# CompOSE Reference Manual

Version 3.01
CompStar Online Supernovæ Equations of State
_“harmonising the concert of nuclear physics and astrophysics”_
https://compose.obspm.fr

CompOSE Core Team
(S. Typel  M. Oertel  T. Klähn
D. Chatterjee  V. Dexheimer  C. Ishizuka
M. Mancini  J. Novak  H. Pais  C. Providencia
A. Raduta  M. Servillat  L. Tolos)

###### Abstract

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
http://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.
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.

###### Contents

  1. I Introduction
    1. 0 What CompOSE can do and what not
    2. 1 How to read this document
      1. 1 Recent Changes
    3. 2 Definitions and notation
      1. 1 Units and conventions
      2. 2 Physical conditions
      3. 3 Particle number densities and particle fractions
      4. 4 Particle indexing
      5. 5 Thermodynamic potentials and basic quantities
      6. 6 Thermodynamic coefficients
      7. 7 Microscopic quantities
      8. 8 Transport properties
        1. 1 Viscosities
        2. 2 Conductivities
      9. 9 Contribution of photons
      10. 10 Properties of compact stars
  2. 1 For Contributors
    1. 3 How to prepare EoS tables
      1. 1 Tabulation of quantities
        1. 1 Parameters and parameter ranges
        2. 2 Thermodynamic consistency
      2. 2 Structure of tables
        1. 1 Types of EoS tables
        2. 2 Table with thermodynamic quantities (required)
        3. 3 Table with the composition of matter (optional)
        4. 4 Table with information on microscopic quantities (optional)
        5. 5 Table with information on transport quantities (optional)
        6. 6 Table with information on properties of compact start (optional)
        7. 7 Tables with parameters
        8. 8 Identification of tables
        9. 9 Precision of tabulated quantities
      3. 3 Data sheet
    2. 4 Extending CompOSE
  3. 2 For Users
    1. 5 Models for the equation of state
      1. 1 Nuclear matter properties
      2. 2 Remarks of caution
    2. 6 Online services and data handling
      1. 1 Access to CompOSE
      2. 2 EoS data sheets
      3. 3 Options for using EoS data
      4. 4 Different families of EoS data
      5. 5 Handling and transforming EoS data
        1. 1 Downloading and compiling
        2. 2 Direct use of compose
        3. 3 HDF5 table
        4. 4 Additional output
        5. 5 Error messages
      6. 6 Web tools
  4. 3 Appendix
    1. 7 Technical details
      1. 7.A Interpolation
        1. 1 Interpolation in one dimension
        2. 2 Interpolation in two dimensions
      2. 7.B Matter in beta equilibrium
    2. 8 Structure of input files
      1. 8.A eos.init
      2. 8.B eos.quantities
      3. 8.C eos.parameters
    3. 9 Structure of the compose.f90 program
    4. 10 Organization of the CompOSE team
    5. 11 Citing CompOSE
    6. 12 Acknowledgments

##  Part I Introduction

###  Chapter 0 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 the review [2] for a general introduction.
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 11 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.

###  Chapter 1 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 2 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 1, 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 10. A summary of possible future
extensions of the CompOSE data base are summarized in chapter 4.

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. 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 5. 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 6 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 6 too.

####  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 section 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 section 1 has been changed to ’Access to
CompOSE’ with an adjusted description of the online sevice;

  * •

the description of how to use the compose program in section 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 8;

  * •

subsection 7.4.3 was moved to appendix 9 and the corresponding tables with the
(sub-)routines and functions were updated;

  * •

tables 7.1, 4, and 8 were amended and a new table 3 was introduced;

  * •

new appendices 8 and 9 were introduced;

  * •

the appendices 10 and 12 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 section
3.1;

  * •

update of physical constants in table 3.1;

  * •

distinguishing of particle and antiparticle densities as well as net number
densities for fermions with extended indexing scheme, modified new tables 3.2
and 3.3;

  * •

discussion of conversion from nuclear to astrophysical units in section 3.5;

  * •

additional discussion of calculation of speed of sound in section 3.6;

  * •

addition of new section 3.8 on transport properties;

  * •

addition of new section 3.10 on properties of neutron stars;

  * •

extended classification of EoS tables in section 4,2;

  * •

introduction of new tables with transport properties and information on
neutron stars in section 4.2;

  * •

discussion of different families of tabulated data and EoS models in section
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
section 7.6;

  * •

extension of table 7.2;

  * •

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.

###  Chapter 2 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.

####  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 [3]
(http://physics.nist.gov/cuu/Constants/index.html, www.codata.org) is
recommended, see Tab. 1111The compilation on physical constants and review of
particle properties [10] by the Particle Data Group, pdg.lbl.gov, contains the
CODATA values together with some additional constants concerning the
properties and interaction of elementary particles.. 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 section 5
for examples.

For experimental binding energies of nuclei, the values of the 2016 Atomic
mass evaluation [4, 5] (AME2016) or the more recent 2020 Atomic mass
evaluation [6, 7] (AME2020, http://amdc.impcas.ac.cn) and corresponding
updates are recommended. Ground state spin assignments should be taken from
the Nubase2016 [8] or the Nubase2020 [9] evaluation of nuclear properties
(http://amdc.impcas.ac.cn) and corresponding updates.

Table 1: Recommended values for physical constants from the 2018 CODATA evaluation and the compilation of the Particle Data Group [10].  Quantity | Symbol | Value | Unit
---|---|---|---
speed of light in vacuum | $c$ | $299792458$ | m s-1
Planck’s constant | $\hbar$ | $1.054571817\times 10^{-34}$ | J s
|  | $6.582119569\times 10^{-22}$ | MeV s
|  | $197.3269804$ | MeV fm $c^{-1}$
Boltzmann’s constant | $k_{B}$ | $1.380649\times 10^{-23}$ | J K-1
|  | $8.617333262\times 10^{-11}$ | MeV K-1
gravitational constant | $G$ | $6.67430\times 10^{-11}$ | m3 kg-1 s-2
|  | $6.70883\times 10^{-39}$ | GeV${}^{-2}c^{-4}$
square of elementary charge | $e^{2}$ | $1.439964547$ | MeV fm
fine structure constant | $\alpha=e^{2}/(\hbar c)$ | $1/137.035999084$ | $-$
neutron mass | $m_{n}$ | $939.56542052$ | MeV $c^{-2}$
proton mass | $m_{p}$ | $938.27208816$ | MeV $c^{-2}$
electron mass | $m_{e}$ | $0.51099895000$ | MeV $c^{-2}$
muon mass | $m_{\mu}$ | $105.6583745$ | MeV $c^{-2}$
unified atomic mass unit | $u$ | $931.49410242$ | MeV $c^{-2}$
solar mass | $M_{\odot}$ | $1.98841\times 10^{30}$ | kg
|  | $1.11542\times 10^{60}$ | MeV $c^{-2}$

####  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 section 9.

####  3 Particle number densities and particle fractions

Particle number densities of all particles $i$ are given by

$n_{i}=\frac{N_{i}}{V}\qquad[\mbox{fm}^{-3}]$ (1)

where $N_{i}$ [dimensionless] is the particle number inside the volume $V$
[fm3]. 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 am 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

$n_{b}=\frac{B}{V}=\sum_{i}B_{i}n_{i}\qquad[\mbox{fm}^{-3}]$ (2)

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

$B=\sum_{i}B_{i}\qquad[\mbox{dimensionless}]$ (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 $n_{s}$

$n_{s}=\frac{S}{V}=\sum_{i}S_{i}n_{i}\qquad[\mbox{fm}^{-3}]$ (4)

with the total strangeness

$\displaystyle S=\sum_{i}S_{i}\qquad[\mbox{dimensionless}]$ (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
$N_{le}$, $N_{l\mu}$

$\displaystyle n_{le}=\frac{N_{le}}{V}$ $\displaystyle=$
$\displaystyle\sum_{i}L^{e}_{i}n_{i}\qquad[\mbox{fm}^{-3}]$ (6) $\displaystyle
n_{l\mu}=\frac{N_{l\mu}}{V}$ $\displaystyle=$
$\displaystyle\sum_{i}L^{\mu}_{i}n_{i}\qquad[\mbox{fm}^{-3}]$ (7)

with lepton numbers $L^{e}_{i}$, $L^{\mu}_{i}$ [dimensionless], respectively,
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 $n_{q}$

$n_{q}=\frac{Q}{V}=\sum_{i}{}^{\prime}Q_{i}n_{i}\qquad[\mbox{fm}^{-3}]$ (8)

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

$Q=\sum_{i}{}^{\prime}Q_{i}\qquad[\mbox{dimensionless}]$ (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

$Y_{i}=\frac{n_{i}}{n_{b}}\qquad[\mbox{dimensionless}]$ (10)

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

$\sum_{i}{}B_{i}Y_{i}=1$ (11)

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

$X_{i}=B_{i}Y_{i}\qquad[\mbox{dimensionless}]$ (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 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 interaction
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

$Y_{q}=\frac{n_{q}}{n_{b}}\qquad[\mbox{dimensionless}]$ (13)

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

$Y_{e}=\frac{n_{e}}{n_{b}}\qquad[\mbox{dimensionless}]$ (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

$Y_{q}=Y_{e}+Y_{\mu}=Y_{l}$ (15)

with the muon fraction

$Y_{\mu}=\frac{n_{\mu}}{n_{b}}\qquad[\mbox{dimensionless}]$ (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.

####  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.) particle class |  symbol of particle $i$ | $B_{i}$ | $S_{i}$ | $Q_{i}$ | $L^{e}_{i}$ | $L^{\mu}_{i}$ |  particle index $I_{i}$
---|---|---|---|---|---|---|---
nuclei ($A>1$) | ${}^{A}Z$ | $A$ | $0$ | $Z$ | $0$ | $0$ | $1000000+1000\cdot A+Z$
| ${}^{A}\bar{Z}$ | $-A$ | $0$ | $-Z$ | $0$ | $0$ | $2000000+1000\cdot A+Z$
leptons | $e^{-}$ | $0$ | $0$ | $-1$ | $+1$ | $0$ | 2
| $e^{+}$ | $0$ | $0$ | $+1$ | $-1$ | $0$ | 4
| $\mu^{-}$ | $0$ | $0$ | $-1$ | $0$ | $+1$ | 3
| $\mu^{+}$ | $0$ | $0$ | $+1$ | $0$ | $-1$ | 5
baryons | $n$ | $+1$ | $0$ | $0$ | 0 | 0 | 12
| $\overline{n}$ | $-1$ | $0$ | $0$ | 0 | 0 | 14
| $p$ | $+1$ | $0$ | $+1$ | 0 | 0 | 13
| $\overline{p}$ | $-1$ | $0$ | $-1$ | 0 | 0 | 15
| $\Delta^{-}$ | $+1$ | $0$ | $-1$ | 0 | 0 | 24
| $\overline{\Delta}^{-}$ | $-1$ | $0$ | $+1$ | 0 | 0 | 28
| $\Delta^{0}$ | $+1$ | $0$ | $0$ | 0 | 0 | 25
| $\overline{\Delta}^{0}$ | $-1$ | $0$ | $0$ | 0 | 0 | 29
| $\Delta^{+}$ | $+1$ | $0$ | $+1$ | 0 | 0 | 26
| $\overline{\Delta}^{+}$ | $-1$ | $0$ | $-1$ | 0 | 0 | 30
| $\Delta^{++}$ | $+1$ | $0$ | $+2$ | 0 | 0 | 27
| $\overline{\Delta}^{++}$ | $-1$ | $0$ | $-2$ | 0 | 0 | 31
| $\Lambda$ | $+1$ | $-1$ | $0$ | 0 | 0 | 101
| $\overline{\Lambda}$ | $-1$ | $+1$ | $0$ | 0 | 0 | 102
| $\Sigma^{-}$ | $+1$ | $-1$ | $-1$ | 0 | 0 | 113
| $\overline{\Sigma}^{-}$ | $-1$ | $+1$ | $+1$ | 0 | 0 | 116
| $\Sigma^{0}$ | $+1$ | $-1$ | $0$ | 0 | 0 | 114
| $\overline{\Sigma}^{0}$ | $-1$ | $+1$ | $0$ | 0 | 0 | 117
| $\Sigma^{+}$ | $+1$ | $-1$ | $+1$ | 0 | 0 | 115
| $\overline{\Sigma}^{+}$ | $-1$ | $+1$ | $+1$ | 0 | 0 | 118
| $\Xi^{-}$ | $+1$ | $-2$ | $-1$ | 0 | 0 | 122
| $\overline{\Xi}^{-}$ | $-1$ | $+2$ | $+1$ | 0 | 0 | 124
| $\Xi^{0}$ | $+1$ | $-2$ | $0$ | 0 | 0 | 123
| $\overline{\Xi}^{0}$ | $-1$ | $+2$ | $0$ | 0 | 0 | 125
quarks | $u$ | $1/3$ | $0$ | $+2/3$ | 0 | 0 | 503
| $\overline{u}$ | $-1/3$ | $0$ | $-2/3$ | 0 | 0 | 506
| $d$ | $1/3$ | $0$ | $-1/3$ | 0 | 0 | 504
| $\overline{d}$ | $-1/3$ | $0$ | $+1/3$ | 0 | 0 | 507
| $s$ | $1/3$ | $-1$ | $-1/3$ | 0 | 0 | 505
| $\overline{s}$ | $-1/3$ | $+1$ | $+1/3$ | 0 | 0 | 508
Table 3: Indices $I_{i}$ of nuclei and the most relevant fermions $i$ in dense matter for net quantities. particle class |  symbol of particle $i$ |  particle index $I_{i}$
---|---|---
nuclei ($A>1$) |  ${A}^{Z}$/$\overline{A}^{Z}$ | $1000\cdot A+Z$
leptons |  $e^{-}$/$e^{+}$ | 0
|  $\mu^{-}$/$\mu^{+}$ | 1
baryons |  $n$/$\overline{n}$ | 10
|  $p$/$\overline{p}$ | 11
|  $\Delta^{-}$/$\overline{\Delta}^{-}$ | 20
|  $\Delta^{0}$/$\overline{\Delta}^{0}$ | 21
|  $\Delta^{+}$/$\overline{\Delta}^{+}$ | 22
|  $\Delta^{++}$/$\overline{\Delta}^{++}$ | 23
|  $\Lambda$/$\overline{\Lambda}$ | 100
|  $\Sigma^{-}$/$\overline{\Sigma}^{-}$ | 110
|  $\Sigma^{0}$/$\overline{\Sigma}^{0}$ | 111
|  $\Sigma^{+}$/$\overline{\Sigma}^{+}$ | 112
|  $\Xi^{-}$/$\overline{\Xi}^{-}$ | 120
|  $\Xi^{0}$/$\overline{\Xi}^{0}$ | 121
quarks |  $u$/$\overline{u}$ | 500
|  $d$/$\overline{d}$ | 501
|  $s$/$\overline{s}$ | 502

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. particle class |  symbol of particle $i$ | $B_{i}$ | $S_{i}$ | $Q_{i}$ | $L^{e}_{i}$ | $L^{\mu}_{i}$ |  particle index $I_{i}$
---|---|---|---|---|---|---|---
mesons | $\omega$ | $0$ | $0$ | $0$ | 0 | 0 | 200
| $\sigma$ | $0$ | $0$ | $0$ | 0 | 0 | 210
| $\eta$ | $0$ | $0$ | $0$ | 0 | 0 | 220
| $\eta^{\prime}$ | $0$ | $0$ | $0$ | 0 | 0 | 230
| $\rho^{-}$ | $0$ | $0$ | $-1$ | 0 | 0 | 300
| $\rho^{0}$ | $0$ | $0$ | $0$ | 0 | 0 | 301
| $\rho^{+}$ | $0$ | $0$ | $+1$ | 0 | 0 | 302
| $\delta^{-}$ | $0$ | $0$ | $-1$ | 0 | 0 | 310
| $\delta^{0}$ | $0$ | $0$ | $0$ | 0 | 0 | 311
| $\delta^{+}$ | $0$ | $0$ | $+1$ | 0 | 0 | 312
| $\pi^{-}$ | $0$ | $0$ | $-1$ | 0 | 0 | 320
| $\pi^{0}$ | $0$ | $0$ | $0$ | 0 | 0 | 321
| $\pi^{+}$ | $0$ | $0$ | $+1$ | 0 | 0 | 322
| $\phi$ | $0$ | $0$ | $0$ | 0 | 0 | 400
| $\sigma_{s}$ | $0$ | $0$ | $0$ | 0 | 0 | 410
| $K^{-}$ | $0$ | $-1$ | $-1$ | 0 | 0 | 420
| $K^{0}$ | $0$ | $+1$ | $0$ | 0 | 0 | 421
| $\bar{K}^{0}$ | $0$ | $-1$ | $0$ | 0 | 0 | 422
| $K^{+}$ | $0$ | $+1$ | $+1$ | 0 | 0 | 423
photon | $\gamma$ | $0$ | $0$ | $0$ | 0 | 0 | 600
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. correlation class | particles | channel | $B_{i}$ | $S_{i}$ | $Q_{i}$ |  index $I_{i}$ of correlation
---|---|---|---|---|---|---
two-body | $nn$ | ${}^{1}S_{0}$ | $2$ | $0$ | $0$ | 700
| $np$ | ${}^{1}S_{0}$ | $2$ | $0$ | $1$ | 701
| $pp$ | ${}^{1}S_{0}$ | $2$ | $0$ | $2$ | 702
| $np$ | ${}^{3}S_{1}$ | $2$ | $0$ | $1$ | 703

####  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

$\xi_{i}=\left.\frac{\partial\Xi}{\partial x_{i}}\right|_{x_{k},k\neq
i;\xi_{j}}=\xi_{i}(\\{x_{i}\\},\\{\xi_{j}\\})$ (17)

and

$x_{j}=-\left.\frac{\partial\Xi}{\partial\xi_{j}}\right|_{\xi_{k},k\neq
j;x_{i}}=x_{j}(\\{x_{i}\\},\\{\xi_{j}\\})$ (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

$\Xi(\\{x_{i}\\},\\{\xi_{j}\\})=\sum_{i=1}^{n_{1}}\xi_{i}x_{i}$ (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

$\left.\frac{\partial\xi_{l}}{\partial x_{k}}\right|_{x_{j},j\neq
k}=\left.\frac{\partial\xi_{k}}{\partial x_{l}}\right|_{x_{j},j\neq l}$ (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 $V$ [fm3] 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

$f(T,n_{i})=\frac{F}{V}\qquad[\mbox{MeV~{}fm}^{-3}]$ (21)

and the entropy density

$s(T,n_{i})=-\left.\frac{\partial f}{\partial
T}\right|_{n_{i}}\qquad[\mbox{fm}^{-3}]$ (22)

with the entropy

$S(T,V,N_{i})=Vs(T,n_{i})=-\left.\frac{\partial F}{\partial
T}\right|_{V,N_{i}}\qquad[\mbox{dimensionless}]\>.$ (23)

The chemical potential of a particle $i$ is given by

$\mu_{i}=\left.\frac{\partial F}{\partial N_{i}}\right|_{T,V,N_{j},j\neq
i}=\left.\frac{\partial f}{\partial n_{i}}\right|_{T,n_{j},j\neq
i}\qquad[\mbox{MeV}]$ (24)

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

$p=-\left.\frac{\partial F}{\partial
V}\right|_{T,N_{i}}=n_{b}^{2}\left.\frac{\partial(f/n_{b})}{\partial
n_{b}}\right|_{T,Y_{q}}=\sum_{i}\mu_{i}n_{i}-f\qquad[\mbox{MeV~{}fm}^{-3}]\>.$
(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

$\mu_{i}=B_{i}\mu_{b}+Q_{i}\mu_{q}+S_{i}\mu_{s}+L^{e}_{i}\mu_{le}+L^{\mu}_{i}\mu_{l\mu}\qquad[\mbox{MeV}]\>,$
(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.

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

$E=E(S,V,N_{i})=F+TS\qquad[\mbox{MeV}]\>,$ (27)

the free enthalpy (Gibbs potential)

$G=G(T,p,N_{i})=F+pV=\sum_{i}\mu_{i}N_{i}\qquad[\mbox{MeV}]\>,$ (28)

the enthalpy

$H=H(S,p,N_{i})=E+pV\qquad[\mbox{MeV}]\>,$ (29)

and the grand canonical potential

$\Omega=\Omega(T,V,\mu_{i})=F-\sum_{i}\mu_{i}N_{i}=-pV\qquad[\mbox{MeV}]$ (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

$\mathcal{F}=\frac{F}{N_{b}}=\frac{f}{n_{b}}\qquad[\mbox{MeV}]\>,$ (31)

the internal energy per baryon

$\mathcal{E}=\frac{E}{N_{b}}=\frac{e}{n_{b}}\qquad[\mbox{MeV}]\>,$ (32)

the enthalpy per baryon

$\mathcal{H}=\frac{H}{N_{b}}=\frac{h}{n_{b}}\qquad[\mbox{MeV}]\>,$ (33)

and the free enthalpy per baryon

$\mathcal{G}=\frac{G}{N_{b}}=\frac{g}{n_{b}}\qquad[\mbox{MeV}]$ (34)

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

$N_{b}=n_{b}V\qquad[\mbox{dimensionless}]\>.$ (35)

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

$\displaystyle T$ $\displaystyle=$ $\displaystyle\left.\frac{\partial
E}{\partial S}\right|_{V,N_{i}}=\left.\frac{\partial H}{\partial
S}\right|_{p,N_{i}}\>,$ (36) $\displaystyle S$ $\displaystyle=$
$\displaystyle-\left.\frac{\partial F}{\partial
T}\right|_{V,N_{i}}=-\left.\frac{\partial G}{\partial
T}\right|_{p,N_{i}}=-\left.\frac{\partial\Omega}{\partial
T}\right|_{V,\mu_{i}}\>,$ (37) $\displaystyle-p$ $\displaystyle=$
$\displaystyle\left.\frac{\partial E}{\partial
V}\right|_{S,N_{i}}=\left.\frac{\partial F}{\partial V}\right|_{T,N_{i}}\>,$
(38) $\displaystyle V$ $\displaystyle=$ $\displaystyle\left.\frac{\partial
G}{\partial p}\right|_{T,N_{i}}=\left.\frac{\partial H}{\partial
p}\right|_{S,N_{i}}=\left.\frac{\partial\Omega}{\partial
p}\right|_{T,\mu_{i}}\>,$ (39) $\displaystyle\mu_{i}$ $\displaystyle=$
$\displaystyle\left.\frac{\partial E}{\partial N_{i}}\right|_{S,V,N_{j\neq
i}}=\left.\frac{\partial F}{\partial N_{i}}\right|_{T,V,N_{j\neq
i}}=\left.\frac{\partial G}{\partial N_{i}}\right|_{T,p,N_{j\neq
i}}=\left.\frac{\partial H}{\partial N_{i}}\right|_{S,p,N_{j\neq i}}\>,$ (40)
$\displaystyle N_{i}$ $\displaystyle=$
$\displaystyle-\left.\frac{\partial\Omega}{\partial\mu_{i}}\right|_{S,V,\mu_{j\neq
i}}\>.$ (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. 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. 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. 3.

The pressure $p$ in MeV fm-3 can be expressed in the CGS unit dyn/cm2 ($1$ dyn
= $1$ erg/cm) by a multiplication with the same numerical factor
$1.602176634\cdot 10^{33}$ as for the energy density.

####  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

$c_{V}=\frac{T}{N_{b}}\left.\frac{dS}{dT}\right|_{V,N_{b},N_{q}}=-T\left.\frac{\partial^{2}\mathcal{F}}{\partial
T^{2}}\right|_{n_{b},Y_{q}}\qquad[\mbox{dimensionless}]\>,$ (42)

the tension coefficient at constant volume

$\beta_{V}=\left.\frac{dp}{dT}\right|_{V,N_{b},N_{q}}=\left.\frac{dS}{dV}\right|_{T,N_{b},N_{q}}=n_{b}^{2}\left.\frac{\partial^{2}\mathcal{F}}{\partial
T\partial n_{b}}\right|_{Y_{q}}\qquad[\mbox{fm}^{-3}]\>,$ (43)

the isothermal compressibility

$\displaystyle\kappa_{T}$ $\displaystyle=$
$\displaystyle-\frac{1}{V}\left.\frac{dV}{dp}\right|_{T,N_{b},N_{q}}=\left(n_{b}\left.\frac{\partial
p}{\partial n_{b}}\right|_{T,Y_{q}}\right)^{-1}$ $\displaystyle=$
$\displaystyle\left(n_{b}^{2}\left.\frac{\partial^{2}(\mathcal{F}n_{b})}{\partial
n_{b}^{2}}\right|_{T,Y_{q}}\right)^{-1}\qquad[\mbox{MeV}^{-1}\mbox{fm}^{3}]\>,$

the expansion coefficient at constant pressure

$\alpha_{p}=\frac{1}{V}\left.\frac{dV}{dT}\right|_{p,N_{b},N_{q}}=\kappa_{T}\beta_{V}\qquad[\mbox{MeV}^{-1}]\>,$
(45)

the specific heat capacity at constant pressure

$c_{p}=\frac{T}{N_{b}}\left.\frac{dS}{dT}\right|_{p,N_{b},N_{q}}=c_{V}+\frac{T}{n_{b}}\alpha_{p}\beta_{V}\qquad[\mbox{dimensionless}]\>,$
(46)

the thermodynamic adiabatic index

$\Gamma=\frac{c_{p}}{c_{V}}\qquad[\mbox{dimensionless}]$ (47)

and the adiabatic compressibility

$\kappa_{S}=-\frac{1}{V}\left.\frac{dV}{dp}\right|_{S,N_{b},N_{q}}=\frac{\kappa_{T}}{\Gamma}\qquad[\mbox{MeV}^{-1}\mbox{fm}^{3}]\>.$
(48)

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

$\tilde{\Gamma}=\left.\frac{\partial\ln p}{\partial\ln
n_{b}}\right|_{S}=\frac{1}{p\kappa_{S}}\qquad[\mbox{dimensionless}]$ (49)

is used in applications with polytrope equations of state. A tilde is
introduced here to distinguish it from (47). A further definition is given by

$\gamma=\left.\frac{\partial\ln p}{\partial\ln
e}\right|_{S}\qquad[\mbox{dimensionless}]$ (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

$c_{s}^{2}=\left.\frac{dp}{de}\right|_{S,N_{b},N_{q}}=\frac{1}{h\kappa_{S}}=\frac{p}{h}\tilde{\Gamma}=\frac{p}{e}\gamma\qquad[c^{2}]$
(51)

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

$c_{s}^{2}=\left.\frac{n_{b}}{h}\frac{\partial p}{\partial
n_{b}}\right|_{Y_{q},e}+\left.\frac{\partial p}{\partial
e}\right|_{n_{b},n_{q}}\qquad[c^{2}]$ (52)

can be used for 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

$c_{s}^{2}=\left.\frac{dp}{de}\right|_{T=0,\beta~{}eq.}=\frac{n_{b}}{\mu_{b}}\left.\frac{d\mu_{b}}{dn_{b}}\right|_{T=0,\beta~{}eq.}\qquad[c^{2}]$
(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 $n_{q}$.

####  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 parametrizations 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

$m^{L}_{i}\qquad[\mbox{MeV}]$ (54)

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

$\frac{1}{m^{L}_{i}}=\frac{1}{p^{F}_{i}}\left.\frac{dE_{i}}{dp}\right|_{p=p_{i}^{F}}\>,$
(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

$T_{i}=\frac{p_{i}^{2}}{2m^{L}_{i}}\qquad[\mbox{MeV}]$ (56)

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

$U_{i}\qquad[\mbox{MeV}]$ (57)

contributes to the single-particle energy

$E_{i}=T_{i}+U_{i}+m_{i}$ (58)

in nonrelativistic models. The effective Dirac mass

$m^{D}_{i}=m_{i}-S_{i}\qquad[\mbox{MeV}]$ (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

$m^{L}_{i}=\sqrt{(m^{D}_{i})^{2}+(p^{F}_{i})^{2}}\>.$ (60)

The effective Dirac mass depends on the scalar self-energy

$S_{i}=\Sigma_{i}\qquad[\mbox{MeV}]$ (61)

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

$V_{i}=\Sigma^{0}_{i}\qquad[\mbox{MeV}]$ (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

$\Delta_{i}\qquad[\mbox{MeV}]\>,$ (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.

####  8 Transport properties

Besides the thermodynamic quantities and coefficients as described in sections
(5) and (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.

#####  1 Viscosities

Corrections to the equations of ideal hydrodynamics [21] can be introduced by
considering the energy-momentum tensor (in relativistic notation neglecting
heat flow)

$T^{\mu\nu}=T^{\mu\nu}_{\mathrm{ideal}}+\Pi^{\mu\nu}\qquad[\mbox{MeV~{}fm${}^{-3}$}]$
(64)

with the ideal contribution

$T^{\mu\nu}_{\mathrm{ideal}}=(e+p)u^{\mu}u^{\nu}-pg^{\mu\nu}=eu^{\mu}u^{\nu}-p\Delta^{\mu\nu}\qquad[\mbox{MeV~{}fm${}^{-3}$}]$
(65)

where

$\Delta^{\mu\nu}=g^{\mu\nu}-u^{\mu}u^{\nu}\qquad[\mbox{dimensionless}]$ (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

$\Pi^{\mu\nu}=\pi^{\mu\nu}-\Phi\Delta^{\mu\nu}\qquad[\mbox{MeV~{}fm${}^{-3}$}]$
(67)

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

$\pi^{\mu\nu}=\eta\>\left(\nabla^{\mu}u^{\nu}+\nabla^{\nu}u^{\mu}-\frac{2}{3}\Delta^{\mu\nu}\nabla_{\alpha}u^{\alpha}\right)\qquad[\mbox{MeV~{}fm${}^{-3}$}]$
(68)

and

$\Phi=\zeta\>\nabla_{\alpha}u^{\alpha}\qquad[\mbox{MeV~{}fm${}^{-3}$}]$ (69)

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

#####  2 Conductivities

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

$\vec{q}=-\kappa\>\vec{\nabla}T\qquad[\mbox{MeV~{}fm${}^{-3}$}]$ (70)

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

$\vec{\jmath}=\sigma\>\vec{E}\qquad[\mbox{$e$~{}fm${}^{-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.

####  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 can be simply added to the
thermodynamic quantities of the other constituents. We have the photon free
energy density

$f_{\gamma}(T)=-\frac{\pi^{2}}{45}T^{4}\qquad[\mbox{MeV~{}fm}^{-3}]$ (72)

the photon entropy density

$s_{\gamma}(T)=\frac{4\pi^{2}}{45}T^{3}\qquad[\mbox{fm}^{-3}]$ (73)

and the photon pressure

$p_{\gamma}(T)=\frac{\pi^{2}}{45}T^{4}\qquad[\mbox{MeV~{}fm}^{-3}]$ (74)

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

$\mu_{\gamma}=0\qquad[\mbox{MeV}]$ (75)

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

####  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 [11, 12]

$\frac{dp}{dr}=-G\frac{M(r)e(r)}{r^{2}}\left[1+\frac{p(r)}{e(r)}\right]\left[1+\frac{4\pi
r^{3}p(r)}{M(r)}\right]\left[1-\frac{2GM(r)}{r}\right]^{-1}$ (76)

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

$M(r)=4\pi\int_{0}^{r}dr^{\prime}\>\left(r^{\prime}\right)^{2}\>e(r^{\prime})\qquad[\mbox{M}_{\odot}]$
(77)

that depends on the (internal) energy density $e$. Equation (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 [13, 14]. This quantity is
defined as

$\Lambda=\frac{2}{3}k_{2}C^{-5}\qquad[\mbox{dimensionless}]$ (78)

with the compactness

$C=\frac{GM(R)}{R}\qquad[\mbox{dimensionless}]$ (79)

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

$\displaystyle k_{2}$ $\displaystyle=$
$\displaystyle\frac{8}{5}C^{5}\left(1-2C\right)^{2}\left[2+2C\left(y-1\right)-y\right]$
$\displaystyle\times\left\\{2C\left[6-3y+3C\left(5y-8\right)\right]+4C^{3}\left[13-11y+C\left(3y-2\right)+2C^{2}\left(1+y\right)\right]\right.$
$\displaystyle\left.+3\left(1-2C\right)^{2}\left[2-y+2C\left(y-1\right)\right]\ln\left(1-2C\right)\right\\}^{-1}\qquad[\mbox{dimensionless}]$

when the logarithmic derivative

$y=\frac{d\ln H}{d\ln R}\>,\qquad[\mbox{dimensionless}]$ (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

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{d^{2}H}{dr^{2}}+\frac{dH}{dr}\left(\frac{2}{r}+\frac{d\Phi}{dr}-\frac{d\lambda}{dr}\right)$
$\displaystyle+H\left[-\frac{6}{r^{2}}e^{2\lambda}-2\left(\frac{d\Phi}{dr}\right)^{2}+2\frac{d\Phi^{2}}{dr^{2}}+\frac{3}{r}\frac{d\lambda}{dr}+\frac{7}{r}\frac{d\Phi}{dr}+2\frac{d\Phi}{dr}\frac{d\lambda}{dr}+\frac{f}{r}\left(\frac{d\Phi}{dr}+\frac{d\lambda}{dr}\right)\right]$

simultaneously with (76) for $dp/dr$ using the potential

$e^{2\lambda}=\left[1-\frac{2GM(r)}{r}\right]^{-1}\qquad[\mbox{dimensionless}]$
(83)

of the Schwarzschild metric and the auxiliary quantities

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle-\frac{1}{\varepsilon(r)+p(r)}\frac{dp}{dr}\qquad[\mbox{km}^{-1}]\>,$
(84) $\displaystyle f$ $\displaystyle=$
$\displaystyle\frac{de}{dp}=1/c_{s}^{2}\qquad[\mbox{dimensionless}]$ (85)

that are obtained from the compact-star EoS. The boundary condition for $H(r)$
is given by the condition that $H(r)\to a_{0}r^{2}$ for $r\to 0$ 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 [15, 16, 17] and the LORENE
library [18]. The latter has an interface to directly employ EoS data from
CompOSE, see subsection 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

$Y_{q}^{\mathrm{dUrca}}=\frac{1}{9}$ (86)

or

$Y_{q}^{\mathrm{dUrca}}=\left\\{1+\left[1+\left(\frac{Y_{e}}{Y_{e}+Y_{\mu}}\right)^{1/3}\right]^{3}\right\\}^{-1}$
(87)

if both electrons and muons are accounted for [19, 20]. 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

$n\to p+e^{-}+\bar{\nu}_{e}$ (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.

##  Part 1 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.

###  Chapter 3 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 $V$ [fm3] 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.

####  1 Tabulation of quantities

#####  1 Parameters and parameter ranges

Because of the imposed physical conditions, see section 2, the state of the
system is uniquely characterized by only three quantities, see section 5, that
are used as parameters in the EoS tables with the properties of dense matter:

  1. 1.

temperature $T$ [MeV],

  2. 2.

baryon number density $n_{b}$ [fm-3],

  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. 1.

temperature
In this case, two standards are suggested:
a. linear mesh in $\ln T$, i.e.

$T(i_{T})=T^{\textrm{ref}}(f_{T})^{i_{T}-1}\quad f_{T}=10^{1/M_{T}}\quad
i_{T}=N_{T}^{\textrm{min}},N_{T}^{\textrm{min}}+1,\dots,N_{T}^{\textrm{max}}$
(1)

with parameters $T^{\textrm{ref}}>0$ MeV,
$N_{T}^{\textrm{min}}<N_{T}^{\textrm{max}}$ 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.

$T(i_{T})=T^{\textrm{ref}}\frac{\sinh(f_{T}i_{T})}{\sinh(f_{T})}\quad
i_{T}=N_{T}^{\textrm{min}},N_{T}^{\textrm{min}}+1,\dots,N_{T}^{\textrm{max}}$
(2)

with parameters $T^{\textrm{ref}}>0$ MeV,
$N_{T}^{\textrm{min}}<N_{T}^{\textrm{max}}$ 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. 2.

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

$n_{b}(i_{n_{b}})=n_{b}^{\textrm{ref}}(f_{n_{b}})^{i_{n_{b}}-1}\quad
f_{n_{b}}=10^{1/M_{n_{b}}}\quad
i_{n_{b}}=N_{n_{b}}^{\textrm{min}},N_{n_{b}}^{\textrm{min}}+1,\dots,N_{n_{b}}^{\textrm{max}}$
(3)

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

  3. 3.

charge fraction of strongly interacting particles
linear mesh in $Y_{q}$, i.e.

$Y_{q}(i_{Y_{q}})=\frac{i_{Y_{q}}}{M_{Y_{q}}}\quad
i_{Y_{q}}=N_{Y_{q}}^{\textrm{min}},N_{Y_{q}}^{\textrm{min}}+1,\dots,N_{Y_{q}}^{\textrm{max}}$
(4)

with parameters $N_{Y_{q}}^{\textrm{min}}<N_{Y_{q}}^{\textrm{max}}$ and
$M_{Y_{q}}>N_{Y_{q}}^{\textrm{max}}$ 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 section 1. Actually used meshes for particular EoS are
specified in the online documentation for each individual EoS table.

#####  2 Thermodynamic consistency

The equations of state included in the CompOSE data base are required to
fullfill 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. (19),

$f(T,n_{b},Y_{q})=-p+\mu_{b}n_{b}+\mu_{q}\left(n_{q}-n_{le}-n_{l\mu}\right)+\mu_{le}n_{le}+\mu_{l\mu}n_{l\mu}$
(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. (5) it was
assumed that the strangeness chemical potential $\mu_{s}$ is zero.

It is convenient to define the effective lepton density as

$n_{l}=n_{le}+n_{l\mu}\qquad[\mbox{fm}^{-3}]$ (6)

and the effective lepton chemical potential

$\mu_{l}=\frac{\mu_{le}n_{le}+\mu_{l\mu}n_{l\mu}}{n_{l}}\qquad[\mbox{MeV}]$
(7)

such that

$f(T,n_{b},Y_{q})=-p+\left(\mu_{b}+Y_{q}\mu_{q}\right)n_{b}$ (8)

in case of an EoS without leptons and

$f(T,n_{b},Y_{q})=-p+\left(\mu_{b}+Y_{l}\mu_{l}\right)n_{b}$ (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

$\displaystyle s$ $\displaystyle=$ $\displaystyle-\left.\frac{\partial
f}{\partial T}\right|_{n_{b},Y_{q}}\>,$ (10) $\displaystyle p$
$\displaystyle=$ $\displaystyle
n_{b}^{2}\left.\frac{\partial(f/n_{b})}{\partial n_{b}}\right|_{T,Y_{q}}\>,$
(11) $\displaystyle\mu_{b}+Y_{i}\mu_{i}$ $\displaystyle=$
$\displaystyle\left.\frac{\partial f}{\partial n_{b}}\right|_{T,Y_{q}}\>,$
(12) $\displaystyle\mu_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{n_{b}}\left.\frac{\partial f}{\partial
Y_{i}}\right|_{T,n_{b}}$ (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.

For a mixed second partial derivative, the Maxwell relations are

$\displaystyle-n_{b}^{2}\left.\frac{\partial(s/n_{b})}{\partial
n_{b}}\right|_{T,Y_{q}}$ $\displaystyle=$ $\displaystyle\left.\frac{\partial
p}{\partial T}\right|_{n_{b},Y_{q}}\>,$ (14)
$\displaystyle-\left.\frac{\partial s}{\partial n_{x}}\right|_{T,n_{y},y\neq
x}$ $\displaystyle=$ $\displaystyle\left.\frac{\partial\mu_{x}}{\partial
T}\right|_{n_{y}}\>,$ (15) $\displaystyle\left.\frac{\partial p}{\partial
n_{x}}\right|_{T,n_{y},y\neq x}$ $\displaystyle=$ $\displaystyle
n_{b}\left.\frac{\partial\mu_{x}}{\partial n_{b}}\right|_{T,n_{y},y\neq b}\>,$
(16) $\displaystyle\left.\frac{\partial\mu_{x}}{\partial
n_{y}}\right|_{T,n_{z},z\neq y}$ $\displaystyle=$
$\displaystyle\left.\frac{\partial\mu_{y}}{\partial
n_{x}}\right|_{T,n_{z},z\neq x}$ (17)

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

The knowlegde 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
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.

####  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 caclculable in the EoS model.

#####  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. 1.

Three-dimensional tables

These tables depend on all three indepedent 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 1. This choice of discretation corresponds to a table with
$N_{T}^{\textrm{max}}\times N_{n_{b}}^{\textrm{max}}\times
N_{Y_{q}}^{\textrm{max}}=1119720(1462860)$ 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. 2.

Two-dimensional tables

These tables depend on two of the three indepedent 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 2. This choice of discretization corresponds to a table with
$N_{n_{b}}^{\textrm{max}}\times(N_{Y_{q}}^{\textrm{max}}+1)=18361$ 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 3. This choice of discretization corresponds to a table with
$N_{T}^{\textrm{max}}\times N_{n_{b}}^{\textrm{max}}=18963(24381)$ 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 3. This choice of discretization corresponds to a table with
$N_{T}^{\textrm{max}}\times N_{n_{b}}^{\textrm{max}}=18963(24381)$ 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 3. This choice of discretization corresponds to a table with
$N_{T}^{\textrm{max}}\times N_{n_{b}}^{\textrm{max}}=18963(24381)$ 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. 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 4. This choice of discretization corresponds to a table with
$N_{n_{b}}^{\textrm{max}}=301$ 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 4. This choice of discretization corresponds to a table with
$N_{n_{b}}^{\textrm{max}}=301$ 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 4. This choice of discretization corresponds to a table with
$N_{n_{b}}^{\textrm{max}}=301$ 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.

#####  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. 1.

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

  2. 2.

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

  3. 3.

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

  4. 4.

scaled charge chemical potential $Q_{4}=\mu_{q}/m_{n}$ [dimensionless],

  5. 5.

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

  6. 6.

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

  7. 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

$i_{T}\>\>i_{n_{b}}\>\>i_{Y_{q}}\>\>Q_{1}\>\>Q_{2}\>\>Q_{3}\>\>Q_{4}\>\>Q_{5}\>\>Q_{6}\>\>Q_{7}\>\>N_{\textrm{add}}\>\>\underbrace{q_{1}\>\>q_{2}\>\>\dots\>\>q_{N_{\textrm{add}}}}_{N_{\textrm{add}}\>{\textrm{quantities}}}$
(18)

with seven required quantities $Q_{i}$ and $N_{\textrm{add}}$ 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 $Q_{i}$,
$N_{\textrm{add}}$, and $q_{i}$ are used from the row that is read last.

#####  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. 1.

particle fractions $Y_{i}$ [dimensionless],

  2. 2.

average mass number of a representative nucleus $A^{\textrm{av}}$
[dimensionless],

  3. 3.

average charge number of a representative nucleus $Z^{\textrm{av}}$
[dimensionless],

  4. 4.

index encoding the type of phase $I_{\textrm{phase}}$ [dimensionless]
(The correspondence of the phase index $I_{\textrm{phase}}$ 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

$\displaystyle
i_{T}\>\>i_{n_{b}}\>\>i_{Y_{q}}\>\>I_{\textrm{phase}}\>\>N_{\textrm{pairs}}\underbrace{I_{1}\>\>Y_{I_{1}}\>\>\dots\>\>I_{N_{\textrm{pairs}}}\>\>Y_{I_{N_{\textrm{pairs}}}}}_{N_{\textrm{pairs}}\>{\textrm{pairs}}}$
$\displaystyle
N_{\textrm{quad}}\>\>\underbrace{I_{1}\>\>A^{\textrm{av}}_{I_{1}}\>\>Z^{\textrm{av}}_{I_{1}}\>\>Y_{I_{1}}\>\>\dots\>\>I_{N_{\textrm{quad}}}\>\>A^{\textrm{av}}_{I_{N_{\textrm{quad}}}}\>\>Z^{\textrm{av}}_{I_{N_{\textrm{quad}}}}\>\>Y_{I_{N_{\textrm{quad}}}}}_{N_{\textrm{quad}}\>{\textrm{quadruples}}}$

with $N_{\textrm{pairs}}$ 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 equation 10.) In
addition, there are $N_{\textrm{quad}}$ quadruples that contain an index
$I_{i}$ that specifies a group of nuclei $\mathcal{M}_{I_{i}}$ with average
mass number

$A^{\textrm{av}}_{I_{i}}=\frac{\sum_{j\in\mathcal{M}_{I_{i}}}A_{j}Y_{j}}{\sum_{j\in\mathcal{M}_{I_{i}}}Y_{j}}\>,$
(20)

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

$Z^{\textrm{av}}_{I_{i}}=\frac{\sum_{j\in\mathcal{M}_{I_{i}}}Z_{j}Y_{j}}{\sum_{j\in\mathcal{M}_{I_{i}}}Y_{j}}\>,$
(21)

and combined fraction

$Y_{I_{i}}=\frac{\sum_{j\in\mathcal{M}_{I_{i}}}A_{j}Y_{j}}{A^{\textrm{av}}_{I_{i}}}=\sum_{j\in\mathcal{M}_{I_{i}}}Y_{j}\>.$
(22)

The average neutron number

$N^{\textrm{av}}_{I_{i}}=\frac{\sum_{j\in\mathcal{M}_{I_{i}}}N_{j}Y_{j}}{\sum_{j\in\mathcal{M}_{I_{i}}}Y_{j}}\>,$
(23)

is not stored because it is easily found from
$N^{\textrm{av}}_{I_{i}}=A^{\textrm{av}}_{I_{i}}-Z^{\textrm{av}}_{I_{i}}$. In
case that there are no pairs (quadruples) to be stored, the number
$N_{\textrm{pairs}}$ ($N_{\textrm{quad}}$) 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 given in reach row. It is
sufficient to list only those that are relevant, e.g. with nonzero $Y_{i}$.
Others can be omitted.

#####  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. 1.

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

  2. 2.

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

  3. 3.

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

  4. 4.

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

  5. 5.

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

  6. 6.

pairing gap $\Delta_{i}$ [MeV].

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

$i_{T}\>\>i_{n_{b}}\>\>i_{Y_{q}}\>\>N_{\textrm{qty}}\>\>\underbrace{K_{1}\>\>q_{K_{1}}\>\>K_{2}\>\>q_{K_{2}}\>\>\dots\>\>K_{N_{\textrm{qty}}}\>\>q_{K_{N_{\textrm{qty}}}}}_{N_{\textrm{qty}}\>{\textrm{pairs}}}$
(24)

where $N_{\textrm{qty}}$ 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

$K_{i}=1000\>I_{i}+J_{i}$ (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 5.

#####  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.

#####  6 Table with information on properties of compact start (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⊙],

  * •

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⊙, are given in the first and second column of the file,
respectively.

#####  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,
$N_{T}^{\textrm{min}}$ is given, and $N_{T}^{\textrm{max}}$ in the second row.
Then the parameter values $T(N_{T}^{\textrm{min}})$,
$T(N_{T}^{\textrm{min}}+1)$, …, $T(N_{T}^{\textrm{max}})$, are given in every
following row. In total, the table contains
$N_{T}^{\textrm{max}}-N_{T}^{\textrm{min}}+3$ rows. The same storage scheme
applies for the parameters $n_{b}$ and $Y_{q}$.

#####  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).

#####  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.

####  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
section 1) are obtained by using the program compose.f90 (see section 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
$I_{\textrm{phase}}$ for the phases considered in the EoS.

Table 1: Parameter values for the recommended discretization in $T$, $n_{b}$ and $Y_{q}$ for a general purpose EoS table. quantity | reference | minimum | maximum | resolution
---|---|---|---|---
| value(s) | index | index | parameter
$T$, case a |  $T^{\textrm{ref}}=0.1$ MeV | $N_{T}^{\textrm{min}}=1$ | $N_{T}^{\textrm{max}}=81$ | $M_{T}=25$
$T$, case b |  $T^{\textrm{ref}}=0.1$ MeV | $N_{T}^{\textrm{min}}=1$ | $N_{T}^{\textrm{max}}=62$ | $M_{T}=25$
$n_{b}$ |  $n_{b}^{\textrm{ref}}=10^{-12}$ fm-3 | $N_{n_{b}}^{\textrm{min}}=1$ | $N_{n_{b}}^{\textrm{max}}=301$ | $M_{n_{b}}=25$
$Y_{q}$ | N/A | $N_{Y_{q}}^{\textrm{min}}=1$ | $N_{Y_{q}}^{\textrm{max}}=60$ | $M_{Y_{q}}=100$
Table 2: Parameter values for the recommended discretization in $n_{b}$ and $Y_{q}$ for a zero-temperature EoS table. quantity | reference | minimum | maximum | resolution
---|---|---|---|---
| value | index | index | parameter
$n_{b}$ |  $n_{b}^{\textrm{ref}}=10^{-12}$ fm-3 | $N_{n_{b}}^{\textrm{min}}=1$ | $N_{n_{b}}^{\textrm{max}}=301$ | $M_{n_{b}}=25$
$Y_{q}$ | N/A | $N_{Y_{q}}^{\textrm{min}}=0$ | $N_{Y_{q}}^{\textrm{max}}=60$ | $M_{Y_{q}}=100$
Table 3: Parameter values for the recommended discretization in $T$ and $n_{b}$ for EoS tables of symmetric matter, neutron matter or $\beta$-equilibrated matter. quantity | reference | minimum | maximum | resolution
---|---|---|---|---
| value | index | index | parameter
$T$, case a |  $T^{\textrm{ref}}=0.1$ MeV | $N_{T}^{\textrm{min}}=1$ | $N_{T}^{\textrm{max}}=81$ | $M_{T}=25$
$T$, case b |  $T^{\textrm{ref}}=0.1$ MeV | $N_{T}^{\textrm{min}}=0$ | $N_{T}^{\textrm{max}}=62$ | $M_{T}=25$
$n_{b}$ |  $n_{b}^{\textrm{ref}}=10^{-12}$ fm-3 | $N_{n_{b}}^{\textrm{min}}=1$ | $N_{n_{b}}^{\textrm{max}}=301$ | $M_{n_{b}}=25$
Table 4: Parameter values for the recommended discretization in $n_{b}$ for one-dimensional EoS tables. quantity | reference | minimum | maximum | resolution
---|---|---|---|---
| value | index | index | parameter
$n_{b}$ |  $n_{b}^{\textrm{ref}}=10^{-12}$ fm-3 | $N_{n_{b}}^{\textrm{min}}=1$ | $N_{n_{b}}^{\textrm{max}}=301$ | $M_{n_{b}}=25$

###  Chapter 4 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

><EMAIL_ADDRESS>
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.

##  Part 2 For Users

###  Chapter 5 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 reference [2] for a in-depth review.
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 section 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 section
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 [22] and for
nonrelativistic Hartree-Fock models with Gogny-type interactions [23]. Results
for relativistic mean-field models can be found in [24]. See the review [2]
for recent experimental constraints.

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

####  1 Nuclear matter properties

Realistic models of nuclear, i.e. purely hadronic, matter without leptons show
some general features that are related to the occurence 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 $n_{b}^{\textrm{sat}}$ with equal
concentrations of neutrons and protons111This is exactly true only if the mass
difference between neutrons and protons is neglected.. For $T=0$ MeV, the
energy per nucleon $\mathcal{E}$ can be considerd as a function of the baryon
number density $n_{b}$ and the asymmetry

$\alpha=\frac{n_{n}-n_{p}}{n_{n}+n_{p}}=1-2Y_{q}\qquad[\mbox{dimensionless}]$
(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

$\displaystyle\mathcal{E}(n_{b},\alpha)$ $\displaystyle=$ $\displaystyle-
B_{\textrm{sat}}+\frac{1}{2}Kx^{2}+\frac{1}{6}Qx^{3}+\dots$
$\displaystyle+\alpha^{2}\left(J+Lx+\frac{1}{2}K_{\textrm{sym}}x^{2}+\dots\right)+\dots\qquad[\mbox{MeV}]$

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

$x=\frac{n_{b}-n_{b}^{\textrm{sat}}}{3n_{b}^{\textrm{sat}}}\qquad[\mbox{dimensionless}]$
(3)

and the asymmetry $\alpha$. The coefficients $B_{\textrm{sat}}$, $K$,
$Q=-K^{\prime}$, $J$, $L$, $K_{\textrm{sym}}$ [MeV], …and
$n_{b}^{\textrm{sat}}$ [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 equation (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

$E_{\textrm{sym}}(n_{b})=\left.\frac{1}{2}\frac{\partial^{2}\mathcal{E}(n_{b},\alpha)}{\partial\alpha^{2}}\right|_{\alpha=0}\qquad[\mbox{MeV}]$
(4)

that is a function of the baryon number density $n_{b}$ only. In many cases
the quadratic approximation in $\alpha$ in equation (1) is sufficient. Then
the symmetry energy can also be calculated from the finite difference
approximation

$E_{\textrm{sym}}(n_{b})=\frac{1}{2}\left[\mathcal{E}(n_{b},-1)-2\mathcal{E}(n_{b},0)+\mathcal{E}(n_{b},1)\right]\qquad[\mbox{MeV}]$
(5)

comparing symmetric matter with pure neutron and pure proton matter.

In the following, the coefficients appearing in equation (1) are discussed in
more detail.

  * •

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

$p=\left.n_{b}^{2}\frac{d\mathcal{E}(n_{b},0)}{dn_{b}}\right|_{n_{b}=n_{b}^{\textrm{sat}}}=0\qquad[\mbox{MeV~{}fm}^{-3}]$
(6)

and the energy per baryon becomes minimal.

  * •

The binding energy at saturation $B_{\textrm{sat}}$ [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

$K=\left.9n_{b}^{2}\frac{\partial^{2}\mathcal{E}(n_{b},0)}{\partial
n_{b}^{2}}\right|_{n_{b}=n_{b}^{\textrm{sat}}}=\left.9n_{b}\frac{\partial(p/n_{b})}{\partial
n_{b}}\right|_{n_{b}=n_{b}^{\textrm{sat}}}\qquad[\mbox{MeV}]\>.$ (7)

It is related to the isothermal compressibility $\kappa_{T}$ (6) at zero
temperature, zero asymmetry and saturation density by

$K=\frac{9}{\kappa_{T}n_{b}^{\textrm{sat}}}\qquad[\mbox{MeV}]\>.$ (8)

Thus $K$ would be better called compression modulus. It can be derived, e.g.
from studies of isoscalar giant monopole excitations of nuclei. 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 also a recent comprehensive study of giant monopole
resonances [25].

  * •

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

$Q=27n_{b}^{3}\left.\frac{\partial^{3}\mathcal{E}(n_{b},0)}{\partial
n_{b}^{3}}\right|_{n_{b}=n_{b}^{\textrm{sat}}}=27n_{b}^{2}\left.\frac{\partial^{2}(p/n_{b})}{\partial
n_{b}^{2}}\right|_{n_{b}=n_{b}^{\textrm{sat}}}-6K\qquad[\mbox{MeV}]\>.$ (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

$J=E_{\textrm{sym}}(n_{b}^{\textrm{sat}})\qquad[\mbox{MeV}].$ (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

$L=3n_{b}\left.\frac{dE_{\textrm{sym}}(n_{b})}{dn_{b}}\right|_{n_{b}=n_{b}^{\textrm{sat}}}\qquad[\mbox{MeV}].$
(11)

This coefficient is related to the density dependence of the neutron matter
EoS near the saturation density. It is strongly correlated with the neutron
skin thickness of heavy nuclei like 208Pb. 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 $K_{\textrm{sym}}$ [MeV] quantifies the
curvature of the symmetry energy with respect to the density variation at
saturation. It is defined by

$K_{\textrm{sym}}=9n_{b}^{2}\left.\frac{d^{2}E_{\textrm{sym}}}{dn_{b}^{2}}\right|_{n_{b}=n_{b}^{\textrm{sat}}}\qquad[\mbox{MeV}]\>.$
(12)

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

####  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 occurence 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).

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 occurence 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 neutron star modelisation
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.

###  Chapter 6 Online services and data handling

####  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 11). If
you encounter any difficulties, it is possible to contact the CompOSE team by
sending an email to

><EMAIL_ADDRESS>.

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<EMAIL_ADDRESS>
####  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.

####  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. 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 section 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. 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 section 5.

  3. 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(at)obspm.fr if you wish an account to be created for you.

####  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. 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 library111https://lorene.obspm.fr.

  2. 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
$p_{\mathrm{crust}}=p_{\mathrm{core}}$ together with
$\mu_{\mathrm{crust}}=\mu_{\mathrm{core}}$ at the transition.

  3. 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. 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. 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.

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. 1.

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

  2. 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. 3.

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

  4. 4.

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

  5. 5.

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

  6. 6.

Quark models, category for EoS models containing 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. 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. 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. 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. 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. 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. 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 Shen et al.

  3. 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. 4.

Non-unified models (crust-core matched), see the description of the particular
EoS table for details.

  5. 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 section 6).

####  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 7.A. 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.

#####  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 8.

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.

#####  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.

The operation proceeds in the following steps:

  1. 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
subsection 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 subsection 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
[18] 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 8 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. 2.

Running compose

The user has to run the compose code three times:

    1. (a)

to select the output quantities

    2. (b)

to define the tabulation parameters

    3. (c)

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
appendex 8.

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 ($Y^{\textrm{av}}_{I_{i}}$,
$A^{\textrm{av}}_{I_{i}}$, $Z^{\textrm{av}}_{I_{i}}$, and
$N^{\textrm{av}}_{I_{i}}$) are given.

#####  3 HDF5 table

The organization of the data in case of the HFD5 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 7 and 8. 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
$N_{\mathrm{thermo}}$ thermodynamic quantities, see tables 7 and 8 for
details.

#####  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.

#####  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.

Table 1: Thermodynamic quantities in the data tables with their units. index | quantity/ | unit | description
---|---|---|---
$J$ | expression |  |
1 | $p$ |  MeV fm-3 | total pressure
2 | $\mathcal{S}=s/n_{b}$ | dimensionless | total entropy per baryon
3 | $\mu_{b}-m_{n}$ | MeV | baryon chemical potential
|  |  | with respect to neutron mass
4 | $\mu_{q}$ | MeV | charge chemical potential
5 | $\mu_{l}$ | MeV | lepton chemical potential
6 | $\mathcal{F}/m_{n}-1$ | dimensionless | scaled
|  |  | free energy per baryon
7 | $\varepsilon=$ | dimensionless | scaled
| $\mathcal{E}/m_{n}-1$ |  | internal energy per baryon
8 | $\mathcal{H}/m_{n}-1$ | dimensionless | scaled
|  |  | enthalpy per baryon
9 | $\mathcal{G}/m_{n}-1$ | dimensionless | scaled
|  |  | free enthalpy per baryon
10 | $\left.\frac{\partial p}{\partial n_{b}}\right|_{\mathcal{E}}$ | MeV | partial derivative of pressure with
|  |  | respect to baryon number density
11 | $\left.\frac{\partial p}{\partial\mathcal{E}}\right|_{n_{b}}$ |  fm-3 | partial derivative of pressure with
|  |  | respect to internal energy per baryon
12 | $c_{s}^{2}$ | dimensionless | speed of sound squared
13 | $c_{V}$ | dimensionless | specific heat capacity
|  |  | at constant volume
14 | $c_{p}$ | dimensionless | specific heat capacity
|  |  | at constant pressure
15 | $\Gamma=c_{p}/c_{V}$ | dimensionless | adiabatic index
16 | $\alpha_{p}$ |  MeV-1 | expansion coefficient
|  |  | at constant pressure
17 | $\beta_{V}$ |  fm-3 | tension coefficient
|  |  | at constant volume
18 | $\kappa_{T}$ |  fm3/MeV | isothermal compressibility
19 | $\kappa_{S}$ |  fm3/MeV | adiabatic compressibility
Table 2: Additional thermodynamic quantities in the data tables with their
units. (Quantities 25 - 28 are not yet available with the CompOSE code.)

index | quantity/ | unit | description
---|---|---|---
$J$ | expression |  |
20 | $\mathcal{F}$ | MeV | free energy per baryon
21 | $\mathcal{E}$ | MeV |  energy∗ per baryon
22 | $\mathcal{H}$ | MeV | enthalpy per baryon
23 | $\mathcal{G}$ | MeV | free enthalpy per baryon
24 | $e=\mathcal{E}n_{b}$ |  MeV fm-3 | energy density
25 | $\zeta$ |  MeV fm-2 | bulk viscosity
26 | $\eta$ |  MeV fm-2 | shear viscosity
27 | $\kappa$ |  fm-2 | thermal conductivity
28 | $\sigma$ |  fm-1 | electrical conductivity

∗ This corresponds to the ”internal energy” in the thermodynamic language
including the rest mass contribution.

Table 3: Values and derivatives of the free energy per baryon which are stored in the data tables with their units. index $J$ | quantity/ | unit | description
---|---|---|---
| expression |  |
1 | $\mathcal{F}$ | [MeV] | free energy per baryon
2 | $\frac{\partial\mathcal{F}}{\partial T}$ | dimensionless | first temperature derivative
|  |  | of free energy per baryon
3 | $\frac{\partial^{2}\mathcal{F}}{\partial T^{2}}$ |  [MeV-1] | second temperature derivative
|  |  | of free energy per baryon
4 | $\frac{\partial^{2}\mathcal{F}}{\partial T\partial n_{b}}$ |  [fm3] | mixed second derivative
|  |  | of free energy per baryon
5 | $\frac{\partial^{2}\mathcal{F}}{\partial T\partial Y_{q}}$ | dimensionless | mixed second derivative
|  |  | of free energy per baryon
6 | $\frac{\partial\mathcal{F}}{\partial n_{b}}$ |  [MeV fm3] | first baryon density derivative
|  |  | of free energy per baryon
7 | $\frac{\partial^{2}\mathcal{F}}{\partial n_{b}^{2}}$ |  [MeV fm6] | second baryon density derivative
|  |  | of free energy per baryon
8 | $\frac{\partial^{2}\mathcal{F}}{\partial n_{b}\partial Y_{q}}$ |  [MeV fm3] | mixed second derivative
|  |  | of free energy per baryon
9 | $\frac{\partial\mathcal{F}}{\partial Y_{q}}$ | [MeV] | first hadronic charge fraction derivative
|  |  | of free energy per baryon
10 | $\frac{\partial^{2}\mathcal{F}}{\partial Y_{q}^{2}}$ | [MeV] | second hadronic charge fraction derivative
|  |  | of free energy per baryon
Table 4: Quantities containing information on the composition which are stored in the data tables with their units. The particle index $I_{i}$ has been defined in tables 2, 3, 4 and 5. The symbols % denotes the group of nuclei. index $J$ | quantity/ | unit | description
---|---|---|---
| expression |  |
| $I_{\textrm{phase}}$ | dimensionless | phase index
| $Y_{I_{i}}$ | dimensionless |  fraction of particle $I_{i}$
$1$ | $Y^{\textrm{av}}_{\%}$ | dimensionless | combined fraction
|  |  | of group % of nuclei
$2$ | $A^{\textrm{av}}_{\%}$ | dimensionless | average mass number
|  |  | of group % of nuclei
$3$ | $Z^{\textrm{av}}_{\%}$ | dimensionless | average charge number
|  |  | of group % of nuclei
$4$ | $N^{\textrm{av}}_{\%}$ | dimensionless | average neutron number
|  |  | of group % of nuclei
Table 5: Microscopic quantities which are stored in the data tables with their unit. index $J$ | quantity/ | unit | description
---|---|---|---
| expression |  |
40 | $m^{L}_{I_{i}}/m_{I_{i}}$ | dimensionless | effective Landau mass
|  |  | with respect to
|  |  | the particle mass
41 | $m^{D}_{I_{i}}/m_{I_{i}}$ | dimensionless | effective Dirac mass
|  |  | with respect to
|  |  | the particle mass
50 | $U_{I_{i}}$ | MeV | nonrelativistic
|  |  | single-particle potential
51 | $V_{I_{i}}$ | MeV | relativistic
|  |  | vector self-energy
52 | $S_{I_{i}}$ | MeV | relativistic
|  |  | scalar self-energy
60 | $\Delta_{I_{i}}$ | MeV | gap
Table 6: Error quantities which are stored in the data tables with their unit. index $J$ | quantity/ | unit | description
---|---|---|---
| expression |  |
1 | $\Delta\mathcal{F}$ | MeV | absolute error estimate
|  |  | of free energy per baryon
2 | $\Delta\mathcal{F}/\mathcal{F}$ | [dimensionless] | relative error estimate
|  |  | of free energy per baryon
3 | $\Delta\mathcal{E}$ | MeV | absolute error estimate
|  |  | of internal energy per baryon
4 | $\Delta\mathcal{E}/\mathcal{E}$ | [dimensionless] | relative error estimate
|  |  | of internal energy per baryon
5 | $\Delta\left(\frac{p}{n_{b}}\right)$ | MeV | absolute error estimate
|  |  | of pressure-to-density ratio
6 | $\Delta\left(\frac{p}{n_{b}}\right)/\left(\frac{p}{n_{b}}\right)$ | [dimensionless] | relative error estimate
|  |  | of pressure-to-density ratio
7 | $\Delta\mathcal{S}$ | [dimensionless] | absolute error estimate
|  |  | of entropy per baryon
8 | $\Delta\mathcal{S}/\mathcal{S}$ | [dimensionless] | relative error estimate
|  |  | of entropy per baryon

Table 7: 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 $I_{\mathrm{tab}}=0$ and the second one to $I_{\mathrm{tab}}\neq 0$.  name of | quantity/ | unit | description
---|---|---|---
data set | expression |  |
t | $T$ | MeV | temperature
pointst | $\begin{array}[]{c}N_{\textrm{data}}\\\ N_{T}\end{array}$ | dimensionless |  number of points in $T$
nb | $n_{b}$ |  fm-3 | baryon number density
pointsnb | $\begin{array}[]{c}N_{\textrm{data}}\\\ N_{n_{b}}\end{array}$ | dimensionless |  number of points in $n_{b}$
yq | $Y_{q}$ | dimensionless | charge fraction of strongly
|  |  | interacting particles
pointsyq | $\begin{array}[]{l}N_{\textrm{data}}\\\ N_{Y_{q}}\end{array}$ | dimensionless |  number of points in $Y_{q}$
thermo | $Q(N_{\textrm{data}},1,1,N_{\textrm{thermo}})$ | varying | array of thermodynamical
| $Q(N_{n_{b}},N_{T},N_{Y_{q}},N_{\textrm{thermo}})$ |  |  quantities, see tables 1, 2
pointsthermo | $N_{\textrm{thermo}}$ | dimensionless | number of thermodynamic
|  |  |  quantities, see tables 1, 2.
index_thermo | $J(N_{\textrm{thermo}})$ | dimensionless | index identifying the
|  |  | thermodynamic quantities,
|  |  |  see tables 1, 2.
thermo_add | $q(N_{\textrm{data}},1,1,N_{\textrm{add}})$ | varying | array of additional
| $q(N_{n_{b}},N_{T},N_{Y_{q}},N_{\textrm{add}})$ |  |  quantities $q_{\%}$
|  |  |  in file eos.thermo.
pointsadd | $N_{\textrm{add}}$ | dimensionless | number of additional
|  |  | thermodynamic quantities
|  |  |  in file eos.thermo.
index_thermo_add | $I_{\textrm{add}}(N_{\textrm{add}})$ | dimensionless | index identifying the
|  |  | additional quantities
|  |  |  $q_{\%}$ from file eos.thermo,
|  |  |  see eq. (18)
Table 8: 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 $I_{\mathrm{tab}}=0$ and the second one to $I_{\mathrm{tab}}\neq 0$.  name of | quantity/ | unit | description
---|---|---|---
data set | expression |  |