with modified structure) and eos.quantities don’t have to be provided by the user because they are created automatically by running the compose program;

• an option for generating tables for given entropy per baryon instead of the temperature has been added;

• the number of output quantities has been extended.

Major changes in the current version 3.01 are:

• update of references to physical constants and nuclear properties in Sect. 3.1;

• update of physical constants in Table 1;

• distinguishing of particle and antiparticle densities as well as net number densities for fermions with extended indexing scheme, modified new Tables 2 and 3;

• discussion of conversion from nuclear to astrophysical units in Sect. 3.5;

• additional discussion of calculation of speed of sound in Sect. 3.6;

• addition of new Sect. 3.8 on transport properties;

• addition of new Sect. 3.10 on properties of neutron stars;

• extended classification of EoS tables in Sect. 4.2;

• introduction of new tables with transport properties and information on neutron stars in Sect. 4.2;

• discussion of different families of tabulated data and EoS models in Sect. 7.4;

• extended description of using EoS tables with software from CompOSE and LORENE;

• availability of web tools to calculate EoS tables and visualise the results in Sect. 7.6;

• extension of Table 11;

• update of CompOSE team in Appendix D;

• information on citing CompOSE in Appendix E.

Examples on using the compose program are given in a new “quick guide for users” with some concrete examples.

5 Definitions and notation

The equations of state in the CompOSE data base are provided under some common assumptions on the physical conditions of the matter that are specified in this chapter. In order to fix the notation in the present document, the definition of all relevant quantities is given.

5.1 Units and conventions

We use natural units $\hbar =c=k_{B}=1$ throughout this document as is customary in nuclear physics. Energies and temperatures are measured in MeV, lengths in fm. Units are given in parentheses $[\dots ]$ for all quantities when they are defined within this chapter. For conversion among your favourite units, the use of the CODATA values [7] (http://physics.nist.gov/cuu/Constants/index.html, www.codata.org) is recommended, see Table 1.¹ They should also be used in the preparation of new EoS tables to be incorporated in the CompOSE data base.

In astrophysical applications of equations of state it is often customary to change from nuclear units to units that are traditionally used in that field, in particular for temperature, pressure and energy densities. See Sect. 3.5 for examples.

For experimental binding energies of nuclei, the values of the 2016 Atomic mass evaluation [8, 9] (AME2016) or the more recent 2020 Atomic mass evaluation [10, 11] (AME2020, http://amdc.impcas.ac.cn) and corresponding updates are recommended. Ground state spin assignments should be taken from the Nubase2016 [12] or the Nubase2020 [13] evaluation of nuclear properties (http://amdc.impcas.ac.cn) and corresponding updates.

5.2 Physical conditions

Predictions for the properties of dense matter can differ considerably depending on the employed model, the considered constituents and interactions. In the CompOSE database, the equation of state is considered to describe dense matter in thermodynamic equilibrium, a short denotation of the fulfillment of several equilibrium conditions. Each of them corresponds to the existence of an intensive thermodynamic variable that applies to the full system under consideration. First of all there are the thermal and mechanic equilibrium that allow to define the temperature T and the pressure p. Chemical equilibrium means equilibrium with regard to strong, electromagnetic and weak reactions. For the latter, one has to distinguish several cases, i.e., reactions within a generation of particles, e.g., electronic leptons and muonic leptons, and reactions across flavor generations, e.g., strangeness changing reaction. The equilibrium of each type of reaction is linked to the corresponding chemical potential. These are the baryonic ($\mu _{b}$), charge ($\mu _{q}$), electron lepton ($\mu _{le}$), muon lepton ($\mu _{l\mu }$), and strangeness ($\mu _{s}$) chemical potentials.

It is assumed that all the constituents are in chemical equilibrium regardless of the time scales and reaction rates for the relevant conversion reactions mediated by strong and electromagnetic interactions. This condition leads to relations between the chemical potentials of all particles.

In contrast, an equilibrium with respect to weak interaction reactions, in particular $\beta$-equilibrium, is not supposed in general. Similarly, the chemical potentials of the massive leptons of all generations, i.e. electrons and muons (tauons are irrelevant for the considered conditions) cannot be assumed to be equal. There is an option available such that EoS tables can be provided in particular cases which also take $\beta$-equilibrium into account, reducing the number of independent parameters. Assumptions on the relation between the electron and muon chemical potentials are discussed in the description of each model separately. In almost all cases they are assumed to be identical.

For EoS models with strangeness bearing particles, e.g. hyperons or kaons, it is assumed that the strangeness chemical potential vanishes. This means that we assume the strangeness changing weak interactions to be in equilibrium. Note that this is not the case in heavy-ion collisions, where, on the contrary, strangeness is conserved, i.e., there is no net strangeness.

Except for the tables of pure hadronic or quark matter without massive leptons, also local charge neutrality is assumed to hold. Neutrinos and their contribution to thermodynamic properties are never included in the present tables. They are usually treated independently from the EoS in astrophysical simulations because a thermodynamic equilibrium can not be assumed in general.

Photons are usually included in equations of state for astrophysical applications. They only contribute at nonzero temperatures. The treatment of photons is discussed in Sect. 3.9.

5.3 Particle number densities and particle fractions

Particle number densities of all particles i are given by

$$\begin{aligned} n_{i} = \frac{N_{i}}{V} \quad [\hbox {fm}^{-3}] \end{aligned}$$

(3.1)

where $N_{i}$ [dimensionless] is the particle number inside the volume V [fm$^{3}$]. Note that for particles with half-integer spin, $n_{i}$ represents the net particle density, i.e. it is the difference between the particle and antiparticle density. E.g., for electrons we have $n_{e} = n_{e^{-}}-n_{e^{+}}$. For particles with integer spin, e.g. mesons, particle and antiparticle densities are distinguished and given separately. For particles with half-integer spin, it will be possible to give the particle and antiparticle densities separately in an upcoming version of the CompOSE program.

From the individual particle number densities several new composite number densities can be deduced that are convenient to characterize the state of the system. The baryon number density $n_{b}$ is given by

$$\begin{aligned} n_{b} = \frac{B}{V} = \sum _{i} B_{i} n_{i} \quad [\hbox {fm}^{-3}] \end{aligned}$$

(3.2)

with the baryon number $B_{i}$ [dimensionless] for a particle i and the total baryon number

$$\begin{aligned} B = \sum _{i} B_{i} \quad [\hbox {dimensionless}] \end{aligned}$$

(3.3)

inside the volume V. The baryon number of a nucleus i is just the mass number $A_{i}$ [dimensionless] and for a quark i one has $B_{i}= 1/3$. Warning: In many astrophysical applications, a mass density $\varrho = m \, n_{b}$ [MeV fm$^{-3}$] is introduced as a parameter with m representing, e.g., the neutron mass or the atomic mass unit u . But, $\varrho$ is not identical to the total mass ($=$ total internal energy ) density. Hence, $\varrho V$ is not a conserved quantity.

The strangeness number density $n_{s}$

$$\begin{aligned} n_{s} = \frac{S}{V} = \sum _{i} S_{i} n_{i} \quad [\hbox {fm}^{-3}] \end{aligned}$$

(3.4)

with the total strangeness

$$\begin{aligned} S = \sum _{i} S_{i} \quad [\hbox {dimensionless}] \end{aligned}$$

(3.5)

inside the volume V with strangeness numbers $S_{i}$ [dimensionless], as well as the lepton number densities $n_{le}$, $n_{l\mu }$ and number of leptons $N_{le}$, $N_{l\mu }$

$$\begin{aligned} n_{le} = \frac{N_{le}}{V}= & {} \sum _{i} L^{e}_{i} n_{i} \quad [\hbox {fm}^{-3}] \end{aligned}$$

(3.6)

$$\begin{aligned} n_{l\mu } = \frac{N_{l\mu }}{V}= & {} \sum _{i} L^{\mu }_{i} n_{i} \quad [\hbox {fm}^{-3}] \end{aligned}$$

(3.7)

with lepton numbers $L^{e}_{i}$, $L^{\mu }_{i}$ [dimensionless], respectively, are defined similarly to the baryon number density. Since neutrinos are not included in the CompOSE EoS, $n_{le} = n_{e}$ and $n_{l\mu }=n_{\mu }$. We also consider the charge density of strongly interacting particles $n_{q}$

$$\begin{aligned} n_{q} = \frac{Q}{V} = \sum _{i}{}^{\prime } Q_{i} n_{i} \quad [\hbox {fm}^{-3}] \end{aligned}$$

(3.8)

with electric charge numbers $Q_{i}$ [dimensionless] and the total charge

$$\begin{aligned} Q = \sum _{i}{}^{\prime } Q_{i} \quad [\hbox {dimensionless}] \end{aligned}$$

(3.9)

inside the volume V. The prime at the sum symbols indicates that the summation runs over all particles (including quarks) except leptons. For a nucleus ${}^{A_{i}}Z_{i}$, the baryon number $B_{i}$ and the charge number $Q_{i}$ are simply given by the mass number $A_{i}$ and atomic number $Z_{i}$, respectively. Tables 2 and 4 summarize baryon, strangeness, charge and lepton numbers of the most relevant particles that appear in the definition of the composite number densities.

Corresponding to the particle number densities $n_{i}$, the particle number fractions

$$\begin{aligned} Y_{i} = \frac{n_{i}}{n_{b}} \quad [\hbox {dimensionless}] \end{aligned}$$

(3.10)

are defined. Due to this definition with $n_{b}$, we have the normalization condition

$$\begin{aligned} \sum _{i}{} B_{i} Y_{i} = 1 \end{aligned}$$

(3.11)

with a summation over all particles. Note that $Y_{i}$ is not necessarily identical to the particle mass number fraction

$$\begin{aligned} X_{i} = B_{i}Y_{i} \quad [\hbox {dimensionless}] \end{aligned}$$

(3.12)

that is often introduced. We prefer to use $Y_{i}$ and not $X_{i}$ since the latter quantity is zero for all particles with baryon number $B_{i}=0$, e.g. mesons.

Because of the imposed physical conditions (see 3.2), the state of the system is uniquely characterized by the three quantities temperature T [MeV], baryon number density $n_{b}$ [fm$^{-3}$] and charge density of strongly interacting particles $n_{q}$ [fm$^{-3}$]. Charge neutrality implies $n_{q} = n_{le}+n_{l\mu }$ and the strangeness number density $n_{s}$ is fixed by the condition $\mu _{s}=0$. Instead of $n_{q}$, it is more convenient to use the charge fraction of strongly interacting particles

$$\begin{aligned} Y_{q} = \frac{n_{q}}{n_{b}} \quad [\hbox {dimensionless}] \end{aligned}$$

(3.13)

as the third independent quantity. The choice of $Y_{q}$ instead of the electron fraction

$$\begin{aligned} Y_{e} = \frac{n_{e}}{n_{b}} \quad [\hbox {dimensionless}] \end{aligned}$$

(3.14)

as a parameter is motivated by the following facts. In pure hadronic (quark) equations of state (i.e. without leptons) only $Y_{q}$ and not $Y_{e}$ is defined. In EoS models with the condition of charge neutrality that contain electrons as the only considered charged lepton, the charge fraction of strongly interacting particles $Y_{q}$ is identical to the electronic charge fraction $Y_{e}$. In models with electrons and muons, charge neutrality requires

$$\begin{aligned} Y_{q} = Y_{e}+Y_{\mu } = Y_{l} \end{aligned}$$

(3.15)

with the muon fraction

$$\begin{aligned} Y_{\mu } = \frac{n_{\mu }}{n_{b}} \quad [\hbox {dimensionless}] \end{aligned}$$

(3.16)

and the total lepton fraction $Y_{l}$ [dimensionless]. In this case, the balance between the electron and muon densities depends on the assumed relation of the electron and muon chemical potentials.

5.4 Particle indexing

The composition of dense matter can rapidly change with temperature T [MeV], baryon number density $n_{b}$ [fm${}^{-3}$] and the charge fraction of strongly interacting particles $Y_{q}$ [dimensionless]. We introduce an indexing scheme for the particles that allows to identify them uniquely in order to store only the most abundant particles in the EoS tables. In Tables 2, 3, and 4 an overview of indices for most of the relevant particles is presented. Indices of missing particles can be added on request. In the case of fermions, indices of quantities for particles and antiparticles can be given separately as well as net quantities, i.e., net particle number densities as differences of particle and antiparticle densities. Thus in Table 3 indices for net quantities are given without quantum numbers.

In addition to the various particles that can appear in dense matter, there is the possibility of strong two- or even three-particle correlations such as pairing of nucleons or quarks and phenomena such as superfluidity can arise. Hence, an indexing scheme for identifying these channels is introduced, too. In Table 5 the notation is given for these correlations. Indices of missing particles can be added on request. In the case of fermions, quantities can be given for particles and antiparticles separately as well as for the corresponding net quantities, e.g. number densities as differences of particle and antiparticle densities.

Table 2 Baryon ($B_{i}$), strangeness ($S_{i}$), charge ($Q_{i}$), lepton ($L^{e}_{i}$, $L^{\mu }_{i}$) numbers and indices $I_{i}$ of nuclei and the most relevant fermions i in dense matter. (Not yet implemented in the CompOSE program)

5.5 Thermodynamic potentials and basic quantities

All thermodynamic properties of a system are completely determined if a thermodynamic potential is known as a function of its natural variables. In the general case, this can be formulated as follows. The properties can be derived from the thermodynamic potential $\Xi = \Xi (\{x_{i}\},\{\xi _{j}\})$ depending on n natural variables $x_{i}$, $i=1,\dots ,n_{1}$ and $\xi _{j}$, $j = 1\dots , n_{2}$ with $n_{1} + n_{2} = n$. The quantities $x_{i}$ and $\xi _{i}$ represent extensive and intensive variables, respectively. The relations

$$\begin{aligned} \xi _{i} = \left. \frac{\partial \Xi }{\partial x_{i}} \right| _{x_{k}, k \ne i; \xi _{j}} = \xi _{i}(\{x_{i}\},\{\xi _{j}\}) \end{aligned}$$

(3.17)

and

$$\begin{aligned} x_{j} = -\left. \frac{\partial \Xi }{\partial \xi _{j}} \right| _{\xi _{k}, k \ne j; x_{i}} = x_{j}(\{x_{i}\},\{\xi _{j}\}) \end{aligned}$$

(3.18)

are the thermodynamic equations of state and define the variables $\xi _{i}$ that are conjugate to $x_{i}$ as first partial derivatives of $\Xi$ and vice versa. Due to Euler’s theorem on homogeneous functions, the thermodynamic potential is given by the sum

$$\begin{aligned} \Xi (\{x_{i}\},\{\xi _{j}\}) = \sum _{i=1}^{n_{1}} \xi _{i} x_{i} \end{aligned}$$

(3.19)

running over all $i=1,\dots ,n_{1}$ extensive variables that are primary variables of the thermodynamic potential $\Xi$. Thus, the knowledge of all relevant equations of state or first derivatives is sufficient to recover the thermodynamic potential $\Xi$ completely. For the mixed second partial derivatives of a thermodynamic potential $\Xi$, the result is independent of the order of differentiation and the Maxwell relations

$$\begin{aligned} \left. \frac{\partial \xi _{l}}{\partial x_{k}} \right| _{x_{j}, j\ne k} = \left. \frac{\partial \xi _{k}}{\partial x_{l}} \right| _{x_{j}, j \ne l} \end{aligned}$$

(3.20)

are obtained. For a more detailed discussion of these and further aspects we refer the reader to standard text books on thermodynamics.

In most models for the EoS of dense matter, the temperature T [MeV], the volume V [fm$^{3}$] and the individual particle numbers $N_{i}$ [dimensionless] are selected as natural variables. This case corresponds to the (Helmholtz) free energy $F=F(T,V,N_{i})$ [MeV] as the relevant thermodynamic potential that contains all information. Note that we assume that the free energy includes contributions by the rest masses of the particles. Keeping the volume V fixed, it is convenient to define the free energy density

$$\begin{aligned} f(T,n_{i}) = \frac{F}{V} \quad [\hbox {MeV fm}^{-3}] \end{aligned}$$

(3.21)

and the entropy density

$$\begin{aligned} s(T,n_{i}) = - \left. \frac{\partial f}{\partial T}\right| _{n_{i}} \quad [\hbox {fm}^{-3}] \end{aligned}$$

(3.22)

with the entropy

$$\begin{aligned} S(T,V,N_{i}) = V s(T,n_{i}) = - \left. \frac{\partial F}{\partial T}\right| _{V,N_{i}} \quad [\hbox {dimensionless}] .\nonumber \\ \end{aligned}$$

(3.23)

The chemical potential of a particle i is given by

$$\begin{aligned} \mu _{i} = \left. \frac{\partial F}{\partial N_{i}} \right| _{T,V,N_{j},j\ne i} = \left. \frac{\partial f}{\partial n_{i}} \right| _{T,n_{j},j\ne i} \quad [\hbox {MeV}] \end{aligned}$$

(3.24)

including the rest mass $m_{i}$ [MeV]. The pressure is obtained from

$$\begin{aligned} p= & {} - \left. \frac{\partial F}{\partial V} \right| _{T,N_{i}} = n_{b}^{2} \left. \frac{\partial (f/n_{b})}{\partial n_{b}} \right| _{T,Y_{q}} \nonumber \\= & {} \sum _{i} \mu _{i} n_{i} - f \quad [\hbox {MeV fm}^{-3}] \, . \end{aligned}$$

(3.25)

Each of the composite densities $n_{b}$, $n_{s}$, $n_{le}$, $n_{l\mu }$, $n_{q}$ is accompanied by a corresponding chemical potential, i.e. we have the baryon number chemical potential $\mu _{b}$ [MeV], the strangeness number chemical potential $\mu _{s}$ [MeV], the electron lepton number chemical potential $\mu _{le}$ [MeV], the muon lepton number chemical potential $\mu _{l\mu }$ [MeV], and the charge chemical potential $\mu _{q}$ [MeV]. The chemical potential of a particle i is then given by

$$\begin{aligned} \mu _{i} = B_{i} \mu _{b} + Q_{i} \mu _{q} + S_{i} \mu _{s} + L^{e}_{i} \mu _{le} + L^{\mu }_{i} \mu _{l\mu } \quad [\hbox {MeV}] \,, \end{aligned}$$

(3.26)

e.g. $\mu _{n} = \mu _{b}$, $\mu _{p}=\mu _{b}+\mu _{q}$ and $\mu _{e} = \mu _{le}-\mu _{q}$. Note again that we use the relativistic definition of chemical potentials including rest masses. It has to be mentioned that $n_{b}$, $n_{s}$, $n_{le}$ and $n_{l\mu }$ are conjugate to $\mu _{b}$, $\mu _{s}$, $\mu _{le}$ and $\mu _{l\mu }$, respectively. However, this is not true for $n_{q}$ and $\mu _{q}$ if leptons are included in the EoS. In general, it is assumed that $\mu _{s}=0$ since strangeness changing weak interaction reactions are in equilibrium. Nevertheless, the strangeness density $n_{s}$ can be non-zero if particles with strangeness are considered in the EoS.

Table 3 Indices $I_{i}$ of nuclei and the most relevant fermions i in dense matter for net quantities

In addition to the free energy density, several other thermodynamic potentials can be defined by applying Legendre transformations, e.g. the internal energy

$$\begin{aligned} E = E(S,V,N_{i}) = F+TS \quad [\hbox {MeV}] \,, \end{aligned}$$

(3.27)

the free enthalpy (Gibbs potential )

$$\begin{aligned} G = G(T,p,N_{i}) = F+pV = \sum _{i}\mu _{i}N_{i} \quad [\hbox {MeV}] \,, \end{aligned}$$

(3.28)

the enthalpy

$$\begin{aligned} H = H(S,p,N_{i}) = E+pV \quad [\hbox {MeV}] \,, \end{aligned}$$

(3.29)

and the grand canonical potential

$$\begin{aligned} \Omega= & {} \Omega (T,V,\mu _{i}) = F-\sum _{i}\mu _{i}N_{i} \nonumber \\= & {} -pV \quad [\hbox {MeV}] \end{aligned}$$

(3.30)

with the corresponding densities $e=E/V$, $g=G/V$, $h=H/V$ and $\omega = \Omega /V$ [MeV fm${}^{-3}$], respectively.

It is convenient to define the free energy per baryon

$$\begin{aligned} {\mathcal {F}} = \frac{F}{N_{b}} = \frac{f}{n_{b}} \quad [\hbox {MeV}] \,, \end{aligned}$$

(3.31)

the internal energy per baryon

$$\begin{aligned} {\mathcal {E}} = \frac{E}{N_{b}} = \frac{e}{n_{b}} \quad [\hbox {MeV}] \,, \end{aligned}$$

(3.32)

the enthalpy per baryon

$$\begin{aligned} {\mathcal {H}} = \frac{H}{N_{b}} = \frac{h}{n_{b}} \quad [\hbox {MeV}] \,, \end{aligned}$$

(3.33)

and the free enthalpy per baryon

$$\begin{aligned} {\mathcal {G}} = \frac{G}{N_{b}} = \frac{g}{n_{b}} \quad [\hbox {MeV}] \end{aligned}$$

(3.34)

by dividing the corresponding thermodynamic potential by the total number of baryons

$$\begin{aligned} N_{b} = n_{b} V \quad [\hbox {dimensionless}] \, . \end{aligned}$$

(3.35)

Table 4 Baryon ($B_{i}$), strangeness ($S_{i}$), charge ($Q_{i}$), lepton ($L^{e}_{i}$, $L^{\mu }_{i}$) numbers and indices $I_{i}$ of the most relevant bosons i in dense matter

Table 5 Baryon ($B_{i}$), strangeness ($S_{i}$), charge ($Q_{i}$) numbers and indices $I_{i}$ of the most relevant two-particle correlations i in dense matter. The lepton numbers $L^{e}_{i}$ and $L^{\mu }_{i}$ are always zero for these correlations

All equations of state can be obtained from the thermodynamic potentials $E(S,V,N_{i})$, $F(T,V,N_{i})$, $G(T,p,N_{i})$, $H(S,p,N_{i})$ or $\Omega (T,V,\mu _{i})$ by appropriate partial derivatives, in particular

$$\begin{aligned} T= & {} \left. \frac{\partial E}{\partial S} \right| _{V,N_{i}} = \left. \frac{\partial H}{\partial S} \right| _{p,N_{i}} \,, \end{aligned}$$

(3.36)

$$\begin{aligned} S= & {} -\left. \frac{\partial F}{\partial T} \right| _{V,N_{i}} = -\left. \frac{\partial G}{\partial T} \right| _{p,N_{i}} = -\left. \frac{\partial \Omega }{\partial T} \right| _{V,\mu _{i}} \,, \end{aligned}$$

(3.37)

$$\begin{aligned} -p= & {} \left. \frac{\partial E}{\partial V} \right| _{S,N_{i}} = \left. \frac{\partial F}{\partial V} \right| _{T,N_{i}} \,, \end{aligned}$$

(3.38)

$$\begin{aligned} V= & {} \left. \frac{\partial G}{\partial p} \right| _{T,N_{i}} = \left. \frac{\partial H}{\partial p} \right| _{S,N_{i}} = \left. \frac{\partial \Omega }{\partial p} \right| _{T,\mu _{i}} \,, \end{aligned}$$

(3.39)

$$\begin{aligned} \mu _{i}= & {} \left. \frac{\partial E}{\partial N_{i}} \right| _{S,V,N_{j\ne i}} = \left. \frac{\partial F}{\partial N_{i}} \right| _{T,V,N_{j\ne i}} \nonumber \\= & {} \left. \frac{\partial G}{\partial N_{i}} \right| _{T,p,N_{j\ne i}} = \left. \frac{\partial H}{\partial N_{i}} \right| _{S,p,N_{j\ne i}} \,, \end{aligned}$$

(3.40)

$$\begin{aligned} N_{i}= & {} -\left. \frac{\partial \Omega }{\partial \mu _{i}} \right| _{S,V,\mu _{j\ne i}} \, . \end{aligned}$$

(3.41)

The nuclear physics units given above can be converted easily to units that are frequently used in astrophysics. The most relevant cases are the following:

• 1. The temperature T in MeV can be expressed in K (Kelvin) by a division with the Boltzmann constant $k_{B}$ as given in Table 1. E.g., a temperature of 1 MeV corresponds to about $1.16\cdot 10^{10}$ K and, conversely, $10^{10}$ K are about 0.862 MeV.

• 2. Any energy density, e.g. the internal energy density, in MeV fm${}^{-3}$ can be expressed in the CGS unit erg/cm$^{-3}$ (1 erg = $10^{-7}$ J) by a multiplication with the factor $1.602176634\cdot 10^{33}$.

• 3. The pressure p in MeV fm$^{-3}$ can be expressed in the CGS unit dyn/cm$^{2}$ (1 dyn = 1 erg/cm) by a multiplication with the same numerical factor $1.602176634\cdot 10^{33}$ as for the energy density.

5.6 Thermodynamic coefficients

In many applications tabulated values of the thermodynamic potentials and their first derivatives are not sufficient and additional quantities that depend on second derivatives of the thermodynamic potentials are needed. In general, the relevant thermodynamic potential can depend on a large number of independent variables such as temperature, volume and particle densities or chemical potentials. In applications, however, this number often reduces due to physical constraints. E.g., the condition of weak chemical ($\beta$) equilibrium between certain particle species reduces the number of independent chemical potentials. The condition of charge neutrality relates the particle numbers of charged constituents. In the following, we consider only systems that are completely determined by the temperature T, the volume V, the total number of baryons $N_{b}= n_{b} V$ and the charge number $N_{q} = Y_{q} N_{b} = Y_{q} n_{b} V$, or in the case of an equation of state where leptons are present, the number of charged leptons, which are equal to $N_q$ due to charge neutrality. Then we have from the free energy per baryon ${\mathcal {F}}(T,n_{b},Y_{q})$ [MeV] the specific heat capacity at constant volume

$$\begin{aligned} c_{V}= & {} \frac{T}{N_{b}} \left. \frac{dS}{dT} \right| _{V,N_{b},N_{q}} \nonumber \\= & {} -T \left. \frac{\partial ^{2} {\mathcal {F}}}{\partial T^{2}}\right| _{n_{b},Y_{q}} \quad [\hbox {dimensionless}] \,, \end{aligned}$$

(3.42)

the tension coefficient at constant volume

$$\begin{aligned} \beta _{V}= & {} \left. \frac{dp}{dT} \right| _{V,N_{b},N_{q}} = \left. \frac{dS}{dV} \right| _{T,N_{b},N_{q}} \nonumber \\= & {} n_{b}^{2} \left. \frac{\partial ^{2} {\mathcal {F}}}{\partial T \partial n_{b}}\right| _{Y_{q}} \quad [\hbox {fm}^{-3}] \,, \end{aligned}$$

(3.43)

the isothermal compressibility

$$\begin{aligned} \kappa _{T}= & {} -\frac{1}{V} \left. \frac{dV}{dp} \right| _{T,N_{b},N_{q}} = \left( n_{b} \left. \frac{\partial p}{\partial n_{b}} \right| _{T,Y_{q}} \right) ^{-1}\nonumber \\= & {} \left( n_{b}^{2} \left. \frac{\partial ^{2} ({\mathcal {F}}n_{b})}{\partial n_{b}^{2}} \right| _{T,Y_{q}} \right) ^{-1} \quad [\hbox {MeV}^{-1}\hbox {fm}^{3}] \,, \end{aligned}$$

(3.44)

the expansion coefficient at constant pressure

$$\begin{aligned} \alpha _{p} = \frac{1}{V} \left. \frac{dV}{dT} \right| _{p,N_{b},N_{q}} = \kappa _{T} \beta _{V} \quad [\hbox {MeV}^{-1}] \,, \end{aligned}$$

(3.45)

the specific heat capacity at constant pressure

$$\begin{aligned} c_{p}= & {} \frac{T}{N_{b}} \left. \frac{dS}{dT} \right| _{p,N_{b},N_{q}} \nonumber \\= & {} c_{V} + \frac{T}{n_{b}} \alpha _{p} \beta _{V} \quad [\hbox {dimensionless}] \,, \end{aligned}$$

(3.46)

the thermodynamic adiabatic index

$$\begin{aligned} \Gamma = \frac{c_{p}}{c_{V}} \quad [\hbox {dimensionless}] \end{aligned}$$

(3.47)

and the adiabatic compressibility

$$\begin{aligned} \kappa _{S} = - \frac{1}{V} \left. \frac{dV}{dp} \right| _{S,N_{b},N_{q}} = \frac{\kappa _{T}}{\Gamma } \quad [\hbox {MeV}^{-1}\hbox {fm}^{3}] \, . \end{aligned}$$

(3.48)

Different definitions of the adiabatic index are sometimes employed in astrophysical contexts. E.g., the form

$$\begin{aligned} {\tilde{\Gamma }} = \left. \frac{\partial \ln p}{\partial \ln n_{b}}\right| _{S} = \frac{1}{p\kappa _{S}} \quad [\hbox {dimensionless}] \end{aligned}$$

(3.49)

is used in applications with polytropic equations of state. A tilde is introduced here to distinguish it from (3.47). A further definition is given by

$$\begin{aligned} \gamma = \left. \frac{\partial \ln p}{\partial \ln e}\right| _{S} \quad [\hbox {dimensionless}] \end{aligned}$$

(3.50)

with the internal energy density e. The square of the speed of sound (isoscalar longitudinal compression wave) in the medium for a one-fluid flow at finite temperatures is given by the relativistic definition

$$\begin{aligned} c_{s}^{2} = \left. \frac{dp}{de} \right| _{S,N_{b},N_{q}} = \frac{1}{h\kappa _{S}} = \frac{p}{h} {\tilde{\Gamma }} = \frac{p}{e}\gamma \quad [c^{2}] \end{aligned}$$

(3.51)

with the enthalpy density $h=e+p$ [MeV fm$^{-3}$]. Alternatively, the expression

$$\begin{aligned} c_{s}^{2} = \left. \frac{n_{b}}{h} \frac{\partial p}{\partial n_{b}} \right| _{Y_{q},e} + \left. \frac{\partial p}{\partial e} \right| _{n_{b},n_{q}} \quad [c^{2}] \end{aligned}$$

(3.52)

can be used for a constant hadronic charge fraction $Y_{q}$. In the application to cold compact stars, i.e., matter at zero temperature in $\beta$ equilibrium, the EoS depends only on a single variable, e.g., the baryon density $n_{b}$ or the baryon chemical potential $\mu _{b}$. Then the square of the speed of sound can be calculated from

$$\begin{aligned} c_{s}^{2} = \left. \frac{dp}{de} \right| _{T=0,\beta ~eq.} = \frac{n_{b}}{\mu _{b}} \left. \frac{d \mu _{b}}{d n_{b}} \right| _{T=0,\beta ~eq.} \quad [c^{2}] \end{aligned}$$

(3.53)

in the unit of the squared speed of light. All of the above quantities can be calculated with the help of the second derivatives of ${\mathcal {F}}$ or f with respect to the parameters T, $n_{b}$, and $n_{q}$.

5.7 Microscopic quantities

Most EoS models are based on microscopic models that can give information on microscopic particle properties in addition to the thermodynamic and compositional quantities. Only few models are just simple parameterizations of thermodynamic quantities without recurrence to the underlying microphysics. The CompOSE database allows to store these microscopic values, too, such that they can be used in applications. In the present version, we consider a small set of quantities that are discussed in the following. It can be easily extended if required.

In particular, we consider the effective masses of the particles in which case one has to distinguish between different definitions. The effective Landau mass

$$\begin{aligned} m^{L}_{i} \quad [\hbox {MeV}] \end{aligned}$$

(3.54)

of a particle i is related to the single-particle density of states and defined as

$$\begin{aligned} \frac{1}{m^{L}_{i}} = \frac{1}{p^{F}_{i}} \left. \frac{d E_{i}}{d p}\right| _{p = p_{i}^{F}} \,, \end{aligned}$$

(3.55)

where $p^{F}_{i}$ denotes the respective Fermi-momentum and $E_{i}$ is the single-particle energy. In non-relativistic Skyrme-Hartree-Fock models it appears in the kinetic energy contribution

$$\begin{aligned} T_{i} = \frac{p_{i}^{2}}{2m^{L}_{i}} \quad [\hbox {MeV}] \end{aligned}$$

(3.56)

to the total single-particle energy. In addition to the kinetic single-particle energy $T_{i}$, a single-particle potential

$$\begin{aligned} U_{i} \quad [\hbox {MeV}] \end{aligned}$$

(3.57)

contributes to the single-particle energy

$$\begin{aligned} E_{i} = T_{i} + U_{i} + m_{i} \end{aligned}$$

(3.58)

in nonrelativistic models. The effective Dirac mass

$$\begin{aligned} m^{D}_{i} = m_{i} - S_{i} \quad [\hbox {MeV}] \end{aligned}$$

(3.59)

is found in the single-particle Hamiltonian of relativistic models, such as relativistic mean-field approaches. This quantity does not reflect directly the density of single-particle states. At zero temperature the corresponding effective Landau mass can be obtained from

$$\begin{aligned} m^{L}_{i} = \sqrt{(m^{D}_{i})^{2}+(p^{F}_{i})^{2}} \, . \end{aligned}$$

(3.60)

The effective Dirac mass depends on the scalar self-energy

$$\begin{aligned} S_{i} = \Sigma _{i} \quad [\hbox {MeV}] \end{aligned}$$

(3.61)

that enters the relativistic single-particle Hamiltonian together with the vector self-energy

$$\begin{aligned} V_{i} = \Sigma ^{0}_{i}\quad [\hbox {MeV}] \end{aligned}$$

(3.62)

that is the time-component of the general four-vector self-energy $\Sigma ^{\mu }_{i}$.

In certain parameter regions of an EoS, the phenomena of superconductivity or superfluidity can be found. In this case, it is worthwhile to know the size of the pairing gaps

$$\begin{aligned} \Delta _{i} \quad [\hbox {MeV}] \,, \end{aligned}$$

(3.63)

e.g. in the $i=nn({}^{1}S_{0})$ channel in case of neutron pairing . In the general case, it can be energy and momentum dependent and cannot be easily tabulated in the present format. In many cases, it is given for specific conditions defined in the EoS data sheet.

5.8 Transport properties

Besides the thermodynamic quantities and coefficients as described in Sects. (3.5) and (3.6), there are transport coefficients that are required in astrophysical simulations of dynamically evolving systems in non-ideal hydrodynamics that consider dissipative effects. There is no general agreement on the choice of units for the transport coefficients. These have to be specified in the data sheet of each EoS.

5.8.1 Viscosities

Corrections to the equations of ideal hydrodynamics [14] can be introduced by considering the energy–momentum tensor (in relativistic notation neglecting heat flow)

$$\begin{aligned} T^{\mu \nu } = T^{\mu \nu }_\textrm{ideal} + \Pi ^{\mu \nu } \quad [\hbox {MeV fm}^{-3}] \end{aligned}$$

(3.64)

with the ideal contribution

$$\begin{aligned} T^{\mu \nu }_\textrm{ideal}= & {} (e+p)u^{\mu }u^{\nu } - p g^{\mu \nu } \nonumber \\= & {} e u^{\mu }u^{\nu } -p \Delta ^{\mu \nu } \quad [\hbox {MeV fm}^{-3}] \end{aligned}$$

(3.65)

where

$$\begin{aligned} \Delta ^{\mu \nu } = g^{\mu \nu } - u^{\mu }u^{\nu } \quad [\hbox {dimensionless}] \end{aligned}$$

(3.66)

and the correction $\Pi ^{\mu \nu }$. The ideal part contains the energy density e, the pressure p, the four-velocity $u^{\mu }$ and the metric tensor $g^{\mu \nu }$. The correction can be written in lowest order as

$$\begin{aligned} \Pi ^{\mu \nu } = \pi ^{\mu \nu } - \Phi \Delta ^{\mu \nu } \quad [\hbox {MeV fm}^{-3}] \end{aligned}$$

(3.67)

with a symmetric traceless contribution $\pi ^{\mu \nu }$ and a traceful part. These can be expressed in first order as

$$\begin{aligned} \pi ^{\mu \nu } = \eta \, \left( \nabla ^{\mu } u^{\nu } + \nabla ^{\nu } u^{\mu } - \frac{2}{3} \Delta ^{\mu \nu } \nabla _{\alpha } u^{\alpha } \right) \quad [\hbox {MeV fm}^{-3}] \end{aligned}$$

(3.68)

and

$$\begin{aligned} \Phi = \zeta \, \nabla _{\alpha } u^{\alpha } \quad [\hbox {MeV fm}^{-3}] \end{aligned}$$

(3.69)

with the coefficients $\eta$ [MeV fm${}^{-2}$] and $\zeta$ [MeV fm${}^{-2}$] that are called the shear and bulk viscosity, respectively.

5.8.2 Conductivities

The thermal conductivity $\kappa$ [fm${}^{-2}$] relates the heat flux density, i.e., the transported heat energy per area and time,

$$\begin{aligned} \vec {q} = - \kappa \, \vec {\nabla } T \quad [\hbox {MeV fm}^{-3}] \end{aligned}$$

(3.70)

to the gradient of the temperature T. The electrical conductivity $\sigma$ [fm${}^{-1}$] connects the electrical current density

$$\begin{aligned} \vec {\jmath } = \sigma \, \vec {E} \quad [e \hbox { fm}^{-3}] \end{aligned}$$

(3.71)

and the electrical field strength $\vec {E}=-\vec {\nabla } \varphi$ [e fm${}^{-2}$] with the electric potential $\varphi$ [e fm${}^{-1}$] (e is the elementary charge). In general, the thermal and electrical conductivities can be tensors if the direction of the induced currents is not in the direction of the field gradients.

5.9 Contribution of photons

If photons are considered in an EoS, they are treated as a gas of massless bosons with simple analytical expressions for the relevant basic thermodynamic quantities. The contribution of the photon gas can be simply added to the thermodynamic quantities of the other constituents. We have the photon free energy density

$$\begin{aligned} f_{\gamma }(T) = -\frac{\pi ^{2}}{45} T^{4} \quad [\hbox {MeV fm}^{-3}] \end{aligned}$$

(3.72)

the photon entropy density

$$\begin{aligned} s_{\gamma }(T) = \frac{4\pi ^{2}}{45} T^{3} \quad [\hbox {fm}^{-3}] \end{aligned}$$

(3.73)

and the photon pressure

$$\begin{aligned} p_{\gamma }(T) = \frac{\pi ^{2}}{45} T^{4} \quad [\hbox {MeV fm}^{-3}] \end{aligned}$$

(3.74)

depending only on the temperature. The chemical potential of the photon vanishes

$$\begin{aligned} \mu _{\gamma } = 0 \quad [\hbox {MeV}] \end{aligned}$$

(3.75)

since it is its own antiparticle. From these quantities, all other relevant photon thermodynamic quantities can be derived.

5.10 Properties of compact stars

If an EoS model can provide the thermodynamic properties of charge-neutral matter in $\beta$ equilibrium at zero temperature, e.g., a compact-star EoS, it is possible to calculate the radius R, mass M, and tidal deformability $\Lambda$ of a sequence of compact stars. For non-rotating, spherically symmetric stars, the mass–radius relation is obtained by solving the Tolman–Oppenheimer–Volkoff (TOV) equation [15, 16]

$$\begin{aligned} \frac{dp}{dr}= & {} - G \frac{M(r)e(r)}{ r^{2}} \left[ 1 + \frac{p(r)}{e(r)}\right] \nonumber \\{} & {} \times \left[ 1 + \frac{4\pi r^{3} p(r)}{M(r) } \right] \left[ 1 - \frac{2GM(r)}{ r}\right] ^{-1} \end{aligned}$$

(3.76)

with the pressure p and the mass inside the radius r [km]

$$\begin{aligned} M(r) = 4\pi \int _{0}^{r} dr^{\prime } \, \left( r^{\prime }\right) ^{2} \, e(r^{\prime }) \quad [\hbox {M}_{\odot }] \end{aligned}$$

(3.77)

that depends on the (internal) energy density e. Equation (3.76) is a first-order differential equation that can be solved by assuming a particular central density $n_{c}$ with corresponding pressure and energy density at the center of the star and then integrating outward until the pressure vanishes (or is sufficiently small) which defines the surface of the star at a radius R [km]. Thus the radius R and mass M(R) of a particular compact star are obtained. Varying the central density $n_{c}$, the mass-radius relation of a sequence of stars is found.

A compact star will change its shape under the influence of an external gravitational field. The major effect will be a quadrupole deformation that can be quantified with the tidal deformability [17, 18]. This quantity is defined as

$$\begin{aligned} \Lambda = \frac{2}{3} k_{2} C^{-5} \quad [\hbox {dimensionless}] \end{aligned}$$

(3.78)

with the compactness

$$\begin{aligned} C = \frac{GM(R)}{ R} \quad [\hbox {dimensionless}] \end{aligned}$$

(3.79)

and the $l=2$ Love number $k_{2}$. It can be found as

$$\begin{aligned} k_{2}= & {} \frac{8}{5} C^{5} \left( 1 - 2 C \right) ^{2} \left[ 2 + 2 C \left( y-1\right) -y\right] \nonumber \\{} & {} \times \left\{ 2C \left[ 6 - 3 y + 3 C \left( 5 y - 8 \right) \right] \right. \nonumber \\{} & {} \left. + 4 C^{3} \left[ 13 - 11 y + C \left( 3 y - 2 \right) + 2 C^{2} \left( 1 + y \right) \right] \right. \nonumber \\{} & {} + 3 \left( 1 - 2 C \right) ^{2} \left[ 2 - y + 2 C \left( y - 1 \right) \right] \nonumber \\{} & {} \quad \left. \ln \left( 1 - 2 C \right) \right\} ^{-1} \quad [\hbox {dimensionless}] \end{aligned}$$

(3.80)

when the logarithmic derivative

$$\begin{aligned} y = \frac{d \ln H}{d \ln R} \,, \quad [\hbox {dimensionless}] \end{aligned}$$

(3.81)

of a function H(R) is known for a compact star configuration with radius R. The quantity H for a particular star is obtained by solving the second-order differential equation

$$\begin{aligned} 0= & {} \frac{d^{2}H}{dr^{2}} + \frac{dH}{dr} \left( \frac{2}{r} + \frac{d\Phi }{dr} - \frac{d\lambda }{dr} \right) \nonumber \\{} & {} + H \left[ - \frac{6}{r^{2}} e^{2\lambda } - 2 \left( \frac{d\Phi }{dr} \right) ^{2} + 2 \frac{d\Phi ^{2}}{dr^{2}} + \frac{3}{r} \frac{d\lambda }{dr} \right. \nonumber \\{} & {} \left. + \frac{7}{r} \frac{d\Phi }{dr} + 2 \frac{d\Phi }{dr}\frac{d\lambda }{dr} + \frac{1}{r} \frac{de}{dp} \left( \frac{d\Phi }{dr} + \frac{d\lambda }{dr}\right) \right] \end{aligned}$$

(3.82)

simultaneously with (3.76) for dp/dr using the potential

$$\begin{aligned} e^{2\lambda } = \left[ 1 - \frac{2GM(r)}{ r}\right] ^{-1} \quad [\hbox {dimensionless}] \end{aligned}$$

(3.83)

of the Schwarzschild metric and the additional quantities

$$\begin{aligned} \Phi= & {} - \frac{1}{\varepsilon (r)+p(r)} \frac{dp}{dr} \quad [\hbox {km}^{-1}] \,, \end{aligned}$$

(3.84)

$$\begin{aligned} \frac{de}{dp}= & {} 1/c_s^2\quad [\hbox {dimensionless}] \end{aligned}$$

(3.85)

that are obtained from the compact-star EoS. The boundary condition for H(r) is given by the condition that $H(r) \rightarrow a_{0}r^{2}$ for $r \rightarrow 0$ with an arbitrary constant $a_{0}$ which has no influence on the value of the tidal deformability.

Publicly available numerical codes exist to solve Einstein equations, which can in particular be used to obtain stationary models of rotating axisymmetric stars for a given EoS, see, e.g., the RNS code [19,20,21] and the LORENE library [22]. The latter has an interface to directly employ EoS data from CompOSE, see Sect. 7.5.2. It can compute initial data for binary neutron stars, too.

Another interesting quantity for neutron stars with respect to cooling is the threshold density for the operation of the direct Urca process . For a neutron-star EoS with only electrons as leptonic degrees of freedom, the threshold charge fraction is given by

$$\begin{aligned} Y_{q}^{\textrm{dUrca}} = \frac{1}{9} \end{aligned}$$

(3.86)

or

$$\begin{aligned} Y_{q}^{\textrm{dUrca}} = \left\{ 1+\left[ 1+\left( \frac{Y_{e}}{Y_{e}+Y_{\mu }}\right) ^{1/3}\right] ^{3} \right\} ^{-1} \end{aligned}$$

(3.87)

if both electrons and muons are accounted for [23, 24]. The direct Urca process can only operate if $Y_{q}$ is larger than these threshold values. Otherwise the neutrino generating direct Urca reactions such as for example

$$\begin{aligned} n \rightarrow p + e^{-} + {\bar{\nu }}_{e} \end{aligned}$$

(3.88)

are kinematically blocked; energy and momentum conservation cannot be fulfilled simultaneously. From the solution of the TOV equation and the resulting NS masses as function on the central density – together with the range of baryon densities allowing for direct Urca processes – threshold NS masses for the processes can be obtained. Since in general $Y_{q}$ increases with $n_{b}$ in the core, this gives a lower limit on the NS mass for the direct Urca process to operate for a given EoS.

6 Part II. For contributors

The success of the CompOSE data base depends on the support of nuclear physicists providing tables with their favourite EoS. Some well known EoS used extensively in astrophysical applications are already incorporated in the CompOSE data base and are ready for use. However, a larger collection of EoS from different models is highly desirable. In order to be suitable for a simple usage, contributors should follow the guidelines specified in this part of the manual.

7 How to prepare EoS tables

In general, tables with EoS data contain a wealth of information on the thermodynamic properties, the composition of dense matter and on the microphysical properties of the constituents. In order to minimize the memory size, only the essential thermodynamic quantities should be stored in the tables that are used in the CompOSE data base. These quantities are selected such that they are sufficient to calculate a particular thermodynamic potential, here the free energy F [MeV], and its first derivatives with respect to the parameters that correspond to the natural variables of F, i.e. temperature T [MeV], volume V [fm${}^{3}$] and particle numbers $N_{i}$ [dimensionless]. However, it is worthwhile to have redundant information in order to check the consistency of the EoS tables or to simplify the determination of additional quantities. Further thermodynamic quantities relevant for astrophysical applications can, if not directly available, be generated with the help of thermodynamic identities and, for quantities depending on higher derivatives, by using appropriate interpolation schemes. The organization of the data concerning composition etc. depends on the particular EoS. Thus it is required that up to three separate tables are provided with unique formats that contain the thermodynamic quantities (required), the information on the composition of matter (depending on the choice of the constituents of the model, optional) and the microscopic properties of the particles (optional), respectively. Details of the employed discretization mesh of the parameters have to be supplied in additional tables.

Additional information that is not contained in the Eos tables has to be provided by the contributor. It will appear in the data sheet that accompanies each EoS and is available on the CompOSE web site. Predictions for properties of cold, spherical compact stars will be made available in a separate file if it is possible to calculate these quantities in the model. See the new “quick guide for providers” for a summary.

7.1 Tabulation of quantities

7.1.1 Parameters and parameter ranges

Because of the imposed physical conditions, see Sect. 3.2, the state of the system is uniquely characterized by only three quantities, see Sect. 3.5, that are used as parameters in the EoS tables with the properties of dense matter:

• 1. temperature T [MeV],

• 2. baryon number density $n_{b}$ [fm$^{-3}$],

• 3. charge fraction of strongly interacting particles $Y_{q}$ [dimensionless].

In order to be useful in many applications, the parameter ranges should be chosen as wide as possible and the mesh spacing as fine as possible, i.e. the number of grid points as large as possible within reason. The recommended parameter values for the definition of the grid are given in the tables in the end of this chapter. In some cases it might be useful to adapt the resolution of the tables, in particular for the baryon density, to the physical situation and occurring phenomena. E.g. a rapid change in some quantities or a phase transition can occur. For this purpose, several tables for different blocks in density with different resolution can be supplied for a single EoS model.

The recommended discretization scheme for a general purpose EoS table depends on the parameters:

• 1. Temperature

  In this case, two standards are suggested:

  • a. linear mesh in $\ln T$, i.e.

    $$\begin{aligned} T(i_{T})= & {} T^{\text {ref}} (f_{T})^{i_{T}-1} \quad f_{T} = 10^{1/M_{T}} \nonumber \\ i_{T}= & {} N_{T}^{\text {min}},N_{T}^{\text {min}}+1, \dots , N_{T}^{\text {max}} \end{aligned}$$

    (4.1)

    with parameters $T^{\text {ref}} > 0$ MeV, $N_{T}^{\text {min}} < N_{T}^{\text {max}}$ and $M_{T}>1$ points per decade in temperature;

  • b. a mesh that is linear in T at low temperatures and linear in $\ln T$ at high temperatures, i.e.

    $$\begin{aligned} T(i_{T})= & {} T^{\text {ref}} \frac{\sinh (f_{T} i_{T})}{\sinh (f_{T})} \nonumber \\ i_{T}= & {} N_{T}^{\text {min}},N_{T}^{\text {min}}+1, \dots , N_{T}^{\text {max}} \end{aligned}$$

    (4.2)

    with parameters $T^{\text {ref}}>0$ MeV, $N_{T}^{\text {min}} < N_{T}^{\text {max}}$ and $f_{T}>0$. The resolution parameter $M_{T}>0$ determines $f_{T} = \ln (10)/M_{T}$;

    In both cases the index $i_{T}=0$ is reserved for $T=0$ MeV;

• 2. Baryon number density

  linear mesh in $\ln n_{b}$, i.e.

  $$\begin{aligned} n_{b}(i_{n_{b}} )= & {} n_{b}^{\text {ref}} (f_{n_{b}})^{i_{n_{b}}-1} \quad f_{n_{b}} = 10^{1/M_{n_{b}}} \nonumber \\ i_{n_{b}}= & {} N_{n_{b}}^{\text {min}},N_{n_{b}}^{\text {min}}+1, \dots , N_{n_{b}}^{\text {max}} \end{aligned}$$

  (4.3)

  with parameters $n_{b}^{\text {ref}}>0$ fm${}^{-3}$, $N_{n_{b}}^{\text {min}} < N_{n_{b}}^{\text {max}}$ and $M_{n_{b}}>1$ points per decade in the baryon number density;

• 3. Charge fraction of strongly interacting particles

  linear mesh in $Y_{q}$, i.e.

  $$\begin{aligned} Y_{q}(i_{Y_{q}}) = \frac{i_{Y_{q}}}{M_{Y_{q}}} \quad i_{Y_{q}}=N_{Y_{q}}^{\text {min}},N_{Y_{q}}^{\text {min}}+1, \dots , N_{Y_{q}}^{\text {max}}\nonumber \\ \end{aligned}$$

  (4.4)

  with parameters $N_{Y_{q}}^{\text {min}} < N_{Y_{q}}^{\text {max}}$ and $M_{Y_{q}}>N_{Y_{q}}^{\text {max}}$ with a resolution in $Y_{q}$ of $1/M_{Y_{q}}$.

Each grid point in the table is identified with the triple $(i_{T},i_{n_{b}},i_{Y_{q}})$ of indices. The recommended values for the parameters are given in Sect. 4.2.1. Actually used meshes for particular EoS are specified in the online documentation for each individual EoS table.

7.1.2 Thermodynamic consistency

The equations of state included in the CompOSE data base are required to fulfill some basic thermodynamic consistency relations. Of course, the consistency can only hold up to some numerical level. Measures of the thermodynamic consistency are included in the characterisation of the EoS tables on the web site.

Because the free energy density is considered in the present case as the basic thermodynamic potential, the homogeneity condition, cf. Eq. (3.19),

$$\begin{aligned} f(T,n_{b},Y_{q})= & {} -p+\mu _{b} n_{b}+\mu _{q} \left( n_{q}-n_{le} - n_{l\mu }\right) \nonumber \\{} & {} +\mu _{le} n_{le} +\mu _{l\mu } n_{l\mu } \end{aligned}$$

(4.5)

with the pressure p, the chemical potentials $\mu _{b}$, $\mu _{q}$, $\mu _{le}$, $\mu _{l\mu }$ and corresponding densities $n_{b}$, $n_{q}$, $n_{le}$, $n_{l\mu }$ should hold. Note that $n_{q}$ contains only the contribution of strongly interacting particles to the charge density, that a fixed relation between $\mu _{le}$ and $\mu _{l\mu }$ is assumed and that $n_{le}+n_{l\mu }=n_{q} = Y_{q}n_{b}$ due to local charge neutrality if leptons are considered in the equation of state. Additionally, in Eq. (4.5) it was assumed that the strangeness chemical potential $\mu _{s}$ is zero.

It is convenient to define the effective lepton density as

$$\begin{aligned} n_{l} = n_{le} + n_{l\mu } \quad [\hbox {fm}^{-3}] \end{aligned}$$

(4.6)

and the effective lepton chemical potential

$$\begin{aligned} \mu _{l} = \frac{\mu _{le} n_{le} +\mu _{l\mu } n_{l\mu }}{n_{l}} \quad [\hbox {MeV}] \end{aligned}$$

(4.7)

such that

$$\begin{aligned} f(T,n_{b},Y_{q}) = -p + \left( \mu _{b} + Y_{q} \mu _{q} \right) n_{b} \end{aligned}$$

(4.8)

in case of an EoS without leptons and

$$\begin{aligned} f(T,n_{b},Y_{q}) = -p + \left( \mu _{b} + Y_{l} \mu _{l} \right) n_{b} \end{aligned}$$

(4.9)

in the case with leptons and charge neutrality ($Y_{l} = Y_{q}$), respectively. The tabulated values for the entropy density, pressure and chemical potentials should be given by the first partial derivatives as

$$\begin{aligned}{} & {} s = - \left. \frac{\partial f}{\partial T} \right| _{n_{b},Y_{q}} \,, \end{aligned}$$

(4.10)

$$\begin{aligned}{} & {} p = n_{b}^{2} \left. \frac{\partial (f/n_{b}) }{\partial n_{b}} \right| _{T,Y_{q}} \,, \end{aligned}$$

(4.11)

$$\begin{aligned}{} & {} \mu _{b} + Y_{i} \mu _{i} = \left. \frac{\partial f}{\partial n_{b}} \right| _{T,Y_{q}} \,, \end{aligned}$$

(4.12)

$$\begin{aligned}{} & {} \mu _{i} = \frac{1}{n_{b}} \left. \frac{\partial f}{\partial Y_{i}} \right| _{T,n_{b}} \end{aligned}$$

(4.13)

where $i=l(q)$ if leptons are (not) included in the EoS. These relations will be used to derive the first derivatives of f from the tabulated thermodynamic quantities s, p, $\mu _{b}$, $\mu _{q}$, and $\mu _{l}$ in some particular interpolation schemes.

Table 6 Parameter values for the recommended discretization in T, $n_{b}$ and $Y_{q}$ for a general purpose EoS table

Table 7 Parameter values for the recommended discretization in $n_{b}$ and $Y_{q}$ for a zero-temperature EoS table

For a mixed second partial derivative, the Maxwell relations are

$$\begin{aligned}{} & {} - n_{b}^{2} \left. \frac{\partial (s/n_{b})}{\partial n_{b}} \right| _{T,Y_{q}} = \left. \frac{\partial p}{\partial T} \right| _{n_{b},Y_{q}} \,, \end{aligned}$$

(4.14)

$$\begin{aligned}{} & {} - \left. \frac{\partial s}{\partial n_{x}} \right| _{T,n_{y},y \ne x} = \left. \frac{\partial \mu _{x}}{\partial T} \right| _{n_{y}} \,, \end{aligned}$$

(4.15)

$$\begin{aligned}{} & {} \left. \frac{\partial p}{\partial n_{x}} \right| _{T,n_{y},y \ne x} = n_{b} \left. \frac{\partial \mu _{x}}{\partial n_{b}} \right| _{T,n_{y}, y \ne b} \,, \end{aligned}$$

(4.16)

$$\begin{aligned}{} & {} \left. \frac{\partial \mu _{x}}{\partial n_{y}} \right| _{T,n_{z},z \ne y} = \left. \frac{\partial \mu _{y}}{\partial n_{x}} \right| _{T,n_{z}, z \ne x} \end{aligned}$$

(4.17)

where $x,y,z \in \left\{ b,q,le,l\mu \right\}$ should hold.

The knowledge of three functions $p=p(T,n_{b},Y_{q})$, $\mu _{b} =\mu _{b}(T,n_{b},Y_{q})$, and $\mu _{q} =\mu _{q}(T,n_{b},Y_{q})$ or $\mu _{l} =\mu _{l}(T,n_{b},Y_{q})$ is sufficient to recover the free energy density $f =f(T,n_{b},Y_{q})$ for the particular physical conditions of section 3.2 ($\mu _{s} = 0$ and $n_{q}=n_{l}$). Further quantities can be derived by partial derivatives with respect to the parameters T, $n_{b}$ and $Y_{q}$.

Thus it would be convenient to store only these three quantities in the EoS table for the thermodynamic properties in order to reduce the memory size. However, we require that the pressure, the entropy, the three chemical potentials and also the free and internal energy densities should be provided independently in the EoS tables for checking purposes. It is also allowed to store more than these seven basic quantities.

Table 8 Parameter values for the recommended discretization in T and $n_{b}$ for EoS tables of symmetric matter, neutron matter or $\beta$-equilibrated matter

Table 9 Parameter values for the recommended discretization in $n_{b}$ for one-dimensional EoS tables

7.2 Structure of tables

The CompOSE data base contains for each EoS model data tables that specify the used parameter grid and tables that contain the thermodynamic properties, the compositional information and the microscopic information, respectively. Properties of a sequence of cold, spherical compact stars will be collected in a separate file if calculable in the EoS model.

7.2.1 Types of EoS tables

In general, three types of EoS input tables should be available for each EoS model in order to address different applications:

• 1. Three-dimensional tables

  These tables depend on all three independent parameters T, $n_{b}$ and $Y_{q}$. There is only one case:

  • General purpose EoS table

    The recommended parameter values for the definition of the grid are given in Table 6. This choice of discretation corresponds to a table with $N_{T}^{\text {max}} \times N_{n_{b}}^{\text {max}} \times N_{Y_{q}}^{\text {max}} = 1119720 (1462860)$ data points for the case b(a) in the T grid. The table includes neither points with $T=0$ MeV nor with $Y_{q}=0$.

• 2. Two-dimensional tables

  These tables depend on two of the three independent parameters T, $n_{b}$ and $Y_{q}$. Four cases are offered here:

  • Zero-temperature EoS table

    The recommended parameter values for the definition of the grid are given in Table 7. This choice of discretization corresponds to a table with $N_{n_{b}}^{\text {max}} \times (N_{Y_{q}}^{\text {max}}+1) = 18361$ data points. Only $T=0$ MeV is considered. Data points for $Y_{q}=0$, i.e. pure neutron matter, are included in this EoS table.

  • Symmetric matter EoS table

    The recommended parameter values for the definition of the grid are given in Table 8. This choice of discretization corresponds to a table with $N_{T}^{\text {max}} \times N_{n_{b}}^{\text {max}} = 18963 (24381)$ data points for the case b(a) in the T grid. Only $Y_{q}=0.5$ is considered. $T=0$ MeV, i.e. the zero-temperature case, is (not) included in this EoS table for case b(a).

  • Neutron matter EoS table

    The recommended parameter values for the definition of the grid are given in Table 8. This choice of discretization corresponds to a table with $N_{T}^{\text {max}} \times N_{n_{b}}^{\text {max}} = 18963 (24381)$ data points for case b(a). Only $Y_{q}=0$ is considered. $T=0$ MeV, i.e. the zero-temperature case, is (not) included in this EoS table for case b(a).

  • EoS table of $\beta$-equilibrated matter

    The recommended parameter values for the definition of the grid are given in Table 8. This choice of discretization corresponds to a table with $N_{T}^{\text {max}} \times N_{n_{b}}^{\text {max}} = 18963 (24381)$ data points for the case b(a) in the T grid. The charge fraction of strongly interacting particles $Y_{q}$ is determined by the conditions of charge neutrality and $\beta$ equilibrium with leptons. $T=0$ MeV, i.e. the zero-temperature case, is (not) included in this EoS table for case b(a).

• 3. One-dimensional tables

  These tables depend only on the parameter $n_{b}$. Three cases are offered here:

  • EoS table of cold symmetric matter

    The recommended parameter values for the definition of the grid are given in Table 9. This choice of discretization corresponds to a table with $N_{n_{b}}^{\text {max}} = 301$ data points. Only $Y_{q}=0.5$ and $T=0$ MeV is considered.

  • EoS table of cold neutron matter

    The recommended parameter values for the definition of the grid are given in Table 9. This choice of discretization corresponds to a table with $N_{n_{b}}^{\text {max}} = 301$ data points. Only $Y_{q}=0.0$ and $T=0$ is considered.

  • EoS table of cold $\beta$-equilibrated matter

    The recommended parameter values for the definition of the grid are given in Table 9. This choice of discretization corresponds to a table with $N_{n_{b}}^{\text {max}} = 301$ data points. Only $T=0$ MeV is considered and the charge fraction of strongly interacting particles $Y_{q}$ is determined by the conditions of $\beta$-equilibrium and local charge neutrality with leptons.

Every set of EoS tables contains up to three tables: the first that specifies the thermodynamic state of the system (always required), the second that contains information on the composition of the matter (optional), the third that gives microscopic information (optional).

The format of the thermodynamic, the compositional and the microscopic table, respectively, has to follow the following prescriptions.

7.2.2 Table with thermodynamic quantities (required)

The first row of this table contains three entries: the mass of the neutron $m_{n}$ and that of the proton $m_{p}$ in MeV of the particular EoS model and then an integer $I_{l}$ that indicates whether the EoS contains leptons or not. If electrons (and/or muons) are considered in the EoS, one sets $I_{l}=1$. For other values of $I_{l}$, it is assumed that there are no electrons (and/or muons) and the lepton chemical potential $\mu _{l}$ should be set to zero. The neutron mass $m_{n}$ is used for scaling certain quantities as specified below. The following rows in this table contain information on the particular grid point in the table and the relevant thermodynamic quantities. The actually tabulated quantities are given by

• 1. pressure divided by the baryon number density $Q_{1}= p/n_{b}$ [MeV],

• 2. entropy per baryon (or entropy density per baryon density) $Q_{2} = s/n_{b}$ [dimensionless],

• 3. scaled and shifted baryon chemical potential $Q_{3} =\mu _{b}/m_{n}-1$ [dimensionless],

• 4. scaled charge chemical potential $Q_{4} = \mu _{q}/m_{n}$ [dimensionless],

• 5. scaled effective lepton chemical potential $Q_{5} = \mu _{l}/m_{n}$ [dimensionless],

• 6. scaled free energy per baryon $Q_{6} = f/(n_{b}m_{n})-1$ [dimensionless],

• 7. scaled internal energy per baryon $Q_{7} = e/(n_{b}m_{n})-1$ [dimensionless].

For an ideal gas $Q_{1}$ is just the temperature T. The traditional scaling of the chemical potentials and the energies per baryon with the neutron mass $m_{n}$ has been introduced.

Each line in the thermodynamic table (except the first) has to contain the following quantities

$$\begin{aligned}{} & {} i_{T} \,\, i_{n_{b}} \,\, i_{Y_{q}} \,\, Q_{1} \,\, Q_{2} \,\, Q_{3} \,\, Q_{4} \,\, Q_{5} \,\, Q_{6} \,\, Q_{7} \,\,\nonumber \\{} & {} \quad N_{\text {add}} \,\, \underbrace{q_{1} \,\, q_{2} \,\, \dots \,\, q_{N_{\text {add}}}}_{N_{\text {add}} \, {\text {quantities}}} \end{aligned}$$

(4.18)

with seven required quantities $Q_{i}$ and $N_{\text {add}}$ additional optional thermodynamic quantities $q_{i}$. The meaning of the additional quantities is specified in the data sheet for each EoS table. It is assumed that the order of all quantities $q_{i}$ does not change within an EoS table. Since the triple of table indices $(i_{T},i_{n_{b}},i_{Y_{q}})$ is explicitly specified, the order of tabulation with respect to the grid points is not relevant. If the table contains rows with identical indices $i_{T}$, $i_{n_{b}}$, $i_{Y_{q}}$ then table entries for the quantities $Q_{i}$, $N_{\text {add}}$, and $q_{i}$ are used from the row that is read last.

7.2.3 Table with the composition of matter (optional)

The predicted chemical composition of matter depends strongly on the employed theoretical model. In principle, the particle content and hence the data that are required to be stored can depend on the particular point in the parameter space, too. Thus the format of the table has to be adapted to this situation. The essential information is contained in the particle fractions $Y_{i}$ but in some cases additional information, e.g. on the type of phase, might be useful.

The tabulated quantities in the EoS composition table are typically given by

• 1. particle fractions $Y_{i}$ [dimensionless],

• 2. average mass number of a representative nucleus $A^{\text {av}}$ [dimensionless],

• 3. average charge number of a representative nucleus $Z^{\text {av}}$ [dimensionless],

• 4. index encoding the type of phase $I_{\text {phase}}$ [dimensionless]

  (The correspondence of the phase index $I_{\text {phase}}$ with the actual structure of the phase is defined for each EoS model on the corresponding web page.).

In the following a standard format for the tabulation of compositional information is defined. It is suitable for most of the presently available EoS tables of hadronic and quark EoS models. It allows to change from point to point the particle fractions that are given in the table. Thus, it is possible to select only those particles that are most abundant under the considered local conditions. Each row of the table contains the following entries

$$\begin{aligned}{} & {} i_{T} \,\, i_{n_{b}} \,\, i_{Y_{q}} \,\, I_{\text {phase}} \,\, N_{\text {pairs}} \underbrace{I_{1} \,\, Y_{I_{1}} \,\, \dots \,\, I_{N_{\text {pairs}}} \,\, Y_{I_{N_{\text {pairs}}}}}_{N_{\text {pairs}} \, {\text {pairs}}} \nonumber \\ \quad{} & {} N_{\text {quad}} \,\, \underbrace{I_{1} \,\, A^{\text {av}}_{I_{1}}\,\, Z^{\text {av}}_{I_{1}} \,\, Y_{I_{1}} \,\, \dots \,\, I_{N_{\text {quad}}} \,\, A^{\text {av}}_{I_{N_{\text {quad}}}}\,\, Z^{\text {av}}_{I_{N_{\text {quad}}}} \,\, Y_{I_{N_{\text {quad}}}}}_{N_{\text {quad}} \, {\text {quadruples}}}\nonumber \\ \end{aligned}$$

(4.19)

with $N_{\text {pairs}}$ pairs that combine the particle index $I_{i}$ as defined in Tables 2, 3, 4 and 5 and the corresponding particle fraction $Y_{i}$. (Note the definition of particle fractions in Eq.  3.10.) In addition, there are $N_{\text {quad}}$ quadruples that contain an index $I_{i}$ that specifies a group of nuclei ${\mathcal {M}}_{I_{i}}$ with average mass number

$$\begin{aligned} A^{\text {av}}_{I_{i}} = \frac{\sum _{j\in {\mathcal {M}}_{I_{i}}} A_{j} Y_{j}}{\sum _{j\in {\mathcal {M}}_{I_{i}}} Y_{j}} \,, \end{aligned}$$

(4.20)

average charge number $Z^{\text {av}}_{I_{i}}$

$$\begin{aligned} Z^{\text {av}}_{I_{i}} = \frac{\sum _{j\in {\mathcal {M}}_{I_{i}}} Z_{j} Y_{j}}{\sum _{j\in {\mathcal {M}}_{I_{i}}} Y_{j}} \,, \end{aligned}$$

(4.21)

and combined fraction

$$\begin{aligned} Y_{I_{i}} = \frac{\sum _{j\in {\mathcal {M}}_{I_{i}}} A_{j}Y_{j}}{A^{\text {av}}_{I_{i}}} = \sum _{j\in {\mathcal {M}}_{I_{i}}} Y_{j} \, . \end{aligned}$$

(4.22)

The average neutron number

$$\begin{aligned} N^{\text {av}}_{I_{i}} = \frac{\sum _{j\in {\mathcal {M}}_{I_{i}}} N_{j} Y_{j}}{\sum _{j\in {\mathcal {M}}_{I_{i}}} Y_{j}} \,, \end{aligned}$$

(4.23)

is not stored because it is easily found from $N^{\text {av}}_{I_{i}} = A^{\text {av}}_{I_{i}} - Z^{\text {av}}_{I_{i}}$. In case that there are no pairs (quadruples) to be stored, the number $N_{\text {pairs}}$ ($N_{\text {quad}}$) is set to zero. The average mass and charge numbers correspond to those of a representative heavy nucleus if there is only one group of nuclei except the lightest that are considered explicitly. However, it is also possible to define several subsets of nuclei with corresponding average mass numbers, charge numbers and fractions. The correlation between the index $I_{i}$ and the set of nuclei ${\mathcal {M}}_{I_{i}}$ is defined for each EoS model separately and given in the data sheet on the CompOSE web pages. Since the composition of matter can vary rapidly with a change of the parameters, it is not required that the fractions of all particles or particle sets have to be given in each row. It is sufficient to list only those that are relevant, e.g. with nonzero $Y_{i}$. Others can be omitted.

7.2.4 Table with information on microscopic quantities (optional)

Besides the information on thermodynamic and compositional quantities, most EoS models can provide additional information on microscopic quantities of an individual particle i, e.g.

• 1. Landau effective mass divided by the particle mass $m^{L}_{i}/m_{i}$ [dimensionless],

• 2. Dirac effective mass divided by the particle mass $m^{D}_{i}/m_{i}$ [dimensionless],

• 3. non-relativistic single-particle potential $U_{i}$ [MeV],

• 4. relativistic vector self-energy $V_{i}$ [MeV],

• 5. relativistic scalar self-energy $S_{i}$ [MeV],

• 6. pairing gap $\Delta _{i}$ [MeV].

These quantities are stored in an additional table if available. In this case, each line has the format

$$\begin{aligned} i_{T} \,\, i_{n_{b}} \,\, i_{Y_{q}} \,\, N_{\text {qty}} \,\, \underbrace{K_{1} \,\, q_{K_{1}} \,\, K_{2} \,\, q_{K_{2}} \,\, \dots \,\, K_{N_{\text {qty}}} \,\, q_{K_{N_{\text {qty}}}}}_{N_{\text {qty}} \, {\text {pairs}}}\nonumber \\ \end{aligned}$$

(4.24)

where $N_{\text {qty}}$ is the number of stored quantities $q_{K_{i}}$. The composite index $K_{i}$ identifies uniquely both the particle or correlation and the physical quantity. It is formed as

$$\begin{aligned} K_{i} = 1000 \, I_{i} + J_{i} \end{aligned}$$

(4.25)

with the particle or correlation index $I_{i}$ from Tables 2, 3, 4 or 5, respectively, and the quantity index $J_{i}$ from Table 14.

7.2.5 Table with information on transport quantities (optional)

The structure of this table will be defined after the CompOSE code has been updated to handle transport quantities.

7.2.6 Table with information on properties of compact stars (optional)

If an EoS model can predict the properties of cold, spherical compact stars, e.g., the mass–radius relation and corresponding tidal deformabilities, these data will be provided in a separate file. The first line of this file identifies the given quantities and their units, followed by their actual values in the subsequent lines. The exact structure depends on the available data. Most relevant are

• radius R [km],

• mass M [M${}_{\odot }$],

• tidal deformability $\Lambda$ [dimensionless],

• central density $n_{c}$ [fm${}^{-3}$].

It is required that the radius R and the mass M using the units km and solar masses, M${}_{\odot }$, are given in the first and second column of the file, respectively.

7.2.7 Tables with parameters

The mesh points for the general purpose EoS are stored in three individual data files that are used as an input in addition to the three tables containing the thermodynamic, compositional and microphysical information. It is assumed that the discretization scheme for the relevant parameters in the zero-temperature and in the neutron-matter table is identical to the grid for the general purpose table. Parameter values should be given with at least eight significant digits. For the parameter temperature, the corresponding table contains a single entry in each row. In the first row, $N_{T}^{\text {min}}$ is given, and $N_{T}^{\text {max}}$ in the second row. Then the parameter values $T(N_{T}^{\text {min}})$, $T(N_{T}^{\text {min}}+1)$, ..., $T(N_{T}^{\text {max}})$, are given in every following row. In total, the table contains $N_{T}^{\text {max}}-N_{T}^{\text {min}}+3$ rows. The same storage scheme applies for the parameters $n_{b}$ and $Y_{q}$.

7.2.8 Identification of tables

Each Eos table is identified by a unique name and a unique extension that correspond to the model and type of table, respectively. Using the generic name eos for a particular EoS model, there are always three parameter tables

• eos.t (required)

• eos.nb (required)

• eos.yq (required)

that define the discretization mesh for the three parameters T, $n_{b}$ and $Y_{q}$ as discussed in the previous section. All three tables are required even for EoS tables with less than three dimensions.

The thermodynamic, compositional and microscopic quantities are stored in three tables with the generic names

• eos.thermo (required)

• eos.compo (optional)

• eos.micro (optional)

where only the first one is required for every EoS model since information on the composition or microscopic details are not always available.

If the EoS model allows to calculate properties of cold, spherical compact stars, these data will be stored in a file with the name

• eos.mr (optional).

7.2.9 Precision of tabulated quantities

EoS quantities of a particular model have a limited precision in practical calculations. Machine precision is hardly ever reached. The format of the tabulated data should reflect this fact and only the significant number of digits should be given to avoid excessive memory size. Usually eight significant digits are sufficient since the error in the interpolation can be considerably larger.

7.3 Data sheet

Information on an EoS that is not provided directly in the tables but relevant for the characterisation of the particular model can be found in a data sheet that comes along with the tables on the CompOSE web pages. Some quantities are extracted automatically from the tabulated data, others need to be specified by the contributor. E.g., characteristic nuclear matter parameters (see Sect. 6.1) are obtained by using the program compose.f90 (see Sect. 7.5) if an EoS table for pure nuclear matter without electrons or muons is available. Similarly, information on the parameter grid, type and dimension of the EoS table, considered particles and additionally stored quantities are extracted by the program. The EoS of stellar matter in $\beta$-equilibrium is generated by using compose.f90 if the EoS table for matter with electrons or muons is provided by the contributor. From this extracted EoS table, characteristic properties of neutron stars, e.g. maximum masses or the threshold of the direct Urca process, are calculated and given in the data sheet. It also contains information on the considered constituent particles in the model. The contributor has to supply a short characterisation of the EoS model and relevant references. He/she also needs to define the meaning of the index $I_{\text {phase}}$ for the phases considered in the EoS.

8 Extending CompOSE

CompOSE is meant to be under constant development. The main aim is to enlarge the data base by adding EoS tables of more and more models. The most simple way is to convert tables of existing model calculations to the generic CompOSE scheme of tabulation in the data files. A more involved task is to develop a new EoS model and to provide the results in the appropriate format. In either case, we ask you to contact the CompOSE core team by sending an email to

develop.compose@obspm.fr.

All questions will be clarified by them on how your results can be incorporated in the CompOSE data base and made accessible to the public.

For the future, it is foreseen that the functionality of the CompOSE database will be extended in order to make even more results of EoS model calculations available that are not included in the present sets of data. It is also envisioned to add more quantities to the data tables that might be relevant for astrophysical applications and beyond, in particular transport quantities.

9 Part III. For users

10 Models for the equation of state

There is a large number of model equations of state. They fall into different classes depending on the theoretical approach. The main difficulty in generating an equation of state is the description of the strongly interacting constituents, hadrons and/or quarks. See references [2,3,4,5] for in-depth reviews. Some EoS models are constructed for specific tasks motivated by physical conditions and the corresponding data tables cover only limited ranges of the variables. For instance, in calculating properties of cold neutron stars, i.e., for matter at zero temperature in $\beta$ equilibrium, a separation of low-density crust and a high-density core EoS is often introduced using different theoretical models and numerical methods. The former case is characterized by the occurrence of inhomogeneous matter with the formation of lattice structures of ions in a background of electrons (and neutrons above the neutron-drip density). With increasing density, and after a transition via various so-called pasta phases, homogeneous matter is reached that is described by a core EoS with possibly new particle species appearing. Also for finite-temperature EoS, different models are sometimes employed in different regions in the space of variables, e.g., models with or without cluster degrees of freedom or hadronic and quark-matter models. The EoS tables on the CompOSE website are not classified according to the model but to their structure of tabulation and possible application.

Details and characteristic parameters of each particular EoS can be found in a data sheet that accompanies the data tables on the CompOSE web site

https://compose.obspm.fr.

For electrons and muons, a uniform Fermi gas model is employed in many cases. This assumption simplifies the calculation significantly in case of spatially inhomogeneous charge distributions. Details on the treatment of electrons and muons are given for each EoS separately on the data sheet.

At finite temperatures, photons contribute to the thermodynamic properties of the system. Their treatment is discussed in Sect. 3.9. For each EoS in the database, a remark is given whether photons are considered or not.

The definition of commonly used nuclear matter parameters is given in Sect. 6.1. They characterize the essential properties of nuclear matter and can serve in a first comparison of the various models. Be, however, careful since these parameters are only valid in the vicinity of saturation for symmetric, i.e. same number of protons and neutrons, matter. Specific values of these parameters are given in the data sheet for the individual models and in a summary table on the CompOSE website. Collections of nuclear matter parameters are available for widely used Skyrme Hartree-Fock models [25] and for nonrelativistic Hartree–Fock models with Gogny-type interactions [26]. Results for relativistic mean-field models can be found in [27]. See the review [2] for recent experimental constraints.

In the end of this section, further remarks of caution are given.

10.1 Nuclear matter properties

Realistic models of nuclear, i.e. purely hadronic, matter without leptons show some general features that are related to the occurrence of the saturation phenomenon. Due to the symmetries of the strong interaction, uniform nuclear matter at zero temperature reaches a state with the largest binding energy per nucleon at a finite saturation density $n_{b}^{\text {sat}}$ with equal concentrations of neutrons and protons.² For $T=0$ MeV, the energy per nucleon ${\mathcal {E}}$ can be considered as a function of the baryon number density $n_{b}$ and the asymmetry

$$\begin{aligned} \alpha = \frac{n_{n}-n_{p}}{n_{n}+n_{p}} = 1-2Y_{q} \quad [\hbox {dimensionless}] \end{aligned}$$

(6.1)

that vanishes for symmetric nuclear matter, i.e. $\alpha = 0$ and $Y_{q} = 0.5$. It can be expanded in a power series around the saturation point as

$$\begin{aligned} {\mathcal {E}}(n_{b},\alpha )= & {} -B_{\text {sat}} + \frac{1}{2} K x^{2} + \frac{1}{6} Q x^{3} + \dots + \alpha ^{2} \nonumber \\{} & {} \times \left( J + L x + \frac{1}{2} K_{\text {sym}} x^{2} + \dots \right) + \dots \quad [\hbox {MeV}]\nonumber \\ \end{aligned}$$

(6.2)

with the relative deviation of the actual density from the saturation density

$$\begin{aligned} x = \frac{n_{b}-n_{b}^{\text {sat}}}{3n_{b}^{\text {sat}}} \quad [\hbox {dimensionless}] \end{aligned}$$

(6.3)

and the asymmetry $\alpha$. The coefficients $B_{\text {sat}}$, K, $Q=-K^{\prime }$, J, L, $K_{\text {sym}}$ [MeV], ...and $n_{b}^{\text {sat}}$ [fm${}^{-3}$] characterize the behaviour of the EoS. They can, of course, only give an indication of the general features of the EoS, since these quantities are relevant essentially in the vicinity of the saturation density at zero temperature and close to symmetric nuclear matter (equal number of protons and neutrons). Extrapolations based on this polynomial expansion might be dangerous. The numerical factor in Eq. (6.3) is of historical origin since the Fermi momentum $k_{F}$ [fm$^{-1}$] instead of the density $n_{b} = 2k_{F}^{3}/(3\pi ^{2})$ of symmetric nuclear matter at zero temperature was originally used as the expansion parameter. Note that contributions linear in $\alpha$ and linear in x for $\alpha =0$ are absent in the expansion due to the condition of minimum energy per baryon.

In order to study the dependence on the asymmetry $\alpha$, it is also convenient to introduce the symmetry energy

$$\begin{aligned} E_{\text {sym}}(n_{b}) = \left. \frac{1}{2} \frac{\partial ^{2} {\mathcal {E}}(n_{b},\alpha )}{\partial \alpha ^{2}} \right| _{\alpha = 0} \quad [\hbox {MeV}] \end{aligned}$$

(6.4)

that is a function of the baryon number density $n_{b}$ only. In many cases the quadratic approximation in $\alpha$ in Eq.  (6.2) is sufficient. Then the symmetry energy can also be calculated from the finite difference approximation

$$\begin{aligned} E_{\text {sym}}(n_{b}) = \frac{1}{2} \left[ {\mathcal {E}}(n_{b},-1) - 2 {\mathcal {E}}(n_{b},0) + {\mathcal {E}}(n_{b},1)\right] \!\! \quad [\hbox {MeV}]\nonumber \\ \end{aligned}$$

(6.5)

comparing symmetric matter with pure neutron and pure proton matter.

In the following, the coefficients appearing in Eq.  (6.2) are discussed in more detail.

• The saturation density $n_{b}^{\text {sat}}$ [fm${}^{-3}$] of symmetric nuclear matter is defined by the condition that the pressure vanishes, i.e.

  $$\begin{aligned} p = \left. n_{b}^{2} \frac{d{\mathcal {E}}(n_{b},0)}{dn_{b}} \right| _{n_{b}=n_{b}^{\text {sat}}} = 0 \quad [\hbox {MeV fm}^{-3}] \end{aligned}$$

  (6.6)

  and the energy per baryon becomes minimal.

• The binding energy at saturation $B_{\text {sat}}$ [MeV] of symmetric nuclear matter can be obtained from Bethe–Weizsäcker mass formulas for nuclei ${}^{A}{Z}$ by an extrapolation of $A=2Z$ to infinity.

• The incompressibility of bulk nuclear matter K [MeV] quantifies the curvature of the binding energy per baryon with respect to the density variation at saturation since it is defined by

  $$\begin{aligned} K= & {} \left. 9 n_{b}^{2} \frac{\partial ^{2} {\mathcal {E}}(n_{b},0)}{\partial n_{b}^{2}} \right| _{n_{b}=n_{b}^{\text {sat}}} \nonumber \\= & {} \left. 9 n_{b} \frac{\partial (p/n_b)}{\partial n_{b}} \right| _{n_{b}=n_{b}^{\text {sat}}} \quad [\hbox {MeV}] \, . \end{aligned}$$

  (6.7)

  It is related to the isothermal compressibility $\kappa _{T}$ (3.44) at zero temperature, zero asymmetry and saturation density by

  $$\begin{aligned} K = \frac{9}{\kappa _{T} n_{b}^{\text {sat}}} \quad [\hbox {MeV}] \, . \end{aligned}$$

  (6.8)

  Thus K would be better called compression modulus. It can be derived, e.g. from studies of isoscalar giant monopole excitations of nuclei. Since the heavy nuclei under study are not isospin symmetric, but have a neutron excess, and are charged, their monopole excitations and thus their incompressibility $K_{A}$ depend not only the isoscalar quantity K, but also on isospin $K_{\tau }$ and Coulomb $K_{\text {coul}}$, see, e.g., Eq. (2) in reference [28]. Thus the extraction from data on isoscalar giant monopole resonances depends on the density dependence of the nuclear symmetry energy, a quantity under intensive debate in recent years. See the recent comprehensive study of giant monopole resonances [28].

• The skewness coefficient of bulk nuclear matter $Q=-K^{\prime }$ [MeV] is defined by the third derivative of the energy per baryon as

  $$\begin{aligned} Q= & {} 27 n_{b}^{3} \left. \frac{\partial ^{3} {\mathcal {E}}(n_{b},0)}{\partial n_{b}^{3}} \right| _{n_{b} = n_{b}^{\text {sat}}} \nonumber \\= & {} 27 n_{b}^{2} \left. \frac{\partial ^{2} (p/n_{b})}{\partial n_{b}^{2}} \right| _{n_{b} = n_{b}^{\text {sat}}} - 6 K \quad [\hbox {MeV}] \, . \end{aligned}$$

  (6.9)

  The value of Q in combination with K determines the surface properties of nuclei, e.g. the ratio of the surface tension to the surface thickness. Thus, in order to fix Q in a particular model, it is important to include quantities like radii and the surface thickness in the procedure to fit the model parameters.

• The symmetry energy at saturation J [MeV] is just given by

  $$\begin{aligned} J = E_{\text {sym}}(n_{b}^{\text {sat}}) \quad [\hbox {MeV}]. \end{aligned}$$

  (6.10)

  This coefficient mainly determines the isospin dependence of the binding energy of nuclei. It is important in predicting masses of exotic nuclei far away from the valley of stability in the chart of nuclei.

• The symmetry energy slope coefficient L [MeV] that is obtained from

  $$\begin{aligned} L = 3 n_{b} \left. \frac{dE_{\text {sym}}(n_{b})}{dn_{b}} \right| _{n_{b} = n_{b}^{\text {sat}}} \quad [\hbox {MeV}]. \end{aligned}$$

  (6.11)

  This coefficient is related to the density dependence of the neutron matter EoS near the saturation density. The correlation of L with the neutron skin thickness of heavy nuclei like ${}^{208}$Pb is one of the strongest between quantities of nuclei and nuclear matter.

  Unfortunately, experimentally determined values of the neutron skin thickness are not very precise so far and a large range of values for L is found by comparing different EoS models.

• The symmetry incompressibility $K_{\text {sym}}$ [MeV] quantifies the curvature of the symmetry energy with respect to the density variation at saturation. It is defined by

  $$\begin{aligned} K_{\text {sym}} = 9 n_{b}^{2} \left. \frac{d^{2} E_{\text {sym}}}{d n_{b}^{2}} \right| _{n_{b} = n_{b}^{\text {sat}}} \quad [\hbox {MeV}] \, . \end{aligned}$$

  (6.12)

For each EoS in the CompOSE data base, the actual values of the nuclear matter coefficients are given if available.

10.2 Remarks of caution

We provide equations of state tables in the CompOSE data base that cover large ranges in temperature, baryon number density and charge fraction of strongly interacting particles. However, we cannot guarantee that the physical description of matter under these conditions is close to reality. The user has to judge whether a particular EoS is suitable for her or his application.

The user also has to keep in mind that the thermodynamic and compositional properties of nuclear matter without electrons (and muons) and dense stellar matter with electrons (and muons) are rather different because in the former case the Coulomb interaction is artificially neglected. This can have drastic consequences for the occurrence and type of phase transitions. Thus it is not recommended to generate a dense matter EoS from a nuclear matter EoS simply by adding the contribution from electrons (and muons).

It is not guaranteed that all models used for providing an EoS always obey all physical principles or limits. A specific example is the issue of causality. Some EoS, in particular from non-relativistic approaches, can reach parts of the space of variables where the speed of sound exceeds the speed of light. This can be checked by an explicit calculation of the speed of sound with the help of the CompOSE code, thus yielding boundaries for the applicability of the model. Corresponding warnings are sometimes mentioned in the data sheet accompanying every EoS table.

In many cases it is desirable to extend a particular EoS to densities and temperatures below or above the available or recommended parameter ranges. In these regimes, an equation of state depending only on temperature, charge fraction of strongly interacting particles and baryonic density to describe matter might not be sufficient, e.g. at very low densities and temperatures where the time scales of thermodynamic equilibration reaction rates become large and full reaction networks have to be considered. However, for many purposes a detailed description of matter in these regime is not necessary, such that we decided to furnish tables for these conditions, too.

Our knowledge about the QCD phase diagram suggests that there could be a transition from a hadronic phase to a quark–gluon plasma within the range of densities and temperatures reachable in core-collapse supernovae, hence within the range of our tables. Of course, there are lots of uncertainties about this phase transition, so that its occurrence cannot be affirmed, but the possibility has to be kept in mind when employing a purely hadronic EoS up to densities well above nuclear matter saturation density and temperatures as high as several tens of MeV. Within CompOSE there will be tables available including this phase transition. Even without thinking about a QCD phase transition, other forms of (exotic) matter shall appear at high densities and temperatures. Already for cold matter EoS used for modelling neutron stars for a long time, hyperons, pions and kaons have been considered. At temperatures above about 20 MeV, this point becomes even more crucial. Thus using a purely nuclear EoS in this regime can be a first approximation, technically very appealing, but again it has to be kept in mind that probably the considered physics is too poorly described.

11 Online services and data handling

11.1 Access to CompOSE

The EoS data and routines provided by CompOSE can be downloaded from the web site

https://compose.obspm.fr .

The tables and codes are offered free of charge but come without warranty. Please acknowledge and give proper reference to CompOSE if you use our service for your applications, presentations and publications (see appendix E). If you encounter any difficulties, it is possible to contact the CompOSE team by sending an email to

develop.compose@obspm.fr.

To subscribe to the CompOSE newsletter, distributed via the email list “compose.info”, where changes and updates will be announced, send an e-mail with subject “Subscribe” to develop.compose@obspm.fr.

11.2 EoS data sheets

Each EoS of the CompOSE data base is accompanied with a data sheet available for download from the web site. The data sheet provides essential information that allows the user to decide whether the EoS is suitable for her/his application. It contains information on the origin and creation of the EoS table, a short abstract of the physical model, references, the parameter ranges and considered particle species, a summary of the available data files and fundamental quantities that characterize the EoS, such as nuclear matter and neutron star properties, if available.

11.3 Options for using EoS data

There are different options for downloading and/or generating EoS data and tables. It is possible to obtain EoS tables for different models and for different purposes. The original EoS tables are given as plain ASCII files that allow every user to read the data without the need for further codes. The general idea is that a user chooses a particular model from those available on the CompOSE web site. Then, there are three different ways of accessing EoS data:

• 1. The user downloads data tables of an available EoS in the original form and uses her or his own routines to handle the data. For every model EoS there are three tables that contain the details of the discretization of the parameters: eos.t, eos.nb and eos.yq . The original EoS data are stored as functions of temperature T, baryon number density $n_{b}$ and charge fraction of strongly interacting particles $Y_{q}$ . The last quantity is identical to the electron fraction $Y_{e}$ if electrons are the only charged leptons considered in the EoS model. The actual data on the thermodynamic, compositional and microscopic properties are stored in the files eos.thermo, eos.compo and eos.micro, respectively, as far as available for that particular model. See Sect. 4.2 for the format of all the tables. If available for an EoS, the file eos.mr contains information of the properties of compact stars and can be read directly without the need to use the CompOSE software.

• 2. The user downloads the original data tables for a particular EoS as described above and, in addition, several files that contain routines for reading, testing, interpolating and transforming the data. These codes serve four major purposes:

  • to interpolate the original EoS data tables in order to obtain the quantities at parameter values different from the original tabulation,

  • to calculate additional quantities that are not given in the original data files,

  • to select those quantities that are relevant for a particular application and to store them in separate data files in a format more convenient for the user,

  • to provide EoS data tables in the advanced HDF5 format (see http://www.hdfgroup.org/HDF5/) that is widely used in the astrophysics community.

  Details on how to work with the subroutines are described in the Sect. 7.5.

• 3. Using the tools provided on the webpage to manipulate and visualise EoS data. In contrast to downloading data and the CompOSE software, the access to running the web tools is password restricted. Please contact develop.compose@obspm.fr if you wish an account to be created for you.

11.4 Different families of EoS data

There is now a large number of different EoS models for which data is available on the web page for free download. In order to more easily find the data for a particular application, the different models have been ordered into different categories:

• 1. Cold Neutron Star EoS: this category contains models designed to describe cold (zero temperature) matter in weak ($\beta$) equilibrium. They all contain the contribution of electrons (and for some of them muons, too) and are directly applicable to construct cold neutron star models, e.g. by using the eos_compose class with the LORENE library.³

• 2. Neutron Star Crust EoS: category for EoS that describe the non-homogeneous matter of the outer crust and or inner crust, e.g. the Baym–Pethick–Sutherland (BPS) EoS. The set of crust EoS is divided into outer crust and inner crust and may be used, together with a core EoS, to construct a complete EoS. Some EoS account for the appearance of pasta phases in the transition region from the inner crust to the homogeneous core. In order for radii of low mass NS not to be affected by the crust EoS, the lowest considered density in the table should be sufficiently small, i.e., it has to reach small densities of the order of $10^{-9}$ fm$^{-3}$, where iron group nuclei appear in the outer crust. This is not the case for all models. If the model provides at least an inner crust, any outer crust model can be matched, the exact choice of the outer crust EoS does not much affect global compact star properties such as radius or mass. On the contrary, if the model only contains a core EoS, then some attention should be paid to the choice of the inner crust EoS for matching and the matching conditions, since in particular radii and tidal deformabilities can depend on that choice. The choice is optimal if the symmetry energy properties are similar to the ones of the core EoS. The crust thickness is sensitive to the crust-core transition density which in turn is sensitive to the medium modifications of the cluster properties and the treatment of the pasta phases. A small routine that allows for a thermodynamically consistent matching of the crust EoS to the core EoS will be made available in the near future on the CompOSE web page, e.g. it checks that $p_{\textrm{crust}} = p_{\textrm{ core}}$ together with $\mu _{\textrm{crust}} = \mu _{\textrm{core}}$ at the transition.

• 3. Cold Matter EoS: this category contains models at zero temperature for different charge fractions. It is divided into different entrances according to the underlying physical approach.

• 4. Neutron Matter EoS: this category contains models for neutron matter, i.e. for $Y_q$ = 0 (the net abundances of charged particles and antiparticles is zero) at both $T=0$ and finite temperature.

• 5. General Purpose EoS: this category contains models which cover a large range of different temperatures, charge fractions and baryon number densities, as required in particular for simulations of core-collapse supernovae and binary neutron star mergers. For convenience, most models are provided in two versions, one with the contribution of electrons, positrons and photons included and one containing only the baryonic part.

Note that the “Crust EoS” category contains EoS models that describe only the crust, while the category “Cold Neutron Star EoS” may contain models which describe only the core EoS, or a complete EoS with crust and core. In the first case a crust EoS should be matched to the core EoS as available in the category “Crust EoS” before an application to neutron stars. In addition to this very general distinction between EoS models, based mainly on distinguishing the different possible applications of the EoS data, there are sub-families. These sub-families first describe the particular assumptions for the particle content of matter:

• 1. Nucleonic models, EoS models that assume matter to be composed of nucleons, charged leptons and potentially nuclear clusters,

• 2. Hybrid (quark-hadron) models, category for EoS models with a transition from hadronic matter at low densities to quark matter at high densities,

• 3. Models with hyperons, category for EoS models allowing for hyperons (strange baryons) to be populated,

• 4. Models with hyperons and $\Delta$-resonances, category for EoS models allowing for hyperons (strange baryons) and $\Delta$-resonances to be populated,

• 5. Models with condensates, category for EoS models allowing for kaon or pion condensates to set in,

• 6. Quark models, category for core EoS models containing only quark matter.

Other sub-families might be added in the future. Then, there are sub-families distinguishing a particular framework to determine the EoS (see the review [2]). This concerns first the description of homogeneous matter, i.e. the interaction of nucleons, hyperons etc, and the corresponding many-body treatment:

• 1. Non-relativistic density functional models, EoS, where the interaction of baryons is based on non-relativistic density functional theory, in particular this concerns Skyrme-like interactions, e.g., SLy4, or Gogny-type interactions. In these models, the total energy density of homogeneous baryonic matter is given in terms of the sum of a kinetic and a potential energy density functional in the space defined by the different particle species number densities: proton and neutron densities for nucleonic matter and proton, neutron and hyperons for hyperonic matter.

• 2. Relativistic density functional models, EoS based on covariant density functional theory, e.g. the STOS EoS (H. Shen et al). These models are generally applied in the mean-field approximation. Baryons, nucleons and, when included, hyperons and or Delta-resonances, are described by fermionic fields that interact with purely phenomenological mesonic fields introduced in order to describe the nuclear interaction. The quantum numbers of the interaction channel give the name to the mesons. The most frequently used are the scalar and vector-isoscalar mesons named, respectively, $\sigma$ and $\omega$-mesons, and the vector-isovector $\rho$-meson. Some models also include the scalar-isovector $\delta$-meson, and when hyperons are included also the scalar and vector-isoscalar $\sigma ^*$ and $\phi$-mesons with hidden strangeness (${\bar{s}} s$ mesons) may be included. These models are divided into two categories: models with constant couplings and non-linear mesonic terms (TM1 and SFHo are two examples) and models with density dependent couplings and only quadratic mesonic terms (such as DD2 and DDME2). If the actually used values of the couplings differ from those of the original publications, these new values are provided in the data sheet.

• 3. Microscopic calculations, EoS based on ab initio calculations of nuclear matter, e.g. the BHF technique or variational calculations or Monte Carlo methods. Within these approaches different many-body methods are used to include correlations in the many-body system starting from a few-body interaction that has been fitted to observables in nucleon-nucleon scattering in vacuum and properties of bound few-nucleon systems,

• 4. Holographic models, EoS based on a holographic approach to determine homogeneous matter.

In a second step, different treatments of inhomogeneous matter are distinguished:

• 1. NSE models, using a nuclear statistical equilibrium approach to describe inhomogeneous matter at nonzero temperature. These category concerns mainly the general purpose EoS models, e.g. the models by Hempel & Schaffner-Bielich. Cold neutron star EoS obtained from the lowest temperature entry of a general purpose without a dedicated description of the crust are ranged in this category, too.

• 2. SNA models, using the single nucleus approximation to describe inhomogeneous matter at nonzero temperature. These concern the general purpose EoS models, e.g. the famous EoS by Lattimer & Swesty or H. Shen et al.

• 3. Unified EoS For “Cold neutron star EoS” this keyword indicates models where the same nuclear interaction has been used to describe the neutron star crust and core, all other models have a crust model matched to a core EoS and enter the category

• 4. Non-unified models (crust-core matched), see the description of the particular EoS table and the references to the original work for details.

• 5. Thomas–Fermi calculations concern the NS crust models and indicate the method to determine the crust properties.

Let us mention that a very convenient way to find a particular table is the bibliography search, where a function exists to search for tables related to publications of one particular author (see Sect. 7.6).

11.5 Handling and transforming EoS data

The handling of EoS data is considerably simplified by using the software provided on the CompOSE web page. In the most recent version of the compose code, the generation of the necessary parameter files to define output quantities and format is guided via terminal input. There are only few steps required in order to generate customized data tables from the original sets of EoS tables.

All quantities describing the thermodynamic, compositional and microscopic properties of the matter are found for arbitrary values of the table parameters with an interpolation scheme that is described in detail in Appendix A.1. It is possible to create tables with different mesh settings and ranges. Of course, the ranges should be chosen only within the ranges of the basis table for each model. If they are outside these ranges, error messages will indicate the problems.

11.5.1 Downloading and compiling

There are four Fortran90 files that are needed in every case to create the executable compose program. The file composemodules.f90 contains all the necessary modules and the file compose.f90 all required (sub-)routines and functions in order to read, interpolate and write the EoS data in ASCII format. The file out_to_json.f90 is only relevant for the program version used on the CompOSE webpage. All necessary files are most simply obtained by downloading the file code.zip.

Using the present version of the program compose, it is not necessary to provide input files beyond those that were downloaded from the CompOSE website. The input files that define the selected variables and tabulation scheme are created automatically. Their structure is specified in Appendix B.

If output of the EoS data in HDF5-format is needed, the files hdf5compose.f90 and hdf5writecompose.c are required in addition to compose.f90 and composemodules.f90 .

All the files can be downloaded from the corresponding CompOSE web page and they have to be compiled with an appropriate Fortran90 or C compiler. The program was written using the GNU compilers gfortran and gcc and the use of these compilers is encouraged. A sample Makefile is available on the CompOSE web page. It contains in line 26 a switch to select the compilation without (HDF = 0, default setting) and with (HDF = 1) the option for the HDF5 output. For using the HDF5 routines, the HDF5 library has to be installed. Compiling the program files with the provided Makefile generates an executable with the name compose.

11.5.2 Direct use of compose

In this subsection, the details are described how customized EoS tables from the EoS models on the CompOSE web site can be obtained by running the program compose .

Table 10 Thermodynamic quantities which can be extracted with the compose code from the EoS data with their identifiers in eos.quantities and their units

Table 11 Additional thermodynamic quantities which can be extracted with the compose code from the EoS data with their identifiers in eos.quantities and their units. (Quantities 25–28 are not yet available with the compose code.)

Table 12 Values and derivatives of the free energy per baryon which can be extracted with the compose code from the EoS data with their identifiers in eos.quantities and their units

Table 13 Quantities containing information on the composition which can be extracted with the compose code from the EoS data with their identifiers in eos.quantities and their units. The particle index $I_{i}$ has been defined in Tables 2, 3, 4 and 5. The symbols % denotes the group of nuclei

Table 14 Microscopic quantities which can be extracted with the compose code from the EoS data with their identifiers in eos.quantities and their units

Table 15 Error quantities which can be extracted with the compose code from the EoS data with their identifiers in eos.quantities and their units

Table 16 Thermodynamic quantities which are stored in the HDF5 data file together with their units and the name of the corresponding data set. If there are two lines in the second column, the first one always corresponds to $I_{\textrm{tab}} = 0$ and the second one to $I_{\textrm{tab}} \ne 0$

Table 17 Compositional and microscopic quantities which are stored in the HDF5 data file together with their units and the name of the corresponding data set. If there are two lines in the second column, the first one always corresponds to $I_{\textrm{tab}} = 0$ and the second one to $I_{\textrm{tab}} \ne 0$

The operation proceeds in the following steps:

• 1. Selection of EoS

  In a first step the user selects the EoS that is appropriate for her/his application from the list given on the CompOSE web pages. Often, there are different types of tables available for a single EoS model as described in Sect. 4.2. The tabulated data can depend on one, two or three of the independent parameters T, $n_{b}$ and $Y_{q}$. There is an option to replace the temperature T by the entropy per baryon ${\mathcal {S}}$ as parameter when running the standard version of compose . There are at least four different files needed for the successful application of the program compose as specified in Sect. 4.2.8: three files containing information on the parameter grid and at least one file with the thermodynamic data.

  The downloaded parameter and EoS files are always identified with the same name and a file name extension that specifies the type of stored quantities. The program compose uses these generic names for the input files. Thus the downloaded files are called “eos.t”, “eos.nb”, “eos.yq”, “eos.thermo”, “eos.compo” (if available), “eos.micro” (if available), and “eos.init” (if available).

  For some EoS, additional files are available: eos.mr, the properties of a cold $\beta$-equilibrated spherically symmetric compact star, i.e. mass and radius of a spherically symmetric star, together with optionally tidal deformability and corresponding central density, obtained with the particular EoS model; eos.thermo.ns and eos.nb.ns, baryon number densities and thermodynamic quantities for cold $\beta$-equilibrated matter for direct use within LORENE [22] via the eoscompose class to compute, e.g., initial data for a binary neutron star. The latter files exist only for general purpose EoS tables and have been extracted from the lowest temperature entry of the corresponding table, i.e., in general for a nonzero (but very small) temperature. eos.thermo.ns contains as additional quantity the electron fraction $Y_e$ obtained in $\beta$-equilibrium and the enthalpy density.

  Note that for some of the general purpose EoS tables, the information in the file eos.thermo.ns is not sufficient for building compact stars up to the maximum mass. The reason is that for densities exceeding a certain value, $Y_{q}$ corresponding to $\beta$-equilibrated matter becomes lower than the lowest value considered in the table, typically $Y_q = 0.01$ and thus at high densities no $\beta$-equilibrium solution is found. In this case, for some of these models, corresponding cold Neutron Star Matter tables are provided containing the complementary information.

  If the user does not run the program compose in the standard version (see 2) he or she has to supply two files (in ASCII format) in addition to the downloaded files. They specify the parameter values and the quantities that will be stored in the customized EoS output table. See Appendix B for details of their structure. The input files eos.parameters and eos.quantities are created automatically when the standard version of the program compose is used. Thus it is recommended to run the standard version, in particular for first-time users of CompOSE.

• 2. Running compose

  The user has to run the compose code three times, called “tasks” during the execution:

  • 1: to select the output quantities

  • 2: to define the tabulation parameters

  • 3: to generate the file eos.table with the customized EoS table.

  In each case, the program explicitly asks questions and gives information for possible answers. The files eos.quantities and eos.parameters are created automatically. See the “quick guide for users” for examples.

  In the case that there is no file eos.init provided on the webpage for a particular EoS, it is created automatically. During the execution of the compose program, the relevant input data files are read: first the files with the parameter grids eos.t, eos.nb and eos.yq, second the EoS data files eos.thermo, eos.compo (if available), and eos.micro (if available). The read data are analyzed, checked for consistency and the information is stored in the file eos.init .

  Running the compose code for task 1, a file eos.quantities is created. When task 2 is selected, a file eos.parameters is generated. The structure of the files eos.init, eos.quantities, and eos.parameters is described in detail in Appendix  B. A list of possible output quantities to be specified in the file eos.quantities is given in Tables 10, 11 and 12 for the thermodynamic quantities, in Table 13 for the composition information, in Table 14 for the microscopic quantities and in Table 15 for the information about errors. As an example, let us mention that asking for the output quantity with number 22 among the thermodynamic quantities will print the specific enthalpy ${\mathcal {H}}=(e+p)/n_{b}$, a quantity of particular interest for many neutron star applications.

  Each row of the output file eos.table contains the three parameter values of the user-chosen grid and then the selected quantities. Note that for each quadruple index $I_{i}$ four quantities ($Y^{\text {av}}_{I_{i}}$, $A^{\text {av}}_{I_{i}}$, $Z^{\text {av}}_{I_{i}}$, and $N^{\text {av}}_{I_{i}}$) are given.

11.5.3 HDF5 table

The organization of the data in case of the HDF5 table is different as compared to the ASCII table. All data are stored in a single data file that is denominated eoscompose.h5. The names of the data sets are given by the identifiers in Tables 16 and 17. Note, too, that the values of the different quantities at all grid points are stored together in one data set. The data sets can in addition contain a group of quantities, e.g., the $N_{\textrm{thermo}}$ thermodynamic quantities, see Tables 16 and 17 for details.

11.5.4 Additional output

During the execution of task three of the program compose, a file eos.report is generated that contains information on the EoS derived from the tabulated data. The information is used for the creation of the data sheet that accompanies each EoS on the CompOSE web pages . If possible, a second file eos.beta is produced with the EoS of $\beta$-equilibrated matter for the lowest available temperature for all grid points in baryonic density that are available for the specific EoS and where a solution is found. Each row consists of four quantities, the density $n_{b}$, the charge fraction $Y_{q}$, the (free) energy density f (including rest mass contributions) and the pressure p. These data are used to derive characteristic neutron star parameters for the data sheet. Finally, a third file eos.info.json is created that is used for the interactive web version of the program.

11.5.5 Error messages

During the execution of the subroutines, errors can occur due to several reasons, e.g. parameters out of range etc. In this case, the execution is stopped and a corresponding error message is generated.

11.6 Web tools

The web tools are constructed as an interface to the CompOSE software, i.e. all features of the code described in the previous section are available. The obtained parameter files eos.parameters, eos.quantities as well as the output table eos.table can be downloaded. In addition, it is possible to visualise the results, choosing the axis from the different computed quantities on a linear or a logarithmic scale.

12 Error reporting

During the development of CompOSE, errors in the data tables, computer codes, EoS documentation or the manual can occur. Also, the user can encounter other difficulties. The developers of CompOSE are grateful for any hints on such cases. If you spot any error or detect a problem, we ask you to contact the CompOSE core team by sending an email to

develop.compose@obspm.fr.

The CompOSE developers will investigate how errors can be rectified and make the appropriate changes.

Appendices

Part IV. Appendix

Technical details

The basic EoS tables only give a selected set of thermodynamic quantities at the grid points that are identified with the index triple ($i_{T},i_{n_{b}},i_{Y_{q}}$). From the stored quantities $Q_{i}$, $i=1,\dots ,6$, see Sect. 4.2.2, further relevant thermodynamic quantities can be derived by a smooth interpolation for all possible values of the parameters T, $n_{b}$ and $Y_{q}$ within the tabulated ranges. For these quantities, thermodynamic consistency should be respected as far as possible. When the thermodynamic quantities are interpolated separately, however, this condition is usually not exactly fulfilled. On the other hand, a separate interpolation often leads to smoother dependencies of the thermodynamic quantities on the parameters avoiding unphysical oscillations. In addition, different ways of determining a single quantity gives the opportunity to estimate the error in the interpolation. Thus, in the present code compose.f90 the strategy of a direct interpolation of individual quantities is followed.

1.1 Interpolation

The interpolation scheme for a quantity in the thermodynamic, compositional and microscopic EoS tables generally proceeds in several steps. In the following, the procedure will be explained for a three-dimensional general purpose EoS table. The interpolation scheme follows the method proposed in Ref. [29] using polynomials of sufficiently high order.

In the sequel, the generic symbol Q will be used for any tabulated quantity. Its values are given at the grid points that are specified by a triple of indices $(i_{T},i_{n_{b}},i_{Y_{q}})=(i,j,k)$, see Sect. 4.1.1, corresponding to temperature, baryon number density and charge fraction of strongly interacting particles. Thus all values $Q(T(i),n_{b}(j),Y_{q}(k))$ are known at the grid points. In order to calculate the quantity Q at given values of T, $n_{b}$, and $Y_{q}$, first the values of the indices $i = i_{T}$, $j = i_{n_{b}}$ and $k = i_{Y_{q}}$ are determined such that

$$\begin{aligned} T(i) \le&T&< T (i+1) \,, \end{aligned}$$

(A.1)

$$\begin{aligned} n_{b}(j) \le&n_{b}&< n_{b}(j+1) \,, \end{aligned}$$

(A.2)

$$\begin{aligned} Y_{q}(k) \le&Y_{q}&< Y_{q}(k+1) \, . \end{aligned}$$

(A.3)

Then the interpolation parameters

$$\begin{aligned} \xi= & {} \frac{T-T(i)}{T(i+1)-T(i)} \end{aligned}$$

(A.4)

$$\begin{aligned} \eta= & {} \frac{n_{b}-n_{b}(j)}{n_{b}(j+1)-n_{b}(j)} \end{aligned}$$

(A.5)

$$\begin{aligned} \zeta= & {} \frac{Y_{q}-Y_{q}(k)}{Y_{q}(k+1)-Y_{q}(k)} \end{aligned}$$

(A.6)

are introduced with

$$\begin{aligned} 0 \le \xi< 1 \quad 0 \le \eta< 1 \quad 0 \le \zeta < 1 \, . \end{aligned}$$

(A.7)

For the interpolation of a quantity Q at given $(T,n_{b},Y_{q})$ we need the tabulated values at least at the eight corners of the cube with grid points (i, j, k), $(i+1,j,k)$, $(i,j+1,k)$, $(i,j,k+1)$, $(i+1,j+1,k)$, $(i+1,j,k+1)$, $(i,j+1,k+1)$, $(i+1,j+1,k+1)$.

The interpolation proceeds in two steps: first, an interpolation in the variable $Y_{q}$ such that the three-dimensional grid is mapped to a two-dimensional grid with four corners of each square; second, a two-dimensional interpolation in the variables T and $n_{b}$.

For the interpolation in $Y_{q}$ in the three-dimensional cube as defined above, four separate one-dimensional interpolations along the lines that connect the grid points (i, j, k) and $(i,j,k+1)$, $(i+1,j,k)$ and $(i+1,j,k+1)$, $(i,j+1,k)$ and $(i,j+1,k+1)$, $(i+1,j+1,k)$ and $(i+1,j+1,k+1)$ have to be performed. After this first step, the interpolation proceeds in the parameters T and $n_{b}$ by a two-dimensional scheme as discussed in Sect. A.1.2.

The interpolation in the variables T, $n_{b}$ and $Y_{q}$ can be of different order in general, defined by the value I of the variables ipl_t, ipl_n and ipl_y, see Sect. 7.5. In the following, the case of highest order ($I=3$) is considered first.

1.1.1 Interpolation in one dimension

The order $I=3$ of the interpolation requires that the function values, its first and second derivatives are continuous at the two corner points of each line, i.e. six values have to be reproduced by the interpolation polynomial. Hence, a polynomial of at least fifth degree has to be used. The six independent coefficients $q_{n}$ of a single polynomial

$$\begin{aligned} Q(T(i),n_{b}(j),Y_{q}) = \sum _{n=0}^{5} q_{n} \zeta ^{n} \end{aligned}$$

(A.8)

can be determined from the function values and derivatives at the corner points directly. One finds

$$\begin{aligned} q_{0} = Q_{ijk}^{(0)} \quad q_{1} = Q_{ijk}^{(1)} \quad q_{2} = \frac{1}{2} Q_{ijk}^{(2)} \end{aligned}$$

(A.9)

with

$$\begin{aligned} Q^{(n)}_{ijk} = \left[ Y_{q}(k+1)-Y_{q}(k)\right] ^{n}\left. \frac{\partial ^{n}Q}{\partial Y_{q}^{n}}\right| _{T(i),n_{b}(j),Y_{q}(k)} \end{aligned}$$

(A.10)

for $n=0,1,2$, and

$$\begin{aligned} Q_{ijk}^{(0)} = \left. \frac{\partial ^{0}Q}{\partial Y_{q}^{0}}\right| _{T(i),n_{b}(j),Y_{q}(k)} = Q(T(i),n_{b}(j),Y_{q}(k)) \end{aligned}$$

(A.11)

in particular. The remaining three coefficients are given by

$$\begin{aligned} q_{3}= & {} 10 A -4 B + \frac{1}{2} C \end{aligned}$$

(A.12)

$$\begin{aligned} q_{4}= & {} -15 A + 7 B - C \end{aligned}$$

(A.13)

$$\begin{aligned} q_{5}= & {} 6 A - 3 B + \frac{1}{2}C \end{aligned}$$

(A.14)

with

$$\begin{aligned} A= & {} Q_{ijk+1}^{(0)}- Q_{ijk}^{(0)} - Q_{ijk}^{(1)} - \frac{1}{2} Q_{ijk}^{(2)} \end{aligned}$$

(A.15)

$$\begin{aligned} B= & {} Q_{ijk+1}^{(1)} - Q_{ijk}^{(1)} - Q_{ijk}^{(2)} \end{aligned}$$

(A.16)

$$\begin{aligned} C= & {} Q_{ijk+1}^{(2)} - Q_{ijk}^{(2)} \, . \end{aligned}$$

(A.17)

Alternatively, the approach of Ref. [29] can be followed. The value of the quantity Q at given $\zeta$ is found with the help of the quintic basis functions

$$\begin{aligned} \psi _{0}^{(0)}(z)= & {} 1 - 10 z^{3} + 15 z^{4} - 6 z^{5} \end{aligned}$$

(A.18)

$$\begin{aligned} \psi _{1}^{(0)}(z)= & {} z - 6 z^{3} + 8 z^{4} - 3 z^{5} \end{aligned}$$

(A.19)

$$\begin{aligned} \psi _{2}^{(0)}(z)= & {} \frac{1}{2} \left( z^{2} - 3 z^{3} + 3 z^{4} - z^{5} \right) \end{aligned}$$

(A.20)

that have the properties

$$\begin{aligned} \psi _{m}^{(n)}(0) = \delta _{nm} \quad \psi _{m}^{(n)}(1) = 0 \end{aligned}$$

(A.21)

for

$$\begin{aligned} \psi _{m}^{(n)}(z) = \frac{d^{n}\psi _{m}^{(0)}}{dz^{n}} \end{aligned}$$

(A.22)

with $n,m \in \{0,1,2\}$. Then one has

$$\begin{aligned}{} & {} Q(T(i),n_{b}(j),Y_{q}) \nonumber \\{} & {} \quad = \sum _{n=0}^{2} \left[ Q^{(n)}_{ijk} \psi _{n}^{(0)}(\zeta ) + (-1)^{n} Q^{(n)}_{ijk+1} \psi _{n}^{(0)}(1-\zeta )\right] \end{aligned}$$

(A.23)

with six interpolation coefficients that are given by directly by the function values and derivatives at the corner points defined by the indices. Derivatives are easily found as

$$\begin{aligned}{} & {} \frac{\partial ^{s}}{\partial Y_{q}^{s}} Q(T(i),n_{b}(j),Y_{q}) \nonumber \\{} & {} \quad = \frac{1}{\left[ Y_{q}(k+1)-Y_{q}(k)\right] ^{s}} \nonumber \\{} & {} \qquad \sum _{n=0}^{2} \left[ Q^{(n)}_{ijk} \psi _{n}^{(s)}(\zeta ) + (-1)^{n+s} Q^{(n)}_{ijk+1} \psi _{n}^{(s)}(1-\zeta )\right] \, . \end{aligned}$$

(A.24)

The numerical determination of the derivatives depends on the order of interpolation. For $I=3$, centered five-point finite difference formulas are used to calculate the first and second derivative of a function. Close to the boundaries of the EoS, centered difference formulas cannot be applied. In this case, off-center formulas are used. In the case $I=2$, three-point finite difference formulas are employed. A continuity of the second derivatives at the corner points is not required. Only the function and the first derivative are demanded to be continuous, corresponding to four independent quantities. Hence, a polynomial of third degree is sufficient and $q_{4} = q_{5} = 0$ in Eq. (A.8). Then, the coefficients of the second derivatives have to be defined as

$$\begin{aligned} Q_{ijk}^{(2)}= & {} 6 \left[ Q_{ijk+1}^{(0)} - Q_{ijk}^{(0)} \right] - 2 Q_{ijk+1}^{(1)} - 4 Q_{ijk}^{(1)} \end{aligned}$$

(A.25)

$$\begin{aligned} Q_{ijk+1}^{(2)}= & {} - 6 \left[ Q_{ijk+1}^{(0)} - Q_{ijk}^{(0)} \right] + 4 Q_{ijk+1}^{(1)} + 2 Q_{ijk}^{(1)} \, .\nonumber \\ \end{aligned}$$

(A.26)

For $I=1$, the function has to be continuous at the grid points but there is no condition on the derivatives. In this case, one sets

$$\begin{aligned} Q_{ijk}^{(1)} = Q_{ijk+1}^{(1)} = Q_{ijk+1}^{(0)}-Q_{ijk}^{(0)} \quad Q_{ijk}^{(2)} = Q_{ijk+1}^{(2)} = 0 \ \nonumber \\ \end{aligned}$$

(A.27)

for given function values $Q_{ijk}^{(0)}$ and $Q_{ijk+1}^{(0)}$. Then the polynomial (A.8) reduces to a linear function with $q_{2} = q_{3} = q_{4} = q_{5} = 0$.

1.1.2 Interpolation in two dimensions

The interpolation in two dimensions with conditions on the continuity of the function and its derivatives is more complicated than in the one-dimensional case. For an interpolation order $I=3$ in both variables $\xi$ and $\eta$, there are four function values, eight first derivatives and the twelve second derivatives, hence 24 values in total, that can be used to determine the coefficients of a polynomial in $\xi$ and $\eta$. From this consideration, it would seem to be sufficient to use a polynomial

$$\begin{aligned} Q(T,n_{b},Y_{q}) = \sum _{m=0}^{4} \sum _{n=0}^{4} q_{mn} \xi ^{m} \eta ^{n} \end{aligned}$$

(A.28)

that includes powers up to four in both variables, resulting in $5 \times 5 = 25$ coefficients $q_{nm}$ in total, i.e. one more than required. However, such a form does not guarantee that the function and the derivatives are continuous not only at the grid points but also along the boundaries of the interpolation square. In fact, a continuity of the function Q, the first derivatives $\partial Q/\partial T$ and $\partial Q/\partial n_{b}$, the second derivatives $\partial ^{2}Q/\partial T^{2}$, $\partial ^{2}Q/\partial T \partial n_{b}$ and $\partial ^{2}Q/\partial n_{b}^{2}$, the third derivatives $\partial ^{3}Q/\partial T^{2} \partial n_{b}$ and $\partial ^{3}Q/\partial T \partial n_{b}^{2}$ and the fourth derivative $\partial ^{4}Q/\partial T^{2} \partial n_{b}^{2}$ at the corners has to be demanded, determining $4\times 9 = 36 = 6\times 6$ coefficients of a polynomial

$$\begin{aligned} Q(T,n_{b},Y_{q}) = \sum _{m=0}^{5} \sum _{n=0}^{5} q_{mn} \xi ^{m} \eta ^{n} \end{aligned}$$

(A.29)

with degree six in each variable.

Instead of determining the coefficients directly as in the case of a one-dimensional interpolation, it is more advantageous to use the biquintic interpolation scheme as in Ref. [29] with

$$\begin{aligned}{} & {} Q(T,n_{b},Y_{q})\nonumber \\{} & {} \quad = \sum _{m=0}^{2} \sum _{n=0}^{2} \left[ Q^{(mn)}_{ij} \psi _{m}^{(0)}(\xi )\psi _{n}^{(0)}(\eta ) \right. \nonumber \\{} & {} \qquad + (-1)^{m} Q^{(mn)}_{i+1j} \psi _{m}^{(0)}(1-\xi )\psi _{n}^{(0)}(\eta ) \nonumber \\{} & {} \qquad + (-1)^{n} Q^{(mn)}_{ij+1} \psi _{m}^{(0)}(\xi )\psi _{n}^{(0)}(1-\eta ) \nonumber \\{} & {} \left. \qquad + (-1)^{m+n} Q^{(mn)}_{i+1j+1} \psi _{m}^{(0)}(1-\xi )\psi _{n}^{(0)}(1-\eta ) \right] \end{aligned}$$

(A.30)

and $4 \times 9 = 36$ coefficients

$$\begin{aligned} Q^{(mn)}_{ij}= & {} \left[ T(i+1)-T(i)\right] ^{m}\nonumber \\{} & {} \left[ n_{b}(j+1)-n_{b}(j)\right] ^{n} \left. \frac{\partial ^{m+n} Q}{\partial T^{m} \partial n_{b}^{n}} \right| _{T(i),n_{b}(j),Y_{q}} \end{aligned}$$

(A.31)

$$\begin{aligned} Q^{(mn)}_{i+1j}= & {} \left[ T(i+1)-T(i)\right] ^{m}\nonumber \\{} & {} \left[ n_{b}(j+1)-n_{b}(j)\right] ^{n} \left. \frac{\partial ^{m+n} Q}{\partial T^{m} \partial n_{b}^{n}} \right| _{T(i+1),n_{b}(j),Y_{q}} \end{aligned}$$

(A.32)

$$\begin{aligned} Q^{(mn)}_{ij+1}= & {} \left[ T(i+1)-T(i)\right] ^{m}\nonumber \\{} & {} \left[ n_{b}(j+1)-n_{b}(j)\right] ^{n} \left. \frac{\partial ^{m+n} Q}{\partial T^{m} \partial n_{b}^{n}} \right| _{T(i),n_{b}(j+1),Y_{q}} \end{aligned}$$

(A.33)

$$\begin{aligned} Q^{(mn)}_{i+1j+1}= & {} \left[ T(i+1)-T(i)\right] ^{m}\nonumber \\{} & {} \left[ n_{b}(j+1)-n_{b}(j)\right] ^{n} \left. \frac{\partial ^{m+n} Q}{\partial T^{m} \partial n_{b}^{n}} \right| _{T(i+1),n_{b}(j+1),Y_{q}} \end{aligned}$$

(A.34)

where $m,n \in \{0,1,2\}$. Of course, it is possible to express the coefficients $q_{mn}$ of the polynomial (A.29) through the coefficients $Q_{ij}^{(mn)}$, as done in the code compose.f90. The derivatives are obtained in the same way as in the one-dimensional case for the different interpolations orders I. Derivatives of the function Q with respect to Y and $n_{b}$ can be found with the help of the relation

$$\begin{aligned}{} & {} \frac{\partial ^{s+t}}{\partial T^{s}\partial n_{b}^{t}}Q(T,n_{b},Y_{q}) \nonumber \\{} & {} \quad = \frac{1}{\left[ T(i+1)-T(i)\right] ^{s}} \frac{1}{\left[ n_{b}(j+1)-n_{b}(j) \right] ^{t}}\nonumber \\{} & {} \qquad \times \sum _{m=0}^{2} \sum _{n=0}^{2} \left[ Q^{(mn)}_{ij} \psi _{m}^{(s)}(\xi )\psi _{n}^{(t)}(\eta ) \right. \nonumber \\{} & {} \qquad + (-1)^{m+s} Q^{(mn)}_{i+1j} \psi _{m}^{(s)}(1-\xi )\psi _{n}^{(t)}(\eta ) \nonumber \\{} & {} \qquad + (-1)^{n+t} Q^{(mn)}_{ij+1} \psi _{m}^{(s)}(\xi )\psi _{n}^{(t)}(1-\eta ) \nonumber \\{} & {} \qquad \left. + (-1)^{m+n+s+t} Q^{(mn)}_{i+1j+1} \psi _{m}^{(s)}(1-\xi )\psi _{n}^{(t)}(1-\eta ) \right] \, .\nonumber \\S \end{aligned}$$

(A.35)

Above considerations apply to the interpolation in all three parameters for a general purpose EoS table. For neutron matter, i.e. $Y_{q}=0$ the interpolation in the charge fraction of strongly interacting particles is trivial since $\zeta =0$ and the tabulated values of the quantities can be used directly in the second interpolation step in T and $n_{b}$. A similar procedure applies to the case of an EoS for symmetric nuclear matter of $\beta$-equilibrium. In the case of a zero-temperature table, the second interpolation step reduces from two to one dimension.

1.2 Matter in beta equilibrium

In the case of matter in $\beta$ equilibrium, there is an additional condition on the chemical potentials that reduces the number of independent parameters by one. Requiring that the weak interaction reactions

$$\begin{aligned} p + e^{-} \leftrightarrow n + \nu _{e} \quad p + {\bar{\nu }}_{e} \leftrightarrow n + e^{+} \end{aligned}$$

(A.36)

and, if muons are considered in the model,

$$\begin{aligned} p + \mu ^{-} \leftrightarrow n + \nu _{\mu } \quad p + {\bar{\nu }}_{\mu } \leftrightarrow n + \mu ^{+} \end{aligned}$$

(A.37)

are in weak equilibrium, the relations

$$\begin{aligned} \mu _{p} + \mu _{e} = \mu _{n} + \mu _{\nu _{e}} \end{aligned}$$

(A.38)

and

$$\begin{aligned} \mu _{p} + \mu _{\mu } = \mu _{n} + \mu _{\nu _{\mu }} \,, \end{aligned}$$

(A.39)

respectively, should hold. The chemical potentials of the neutrinos are considered to be zero, i.e. $\mu _{\nu _{e}}=\mu _{\nu _{\mu }}=0$. Because $\mu _{p} = \mu _{n}+\mu _{q}$, $\mu _{e} = \mu _{le}-\mu _{q}$ and $\mu _{\mu } = \mu _{l\mu }-\mu _{q}$, the constraint can be formulated as

$$\begin{aligned} \mu _{l} = 0 \end{aligned}$$

(A.40)

with the effective lepton chemical potential $\mu _{l} = \mu _{le} = \mu _{l\mu }$. For given temperature T and baryon number density $n_{b}$ there is a unique charge fraction of strongly interacting particles $Y_{q}$ that is found by determining the zero of the function

$$\begin{aligned} f(Y_{q}) = \mu _{l}(Y_{q}) \, . \end{aligned}$$

(A.41)

Structure of input files

During the execution of the program compose for tasks 1 and 2, three files (in ASCII format) are generated that are used for task 3 as input files. The structure of these files is described below. Experienced users of compose can modify the files eos.parameters and eos.quantities according to their needs when producing a new EoS table with modified parameters by running only task 3. The file eos.init has to be left untouched.

1.1 eos.init

There are 11 rows in the file eos.init. The first three numbers in the first row indicate whether the EoS dependence on the temperature, the baryon density, and the hadronic charge fraction (index = 1) or not (index = 0). The following two integers are presently set to zero and will be used only for future extensions of the compose code. The next three rows give the minimum and maximum values for the variables T, $n_{b}$, and $Y_{q}$. In the fifth row the number zero appears twice in the present version as dummy floating-point variable. Row six contained four integers. The first indicates whether leptons are included in the EoS (1: yes, 0: no). The second number gives the number of additional quantities in the file eos.thermo and the third number is the total number of thermodynamic quantities in each row of the file eos.thermo. The last integer indicates whether a calculation of an EoS table for the condition of $\beta$ equilibrium is possible (1) or not (0). Row seven gives the number of pairs and quadruples in the file eos.compo and the following two lines contain the corresponding indices according to Tables 2, 3, 4 and 5 for pairs (first line) and according to the data sheet for quadruples (second line). In row ten the number of microscopic quantities is given and their indices are listed in the last row of the file eos.init.

1.2 eos.quantities

This file contains 18 rows. They determine which and how quantities will be stored in the final EoS table with the name eos.table. Rows with odd number (one, three, ...) are comment lines beginning with the character #. The second and fourth lines define the thermodynamic quantities that will be stored in the output file eos.table. The first entry $N_{\text {thermo}}$ in row two specifies the number of thermodynamic quantities that are selected from Tables 10 and 11 followed by the number $N_{\text {add}}$ of quantities that are selected from the stored quantities of the EoS file eos.thermo in addition to the seven standard quantities as described in Sect. 4.2.2. The last number $N_{\text {deriv}}$ in row two is the number of quantities from Table 12 that contains the free energy per baryon and various derivatives of it. The fourth line of the file eos.quantities lists the $N_{\text {thermo}}$ indices (if $N_{\text {thermo}} > 0$) from the second column of Tables 10 and 11. that define the available thermodynamic quantities, the $N_{\text {add}}$ indices (if $N_{\text {add}} > 0$) of the additionally stored quantities and the $N_{\text {deriv}}$ indices (if $N_{\text {deriv}} > 0$) from the first column of Table 12. E.g., the first four lines

# number of regular, additional and derivative quantities

3 1 1

# indices of regular, additional and derivative quantities

6 1 2 1 1

of an example file eos.quantities denote that the quantities ${\mathcal {F}}/m_{n}-1$, p, ${\mathcal {S}}$, $q_{1}$ and ${\mathcal {F}}$ will appear in the output file eos.table or eoscompose.h5 . Regular thermodynamic quantities are those that are obtained by a direct interpolation of the tabulated quantities ${\mathcal {Q}}_{1}$, ..., ${\mathcal {Q}}_{7}$ stored in the file eos.thermo or those that are obtained from them by applying the thermodynamic relations of Sect. 3.6. The indices of the regular thermodynamic quantities can have values of 1, 2, ..., 23 corresponding to the definition in Tables 10 and 11. Additional thermodynamic quantities are those that are denoted $q_{1}$, ..., $q_{N_{\text {add}}}$ in the file eos.thermo. Their number $N_{\text {add}}$ and meaning depends on the specific EoS table. Their values are found by a direct interpolation. The values for ${\mathcal {F}}$ and its derivatives are found from a direct interpolation and subsequent derivatives of the interpolating function. If $N_{\text {thermo}}$, $N_{\text {add}}$ and $N_{\text {deriv}}$ are zero the fourth line of the file eos.quantities is empty.

The sixth row of the file eos.quantities contains the numbers $N_{p}$ and $N_{q}$ of pairs and quadruples that are selected from the compositional quantities that are stored in the file eos.compo as described in Sect. 4.2.3. In the eighth line the $N_{p}$ particle indices $I_{1}$, ..., $I_{N_{p}}$ as defined in Tables 2, 3, 4 and 5 are followed by the $N_{q}$ indices that identify a particular group of nuclei, see Sect. 4.2.3. E.g., the four lines

# number of pairs and quadruples for composition data

3 1

# indices of pairs and quadruples for composition data

10 11 0 1

denote that the fractions $Y_{n}$, $Y_{p}$, $Y_{e}$ are stored in the file eos.table followed by $Y^{\text {av}}_{1}$, $A^{\text {av}}_{1}$, $Z^{\text {av}}_{1}$, and $N^{\text {av}}_{1}$ of the first set of nuclei. If both $N_{p}$ and $N_{q}$ are zero the eighth line of the file eos.quantities is empty.

The tenth line of the file eos.quantities contains the number $N_{\text {mic}}$ of microscopic quantities in the file eos.micro to be stored in the output file eos.table. Then line 12 is a list of $N_{\text {mic}}$ composite indices $K_{i}$ as defined in Sect.  4.2.4. E.g., the four lines

# number of microscopic quantities

2

# indices of microscopic quantities

10050 11050

denote that the nonrelativistic single-particle potentials $U_{n}$ and $U_{p}$ are stored in the file eos.table. If $N_{\text {mic}}$ is zero the twelfth line of the file eos.quantities is empty.

In addition to the quantities considered above, error estimates for the interpolation of the thermodynamic quantities ${\mathcal {F}}$, ${\mathcal {E}}$, $p/n_{b}$ and ${\mathcal {S}}$ are available. The free energy per baryon ${\mathcal {F}}$ can be obtained by direct interpolation or by using the homogeneity condition (4.5) with the interpolated values for the pressure and chemical potentials. An estimate for the absolute error in ${\mathcal {F}}$ is then given by

$$\begin{aligned} \Delta {\mathcal {F}} = {\mathcal {F}} + \frac{p}{n_{b}} - \left( \mu _{b} + Y_{q} \mu _{q}\right) \end{aligned}$$

(B.1)

in case of an EoS without leptons and

$$\begin{aligned} \Delta {\mathcal {F}} = {\mathcal {F}} + \frac{p}{n_{b}} - \left( \mu _{b} + Y_{l} \mu _{l}\right) \end{aligned}$$

(B.2)

in case of an EoS with leptons and charge neutrality. The relative error is then given by $\Delta {\mathcal {F}}/{\mathcal {F}}-1$. The consideration above also apply to the internal energy per baryon ${\mathcal {E}}$. An estimate for the absolute error in $p/n_{b}$ is obtained by comparing the directly interpolated pressure p with that derived from the first derivative of the free energy per baryon, i.e.

$$\begin{aligned} \Delta \frac{p}{n_{b}} = \frac{p}{n_{b}} - n_{b} \left. \frac{\partial {\mathcal {F}}}{\partial n_{b}} \right| _{T,Y_{q}} \, . \end{aligned}$$

(B.3)

Similarly, an error estimate for the entropy per baryon is calculated from

$$\begin{aligned} \Delta {\mathcal {S}} = {\mathcal {S}} - \left. \frac{\partial {\mathcal {F}}}{\partial T} \right| _{n_{b},Y_{q}} \, . \end{aligned}$$

(B.4)

The sixteenth line of the file eos.quantities defines the number $N_{\text {err}}$ of error quantities to be stored in the output file. Line 18 is a list of the $N_{\text {err}}$ indices J as defined in Table 15. E.g., the four lines

# number of error quantities

2

# indices of error quantities

1 2

denote that the estimates $\Delta {\mathcal {F}}$ and $\Delta {\mathcal {F}}/{\mathcal {F}}$ are stored in the file eos.table. If $N_{\text {err}}$ is zero the eighteenth line of the file eos.quantities is empty.

The last row of the file eos.quantities defines the format of the output file by a single integer I. There are two methods that are used for storing the EoS data. The first ($I=1$) is the simple ASCII format using the file name eos.table that easily allows to read the data without the need for further codes. The second method ($I \ne 1$) uses the more advanced HDF5 format (see http://www.hdfgroup.org/HDF5/) that is widely used in the astrophysics community. In this case, the output file carries the name eoscompose.h5 and each data set is designated with a particular identifier $ of the stored quantity Q. Tables 16 and 17 give a list of all tabulated quantities and corresponding identifiers $.

Note that $\varepsilon ={\mathcal {E}}/m_{n}-1$ in Table 10 is not the specific internal energy, since the total mass density is not given by $m_{n} n_{b}$. We do not store ${\mathcal {E}}/m_{n}$ because it is in many regimes largely dominated by the rest mass contribution, such that it is numerically difficult to keep trace of small variations in the internal energy in that case. The same argument applies to the other thermodynamic quantities ${\mathcal {F}}$, ${\mathcal {G}}$ and ${\mathcal {H}}$. The present choice of storing $\varepsilon$ is also motivated by the fact that the total mass density is not a conserved quantity throughout the hydrodynamic evolution, but only the baryon number density, $n_{b}$.

The individual chemical potentials for all the particles (baryons, mesons, nuclei, quarks, ...) can be calculated from the three ones given in the tables. Note that we use the relativistic definition of the chemical potentials.

Particle fractions $Y_{i}$ are stored in the data sets with identifier $y\#$ where $\#$ stands for the particle index $I_{i}$ as defined in Tables 2, 3, 4 and 5. E.g., the data identifier y20 and y400 contain the particle fractions of $\Delta ^{-}$ and $K^{-}$ particles, respectively. Similarly, yav1 stands for the fraction of the first group of nuclei.

Combining the examples above, the file eos.quantities assumes the form

# number of regular, additional and derivative quantities

3 1 1

# indices of regular, additional and derivative quantities

6 1 2 1 1

# number of pairs and quadruples for composition data

3 1

# indices of pairs and quadruples for composition data

10 11 0 1

# number of microscopic quantities

2

# indices of microscopic quantities

10050 11050

# number of error quantities

2

# indices of error quantities

1 2

and, consequently, each line of the output file eos.table will contain the following entries:

$$\begin{aligned} T \quad n_{b} \quad Y_{q}{} & {} {\mathcal {F}}/m_{n}-1 \quad p \quad {\mathcal {S}} \quad q_{1} \quad {\mathcal {F}} \quad Y_{n} \quad Y_{p} \quad Y_{e} \quad \\{} & {} Y^{\text {av}}_{1} \quad A^{\text {av}}_{1} \quad Z^{\text {av}}_{1} \quad N^{\text {av}}_{1} \quad U_{n} \quad U_{p} \quad \Delta {\mathcal {F}} \quad \Delta {\mathcal {F}}/{\mathcal {F}} \end{aligned}$$

i.e., in total 19 quantities.

1.3 eos.parameters

There are at least ten lines in the file eos.parameters . Rows one, three, five and seven of the file are comment lines indicated by the first character #. The second row of this file contains three integer numbers $I_{T}$, $I_{n_{b}}$ and $I_{Y_{q}}$ that specify the interpolation scheme for the temperature, baryon number density and charge fraction of strongly interacting particles, respectively. Presently, there is the choice between first order ($I_{x}=1$, $x=T,n_{b},Y_{q}$), second order ($I_{x}=2$) and third order ($I_{x}=3$) interpolation available with continuity of all quantities Q(x), all quantities Q(x) and their first derivative $\partial Q/\partial x$ or all quantities Q(x), their first and second derivatives $\partial Q/\partial x$ and $\partial ^{2} Q/\partial x^{2}$, respectively. See Appendix A.1 for details on the interpolation scheme.

The fourth row of the file eos.parameters contains two integers $I_{\beta }$ and $I_{\text {entr}}$ . The first determines whether the EoS in the output file eos.table will be generated for matter in $\beta$ equilibrium ($I_{\beta } = 1$) or not ($I_{\beta } \ne 1$). Of course, this option is only effective for EoS tables that include electrons (and muons) and depend on the parameter $Y_{q}$, e.g. three-dimensional general purpose EoS tables and two-dimensional zero-temperature EoS tables. The second integer determines whether the EoS in the output file eos.table will be generated for given entropy per baryon ${\mathcal {S}}$ ($I_{\text {entr}} = 1$) or not ($I_{\text {entr}} \ne 1$). If yes, the input parameters that usually define the temperature values will be interpreted as defining the entropy per baryon. This option can only be used for EoS that depend on the temperature. However it is not guaranteed that there are solutions, found by interpolation, for the given parameter values.

In the sixth row of the file, the tabulation scheme for the parameters is defined by the integer $I_{\text {tab}}$ . For $I_{\text {tab}} = 0$ the the parameter values of T (or ${\mathcal {S}}$), $n_{b}$ and $Y_{q}$ are listed explicitly in the file eos.parameters as follows: The eighth row contains the number of data points $N_{\text {data}}$ to be generated and each line of the next $N_{\text {data}}$ rows contains the three parameter values in the form

$$\begin{aligned} T \,\, n_{b} \,\, Y_{q} \end{aligned}$$

(B.5)

or

$$\begin{aligned} {\mathcal {S}} \,\, n_{b} \,\, Y_{q} \end{aligned}$$

(B.6)

that define the grid points . The EoS data that will be stored in the final EoS table will appear in the same sequence of the parameters as given in the file eos.parameters . For $I_{\beta } =1$, $Y_{q}$ can be set to any value. For two- and one-dimensional EoS tables, the parameters that are not used can be set to arbitrary values, e.q. for a zero-temperature EoS the first entry in each row can have any finite value. An example file eos.parameters for the tabulation scheme with $I_{\text {tab}}=0$ is given by

# interpolation rules for T, n_b and Y_q

3 3 3

# calculation of beta equilibrium (1: yes, else: no) and for given entropy (1: yes, else: no)

0 0

# tabulation scheme of parameter values (see manual)

0

# parameter values depending on tabulation scheme

5

0.1 0.1 0.5

0.1 0.2 0.5

0.1 0.3 0.5

0.1 0.4 0.5

0.1 0.5 0.5

for generating a table using third order interpolations in T, $n_{b}$ and $Y_{q}$. Five data points are calculated for constant $T=0.1$ MeV and $Y_{q} = 0.5$ at densities of $n_{b} = 0.1$, 0.2, 0.3, 0.4, and 0.5 fm$^{-3}$.

For $I_{\text {tab}} \ne 0$ the parameter values are generated in a similar way as suggested in Sect. 4.1.1. In this case, only four additional rows follow the comment line below the row with the specification of $I_{\text {tab}}$. Each column of these rows (in the order temperature, density, charge fraction) contains the four quantities

$$\begin{aligned} \begin{array}{c} p_{\text {min}} \\ p_{\text {max}} \\ N_{p} \\ I_{p} \end{array} \end{aligned}$$

(B.7)

where p stands for the parameters T, $n_{b}$ or $Y_{q}$, respectively. The minimum and maximum values of the parameter p in the EoS table are denoted by $p_{\text {min}}$ and $p_{\text {max}}$, respectively. $N_{p}>0$ is the number of data points in the mesh of the parameter p and $I_{p}$ defines the discretization scheme. If $I_{p} = 0$, the individual points are given by a linear interpolation

$$\begin{aligned} p_{i} = p_{\text {min}} + (p_{\text {max}} - p_{\text {min}}) \frac{i-1}{N_{p}-1} \end{aligned}$$

(B.8)

for $i=1,\dots ,N_{p}$ with $N_{p}>1$ and

$$\begin{aligned} p_{1} = p_{\text {min}} \end{aligned}$$

(B.9)

for $N_{p}=1$. For $I_{p}\ne 0$, a logarithmic scaling is used with

$$\begin{aligned} p_{i} =p_{\text {min}} \left( \frac{p_{\text {max}}}{p_{\text {min}}}\right) ^{\frac{i-1}{N_{p}-1}} \end{aligned}$$

(B.10)

for $i=1,\dots ,N_{p}$ with $N_{p}>1$ and

$$\begin{aligned} p_{1} = p_{\text {min}} \end{aligned}$$

(B.11)

for $N_{p}=1$. For this tabulation scheme an example file eos.parameters has the form

# interpolation rules for T, n_b and Y_q

3 3 3

# calculation of beta equilibrium (1: yes, else: no) and for given entropy (1: yes, else: no)

0 0

# tabulation scheme of parameter values (see manual)

1

# parameter values depending on tabulation scheme

5.0 0.01 0.3

5.0 1.00 0.3

1 201 1

0 1 0

Here, $T=5$ MeV and $Y_{q}=0.3$ are kept constant and the densities $n_{b}$ cover the range from 0.01 fm$^{-3}$ to 1.0 fm$^{-3}$ in 200 intervals with a logarithmic scaling.

In case the option $I_{\beta }=1$ is set in the third row of the file eos.parameters, the specification of the parameter grid in $Y_{q}$ is irrelevant.

Table 18 Modules, (sub)routines and functions in the file compose.f90 with their dependencies (continued on next page)

Table 19 Modules, (sub)routines and functions in the file compose.f90 with their dependencies (continued from previous page)

Table 20 Modules, (sub)routines and functions in the file compose.f90 with their dependencies (continued from previous page)

Structure of the compose.f90 program

(Remark: This appendix is not yet updated from version 2.00 of the manual.)

Instead of using the code compose.f90 as given, the user can employ the subroutines directly in her/his program. The file compose.f90 contains one main program, 31 subroutines and one function. The structure of the program with the dependencies of the subroutines and functions is depicted in Tables 18, 19 and 20. The file composemodules.f90 contains two modules .

The user needs to call the following three subroutines

init_eos_table(iwr)

define_eos_table(iwr)

get_eos(t,n,y,b,ipl,i_beta,i_entr)

from the file compose.f90 in order to generate EoS data. The subroutine

init_terminal(iwr)

is called only from the main program for running the standard terminal version. It is usually not relevant for users that want to use subroutines within their own programs. The first two mentioned subroutines depend on a single integer parameter iwr. For iwr = 1 a progress report during the execution of the subroutine will be written to the terminal. Otherwise, this action will be suppressed. The subroutine get_eos has three double precision parameters (t, n, y) and three integer parameters (ipl, i_beta, i_entr) as arguments, where the first is a vector of dimension three. Be sure to include all (sub)routines, functions and the two modules eos_tables and compose_internal in your code.

The subroutine init_eos_table has to be called only once in order to initialize all relevant tables and quantities. The files eos.t, eos.nb, eos.yq and eos.thermo have to exist in order that init_eos_table is executed properly. The files eos.compo and eos.micro are optional.

The subroutine define_eos_table defines the quantities that are interpolated and that will be stored in the EoS table. It reads the file eos.quantities. The subroutine needs to be called only once. Instead of calling define_eos_table in the user’s program, it is possible that the relevant parameters are defined directly in the user’s code. Thus it is possible to change the selection of quantities that are interpolated during the execution of the user’s program. In the following, the variable names to be specified by the user are given according to the structure of the input file eos.quantities :

# number of regular, additional and derivative quantities

n_qty    n_add    n_df

# indices of regular, additional and derivative quantities

idx_qty(1) ...idx_qty(n_qty) idx_add(1) ...idx_add(n_add) idx_df(1) ...idx_df(n_df)

# number of pairs and quadruples for composition data

n_p    n_q

# indices of pairs and quadruples for composition data

idx_p(1)    ...   idx_p(n_p)    idx_q(1)    ...   idx_q(n_q)

# number of microscopic quantities

n_m

# indices of microscopic quantities

idx_m(1)    ...   idx_m(n_m)

# number of error quantities

n_err

# indices of error quantities

idx_err(1)    ...   idx_err(n_err)

# format of output file

iout

Note that all variables are integers. The appearing vectors with fixed size are defined in the module compose_internal:

idx_qty(dim_qtyt)    with    dim_qtyt = 23

idx_err(dim_qtye)    with    dim_qtye = 8.

where the dimensions are defined in the module eos_tables.

Finally, the subroutine get_eos can be called as often as needed with the appropriate arguments. The double precision variables t, n and y define the temperature T [MeV], baryon number density $n_{b}$ [fm${}^{-3}$] and the charge fraction of strongly interacting particles $Y_{q}$ [dimensionless], respectively, where the EoS is evaluated. The integer indices ipl_t, ipl_n and ipl_y define the interpolation rule in T, $n_{b}$ and $Y_{q}$, respectively. For ipl_t = 1 all selected quantities are interpolated linearly such that the interpolated values agree with the tabulated values at the grid points of the EoS table. For ipl_t = 2 the interpolated values of a quantity and its first derivative with respect to temperature agree with those at the grid points. For ipl_t = 3 a continuity of also the second derivatives at the grid points are demanded. Similarly, the interpolation rules for $n_{b}$ and $Y_{q}$ are determined. If the index is outside the range [1, 3] it is set to 3. See Appendix A.1 for details of the interpolation scheme.

The results of the interpolation are stored in seven double precision vectors/arrays with dimensions defined in the module eos_tables or defined according to the EoS input files:

eos_thermo(dim_qtyt)    with    dim_qtyt = 23

eos_thermo_add(dim_qty)

eos_compo_p(dim_qtyp)

eos_compo_q(dim_qtyq,3)

eos_micro(dim_qtym)

eos_err(dim_qtye)    with    dim_qtye = 8

eos_df(dim_df)    with    dim_df = 10 .

The vector index in eos_thermo corresponds to the index J given in Tables 10 and 11. The vector index in eos_thermo_add is just the index $i=1,\dots ,N_{\text {add}}$ of the additional quantities stored in each row of the file eos.thermo. The (first) index of the vectors/arrays eos_compo_p, eos_compo_q and eos_compo_m corresponds to the index of the vectors idx_p, idx_q and idx_m that are defined in the file eos.quantities or by the user before the subroutine get_eos is called. The second index of the array eos_compo_q is correlated with the index J as given in Table 13. The vector index in eos_err corresponds to the index J given in Table 15. The vector index in eos_df corresponds to the index J given in Table 12.

Organization of the CompOSE team

CompOSE is developed in close collaboration with the communities the service is meant for. These are mainly physicists who develop equations of state or simulate astrophysical phenomena numerically on the computer. Direct interaction with the users is an essential part in the concept of the project.

The following persons are engaged in the preparation of the web site, the manual and the tools, numerical codes and tables.

• Debarati Chatterjee IUCAA, Pune (India),

• Veronica Dexheimer Kent State University (Ohio, USA),

• Chikako Ishizuka Tokyo Institute of Technology (Japan),

• Thomas Klähn California State University Long Beach (California, USA),

• Marco Mancini Tours University (France),

• Jérôme Novak LUTH, Observatoire de Paris (France),

• Micaela Oertel LUTH, Observatoire de Paris (France),

• Helena Pais Coimbra University (Portugal)

• Constança Providencia Coimbra University (Portugal),

• Adriana Raduta NIPNE, Bucharest (Romania)

• Mathieu Servillat LUTH, Observatoire de Paris (France),

• Laura Tolos ICE (CSIC-IEEC), Barcelona (Spain),

• Stefan Typel Technische Universität Darmstadt and GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt (Germany).

Citing CompOSE

If you make use of the tables provided in CompOSE, you will be guided on the CompOSE web pages to the scientific publications where the particular EoS models have been described in detail. Please cite them when using the tables for scientific purposes together with a reference to the CompOSE website (https://compose.obspm.fr) and/or the original CompOSE publications: